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new file mode 100644
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new file mode 100644
index 0000000..4f0f8d4
--- /dev/null
+++ b/_app/error.svelte-bcbf8e93.js
@@ -0,0 +1 @@
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new file mode 100644
index 0000000..6743374
--- /dev/null
+++ b/_app/manifest.json
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+{
+  ".svelte-kit/runtime/client/start.js": {
+    "file": "start-e1e0ba09.js",
+    "src": ".svelte-kit/runtime/client/start.js",
+    "isEntry": true,
+    "imports": [
+      "_vendor-80631b75.js"
+    ],
+    "dynamicImports": [
+      "src/routes/__layout.svelte",
+      ".svelte-kit/runtime/components/error.svelte",
+      "src/routes/index.svelte"
+    ]
+  },
+  "src/routes/__layout.svelte": {
+    "file": "pages/__layout.svelte-3f8aeddf.js",
+    "src": "src/routes/__layout.svelte",
+    "isEntry": true,
+    "isDynamicEntry": true,
+    "imports": [
+      "_vendor-80631b75.js"
+    ],
+    "css": [
+      "assets/pages/__layout.svelte-c9f861a9.css"
+    ]
+  },
+  ".svelte-kit/runtime/components/error.svelte": {
+    "file": "error.svelte-bcbf8e93.js",
+    "src": ".svelte-kit/runtime/components/error.svelte",
+    "isEntry": true,
+    "isDynamicEntry": true,
+    "imports": [
+      "_vendor-80631b75.js"
+    ]
+  },
+  "src/routes/index.svelte": {
+    "file": "pages/index.svelte-255aa6ea.js",
+    "src": "src/routes/index.svelte",
+    "isEntry": true,
+    "isDynamicEntry": true,
+    "imports": [
+      "_vendor-80631b75.js"
+    ],
+    "css": [
+      "assets/pages/index.svelte-7f4057b4.css"
+    ]
+  },
+  "_vendor-80631b75.js": {
+    "file": "chunks/vendor-80631b75.js"
+  }
+}
\ No newline at end of file
diff --git a/_app/pages/__layout.svelte-3f8aeddf.js b/_app/pages/__layout.svelte-3f8aeddf.js
new file mode 100644
index 0000000..13ceb37
--- /dev/null
+++ b/_app/pages/__layout.svelte-3f8aeddf.js
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+    author = {Reed, Patrick M. and Hadjimichael, Antonia and Malek, Keyvan and Karimi, Tina and Vernon, Chris R. and Srikrishnan, Vivek and Gupta, Rohini S. and Gold, David F. and Lee, Ben and Keller, Klaus and Thurber, Travis B. and Rice, Jennie S.},
+    doi = {10.5281/zenodo.6110623},
+    publisher = {Zenodo},
+    title = {{Addressing Uncertainty in Multisector Dynamics Research}},
+    url = {https://uc-ebook.org},
+    year = {2022}
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fetch(`${t.pathname}/__data.json`,{headers:{"x-sveltekit-noredirect":"true"}});if(B.ok){const X=B.headers.get("x-sveltekit-location");if(X)return{redirect:X,props:{},state:this.current};T=await B.json()}else E=B.status,R=new Error("Failed to load data")}if(R||(g=await this._load_node({module:m,url:t,params:h,props:T,stuff:_})),g&&g.loaded){if(g.loaded.fallthrough)return;if(g.loaded.error&&(E=g.loaded.status,R=g.loaded.error),g.loaded.redirect)return{redirect:g.loaded.redirect,props:{},state:this.current};g.loaded.stuff&&(N=!0)}}else g=k}catch(m){E=500,R=oe(m)}if(R){for(;p--;)if(n[p]){let m,k,q=p;for(;!(k=d[q]);)q-=1;try{if(m=await this._load_node({status:E,error:R,module:await n[p](),url:t,params:h,stuff:k.stuff}),m&&m.loaded&&m.loaded.error)continue;m&&m.loaded&&m.loaded.stuff&&(_=y(y({},_),m.loaded.stuff)),d=d.slice(0,q+1).concat(m);break e}catch{continue}}return await this._load_error({status:E,error:R,url:t})}else g&&g.loaded&&g.loaded.stuff&&(_=y(y({},_),g.loaded.stuff)),d.push(g)}return await this._get_navigation_result_from_branch({url:t,params:h,stuff:_,branch:d,status:E,error:R})}async _load_error({status:e,error:t,url:r}){var c,f;const l={},i=await this._load_node({module:await this.fallback[0],url:r,params:l,stuff:{}}),a=await this._load_node({status:e,error:t,module:await this.fallback[1],url:r,params:l,stuff:i&&i.loaded&&i.loaded.stuff||{}}),s=[i,a],n=y(y({},(c=i==null?void 0:i.loaded)==null?void 0:c.stuff),(f=a==null?void 0:a.loaded)==null?void 0:f.stuff);return await this._get_navigation_result_from_branch({url:r,params:l,stuff:n,branch:s,status:e,error:t})}}async function Ye({paths:o,target:e,session:t,route:r,spa:l,trailing_slash:i,hydrate:a}){const s=new ze({Root:Oe,fallback:Ie,target:e,session:t}),n=r?new Be({base:o.base,routes:Ve,trailing_slash:i,renderer:s}):null;qe(o),a&&await s.start(a),n&&(l&&n.goto(location.href,{replaceState:!0},[]),n.init_listeners()),dispatchEvent(new CustomEvent("sveltekit:start"))}export{Ye as start};
diff --git a/_app/version.json b/_app/version.json
new file mode 100644
index 0000000..4993142
--- /dev/null
+++ b/_app/version.json
@@ -0,0 +1 @@
+{"version":"1720790592397"}
\ No newline at end of file
diff --git a/dev/CNAME b/dev/CNAME
new file mode 100644
index 0000000..2757fda
--- /dev/null
+++ b/dev/CNAME
@@ -0,0 +1 @@
+uc-ebook.org
\ No newline at end of file
diff --git a/dev/_app/assets/pages/__layout.svelte-c9f861a9.css b/dev/_app/assets/pages/__layout.svelte-c9f861a9.css
new file mode 100644
index 0000000..2af3d3b
--- /dev/null
+++ b/dev/_app/assets/pages/__layout.svelte-c9f861a9.css
@@ -0,0 +1 @@
+*,:before,:after{box-sizing:border-box;border-width:0;border-style:solid;border-color:#e5e7eb}:before,:after{--tw-content: ""}html{line-height:1.5;-webkit-text-size-adjust:100%;-moz-tab-size:4;-o-tab-size:4;tab-size:4;font-family:ui-sans-serif,system-ui,-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Helvetica Neue,Arial,Noto Sans,sans-serif,"Apple Color Emoji","Segoe UI Emoji",Segoe UI Symbol,"Noto Color Emoji"}body{margin:0;line-height:inherit}hr{height:0;color:inherit;border-top-width:1px}abbr:where([title]){-webkit-text-decoration:underline dotted;text-decoration:underline dotted}h1,h2,h3,h4,h5,h6{font-size:inherit;font-weight:inherit}a{color:inherit;text-decoration:inherit}b,strong{font-weight:bolder}code,kbd,samp,pre{font-family:ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,Liberation Mono,Courier 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}.m-4{margin:1rem}.mx-6{margin-left:1.5rem;margin-right:1.5rem}.my-6{margin-top:1.5rem;margin-bottom:1.5rem}.my-3{margin-top:.75rem;margin-bottom:.75rem}.mx-3{margin-left:.75rem;margin-right:.75rem}.mb-4{margin-bottom:1rem}.mt-6{margin-top:1.5rem}.mb-12{margin-bottom:3rem}.mb-3{margin-bottom:.75rem}.mt-3{margin-top:.75rem}.flex{display:flex}.hidden{display:none}.h-16{height:4rem}.h-20{height:5rem}.h-36{height:9rem}.w-full{width:100%}.w-80{width:20rem}.w-6{width:1.5rem}.w-64{width:16rem}.w-20{width:5rem}.max-w-screen-xl{max-width:1280px}.max-w-xs{max-width:20rem}.max-w-4xl{max-width:56rem}.flex-none{flex:none}.flex-row{flex-direction:row}.flex-col{flex-direction:column}.flex-wrap{flex-wrap:wrap}.items-start{align-items:flex-start}.items-center{align-items:center}.items-stretch{align-items:stretch}.justify-start{justify-content:flex-start}.justify-center{justify-content:center}.justify-around{justify-content:space-around}.overflow-auto{overflow:auto}.rounded-md{border-radius:.375rem}.rounded-lg{border-radius:.5rem}.rounded-l-md{border-top-left-radius:.375rem;border-bottom-left-radius:.375rem}.rounded-r-md{border-top-right-radius:.375rem;border-bottom-right-radius:.375rem}.rounded-br-md{border-bottom-right-radius:.375rem}.border-y-4{border-top-width:4px;border-bottom-width:4px}.border-double{border-style:double}.border-slate-50{--tw-border-opacity: 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new file mode 100644
index 0000000..4f0f8d4
--- /dev/null
+++ b/dev/_app/error.svelte-bcbf8e93.js
@@ -0,0 +1 @@
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new file mode 100644
index 0000000..1c2a870
--- /dev/null
+++ b/dev/_app/manifest.json
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+{
+  ".svelte-kit/runtime/client/start.js": {
+    "file": "start-1815c519.js",
+    "src": ".svelte-kit/runtime/client/start.js",
+    "isEntry": true,
+    "imports": [
+      "_vendor-80631b75.js"
+    ],
+    "dynamicImports": [
+      "src/routes/__layout.svelte",
+      ".svelte-kit/runtime/components/error.svelte",
+      "src/routes/index.svelte"
+    ]
+  },
+  "src/routes/__layout.svelte": {
+    "file": "pages/__layout.svelte-3f8aeddf.js",
+    "src": "src/routes/__layout.svelte",
+    "isEntry": true,
+    "isDynamicEntry": true,
+    "imports": [
+      "_vendor-80631b75.js"
+    ],
+    "css": [
+      "assets/pages/__layout.svelte-c9f861a9.css"
+    ]
+  },
+  ".svelte-kit/runtime/components/error.svelte": {
+    "file": "error.svelte-bcbf8e93.js",
+    "src": ".svelte-kit/runtime/components/error.svelte",
+    "isEntry": true,
+    "isDynamicEntry": true,
+    "imports": [
+      "_vendor-80631b75.js"
+    ]
+  },
+  "src/routes/index.svelte": {
+    "file": "pages/index.svelte-de143cc1.js",
+    "src": "src/routes/index.svelte",
+    "isEntry": true,
+    "isDynamicEntry": true,
+    "imports": [
+      "_vendor-80631b75.js"
+    ],
+    "css": [
+      "assets/pages/index.svelte-7f4057b4.css"
+    ]
+  },
+  "_vendor-80631b75.js": {
+    "file": "chunks/vendor-80631b75.js"
+  }
+}
\ No newline at end of file
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new file mode 100644
index 0000000..13ceb37
--- /dev/null
+++ b/dev/_app/pages/__layout.svelte-3f8aeddf.js
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new file mode 100644
index 0000000..01fd3e4
--- /dev/null
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+    author = {Reed, Patrick M. and Hadjimichael, Antonia and Malek, Keyvan and Karimi, Tina and Vernon, Chris R. and Srikrishnan, Vivek and Gupta, Rohini S. and Gold, David F. and Lee, Ben and Keller, Klaus and Thurber, Travis B. and Rice, Jennie S.},
+    doi = {10.5281/zenodo.6110623},
+    publisher = {Zenodo},
+    title = {{Addressing Uncertainty in Multisector Dynamics Research}},
+    url = {https://uc-ebook.org},
+    year = {2022}
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diff --git a/dev/docs/addressinguncertaintyinmultisectordynamicsresearch.pdf b/dev/docs/addressinguncertaintyinmultisectordynamicsresearch.pdf
new file mode 100644
index 0000000..8cfdf3d
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diff --git a/dev/docs/html/.buildinfo b/dev/docs/html/.buildinfo
new file mode 100644
index 0000000..3780467
--- /dev/null
+++ b/dev/docs/html/.buildinfo
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+# Sphinx build info version 1
+# This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done.
+config: 46bbe39cebfe7ce3eff669bf2e2f482d
+tags: 645f666f9bcd5a90fca523b33c5a78b7
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diff --git a/dev/docs/html/1_introduction.html b/dev/docs/html/1_introduction.html
new file mode 100644
index 0000000..3c12aef
--- /dev/null
+++ b/dev/docs/html/1_introduction.html
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+
+<!DOCTYPE html>
+
+
+<html lang="en" data-content_root="./" >
+
+  <head>
+    <meta charset="utf-8" />
+    <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
+
+    <title>1. Introduction &#8212; Addressing Uncertainty in MultiSector Dynamics Research  documentation</title>
+  
+  
+  
+  <script data-cfasync="false">
+    document.documentElement.dataset.mode = localStorage.getItem("mode") || "";
+    document.documentElement.dataset.theme = localStorage.getItem("theme") || "";
+  </script>
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+  <!-- Loaded before other Sphinx assets -->
+  <link href="_static/styles/theme.css?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
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+<link rel="preload" as="font" type="font/woff2" crossorigin href="_static/vendor/fontawesome/6.5.2/webfonts/fa-brands-400.woff2" />
+<link rel="preload" as="font" type="font/woff2" crossorigin href="_static/vendor/fontawesome/6.5.2/webfonts/fa-regular-400.woff2" />
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+  <!-- Pre-loaded scripts that we'll load fully later -->
+  <link rel="preload" as="script" href="_static/scripts/bootstrap.js?digest=dfe6caa3a7d634c4db9b" />
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+  <section id="introduction">
+<span id="id1"></span><h1><span class="section-number">1. </span>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h1>
+<p>This guidance text has been developed in support of the Integrated Multisector Multiscale Modeling (IM3) Science Focus Area’s objective to formally integrate uncertainty into its research tasks. IM3 is focused on innovative modeling to explore how human and natural system landscapes in the United States co-evolve in response to short-term shocks and long-term influences. The project’s challenging scope is to advance our ability to study the interactions between energy, water, land, and urban systems, at scales ranging from local (~1km) to the contiguous United States, while consistently addressing influences such as population change, technology change, heat waves, and drought. Uncertainty and careful model-driven scientific insights are central to the project’s science objectives shown below.</p>
+<p><strong>IM3 Key MSD Science Objectives:</strong></p>
+<p><em>Develop flexible, open-source, and integrated modeling capabilities that capture the structure, dynamic behavior, and emergent properties of the multiscale interactions within and between human and natural systems.</em></p>
+<p><em>Use these capabilities to study the evolution, vulnerability, and resilience of interacting human and natural systems and landscapes from local to continental scales, including their responses to the compounding effects of long-term influences and short-term shocks.</em></p>
+<p><em>Understand the implications of uncertainty in data, observations, models, and model coupling approaches for projections of human-natural system dynamics.</em></p>
+<p>Addressing the objectives above poses a strong transdisciplinary challenge that depends on a diversity of models and, more specifically, a consistent framing for making model-based science inferences. The term transdisciplinary science as used here formally implies a deep integration of disciplines to aid our hypothesis-driven understanding of coupled human-natural systems–bridging differences in theory, hypothesis generation, modeling, and modes of inference <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id2" title="National Research Council and others. Convergence: Facilitating transdisciplinary integration of life sciences, physical sciences, engineering, and beyond. National Academies Press, 2014.">2</a>]</span>. The IM3 MSD research foci and questions require a deep integration across disciplines, where new modes of analysis can emerge that rapidly synthesize and exploit advances for making decision-relevant insights that at minimum acknowledge uncertainty and more ideally promote a rigorous quantitative mapping of its effects on the generality of claimed scientific insights. More broadly, diverse scientific disciplines engaged in the science of coupled human-natural systems, ranging from natural sciences to engineering and economics, employ a diversity of numerical computer models to study and understand their underlying systems of focus. The utility of these computer models hinges on their ability to represent the underlying real systems with sufficient fidelity and enable the inference of novel insights. This is particularly challenging in the case of coupled human-natural systems where there exists a multitude of interdependent human and natural processes taking place that could potentially be represented. These processes usually translate into modeled representations that are highly complex, non-linear, and exhibit strong interactions and threshold behaviors <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id3" title="Sondoss Elsawah, Tatiana Filatova, Anthony J Jakeman, Albert J Kettner, Moira L Zellner, Ioannis N Athanasiadis, Serena H Hamilton, Robert L Axtell, Daniel G Brown, Jonathan M Gilligan, and others. Eight grand challenges in socio-environmental systems modeling. Socio-Environmental Systems Modelling, 2:16226–16226, 2020.">3</a>, <a class="reference internal" href="R.Bibliography.html#id4" title="Yacov Y Haimes. Risk modeling of interdependent complex systems of systems: theory and practice. Risk analysis, 38(1):84–98, 2018.">4</a>, <a class="reference internal" href="R.Bibliography.html#id5" title="Dirk Helbing. Globally networked risks and how to respond. Nature, 497(7447):51–59, 2013.">5</a>]</span>. Model complexity and detail have also been increasing as a result of our improving understanding of these processes, the availability of data, and the rapid growth in computing power <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id6" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">6</a>]</span>. As model complexity grows, modelers need to specify a lot more information than before: additional model inputs and relationships as more processes are represented, higher resolution data as more observations are collected, new coupling relationships and interactions as diverse models are being used in combination to answer multisector questions (e.g., the land-water-energy nexus). Typically, not all of this information is well known, nor is the impact of these many uncertainties on model outputs well understood. It is further especially difficult to distinguish the effects of individual as well as interacting sources of uncertainty when modeling coupled systems with multisector and multiscale dynamics <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id7" title="Daniel Wirtz and Wolfgang Nowak. The rocky road to extended simulation frameworks covering uncertainty, inversion, optimization and control. Environmental Modelling &amp; Software, 93:180–192, 2017.">7</a>]</span>.</p>
+<p>Given the challenge and opportunity posed by the disciplinary diversity of IM3, we utilized an informal team-wide survey to understand how the various disciplines typically address uncertainty, emphasizing key literature examples and domain-specific reviews. The feedback received provided perspectives across diverse areas within the Earth sciences, different engineering fields, as well as economics. Although our synthesis of this survey information highlighted some commonality across areas (e.g., the frequent use of scenario-based modeling), we identified key differences in vocabulary, the frequency with which formal uncertainty analysis appears in the disciplinary literature, and technical approaches. The IM3 team’s responses captured a very broad conceptual continuum of methodological traditions, ranging from deterministic (no uncertainty) modeling to the theoretical case of fully engaging in modeling sources of uncertainty. Overall, error-driven analyses that focus on replicating prior observed conditions were reported to be the most prevalent types of studies for all disciplines. It was generally less common for studies to strongly engage with analyzing uncertainty via more formal ensemble analyses and design of experiments, though some areas did show significantly higher levels of activity. Another notable finding from our survey was the apparent lack of focus on understanding how model coupling relationships shape uncertainty. Although these observations are limited to the scope of feedback attained in the team-wide IM3 survey responses and the bodies of literature reported by respondents, we believe they reflect challenges that are common across the MSD community.</p>
+<p>In the IM3 uncertainty-related research that has occurred since this survey, we have observed that differences in terminology and interpretation of terminology across modeling teams can be confounding. One of the goals of this eBook is to provide a common language for uncertainty analysis within IM3 and, hopefully, for the broader MSD community. While individual scientific disciplines would be expected to retain their own terminology, by providing explicit definitions of terms we can facilitate the translation of concepts across transdisciplinary science teams. To begin, we use the term Uncertainty Analysis (UA) as an umbrella phrase covering all methods in this eBook. Next, we distinguish the key terms of uncertainty quantification (UQ) and uncertainty characterization (UC). UQ refers to the formal focus on the full specification of likelihoods as well as the distributional forms necessary to infer the joint probabilistic response across all modeled factors of interest <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id11" title="Roger Cooke and others. Experts in uncertainty: opinion and subjective probability in science. Oxford University Press on Demand, 1991.">8</a>]</span>. UC refers to exploratory modeling of alternative hypotheses to understand the co-evolutionary dynamics of influences and stressors, as well as path dependent changes in the form and function of modelled systems <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id12" title="Enayat A Moallemi, Jan Kwakkel, Fjalar J de Haan, and Brett A Bryan. Exploratory modeling for analyzing coupled human-natural systems under uncertainty. Global Environmental Change, 65:102186, 2020.">9</a>, <a class="reference internal" href="R.Bibliography.html#id13" title="Warren E Walker, Poul Harremoës, Jan Rotmans, Jeroen P Van Der Sluijs, Marjolein BA Van Asselt, Peter Janssen, and Martin P Krayer von Krauss. Defining uncertainty: a conceptual basis for uncertainty management in model-based decision support. Integrated assessment, 4(1):5–17, 2003.">10</a>]</span>. As discussed in later sections, the choice of UC or UQ depends on the specific goals of studies, the availability of data, the types of uncertainties (e.g., well-characterized or deep), and the complexity of underlying models as well as computational limits. Definitions of key uncertainty analysis terms used in this eBook appear below, and our Glossary (<code class="xref std std-numref docutils literal notranslate"><span class="pre">glossary</span></code>) contains a complete list of terms.</p>
+<ul class="simple">
+<li><p><strong>Exploratory modeling</strong>: Use of large ensembles of uncertain conditions to discover decision-relevant combinations of uncertain factors</p></li>
+<li><p><strong>Factor</strong>: Any model component that can affect model outputs: inputs, resolution levels, coupling relationships, model relationships and parameters. In models with acceptable model fidelity these factors may represent elements of the real-world system under study.</p></li>
+<li><dl class="simple">
+<dt><strong>Sensitivity analysis</strong>: Model evaluation to understand the factors and processes that most (or least) control a model’s outputs</dt><dd><ul>
+<li><p><strong>Local sensitivity analysis</strong>: Varying uncertain factors around specific reference values</p></li>
+<li><p><strong>Global sensitivity analysis</strong>: Varying uncertain factors throughout their entire feasible value space</p></li>
+</ul>
+</dd>
+</dl>
+</li>
+<li><p><strong>Uncertainty characterization</strong>: Model evaluation under alternative factor hypotheses to explore their implications for model output uncertainty</p></li>
+<li><p><strong>Uncertainty quantification</strong>: Representation of model output uncertainty using probability distributions</p></li>
+</ul>
+<p>At present, there is no singular guide for confronting the computational and conceptual challenges of the multi-model, transdisciplinary workflows that characterize ambitious projects such as IM3 <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id10" title="Andrea Saltelli, Philip B Stark, William Becker, and Pawel Stano. Climate models as economic guides scientific challenge or quixotic quest? Issues in Science and Technology, 31(3):79–84, 2015.">11</a>]</span>. The primary aim of this text is to begin to address this gap and provide guidance for facing these challenges. <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#diagnostic-modeling"><span class="std std-numref">Chapter 2</span></a> provides an overview of diagnostic modeling and the different perspectives for how we should evaluate our models, <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#sensitivity-analysis-the-basics"><span class="std std-numref">Chapter 3</span></a> summarizes basic methods and concepts for sensitivity analysis, and <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#sensitivity-analysis"><span class="std std-numref">Chapter 4</span></a> delves into more technical applications of sensitivity analysis to support diagnostic model evaluation and exploratory modeling. Finally, <a class="reference internal" href="5_conclusion.html#conclusion"><span class="std std-numref">Chapter 5</span></a> provides some concluding remarks across the UC and UQ topics covered in this text. The appendices of this text include a glossary of the key concepts, an overview of UQ methods, and coding-based illustrative examples of key UC concepts discussed in earlier chapters.</p>
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+  <section id="overview-of-model-diagnostics">
+<h1>Overview of model diagnostics<a class="headerlink" href="#overview-of-model-diagnostics" title="Link to this heading">#</a></h1>
+<p>Model diagnostics provide a rich basis for hypothesis testing, model innovation, and improved inferences when classifying what is controlling highly consequential results (e.g., vulnerability or resilience in coupled human-natural systems). <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a>, adapted from <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id6" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">6</a>]</span>, presents idealized illustrations of the relationship between UC and global sensitivity analysis for two coupled simulation models. The figure illustrates how UC can be used to address how uncertainties in various modeling decisions (e.g., data inputs, parameters, model structures, coupling relationships) can be sampled and simulated to yield the empirical model output distribution(s) of interest. Monte Carlo frameworks allow us to sample and propagate (or integrate) the ensemble response of the model(s) of focus. The first step of any UC analysis is the specification of the initial input distributions as illustrated in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a>. The second step is to perform the Monte Carlo simulations. The question can then be raised, which of the modeling assumptions in our Monte Carlo experiment are the most responsible for the resulting output uncertainty. We can answer this question using global sensitivity analysis as illustrated in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a>. Global sensitivity analysis can be defined as a formal Monte Carlo sampling and analysis of modeling choices (structures, parameters, inputs) to quantify their influence on direct model outputs (or output-informed metrics). UC experiments by themselves do not explain why a particular uncertain outcome is produced, but produce distributions of model outcomes, as portrayed by the yellow curve. The pie chart shown in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a> is a conceptual representation of the results of using a global sensitivity analysis to identify those factors that are most dominantly influencing results, either individually or interactively <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id48" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global Sensitivity Analysis: The Primer. Wiley-Interscience, Chichester, England ; Hoboken, NJ, 1 edition edition, February 2008. ISBN 978-0-470-05997-5.">14</a>]</span>.</p>
+<figure class="margin-caption align-center" id="id4">
+<span id="figure-2-1"></span><a class="reference internal image-reference" href="_images/figure2_1_idealized_uc.png"><img alt="Figure 2.1" src="_images/figure2_1_idealized_uc.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Idealized uncertainty characterization and global sensitivity analysis for two coupled simulation models. Uncertainty coming from various sources (e.g., inputs, model structures, coupling relationships) is propagated through the coupled model(s) to generate empirical distributions of outputs of interest (uncertainty characterization). This model output uncertainty can be decomposed to its origins, by means of sensitivity analysis. Figure adapted from <span id="id3">Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id6" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">6</a>]</span>.</span><a class="headerlink" href="#id4" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>UC and global sensitivity analysis are not independent modeling analyses. As illustrated here, any global sensitivity analysis requires an initial UC hypothesis in the form of statistical assumptions and representations for the modeling choices of focus (structural, parametric, and data inputs). Information from these two model diagnostic tools can then be used to inform data needs for future model runs, experiments to reduce the uncertainty present, or the simplification or enhancement of the model where necessary. Together UC and global sensitivity analysis provide a foundation for diagnostic exploratory modeling that has a consistent focus on the assumptions, structural model forms, alternative parameterizations, and input data sets that are used to characterize the behavioral space of one or more models.</p>
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+  <section id="perspectives-on-diagnostic-model-evaluation">
+<span id="perspectives"></span><h1>Perspectives on diagnostic model evaluation<a class="headerlink" href="#perspectives-on-diagnostic-model-evaluation" title="Link to this heading">#</a></h1>
+<p>When we judge or diagnose models, the terms “verification” and “validation” are commonly used. However, their appropriateness in the context of numerical models representing complex coupled human-natural systems is questionable <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id54" title="Keith Beven. Towards a coherent philosophy for modelling the environment. Proceedings of the royal society of London. Series A: mathematical, physical and engineering sciences, 458(2026):2465–2484, 2002.">15</a>, <a class="reference internal" href="R.Bibliography.html#id118" title="Naomi Oreskes, Kristin Shrader-Frechette, and Kenneth Belitz. Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences. Science, 263(5147):641–646, February 1994. URL: https://science.sciencemag.org/content/263/5147/641 (visited on 2020-04-15), doi:10.1126/science.263.5147.641.">16</a>]</span>. The core issue relates to the fact that these systems are often not fully known or perfectly implemented when modeled. Rather, they are defined within specific system framings and boundary conditions in an evolving learning process with the goal of making continual progress towards attaining higher levels of fidelity in capturing behaviors or properties of interest. Evaluating the fidelity of a model’s performance can be highly challenging. For example, the observations used to evaluate the fidelity of parameterized processes are often measured at a finer resolution than what is represented in the model, creating the challenge of how to manage their relative scales when performing evaluation. In other cases, numerical models may neglect or simplify system processes because sufficient data is not available or the physical mechanisms are not fully known. If sufficient agreement between prediction and observation is not achieved, it is challenging to know whether these types of modeling choices are the cause, or if other issues, such as deficiencies in the input parameters and/or other modeling assumptions are the true cause of errors. Even if there is high agreement between prediction and observation, the model cannot necessarily be considered validated, as it is always possible that the right values were produced for the wrong reasons. For example, low error can stem from a situation where different errors in underlying assumptions or parameters cancel each other out (“compensatory errors”). Furthermore, coupled human-natural system models are often subject to “equifinality”, a situation where multiple parameterized formulations can produce similar outputs or equally acceptable representations of the observed data. There is therefore no uniquely “true” or validated model, and the common practice of selecting “the best” deterministic calibration set is more of an assumption than a finding <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id49" title="Keith Beven. Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in water resources, 16(1):41–51, 1993.">17</a>, <a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>]</span>. The situation becomes even more tenuous when observational data is limited in its scope and/or quality to be insufficient to distinguish model representations or their performance differences.</p>
+<p>These limitations on model verification undermine any purely positivist treatment of model validity: that a model should correctly and precisely represent reality to be valid. Under this perspective, closely related to empiricism, statistical tests should be used to compare the model’s output with observations and only through empirical verification can a model or theory be deemed credible. A criticism to this viewpoint (besides the aforementioned challenges for model verification) is that it reduces the justification of a model to the single criterion of predictive ability and accuracy <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id119" title="Yaman Barlas and Stanley Carpenter. Philosophical roots of model validation: Two paradigms. System Dynamics Review, 6(2):148–166, 1990. doi:10.1002/sdr.4260060203.">19</a>]</span>. Authors have argued that this ignores the explanatory power held in models and other procedures, which can also advance scientific knowledge <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id120" title="Stephen Toulmin. From form to function: philosophy and history of science in the 1950s and now. Daedalus, pages 143–162, 1977. Publisher: JSTOR.">20</a>]</span>. These views gave rise to relativist perspectives of science, which instead place more value on model utility in terms of fitness for a specific purpose or inquiry, rather than representational accuracy and predictive ability <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id121" title="George B. Kleindorfer, Liam O'Neill, and Ram Ganeshan. Validation in simulation: Various positions in the philosophy of science. Management Science, 44(8):1087–1099, 1998. Publisher: INFORMS.">21</a>]</span>. This viewpoint appears to be most prevalent among practitioners seeking decision-relevant insights (i.e., inspire new views vs. predict future conditions). The relativist perspective argues for the use of models as heuristics that can enhance our understanding and conceptions of system behaviors or possibilities <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id64" title="Sibel Eker, Elena Rovenskaya, Michael Obersteiner, and Simon Langan. Practice and perspectives in the validation of resource management models. Nature communications, 9(1):1–10, 2018.">22</a>]</span>. In contrast, natural sciences favor a positivist perspective, emphasizing similarity between simulation and observation even in application contexts where it is clear that projections are being made for conditions that have never been observed and the system of focus will have evolved structurally beyond the model representation being employed (e.g., decadal to centennial evolution of human-natural systems).</p>
+<p>These differences in prevalent perspectives are mirrored in how model validation is defined by the two camps: From the relativist perspective, validation is seen as a process of incremental “confidence building” in a model as a mechanism for insight <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id122" title="Yaman Barlas. Formal aspects of model validity and validation in system dynamics. System Dynamics Review: The Journal of the System Dynamics Society, 12(3):183–210, 1996. Publisher: Wiley Online Library.">23</a>]</span>, whereas in natural sciences validation is framed as a way to classify a model as having an acceptable representation of physical reality <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id118" title="Naomi Oreskes, Kristin Shrader-Frechette, and Kenneth Belitz. Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences. Science, 263(5147):641–646, February 1994. URL: https://science.sciencemag.org/content/263/5147/641 (visited on 2020-04-15), doi:10.1126/science.263.5147.641.">16</a>]</span>. Even though the relativist viewpoint does not dismiss the importance of representational accuracy, it does place it within a larger process of establishing confidence through a variety of tools. These tools, not necessarily quantitative, include communicating information between practitioners and modelers, interpreting a multitude of model outputs, and contrasting preferences and viewpoints.</p>
+<p>On the technical side of the argument, differing views on the methodology of model validation appear as early as in the 1960’s. <span id="id9">Naylor and Finger [<a class="reference internal" href="R.Bibliography.html#id123" title="Thomas H. Naylor and Joseph Michael Finger. Verification of computer simulation models. Management science, 14(2):B–92, 1967. Publisher: INFORMS.">24</a>]</span> argue that model validation should not be limited to a single metric or test of performance (e.g., a single error metric), but should rather be extended to multiple tests that reflect different aspects of a model’s structure and behavior. This and similar arguments are made in literature to this day <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id46" title="Hoshin V. Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008. Publisher: Wiley Online Library.">12</a>, <a class="reference internal" href="R.Bibliography.html#id52" title="Keith J Beven. On hypothesis testing in hydrology: why falsification of models is still a really good idea. Wiley Interdisciplinary Reviews: Water, 5(3):e1278, 2018.">25</a>, <a class="reference internal" href="R.Bibliography.html#id51" title="Hoshin V Gupta, Martyn P Clark, Jasper A Vrugt, Gab Abramowitz, and Ming Ye. Towards a comprehensive assessment of model structural adequacy. Water Resources Research, 2012.">26</a>, <a class="reference internal" href="R.Bibliography.html#id124" title="Praveen Kumar. Typology of hydrologic predictability. Water Resources Research, 2011. URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2010WR009769 (visited on 2020-04-15), doi:10.1029/2010WR009769.">27</a>, <a class="reference internal" href="R.Bibliography.html#id125" title="Grey S. Nearing, Benjamin L. Ruddell, Andrew R. Bennett, Cristina Prieto, and Hoshin V. Gupta. Does Information Theory Provide a New Paradigm for Earth Science? Hypothesis Testing. Water Resources Research, 56(2):e2019WR024918, 2020. doi:10.1029/2019WR024918.">28</a>]</span> and are primarily founded on two premises. First, that even though modelers widely recognize that their models are abstractions of the truth, they still make truth claims based on traditional performance metrics that measure the divergence of their model from observation <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id125" title="Grey S. Nearing, Benjamin L. Ruddell, Andrew R. Bennett, Cristina Prieto, and Hoshin V. Gupta. Does Information Theory Provide a New Paradigm for Earth Science? Hypothesis Testing. Water Resources Research, 56(2):e2019WR024918, 2020. doi:10.1029/2019WR024918.">28</a>]</span>. Second, that the natural systems mimicked by the models contain many processes that exhibit significant heterogeneity at various temporal and spatial scales. This heterogeneity is lost when a single performance measure is used, as a result of the inherent loss of process information occurring when transitioning from a highly dimensional and interactive system to the dimension of a single metric <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id54" title="Keith Beven. Towards a coherent philosophy for modelling the environment. Proceedings of the royal society of London. Series A: mathematical, physical and engineering sciences, 458(2026):2465–2484, 2002.">15</a>]</span>. These arguments are further elaborated in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#sensitivity-analysis"><span class="std std-numref">Chapter 4</span></a>.</p>
+<p>Multiple authors have proposed that the traditional reliance on single measures of model performance should be replaced by the evaluation of several model signatures (characteristics) to identify model structural errors and achieve a sufficient assessment of model performance <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id46" title="Hoshin V. Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008. Publisher: Wiley Online Library.">12</a>, <a class="reference internal" href="R.Bibliography.html#id126" title="Hoshin Vijai Gupta, Soroosh Sorooshian, and Patrice Ogou Yapo. Toward improved calibration of hydrologic models: Multiple and noncommensurable measures of information. Water Resources Research, 34(4):751–763, 1998. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/97WR03495 (visited on 2020-04-07), doi:10.1029/97WR03495.">29</a>, <a class="reference internal" href="R.Bibliography.html#id168" title="Francesca Pianosi and Thorsten Wagener. Understanding the time-varying importance of different uncertainty sources in hydrological modelling using global sensitivity analysis. Hydrological Processes, pages 3991–4003, November 2017. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/hyp.10968%4010.1111/%28ISSN%291099-1085.Kieth-Beven, doi:10.1002/hyp.10968&#64;10.1111/(ISSN)1099-1085.Kieth-Beven.">30</a>, <a class="reference internal" href="R.Bibliography.html#id167" title="Charles Rougé, Patrick M. Reed, Danielle S. Grogan, Shan Zuidema, Alexander Prusevich, Stanley Glidden, Jonathan R. Lamontagne, and Richard B. Lammers. Coordination and Control: Limits in Standard Representations of Multi-Reservoir Operations in Hydrological Modeling. Hydrology and Earth System Sciences Discussions, pages 1–37, November 2019. URL: https://www.hydrol-earth-syst-sci-discuss.net/hess-2019-589/, doi:https://doi.org/10.5194/hess-2019-589.">31</a>]</span>. There is however a point of departure here, especially when models are used to produce inferences that can inform decisions. When agencies and practitioners use models of their systems for public decisions, those models have already met sufficient conditions for credibility (e.g., acceptable representational fidelity), but may face broader tests on their salience and legitimacy in informing negotiated decisions <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id64" title="Sibel Eker, Elena Rovenskaya, Michael Obersteiner, and Simon Langan. Practice and perspectives in the validation of resource management models. Nature communications, 9(1):1–10, 2018.">22</a>, <a class="reference internal" href="R.Bibliography.html#id127" title="David W. Cash, William C. Clark, Frank Alcock, Nancy M. Dickson, Noelle Eckley, David H. Guston, Jill Jäger, and Ronald B. Mitchell. Knowledge systems for sustainable development. Proceedings of the national academy of sciences, 100(14):8086–8091, 2003. Publisher: National Acad Sciences.">32</a>, <a class="reference internal" href="R.Bibliography.html#id128" title="Dave D. White, Amber Wutich, Kelli L. Larson, Patricia Gober, Timothy Lant, and Clea Senneville. Credibility, salience, and legitimacy of boundary objects: water managers' assessment of a simulation model in an immersive decision theater. Science and Public Policy, 37(3):219–232, April 2010. Publisher: Oxford Academic. URL: https://academic.oup.com/spp/article/37/3/219/1626552 (visited on 2020-05-12), doi:10.3152/030234210X497726.">33</a>]</span>. This presents a new challenge to model validation, that of selecting decision-relevant performance metrics, reflective of the system’s stakeholders’ viewpoints, so that the most consequential uncertainties are identified and addressed <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id129" title="Andrea Saltelli and Silvio Funtowicz. When all models are wrong. Issues in Science and Technology, 30(2):79–85, 2014. Publisher: JSTOR.">34</a>]</span>. For complex multisector models at the intersection of climatic, hydrologic, agricultural, energy, or other processes, the output space is made up of a multitude of states and variables, with very different levels of salience to the system’s stakeholders and to their goals being achieved <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id203" title="Thorsten Wagener and Francesca Pianosi. What has Global Sensitivity Analysis ever done for us? A systematic review to support scientific advancement and to inform policy-making in earth system modelling. Earth-Science Reviews, 194:1–18, July 2019. URL: https://www.sciencedirect.com/science/article/pii/S0012825218300990 (visited on 2021-08-30), doi:10.1016/j.earscirev.2019.04.006.">35</a>]</span>. This is further complicated when such systems are also institutionally and dynamically complex. As a result, a broader set of qualitative and quantitative performance metrics is necessary to evaluate models of such complex systems, one that embraces the plurality of value systems, agencies and perspectives present. For IM3, even though the goal is to develop better projections of future vulnerability and resilience in co-evolving human-natural systems and not to provide decision support per se, it is critical for our multisector, multiscale model evaluation processes to represent stakeholders’ adaptive decision processes credibly.</p>
+<p>As a final point, when a model is used in a projection mode, its results are also subject to additional uncertainty, as there is no guarantee that the model’s functionality and predictive ability will stay the same as the baseline, where the verification and validation tests were conducted. This challenge requires an additional expansion of the scope of model evaluation: a broader set of uncertain conditions needs to be explored, spanning beyond historical observation and exploring a wide range of unprecedented conditions. This perspective on modeling, termed exploratory <span id="id17">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>]</span>, views models as computational experiments that can be used to explore vast ensembles of potential scenarios to identify those with consequential effects. Exploratory modeling literature explicitly orients experiments toward stakeholder consequences and decision-relevant inferences and shifts the focus from predicting future conditions to <em>discovering</em> which conditions lead to undesirable or desirable consequences.</p>
+<p>This evolution in modeling perspectives can be mirrored by the IM3 family of models in a progression from evaluating models relative to observed history to advanced formalized analyses to make inferences on multisector, multiscale vulnerabilities and resilience. Exploratory modeling approaches can help fashion experiments with large numbers of alternative hypotheses on the co-evolutionary dynamics of influences, stressors, as well as path-dependent changes in the form and function of human-natural systems <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id130" title="Christopher P. Weaver, Robert J. Lempert, Casey Brown, John A. Hall, David Revell, and Daniel Sarewitz. Improving the contribution of climate model information to decision making: the value and demands of robust decision frameworks. Wiley Interdisciplinary Reviews: Climate Change, 4(1):39–60, 2013. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/wcc.202 (visited on 2019-10-01), doi:10.1002/wcc.202.">37</a>]</span>. The aim of this text is to therefore guide the reader through the use of sensitivity analysis (SA) methods across these perspectives on diagnostic and exploratory modeling.</p>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#overview-of-model-diagnostics">2.1. Overview of model diagnostics</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#perspectives-on-diagnostic-model-evaluation">2.2. Perspectives on diagnostic model evaluation</a></li>
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+  <section id="diagnostic-modeling-overview-and-perspectives">
+<span id="diagnostic-modeling"></span><h1><span class="section-number">2. </span>Diagnostic Modeling Overview and Perspectives<a class="headerlink" href="#diagnostic-modeling-overview-and-perspectives" title="Link to this heading">#</a></h1>
+<p>This text prescribes a formal model diagnostic approach that is a deliberative and iterative combination of state-of-the-art UC and global sensitivity analysis techniques that progresses from observed history-based fidelity evaluations to forward looking resilience and vulnerability inferences <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id46" title="Hoshin V. Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008. Publisher: Wiley Online Library.">12</a>, <a class="reference internal" href="R.Bibliography.html#id47" title="Antonia Hadjimichael, Julianne Quinn, and Patrick Reed. Advancing Diagnostic Model Evaluation to Better Understand Water Shortage Mechanisms in Institutionally Complex River Basins. Water Resources Research, 56(10):e2020WR028079, 2020. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR028079 (visited on 2020-10-16), doi:10.1029/2020WR028079.">13</a>]</span>.</p>
+<section id="overview-of-model-diagnostics">
+<h2><span class="section-number">2.1. </span>Overview of model diagnostics<a class="headerlink" href="#overview-of-model-diagnostics" title="Link to this heading">#</a></h2>
+<p>Model diagnostics provide a rich basis for hypothesis testing, model innovation, and improved inferences when classifying what is controlling highly consequential results (e.g., vulnerability or resilience in coupled human-natural systems). <a class="reference internal" href="#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a>, adapted from <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id6" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">6</a>]</span>, presents idealized illustrations of the relationship between UC and global sensitivity analysis for two coupled simulation models. The figure illustrates how UC can be used to address how uncertainties in various modeling decisions (e.g., data inputs, parameters, model structures, coupling relationships) can be sampled and simulated to yield the empirical model output distribution(s) of interest. Monte Carlo frameworks allow us to sample and propagate (or integrate) the ensemble response of the model(s) of focus. The first step of any UC analysis is the specification of the initial input distributions as illustrated in <a class="reference internal" href="#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a>. The second step is to perform the Monte Carlo simulations. The question can then be raised, which of the modeling assumptions in our Monte Carlo experiment are the most responsible for the resulting output uncertainty. We can answer this question using global sensitivity analysis as illustrated in <a class="reference internal" href="#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a>. Global sensitivity analysis can be defined as a formal Monte Carlo sampling and analysis of modeling choices (structures, parameters, inputs) to quantify their influence on direct model outputs (or output-informed metrics). UC experiments by themselves do not explain why a particular uncertain outcome is produced, but produce distributions of model outcomes, as portrayed by the yellow curve. The pie chart shown in <a class="reference internal" href="#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a> is a conceptual representation of the results of using a global sensitivity analysis to identify those factors that are most dominantly influencing results, either individually or interactively <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id48" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global Sensitivity Analysis: The Primer. Wiley-Interscience, Chichester, England ; Hoboken, NJ, 1 edition edition, February 2008. ISBN 978-0-470-05997-5.">14</a>]</span>.</p>
+<figure class="margin-caption align-center" id="id23">
+<span id="figure-2-1"></span><a class="reference internal image-reference" href="_images/figure2_1_idealized_uc.png"><img alt="Figure 2.1" src="_images/figure2_1_idealized_uc.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 2.1 </span><span class="caption-text">Idealized uncertainty characterization and global sensitivity analysis for two coupled simulation models. Uncertainty coming from various sources (e.g., inputs, model structures, coupling relationships) is propagated through the coupled model(s) to generate empirical distributions of outputs of interest (uncertainty characterization). This model output uncertainty can be decomposed to its origins, by means of sensitivity analysis. Figure adapted from <span id="id4">Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id6" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">6</a>]</span>.</span><a class="headerlink" href="#id23" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>UC and global sensitivity analysis are not independent modeling analyses. As illustrated here, any global sensitivity analysis requires an initial UC hypothesis in the form of statistical assumptions and representations for the modeling choices of focus (structural, parametric, and data inputs). Information from these two model diagnostic tools can then be used to inform data needs for future model runs, experiments to reduce the uncertainty present, or the simplification or enhancement of the model where necessary. Together UC and global sensitivity analysis provide a foundation for diagnostic exploratory modeling that has a consistent focus on the assumptions, structural model forms, alternative parameterizations, and input data sets that are used to characterize the behavioral space of one or more models.</p>
+</section>
+<section id="perspectives-on-diagnostic-model-evaluation">
+<span id="perspectives"></span><h2><span class="section-number">2.2. </span>Perspectives on diagnostic model evaluation<a class="headerlink" href="#perspectives-on-diagnostic-model-evaluation" title="Link to this heading">#</a></h2>
+<p>When we judge or diagnose models, the terms “verification” and “validation” are commonly used. However, their appropriateness in the context of numerical models representing complex coupled human-natural systems is questionable <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id54" title="Keith Beven. Towards a coherent philosophy for modelling the environment. Proceedings of the royal society of London. Series A: mathematical, physical and engineering sciences, 458(2026):2465–2484, 2002.">15</a>, <a class="reference internal" href="R.Bibliography.html#id118" title="Naomi Oreskes, Kristin Shrader-Frechette, and Kenneth Belitz. Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences. Science, 263(5147):641–646, February 1994. URL: https://science.sciencemag.org/content/263/5147/641 (visited on 2020-04-15), doi:10.1126/science.263.5147.641.">16</a>]</span>. The core issue relates to the fact that these systems are often not fully known or perfectly implemented when modeled. Rather, they are defined within specific system framings and boundary conditions in an evolving learning process with the goal of making continual progress towards attaining higher levels of fidelity in capturing behaviors or properties of interest. Evaluating the fidelity of a model’s performance can be highly challenging. For example, the observations used to evaluate the fidelity of parameterized processes are often measured at a finer resolution than what is represented in the model, creating the challenge of how to manage their relative scales when performing evaluation. In other cases, numerical models may neglect or simplify system processes because sufficient data is not available or the physical mechanisms are not fully known. If sufficient agreement between prediction and observation is not achieved, it is challenging to know whether these types of modeling choices are the cause, or if other issues, such as deficiencies in the input parameters and/or other modeling assumptions are the true cause of errors. Even if there is high agreement between prediction and observation, the model cannot necessarily be considered validated, as it is always possible that the right values were produced for the wrong reasons. For example, low error can stem from a situation where different errors in underlying assumptions or parameters cancel each other out (“compensatory errors”). Furthermore, coupled human-natural system models are often subject to “equifinality”, a situation where multiple parameterized formulations can produce similar outputs or equally acceptable representations of the observed data. There is therefore no uniquely “true” or validated model, and the common practice of selecting “the best” deterministic calibration set is more of an assumption than a finding <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id49" title="Keith Beven. Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in water resources, 16(1):41–51, 1993.">17</a>, <a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>]</span>. The situation becomes even more tenuous when observational data is limited in its scope and/or quality to be insufficient to distinguish model representations or their performance differences.</p>
+<p>These limitations on model verification undermine any purely positivist treatment of model validity: that a model should correctly and precisely represent reality to be valid. Under this perspective, closely related to empiricism, statistical tests should be used to compare the model’s output with observations and only through empirical verification can a model or theory be deemed credible. A criticism to this viewpoint (besides the aforementioned challenges for model verification) is that it reduces the justification of a model to the single criterion of predictive ability and accuracy <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id119" title="Yaman Barlas and Stanley Carpenter. Philosophical roots of model validation: Two paradigms. System Dynamics Review, 6(2):148–166, 1990. doi:10.1002/sdr.4260060203.">19</a>]</span>. Authors have argued that this ignores the explanatory power held in models and other procedures, which can also advance scientific knowledge <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id120" title="Stephen Toulmin. From form to function: philosophy and history of science in the 1950s and now. Daedalus, pages 143–162, 1977. Publisher: JSTOR.">20</a>]</span>. These views gave rise to relativist perspectives of science, which instead place more value on model utility in terms of fitness for a specific purpose or inquiry, rather than representational accuracy and predictive ability <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id121" title="George B. Kleindorfer, Liam O'Neill, and Ram Ganeshan. Validation in simulation: Various positions in the philosophy of science. Management Science, 44(8):1087–1099, 1998. Publisher: INFORMS.">21</a>]</span>. This viewpoint appears to be most prevalent among practitioners seeking decision-relevant insights (i.e., inspire new views vs. predict future conditions). The relativist perspective argues for the use of models as heuristics that can enhance our understanding and conceptions of system behaviors or possibilities <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id64" title="Sibel Eker, Elena Rovenskaya, Michael Obersteiner, and Simon Langan. Practice and perspectives in the validation of resource management models. Nature communications, 9(1):1–10, 2018.">22</a>]</span>. In contrast, natural sciences favor a positivist perspective, emphasizing similarity between simulation and observation even in application contexts where it is clear that projections are being made for conditions that have never been observed and the system of focus will have evolved structurally beyond the model representation being employed (e.g., decadal to centennial evolution of human-natural systems).</p>
+<p>These differences in prevalent perspectives are mirrored in how model validation is defined by the two camps: From the relativist perspective, validation is seen as a process of incremental “confidence building” in a model as a mechanism for insight <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id122" title="Yaman Barlas. Formal aspects of model validity and validation in system dynamics. System Dynamics Review: The Journal of the System Dynamics Society, 12(3):183–210, 1996. Publisher: Wiley Online Library.">23</a>]</span>, whereas in natural sciences validation is framed as a way to classify a model as having an acceptable representation of physical reality <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id118" title="Naomi Oreskes, Kristin Shrader-Frechette, and Kenneth Belitz. Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences. Science, 263(5147):641–646, February 1994. URL: https://science.sciencemag.org/content/263/5147/641 (visited on 2020-04-15), doi:10.1126/science.263.5147.641.">16</a>]</span>. Even though the relativist viewpoint does not dismiss the importance of representational accuracy, it does place it within a larger process of establishing confidence through a variety of tools. These tools, not necessarily quantitative, include communicating information between practitioners and modelers, interpreting a multitude of model outputs, and contrasting preferences and viewpoints.</p>
+<p>On the technical side of the argument, differing views on the methodology of model validation appear as early as in the 1960’s. <span id="id13">Naylor and Finger [<a class="reference internal" href="R.Bibliography.html#id123" title="Thomas H. Naylor and Joseph Michael Finger. Verification of computer simulation models. Management science, 14(2):B–92, 1967. Publisher: INFORMS.">24</a>]</span> argue that model validation should not be limited to a single metric or test of performance (e.g., a single error metric), but should rather be extended to multiple tests that reflect different aspects of a model’s structure and behavior. This and similar arguments are made in literature to this day <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id46" title="Hoshin V. Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008. Publisher: Wiley Online Library.">12</a>, <a class="reference internal" href="R.Bibliography.html#id52" title="Keith J Beven. On hypothesis testing in hydrology: why falsification of models is still a really good idea. Wiley Interdisciplinary Reviews: Water, 5(3):e1278, 2018.">25</a>, <a class="reference internal" href="R.Bibliography.html#id51" title="Hoshin V Gupta, Martyn P Clark, Jasper A Vrugt, Gab Abramowitz, and Ming Ye. Towards a comprehensive assessment of model structural adequacy. Water Resources Research, 2012.">26</a>, <a class="reference internal" href="R.Bibliography.html#id124" title="Praveen Kumar. Typology of hydrologic predictability. Water Resources Research, 2011. URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2010WR009769 (visited on 2020-04-15), doi:10.1029/2010WR009769.">27</a>, <a class="reference internal" href="R.Bibliography.html#id125" title="Grey S. Nearing, Benjamin L. Ruddell, Andrew R. Bennett, Cristina Prieto, and Hoshin V. Gupta. Does Information Theory Provide a New Paradigm for Earth Science? Hypothesis Testing. Water Resources Research, 56(2):e2019WR024918, 2020. doi:10.1029/2019WR024918.">28</a>]</span> and are primarily founded on two premises. First, that even though modelers widely recognize that their models are abstractions of the truth, they still make truth claims based on traditional performance metrics that measure the divergence of their model from observation <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id125" title="Grey S. Nearing, Benjamin L. Ruddell, Andrew R. Bennett, Cristina Prieto, and Hoshin V. Gupta. Does Information Theory Provide a New Paradigm for Earth Science? Hypothesis Testing. Water Resources Research, 56(2):e2019WR024918, 2020. doi:10.1029/2019WR024918.">28</a>]</span>. Second, that the natural systems mimicked by the models contain many processes that exhibit significant heterogeneity at various temporal and spatial scales. This heterogeneity is lost when a single performance measure is used, as a result of the inherent loss of process information occurring when transitioning from a highly dimensional and interactive system to the dimension of a single metric <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id54" title="Keith Beven. Towards a coherent philosophy for modelling the environment. Proceedings of the royal society of London. Series A: mathematical, physical and engineering sciences, 458(2026):2465–2484, 2002.">15</a>]</span>. These arguments are further elaborated in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#sensitivity-analysis"><span class="std std-numref">Chapter 4</span></a>.</p>
+<p>Multiple authors have proposed that the traditional reliance on single measures of model performance should be replaced by the evaluation of several model signatures (characteristics) to identify model structural errors and achieve a sufficient assessment of model performance <span id="id17">[<a class="reference internal" href="R.Bibliography.html#id46" title="Hoshin V. Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008. Publisher: Wiley Online Library.">12</a>, <a class="reference internal" href="R.Bibliography.html#id126" title="Hoshin Vijai Gupta, Soroosh Sorooshian, and Patrice Ogou Yapo. Toward improved calibration of hydrologic models: Multiple and noncommensurable measures of information. Water Resources Research, 34(4):751–763, 1998. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/97WR03495 (visited on 2020-04-07), doi:10.1029/97WR03495.">29</a>, <a class="reference internal" href="R.Bibliography.html#id168" title="Francesca Pianosi and Thorsten Wagener. Understanding the time-varying importance of different uncertainty sources in hydrological modelling using global sensitivity analysis. Hydrological Processes, pages 3991–4003, November 2017. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/hyp.10968%4010.1111/%28ISSN%291099-1085.Kieth-Beven, doi:10.1002/hyp.10968&#64;10.1111/(ISSN)1099-1085.Kieth-Beven.">30</a>, <a class="reference internal" href="R.Bibliography.html#id167" title="Charles Rougé, Patrick M. Reed, Danielle S. Grogan, Shan Zuidema, Alexander Prusevich, Stanley Glidden, Jonathan R. Lamontagne, and Richard B. Lammers. Coordination and Control: Limits in Standard Representations of Multi-Reservoir Operations in Hydrological Modeling. Hydrology and Earth System Sciences Discussions, pages 1–37, November 2019. URL: https://www.hydrol-earth-syst-sci-discuss.net/hess-2019-589/, doi:https://doi.org/10.5194/hess-2019-589.">31</a>]</span>. There is however a point of departure here, especially when models are used to produce inferences that can inform decisions. When agencies and practitioners use models of their systems for public decisions, those models have already met sufficient conditions for credibility (e.g., acceptable representational fidelity), but may face broader tests on their salience and legitimacy in informing negotiated decisions <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id64" title="Sibel Eker, Elena Rovenskaya, Michael Obersteiner, and Simon Langan. Practice and perspectives in the validation of resource management models. Nature communications, 9(1):1–10, 2018.">22</a>, <a class="reference internal" href="R.Bibliography.html#id127" title="David W. Cash, William C. Clark, Frank Alcock, Nancy M. Dickson, Noelle Eckley, David H. Guston, Jill Jäger, and Ronald B. Mitchell. Knowledge systems for sustainable development. Proceedings of the national academy of sciences, 100(14):8086–8091, 2003. Publisher: National Acad Sciences.">32</a>, <a class="reference internal" href="R.Bibliography.html#id128" title="Dave D. White, Amber Wutich, Kelli L. Larson, Patricia Gober, Timothy Lant, and Clea Senneville. Credibility, salience, and legitimacy of boundary objects: water managers' assessment of a simulation model in an immersive decision theater. Science and Public Policy, 37(3):219–232, April 2010. Publisher: Oxford Academic. URL: https://academic.oup.com/spp/article/37/3/219/1626552 (visited on 2020-05-12), doi:10.3152/030234210X497726.">33</a>]</span>. This presents a new challenge to model validation, that of selecting decision-relevant performance metrics, reflective of the system’s stakeholders’ viewpoints, so that the most consequential uncertainties are identified and addressed <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id129" title="Andrea Saltelli and Silvio Funtowicz. When all models are wrong. Issues in Science and Technology, 30(2):79–85, 2014. Publisher: JSTOR.">34</a>]</span>. For complex multisector models at the intersection of climatic, hydrologic, agricultural, energy, or other processes, the output space is made up of a multitude of states and variables, with very different levels of salience to the system’s stakeholders and to their goals being achieved <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id203" title="Thorsten Wagener and Francesca Pianosi. What has Global Sensitivity Analysis ever done for us? A systematic review to support scientific advancement and to inform policy-making in earth system modelling. Earth-Science Reviews, 194:1–18, July 2019. URL: https://www.sciencedirect.com/science/article/pii/S0012825218300990 (visited on 2021-08-30), doi:10.1016/j.earscirev.2019.04.006.">35</a>]</span>. This is further complicated when such systems are also institutionally and dynamically complex. As a result, a broader set of qualitative and quantitative performance metrics is necessary to evaluate models of such complex systems, one that embraces the plurality of value systems, agencies and perspectives present. For IM3, even though the goal is to develop better projections of future vulnerability and resilience in co-evolving human-natural systems and not to provide decision support per se, it is critical for our multisector, multiscale model evaluation processes to represent stakeholders’ adaptive decision processes credibly.</p>
+<p>As a final point, when a model is used in a projection mode, its results are also subject to additional uncertainty, as there is no guarantee that the model’s functionality and predictive ability will stay the same as the baseline, where the verification and validation tests were conducted. This challenge requires an additional expansion of the scope of model evaluation: a broader set of uncertain conditions needs to be explored, spanning beyond historical observation and exploring a wide range of unprecedented conditions. This perspective on modeling, termed exploratory <span id="id21">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>]</span>, views models as computational experiments that can be used to explore vast ensembles of potential scenarios to identify those with consequential effects. Exploratory modeling literature explicitly orients experiments toward stakeholder consequences and decision-relevant inferences and shifts the focus from predicting future conditions to <em>discovering</em> which conditions lead to undesirable or desirable consequences.</p>
+<p>This evolution in modeling perspectives can be mirrored by the IM3 family of models in a progression from evaluating models relative to observed history to advanced formalized analyses to make inferences on multisector, multiscale vulnerabilities and resilience. Exploratory modeling approaches can help fashion experiments with large numbers of alternative hypotheses on the co-evolutionary dynamics of influences, stressors, as well as path-dependent changes in the form and function of human-natural systems <span id="id22">[<a class="reference internal" href="R.Bibliography.html#id130" title="Christopher P. Weaver, Robert J. Lempert, Casey Brown, John A. Hall, David Revell, and Daniel Sarewitz. Improving the contribution of climate model information to decision making: the value and demands of robust decision frameworks. Wiley Interdisciplinary Reviews: Climate Change, 4(1):39–60, 2013. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/wcc.202 (visited on 2019-10-01), doi:10.1002/wcc.202.">37</a>]</span>. The aim of this text is to therefore guide the reader through the use of sensitivity analysis (SA) methods across these perspectives on diagnostic and exploratory modeling.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The following articles are suggested as fundamental reading for the information presented in this section:</p>
+<ul class="simple">
+<li><p>Naomi Oreskes, Kristin Shrader–Frechette, and Kenneth Belitz. Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences. <em>Science</em>, 263 (5147): 641-646, February 1994. URL: <a class="reference external" href="https://science.sciencemag.org/content/263/5147/641">https://science.sciencemag.org/content/263/5147/641</a>. DOI: <a class="reference external" href="https://doi.org/10.1126/science.263.5147.641">https://doi.org/10.1126/science.263.5147.641</a>.</p></li>
+<li><p>Keith Beven. Towards a coherent philosophy for modelling the environment. <em>Proceedings of the Royal Society of London</em>. Series A: mathematical, physical and engineering sciences, 458 (2026): 2465-2484, 2002.</p></li>
+<li><p>Eker, S., Rovenskaya, E., Obersteiner, M., Langan, S., 2018. Practice and perspectives in the validation of resource management models. <em>Nature Communications</em> 9, 1–10. <a class="reference external" href="https://doi.org/10.1038/s41467-018-07811-9">https://doi.org/10.1038/s41467-018-07811-9</a></p></li>
+</ul>
+<p>The following articles can be used as supplemental reading:</p>
+<ul class="simple">
+<li><p>Canham, C.D., Cole, J.J., Lauenroth, W.K. (Eds.), 2003. Models in Ecosystem Science. Princeton University Press. <a class="reference external" href="https://doi.org/10.2307/j.ctv1dwq0tq">https://doi.org/10.2307/j.ctv1dwq0tq</a></p></li>
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+    <h1>Global Versus Local Sensitivity</h1>
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+  <section id="global-versus-local-sensitivity">
+<span id="global-vs-local"></span><h1>Global Versus Local Sensitivity<a class="headerlink" href="#global-versus-local-sensitivity" title="Link to this heading">#</a></h1>
+<p>Out of the several definitions for sensitivity analysis presented in the literature, the most widely used has been proposed by <span id="id1">Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span> as “the study of how uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input”. In other words, sensitivity analysis explores the relationship between the model’s <span class="math notranslate nohighlight">\(N\)</span> input variables, <span class="math notranslate nohighlight">\(x=[x_1,x_2,...,x_N]\)</span>, and <span class="math notranslate nohighlight">\(M\)</span> output variables, <span class="math notranslate nohighlight">\(y=[y_1,y_2,...,y_M]\)</span> with <span class="math notranslate nohighlight">\(y=g(x)\)</span>, where <span class="math notranslate nohighlight">\(g\)</span> is the model that maps the model inputs to the outputs <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>]</span>.</p>
+<p>Historically, there have been two broad categories of sensitivity analysis techniques: local and global. Local sensitivity analysis is performed by varying model parameters around specific reference values, with the goal of exploring how small input perturbations influence model performance. Due to its ease-of-use and limited computational demands, this approach has been widely used in literature, but has important limitations <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id18" title="O Rakovec, Mary C Hill, MP Clark, AH Weerts, AJ Teuling, and R Uijlenhoet. Distributed evaluation of local sensitivity analysis (delsa), with application to hydrologic models. Water Resources Research, 50(1):409–426, 2014.">40</a>, <a class="reference internal" href="R.Bibliography.html#id19" title="Andrea Saltelli and Paola Annoni. How to avoid a perfunctory sensitivity analysis. Environmental Modelling &amp; Software, 25(12):1508–1517, 2010.">41</a>]</span>. If the model is not linear, the results of local sensitivity analysis can be heavily biased, as they are strongly influenced by independence assumptions and a limited exploration of model inputs (e.g., <span id="id4">Tang <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id20" title="Yong Tang, Patrick Reed, Thibaut Wagener, and K van Werkhoven. Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation. Hydrology and Earth System Sciences, 11(2):793–817, 2007.">42</a>]</span>). If the model’s factors interact, local sensitivity analysis will underestimate their importance, as it does not account for those effects (e.g., <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id23" title="Nicholas AS Hamm, Jim W Hall, and MG Anderson. Variance-based sensitivity analysis of the probability of hydrologically induced slope instability. Computers &amp; geosciences, 32(6):803–817, 2006.">43</a>]</span>). In general, as local sensitivity analysis only partially and locally explores a model’s parametric space, it is not considered a valid approach for nonlinear models <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id24" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">44</a>]</span>. This is illustrated in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (a-b), presenting contour plots of a model response (<span class="math notranslate nohighlight">\(y\)</span>) with an additive linear model (a) and with a nonlinear model (b). In a linear model without interactions between the input terms <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span>, local sensitivity analysis (assuming deviations from some reference values) can produce appropriate sensitivity indices (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (a)). If however, factors <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> interact, the local and partial consideration of the space can not properly account for each factor’s effects on the model response (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (b)), as it is only informative at the reference value where it is applied. In contrast, a global sensitivity analysis varies uncertain factors within the entire feasible space of variable model responses (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (c)). This approach reveals the global effects of each parameter on the model output, including any interactive effects. For models that cannot be proven linear, global sensitivity analysis is preferred and this text is primarily discussing global sensitivity analysis methods. In the text that follows, whenever we use the term sensitivity analysis we are referring to its global application.</p>
+<figure class="margin-caption align-center" id="id7">
+<span id="figure-3-1"></span><a class="reference internal image-reference" href="_images/figure3_1_global_versus_local.png"><img alt="Figure 3.1" src="_images/figure3_1_global_versus_local.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Treatment of a two-dimensional space of variability by local (panels a-b) and global (panel c) sensitivity analyses. Panels depict contour plots with the value of a model response (<span class="math notranslate nohighlight">\(y\)</span>) changing with changes in the values of input terms <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span>. Local sensitivity analysis is only an appropriate approach to sensitivity in the case of linear models without interactions between terms, for example in panel (a), where <span class="math notranslate nohighlight">\(y=3x_1+5x_2\)</span>. In the case of more complex models, for example in panels (b-c), where <span class="math notranslate nohighlight">\(y={1 \above 1pt e^{x^2_1+x^2_2}} + {50 \above 1pt e^{(0.1x_1)^2+(0.1x_2)^3}}\)</span>, local sensitivity will miscalculate sensitivity indices as the assessed changes in the value <span class="math notranslate nohighlight">\(y\)</span> depend on the assumed base values chose for <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> (panel (b)). In these cases, global sensitivity methods should be used instead (panel (c)). The points in panel (c) are generated using a uniform random sample of <span class="math notranslate nohighlight">\(n=50\)</span>, but many other methods are available.</span><a class="headerlink" href="#id7" title="Link to this image">#</a></p>
+</figcaption>
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+    <h1>Why Perform Sensitivity Analysis</h1>
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+  <section id="why-perform-sensitivity-analysis">
+<span id="why-sa"></span><h1>Why Perform Sensitivity Analysis<a class="headerlink" href="#why-perform-sensitivity-analysis" title="Link to this heading">#</a></h1>
+<p>It is important to understand the many ways in which a SA might be of use to your modeling effort. Most commonly, one might be motivated to perform sensitivity analysis for the following reasons:</p>
+<p><em>Model evaluation</em>: Sensitivity analysis can be used to gauge model inferences when assumptions about the structure of the model or its parameterization are dubious or have changed. For instance, consider a numerical model that uses a set of calibrated parameter values to produce outputs, which we then use to inform decisions about the real-world system represented. One might like to know if small changes in these parameter values significantly change this model’s output and the decisions it informs or if, instead, our parameter inferences yield stable model behavior regardless of the uncertainty present in the specific parameterized processes or properties. This can either discredit or lend credence to the model at hand, as well as any inferences drawn that are founded on its accurate representation of the system. Sensitivity analysis can identify which uncertain model factors cause this undesirable model behavior.</p>
+<p><em>Model simplification</em>: Sensitivity analysis can also be used to identify factors or components of the model that appear to have limited effects on direct outputs or metrics of interest. Consider a model that has been developed in an organization for the purposes of a specific research question and is later used in the context of a different application. Some processes represented in significant detail might no longer be of the same importance while consuming significant data or computational resources, as different outputs might be pertinent to the new application. Sensitivity analysis can be used to identify unimportant model components and simplify them to nominal values and reduced model forms. Model complexity and computational costs can therefore be reduced.</p>
+<p><em>Model refinement</em>: Alternatively, sensitivity analysis can reveal the factors or processes that are highly influential to the outputs or metrics of interest, by assessing their relative importance. In the context of model evaluation, this can inform which model components warrant additional investigation or measurement so the uncertainty surrounding them and the resulting model outputs or metrics of interest can be reduced.</p>
+<p><em>Exploratory modeling</em>: When sufficient credence has been established in the model, sensitivity analysis can be applied to a host of other inquiries. Inferences about the factors and processes that most (or least) control a model’s outputs of interest can be extrapolated to the real system they represent and be used in a heuristic manner to inform model-based inferences. On this foundation, a model paired with the advanced techniques presented in this text can be used to “discover” decision relevant and highly consequential outcomes (i.e., scenario discovery, discussed in more detail in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a> <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>, <a class="reference internal" href="R.Bibliography.html#id16" title="Benjamin P Bryant and Robert J Lempert. Thinking inside the box: a participatory, computer-assisted approach to scenario discovery. Technological Forecasting and Social Change, 77(1):34–49, 2010.">45</a>]</span>).</p>
+<p>The nature and context of the model shapes the specific objectives of applying a sensitivity analysis, as well as methods and tools most appropriate and defensible for each application setting <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id203" title="Thorsten Wagener and Francesca Pianosi. What has Global Sensitivity Analysis ever done for us? A systematic review to support scientific advancement and to inform policy-making in earth system modelling. Earth-Science Reviews, 194:1–18, July 2019. URL: https://www.sciencedirect.com/science/article/pii/S0012825218300990 (visited on 2021-08-30), doi:10.1016/j.earscirev.2019.04.006.">35</a>, <a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id25" title="Andrea Saltelli and Stefano Tarantola. On the relative importance of input factors in mathematical models: safety assessment for nuclear waste disposal. Journal of the American Statistical Association, 97(459):702–709, 2002.">46</a>]</span>. The three most common sensitivity analysis modes (<em>Factor Prioritization</em>, <em>Factor Fixing</em>, and <em>Factor Mapping</em>) are presented below, but the reader should be aware that other uses have been proposed in the literature (e.g., <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id26" title="Barry Anderson, Emanuele Borgonovo, Marzio Galeotti, and Roberto Roson. Uncertainty in climate change modeling: can global sensitivity analysis be of help? Risk analysis, 34(2):271–293, 2014.">47</a>, <a class="reference internal" href="R.Bibliography.html#id27" title="Emanuele Borgonovo. Sensitivity analysis with finite changes: an application to modified eoq models. European Journal of Operational Research, 200(1):127–138, 2010.">48</a>]</span>).</p>
+<p><em>Factor prioritization</em>: This sensitivity analysis application mode (also referred to as <em>factor ranking</em>) refers to when one would like to identify the uncertain factors that have the greatest impact on the variability of the output, and which, when fixed to their true value (i.e., if there were no uncertainty regarding their value), would lead to the greatest reduction in output variability <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. Information from this type of analysis can be crucial to model improvement as these factors can become the focus of future measurement campaigns or numerical experiments so that uncertainty in the model output can be reduced. The impact of each uncertain input on the variance of the model output is often used as the criterion for factor prioritization. <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (a) shows the effects of three uncertain variables (<span class="math notranslate nohighlight">\(X_1\)</span>, <span class="math notranslate nohighlight">\(X_2\)</span>, and <span class="math notranslate nohighlight">\(X_3\)</span>) on the variance of output <span class="math notranslate nohighlight">\(Y\)</span>. <span class="math notranslate nohighlight">\(V(E(Y|X_i))\)</span> indicates the variance in <span class="math notranslate nohighlight">\(Y\)</span> if factor <span class="math notranslate nohighlight">\(X_i\)</span> is left to vary freely while all other factors remain fixed to nominal values. In this case, factor <span class="math notranslate nohighlight">\(X_2\)</span> makes the largest contribution to the variability of output <span class="math notranslate nohighlight">\(Y\)</span> and it should therefore be prioritized. In the context of risk analysis, factor prioritization can be used to reduce output variance to below a given tolerable threshold (also known as variance cutting).</p>
+<p><em>Factor fixing</em>: This mode of sensitivity analysis (also referred to as <em>factor screening</em>) aims to identify the model components that have a negligible effect or make no significant contributions to the variability of the outputs or metrics of interest (usually referred to as non-influential <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>). In the stylized example of <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (a), <span class="math notranslate nohighlight">\(X_1\)</span> makes the smallest contribution to the variability of output <span class="math notranslate nohighlight">\(Y\)</span> suggesting that the uncertainty in its value could be negligible and the factor itself fixed in subsequent model executions. Eliminating these factors or processes in the model or fixing them to a nominal value can help reduce model complexity as well as the unnecessary computational burden of subsequent model runs, results processing, or other sensitivity analyses (the fewer uncertain factors considered, the fewer runs are necessary to illuminate their effects on the output). Significance of the outcome can be gauged in a variety of manners, depending on the application. For instance, if applying a variance-based method, a minimum threshold value of contribution to the variance could be considered as a significance ‘cutoff’, and factors with indices below that value can be considered non-influential. Conclusions about factor fixing should be made carefully, considering all of the effects a factor has, individually and in interaction with other factors (explained in more detail in the <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#variance-based-methods"><span class="std std-numref">Chapter 3.4.5</span></a>).</p>
+<p><em>Factor mapping</em>: Finally, factor mapping can be used to pinpoint which values of uncertain factors lead to model outputs within a given range of the output space <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. In the context of model diagnostics, it is possible that the model’s output changes in ways considered impossible based on the represented processes, or other observed evidence. In this situation, factor mapping can be used to identify which uncertain model factors cause this undesirable model behavior by ‘filtering’ model runs that are considered ‘non-behavioral’ <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id29" title="Neil R Edwards, David Cameron, and Jonathan Rougier. Precalibrating an intermediate complexity climate model. Climate dynamics, 37(7):1469–1482, 2011.">50</a>, <a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id31" title="RC Spear and GM Hornberger. Eutrophication in peel inlet—ii. identification of critical uncertainties via generalized sensitivity analysis. Water research, 14(1):43–49, 1980.">52</a>]</span>. In <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (b), region <span class="math notranslate nohighlight">\(B\)</span> of the output space <span class="math notranslate nohighlight">\(Y\)</span> denotes the set of behavioral model outcomes and region <span class="math notranslate nohighlight">\(\bar{B}\)</span> denotes the set of non-behavioral outcomes, resulting from the entirety of input space <span class="math notranslate nohighlight">\(X\)</span>. Factor mapping refers to the process of tracing which factor values of input space <span class="math notranslate nohighlight">\(X\)</span> produce the behavioral model outcomes in the output space.</p>
+<figure class="margin-caption align-center" id="id8">
+<span id="figure-3-2"></span><a class="reference internal image-reference" href="_images/figure3_2_factor_mapping.png"><img alt="Figure 3.2" src="_images/figure3_2_factor_mapping.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Factor prioritization, factor fixing and factor mapping settings of sensitivity analysis.</span><a class="headerlink" href="#id8" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The language used above reflects a use of sensitivity analysis for model fidelity evaluation and refinement. However, as previously mentioned, when a model has been established as a sufficiently accurate representation of the system, sensitivity analysis can produce additional inferences (i.e., exploratory modeling and scenario discovery). For instance, under the factor mapping use, the analyst can now focus on undesirable system states and discover which factors are most responsible for them: for instance, “population growth of above 25% would be responsible for unacceptably high energy demands”. Factor prioritization and factor fixing can be used to make equivalent inferences, such as “growing populations and increasing temperatures are the leading factors for changing energy demands” (prioritizing of factors) or “changing dietary needs are inconsequential to increasing energy demands for this region” (a factor that can be fixed in subsequent model runs). All these inferences hinge on the assumption that the real system’s stakeholders consider the model states faithful enough representations of system states. As elaborated in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>, this view on sensitivity analysis is founded on a relativist perspective on modeling, which tends to place more value on model usefulness rather than strict accuracy of representation in terms of error. As such, sensitivity analysis performed with decision-making relevance in mind will focus on model outputs or metrics that are consequential and decision relevant (e.g., energy demand in the examples above).</p>
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+    <h1>One-At-a-Time (OAT)</h1>
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+  <section id="one-at-a-time-oat">
+<h1>One-At-a-Time (OAT)<a class="headerlink" href="#one-at-a-time-oat" title="Link to this heading">#</a></h1>
+<p>In this approach, only one model factor is changed at a time while all others are kept fixed across each iteration in a sampling sequence. The OAT method assumes that model factors of focus are linearly independent (i.e., there are no interactions) and can analyze how factors individually influence model outputs or metrics of interest. While popular given its ease of implementation, OAT  is ultimately limited in its exploration of a model’s sensitivities <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. It is primarily used with local sensitivity techniques with similar criticisms: applying this sampling scheme on a system with nonlinear and interactive processes will miss important information on the effect uncertain factors have on the model. OAT samplings can be repeated multiple times in a more sophisticated manner and across different locations of the parameter space to overcome some of these challenges, which would increase computational costs and negate the main reasons for its selection. Given these limitations OAT methods could be used as preliminary, low-cost analyses of the factors’ individual effects, but should ultimately be complemented with more sophisticated methods.</p>
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+    <h1>Full and Fractional Factorial Sampling</h1>
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+  <section id="full-and-fractional-factorial-sampling">
+<h1>Full and Fractional Factorial Sampling<a class="headerlink" href="#full-and-fractional-factorial-sampling" title="Link to this heading">#</a></h1>
+<p>In full factorial sampling each factor is treated as being discrete by considering two or more levels (or intervals) of its values. The sampling process then generates samples within each possible combination of levels, corresponding to each parameter. This scheme produces a more comprehensive sampling of the factors’ variability space, as it accounts for all candidate combinations of factor levels (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a)). If the number of levels is the same across all factors, the number of generated samples is estimated using <span class="math notranslate nohighlight">\(n^k\)</span>, where <span class="math notranslate nohighlight">\(n\)</span> is the number of levels and <span class="math notranslate nohighlight">\(k\)</span> is the number of factors. For example, <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a) presents a full factorial sampling of three uncertain factors <span class="math notranslate nohighlight">\((x_1,\)</span> <span class="math notranslate nohighlight">\(x_2,\)</span> and <span class="math notranslate nohighlight">\(x_3)\)</span>, each considered as having four discrete levels. The total number of samples necessary for such an experiment is <span class="math notranslate nohighlight">\(4^3=64\)</span>. As the number of factors increases, the number of simulations necessary will also grow exponentially, making full factorial sampling computationally burdensome (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (b)). As a result, it is common in the literature to apply full factorial sampling at only two levels per factor, typically the two extremes <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id32" title="Douglas C Montgomery. Design and analysis of experiments. John wiley &amp; sons, 2017.">60</a>]</span>. This significantly reduces computational burden but is only considered appropriate in cases where factors can indeed only assume two discrete values (e.g., when testing the effects of epistemic uncertainty and comparing between model structure A and model structure B). In the case of physical parameters on continuous distributions (e.g., when considering the effects of measurement uncertainty in a temperature sensor), discretizing the range of a factor to only extreme levels can bias its estimated importance.</p>
+<p>Fractional factorial sampling is a widely used alternative to full factorial sampling that allows the analyst to significantly reduce the number of simulations by focusing on the main effects of a factor and seeking to avoid model runs that yield redundant response information <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. In other words, if one can reasonably assume that higher-order interactions are negligible, information about the most significant effects and lower-order interactions (e.g., effects from pairs of factors) can be obtained using a fraction of the full factorial design. Traditionally, fractional factorial design has also been limited to two levels <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id32" title="Douglas C Montgomery. Design and analysis of experiments. John wiley &amp; sons, 2017.">60</a>]</span>, referred to as Fractional Factorial designs 2k-p <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id34" title="George EP Box and J Stuart Hunter. The 2 k—p fractional factorial designs. Technometrics, 3(3):311–351, 1961.">61</a>]</span>. Recently, Generalized Fractional Factorial designs have also been proposed that allow for the structured generation of samples at more than two levels per factor <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id33" title="Izabella Surowiec, Ludvig Vikstrom, Gustaf Hector, Erik Johansson, Conny Vikstrom, and Johan Trygg. Generalized subset designs in analytical chemistry. Analytical chemistry, 89(12):6491–6497, 2017.">62</a>]</span>. Consider a case where the modeling team dealing with the problem in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a) cannot afford to perform 64 simulations of their model. They can afford 32 runs for their experiment and instead decide to fractionally sample the variability space of their factors. A potential design of such a sampling strategy is presented in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (c).</p>
+<figure class="margin-caption align-center" id="id6">
+<span id="figure-3-3"></span><a class="reference internal image-reference" href="_images/figure3_3_alternative_designs.png"><img alt="Figure 3.3" src="_images/figure3_3_alternative_designs.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Alternative designs of experiments and their computational costs for three uncertain factors <span class="math notranslate nohighlight">\((x_1,\)</span> <span class="math notranslate nohighlight">\(x_2,\)</span> and <span class="math notranslate nohighlight">\(x_3)\)</span>. (a) Full factorial design sampling of three factors at four levels, at a total of 64 samples; (b) exponential growth of necessary number of samples when applying full factorial design at four levels; (c) fractional factorial design of three factors at four levels, at a total of 32 samples; and (d) Latin Hypercube sample of three factors with uniform distributions, at a total of 32 samples.</span><a class="headerlink" href="#id6" title="Link to this image">#</a></p>
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+  <section id="latin-hypercube-sampling-lhs">
+<h1>Latin Hypercube Sampling (LHS)<a class="headerlink" href="#latin-hypercube-sampling-lhs" title="Link to this heading">#</a></h1>
+<p>Latin hypercube sampling (LHS) <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id91" title="Michael D McKay, Richard J Beckman, and William J Conover. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21(1):239-2451, 1979.">63</a>]</span> is one of the most common methods in space-filling experimental designs. With this sampling technique, for <span class="math notranslate nohighlight">\(N\)</span> uncertain factors, an <span class="math notranslate nohighlight">\(N\)</span>-dimensional hypercube is generated, with each factor divided into an equal number of levels depending on the total number of samples to be generated. Equal numbers of samples are then randomly generated at each level, across all factors. In this manner, latin hypercube design guarantees sampling from every level of the variability space and without any overlaps. When the number of samples generated is much larger than the number of uncertain factors, LHS can be very effective in examining the effects of each factor <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. LHS is an attractive technique, because it guarantees a diverse coverage of the space, through the use of subintervals, without being constrained to discrete levels for each factor - compare <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (c) with <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (d) for the same number of samples.</p>
+<p>LHS is less effective when the number of samples is not much larger than the number of uncertain factors, and the effects of each factor cannot be appropriately distinguished. The samples between factors can also be highly correlated, biasing any subsequent sensitivity analysis results. To address this, the sampling scheme can be modified to control for the correlation in parameters while maximizing the information derived. An example of such modification is through the use of orthogonal arrays <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id21" title="Boxin Tang. Orthogonal array-based latin hypercubes. Journal of the American statistical association, 88(424):1392–1397, 1993.">64</a>]</span>.</p>
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+  <section id="low-discrepancy-sequences">
+<h1>Low-Discrepancy Sequences<a class="headerlink" href="#low-discrepancy-sequences" title="Link to this heading">#</a></h1>
+<p>Low-discrepancy sequences is another sampling technique that employs a pseudo-random generator for Monte Carlo sampling <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>, <a class="reference internal" href="R.Bibliography.html#id92" title="SK Zaremba. The mathematical basis of monte carlo and quasi-monte carlo methods. SIAM review, 10(3):303–314, 1968.">66</a>]</span>. These quasi-Monte Carlo methods eliminate ‘lumpiness’ across samples (i.e, the presence of gaps and clusters) by minimizing discrepancy across the hypercube samples. Discrepancy can be quantitatively measured using the deviations of sampled points from a uniform distribution <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>, <a class="reference internal" href="R.Bibliography.html#id95" title="Sergei Kucherenko, Daniel Albrecht, and Andrea Saltelli. Exploring multi-dimensional spaces: a comparison of latin hypercube and quasi monte carlo sampling techniques. arXiv preprint arXiv:1505.02350, 2015.">67</a>]</span>. Low-discrepancy sequences ensure that the number of samples in any subspace of the variability hypercube is approximately the same. This is not something guaranteed by Latin Hypercube sampling, and even though its design can be improved through optimization with various criteria, such adjustments are limited to small sample sizes and low dimensions <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id95" title="Sergei Kucherenko, Daniel Albrecht, and Andrea Saltelli. Exploring multi-dimensional spaces: a comparison of latin hypercube and quasi monte carlo sampling techniques. arXiv preprint arXiv:1505.02350, 2015.">67</a>, <a class="reference internal" href="R.Bibliography.html#id93" title="Bertrand Iooss, Loïc Boussouf, Vincent Feuillard, and Amandine Marrel. Numerical studies of the metamodel fitting and validation processes. arXiv preprint arXiv:1001.1049, 2010.">68</a>, <a class="reference internal" href="R.Bibliography.html#id94" title="Ruichen Jin, Wei Chen, and Agus Sudjianto. An efficient algorithm for constructing optimal design of computer experiments. In International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, volume 37009, 545–554. 2003.">69</a>, <a class="reference internal" href="R.Bibliography.html#id96" title="Max D Morris and Toby J Mitchell. Exploratory designs for computational experiments. Journal of statistical planning and inference, 43(3):381–402, 1995.">70</a>, <a class="reference internal" href="R.Bibliography.html#id97" title="Jeong-Soo Park. Optimal latin-hypercube designs for computer experiments. Journal of statistical planning and inference, 39(1):95–111, 1994.">71</a>]</span>. In contrast, the Sobol sequence <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id98" title="Ilya M Sobol. Uniformly distributed sequences with an additional uniform property. USSR Computational Mathematics and Mathematical Physics, 16(5):236–242, 1976.">72</a>, <a class="reference internal" href="R.Bibliography.html#id99" title="Il'ya Meerovich Sobol'. On the distribution of points in a cube and the approximate evaluation of integrals. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 7(4):784–802, 1967.">73</a>]</span>, one of the most widely used sampling techniques, utilizes the low-discrepancy approach to uniformly fill the sampled factor space. A core advantage of this style of sampling is that it takes far fewer samples (i.e., simulations) to attain a much lower level of error in estimating model output statistics (e.g., the mean and variance of outputs).</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial which uses Sobol sequence sampling for the purposes of a Sobol sensitivity analysis:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA using SALib Jupyter Notebook</a></p>
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+  <section id="other-types-of-sampling">
+<h1>Other types of sampling<a class="headerlink" href="#other-types-of-sampling" title="Link to this heading">#</a></h1>
+<p>The sampling techniques mentioned so far are general sampling methods useful for a variety of applications beyond sensitivity analysis. There are however techniques that have been developed for specific sensitivity analysis methods. Examples of these methods include the Morris One-At-a-Time <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id132" title="Max D Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.">74</a>]</span>, Fourier Amplitude Sensitivity Test (FAST; <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id100" title="RI Cukier, CM Fortuin, Kurt E Shuler, AG Petschek, and JH Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. i theory. The Journal of chemical physics, 59(8):3873–3878, 1973.">75</a>]</span>), Extended FAST <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id101" title="Andrea Saltelli, Stefano Tarantola, and KP-S Chan. A quantitative model-independent method for global sensitivity analysis of model output. Technometrics, 41(1):39–56, 1999.">76</a>]</span>, and Extended Sobol methods <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>]</span>. For example, the Morris sampling strategy builds a number of trajectories (usually referred to as repetitions and denoted by <span class="math notranslate nohighlight">\(r\)</span>) in the input space each composed of <span class="math notranslate nohighlight">\(N+1\)</span> factor points, where <span class="math notranslate nohighlight">\(N\)</span> is the number of uncertain factors. The first point of the trajectory is selected randomly and the subsequent <span class="math notranslate nohighlight">\(N\)</span> points are generated by moving one factor at a time by a fixed amount. Each factor is perturbed once along the trajectory, while the starting points of all of the trajectories are randomly and uniformly distributed. Several variations of this strategy also exist in the literature; for more details on each approach and their differences the reader is directed to <span id="id5">Pianosi <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>.</p>
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+  <section id="synthetic-generation-of-input-time-series">
+<h1>Synthetic generation of input time series<a class="headerlink" href="#synthetic-generation-of-input-time-series" title="Link to this heading">#</a></h1>
+<p>Models often have input time series or processes with strong temporal and/or spatial correlations (e.g., streamflow, energy demand, pricing of commodities, etc.) that, while they might not immediately come to mind as factors to be examined in sensitivity analysis, can be treated as such. Synthetic input time series are used for a variety of reasons, for example, when observations are not available or are limited, or when past observations are not considered sufficiently representative to capture rare or extreme events of interest <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id103" title="Jonathan D Herman, Harrison B Zeff, Jonathan R Lamontagne, Patrick M Reed, and Gregory W Characklis. Synthetic drought scenario generation to support bottom-up water supply vulnerability assessments. Journal of Water Resources Planning and Management, 142(11):04016050, 2016.">78</a>, <a class="reference internal" href="R.Bibliography.html#id104" title="PCD Milly, Julio Betancourt, Malin Falkenmark, Robert M Hirsch, Zbigniew W Kundzewicz, Dennis P Lettenmaier, and Ronald J Stouffer. Stationarity is dead: whither water management? Earth, 4:20, 2008.">79</a>]</span>. Synthetic generation of input time series provides a valuable tool to consider non-stationarity and incorporate potential stressors, such as climate change impacts into input time series <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id105" title="Edoardo Borgomeo, Christopher L Farmer, and Jim W Hall. Numerical rivers: a synthetic streamflow generator for water resources vulnerability assessments. Water Resources Research, 51(7):5382–5405, 2015.">80</a>]</span>. For example, a century of record will be insufficient to capture very high impact rare extreme events (e.g., persistent multi-year droughts). A large body of statistical literature exists focusing on the topics of synthetic weather <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id74" title="Manuel Herrera, Sukumar Natarajan, David A Coley, Tristan Kershaw, Alfonso P Ramallo-González, Matthew Eames, Daniel Fosas, and Michael Wood. A review of current and future weather data for building simulation. Building Services Engineering Research and Technology, 38(5):602–627, 2017.">81</a>, <a class="reference internal" href="R.Bibliography.html#id75" title="Daniel S Wilks and Robert L Wilby. The weather generation game: a review of stochastic weather models. Progress in physical geography, 23(3):329–357, 1999.">82</a>]</span> and streamflow <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id76" title="JR Lamontagne and JR Stedinger. Generating synthetic streamflow forecasts with specified precision. Journal of Water Resources Planning and Management, 144(4):04018007, 2018.">83</a>, <a class="reference internal" href="R.Bibliography.html#id77" title="Sanghamitra Medda and Kalyan Kumar Bhar. Comparison of single-site and multi-site stochastic models for streamflow generation. Applied Water Science, 9(3):67, 2019.">84</a>]</span> generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound (hot, severe drought) extremes. It is beyond the scope of this text to review these methods, but readers are encouraged to explore the studies cited above as well as the following publications for discussions and comparisons of these methods: <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id103" title="Jonathan D Herman, Harrison B Zeff, Jonathan R Lamontagne, Patrick M Reed, and Gregory W Characklis. Synthetic drought scenario generation to support bottom-up water supply vulnerability assessments. Journal of Water Resources Planning and Management, 142(11):04016050, 2016.">78</a>, <a class="reference internal" href="R.Bibliography.html#id105" title="Edoardo Borgomeo, Christopher L Farmer, and Jim W Hall. Numerical rivers: a synthetic streamflow generator for water resources vulnerability assessments. Water Resources Research, 51(7):5382–5405, 2015.">80</a>, <a class="reference internal" href="R.Bibliography.html#id106" title="Brian R Kirsch, Gregory W Characklis, and Harrison B Zeff. Evaluating the impact of alternative hydro-climate scenarios on transfer agreements: practical improvement for generating synthetic streamflows. Journal of Water Resources Planning and Management, 139(4):396–406, 2013.">85</a>, <a class="reference internal" href="R.Bibliography.html#id107" title="Daniel P Loucks and Eelco Van Beek. Water resource systems planning and management: An introduction to methods, models, and applications. Springer, 2017.">86</a>, <a class="reference internal" href="R.Bibliography.html#id108" title="Scott Steinschneider, Sungwook Wi, and Casey Brown. The integrated effects of climate and hydrologic uncertainty on future flood risk assessments. Hydrological Processes, 29(12):2823–2839, 2015.">87</a>, <a class="reference internal" href="R.Bibliography.html#id109" title="Richard M Vogel. Stochastic watershed models for hydrologic risk management. Water Security, 1:28–35, 2017.">88</a>, <a class="reference internal" href="R.Bibliography.html#id110" title="Richard M Vogel and Jery R Stedinger. The value of stochastic streamflow models in overyear reservoir design applications. Water Resources Research, 24(9):1483–1490, 1988.">89</a>]</span>. The use of these methods for the purposes of exploratory modeling, especially in the context of well-characterized versus deep uncertainty, is further discussed in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a>.</p>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#one-at-a-time-oat">One-At-a-Time (OAT)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#full-and-fractional-factorial-sampling">Full and Fractional Factorial Sampling</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#latin-hypercube-sampling-lhs">Latin Hypercube Sampling (LHS)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#low-discrepancy-sequences">Low-Discrepancy Sequences</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-types-of-sampling">Other types of sampling</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-generation-of-input-time-series">Synthetic generation of input time series</a></li>
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+  <section id="design-of-experiments">
+<span id="id1"></span><h1>Design of Experiments<a class="headerlink" href="#design-of-experiments" title="Link to this heading">#</a></h1>
+<p>Before conducting a sensitivity analysis, the first element that needs to be clarified is the uncertainty space of the model <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id111" title="Jon C Helton, Jay Dean Johnson, Cedric J Sallaberry, and Curt B Storlie. Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliability Engineering &amp; System Safety, 91(10-11):1175–1209, 2006.">53</a>]</span>. In other words, how many and which factors making up the mathematical model are considered uncertain and can potentially affect the model output and the inferences drawn from it. Uncertain factors can be model parameters, model structures, inputs, or alternative model resolution levels (scales), all of which can be assessed through the tools presented in this text. Depending on the kind of factor, its variability can be elicited through various means: expert opinion, values reported in the literature, historical observations, its physical meaning (e.g., population values in a city can never be negative), or through the use of more formal UQ methods (<a class="reference internal" href="A1_Uncertainty_Quantification.html#a1-uncertainty-quantification"><span class="std std-numref">Chapter A</span></a>). The model uncertainty space represents the entire space of variability present in each of the uncertain factors of a model. The complexity of most real-world models means that the response function, <span class="math notranslate nohighlight">\(y=g(x)\)</span>, mapping inputs to outputs, is hardly ever available in an analytical form and therefore analytically computing the sensitivity of the output to each uncertain factor becomes impossible. In these cases, sensitivity analysis is only feasible through numerical procedures that employ different strategies to sample the uncertainty space and calculate sensitivity indices.</p>
+<p>A sampling strategy is often referred to as a <em>design of experiments</em> and represents a methodological choice made before conducting any sensitivity analysis. Experimental design was first introduced by <span id="id3">Fisher [<a class="reference internal" href="R.Bibliography.html#id112" title="Ronald Aylmer Fisher. Design of experiments. Br Med J, 1(3923):554–554, 1936.">54</a>]</span> in the context of laboratory or field-based experiments. Its application in sensitivity analysis is similar to setting up a physical experiment in that it is used to discover the behavior of a system under specific conditions. An ideal design of experiments should provide a framework for the extraction of all plausible information about the impact of each factor on the output of the model. The design of experiments is used to set up a simulation platform with the minimum computational cost to answer specific questions that cannot be readily drawn from the data through analytical or common data mining techniques. Models representing coupled human-natural systems usually have a large number of inputs, state variables and parameters, but not all of them exert fundamental control over the numerical process, despite their uncertainty, nor have substantial impacts on the model output, either independently or through their interactions. Each factor influences the model output in different ways that need to be discovered. For example, the influence of a parameter on model output can be linear or non-linear and can be continuous or only be active during specific times or at particular states of the system <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id53" title="JD Herman, PM Reed, and T Wagener. Time-varying sensitivity analysis clarifies the effects of watershed model formulation on model behavior. Water Resources Research, 49(3):1400–1414, 2013.">55</a>, <a class="reference internal" href="R.Bibliography.html#id113" title="Carolina Massmann, Thorsten Wagener, and Hubert Holzmann. A new approach to visualizing time-varying sensitivity indices for environmental model diagnostics across evaluation time-scales. Environmental modelling &amp; software, 51:190–194, 2014.">56</a>]</span>. An effective and efficient design of experiments allows the analyst to explore these complex relationships and evaluate different behaviors of the model for various scientific questions <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id114" title="An Van Schepdael, Aurélie Carlier, and Liesbet Geris. Sensitivity analysis by design of experiments. In Uncertainty in Biology, pages 327–366. Springer, 2016.">57</a>]</span>. The rest of this section overviews some of the most commonly used designs of experiments. Table 1 summarizes the designs discussed.</p>
+<div class="pst-scrollable-table-container"><table class="table" id="id32">
+<caption><span class="caption-text">Summary of designs of experiments overviewed in this section. * Depends on the sample size.</span><a class="headerlink" href="#id32" title="Link to this table">#</a></caption>
+<thead>
+<tr class="row-odd"><th class="head"><p><em>Design of experiments</em></p></th>
+<th class="head"><p><em>Factor interactions considered</em></p></th>
+<th class="head"><p><em>Treatment of factor domains</em></p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>One-At-a-Time (OAT)</p></td>
+<td><p>No - main effects only</p></td>
+<td><p>Continuous (distributions)</p></td>
+</tr>
+<tr class="row-odd"><td><p>Full Factorial Sampling</p></td>
+<td><p>Yes - including total effects</p></td>
+<td><p>Discrete (levels)</p></td>
+</tr>
+<tr class="row-even"><td><p>Fractional Factorial Sampling</p></td>
+<td><p>Yes - only lower-order effects*</p></td>
+<td><p>Discrete (levels)</p></td>
+</tr>
+<tr class="row-odd"><td><p>Latin Hypercube (LH) Sampling</p></td>
+<td><p>Yes - including total effects*</p></td>
+<td><p>Continuous (distributions)</p></td>
+</tr>
+<tr class="row-even"><td><p>Quasi-Random Sampling with Low-Discrepancy Sequences</p></td>
+<td><p>Yes - including total effects*</p></td>
+<td><p>Continuous (distributions)</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>There are a few different approaches to the design of experiments, closely related to the chosen sensitivity analysis approach, which is in turn shaped by the research motivations, scientific questions, and computational constraints at hand (additional discussion of this can be found at the end of <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#sensitivity-analysis-the-basics"><span class="std std-numref">Chapter 3</span></a>). For example, in a sensitivity analysis using perturbation and derivatives methods, the model input parameters vary from their nominal values one at a time, something that the design of experiments needs to reflect. If, instead, one were to perform sensitivity analysis using a multiple-starts perturbation method, the design of experiments needs to consider that multiple points across the factor space are used. The design of experiments specifically defines two key characteristics of samples that are fed to the numerical model: the number of samples and the range of each factor.</p>
+<p>Generally, sampling can be performed randomly or by applying a stratifying approach. In random sampling, such as Monte Carlo <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id115" title="Nicholas Metropolis and Stanislaw Ulam. The monte carlo method. Journal of the American statistical association, 44(247):335–341, 1949.">58</a>]</span>, samples are randomly generated by a pseudo-random number generator with an a-priori assumption about the distribution of parameters and their possible ranges. Random seeds can also be used to ensure consistency and higher control over the random process. However, this method could leave some gaps in the parameter space and cause clustering in some spaces, especially for a large number of parameters <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id116" title="John Norton. An introduction to sensitivity assessment of simulation models. Environmental Modelling &amp; Software, 69:166–174, 2015.">59</a>]</span>. Most sampling strategies use stratified sampling to mitigate these disadvantages. Stratified sampling techniques divide the domain of each factor into subintervals, often of equal lengths. From each subinterval, an equal number of samples is drawn randomly, or based on the specific locations within the subintervals <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>.</p>
+<section id="one-at-a-time-oat">
+<h2>One-At-a-Time (OAT)<a class="headerlink" href="#one-at-a-time-oat" title="Link to this heading">#</a></h2>
+<p>In this approach, only one model factor is changed at a time while all others are kept fixed across each iteration in a sampling sequence. The OAT method assumes that model factors of focus are linearly independent (i.e., there are no interactions) and can analyze how factors individually influence model outputs or metrics of interest. While popular given its ease of implementation, OAT  is ultimately limited in its exploration of a model’s sensitivities <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. It is primarily used with local sensitivity techniques with similar criticisms: applying this sampling scheme on a system with nonlinear and interactive processes will miss important information on the effect uncertain factors have on the model. OAT samplings can be repeated multiple times in a more sophisticated manner and across different locations of the parameter space to overcome some of these challenges, which would increase computational costs and negate the main reasons for its selection. Given these limitations OAT methods could be used as preliminary, low-cost analyses of the factors’ individual effects, but should ultimately be complemented with more sophisticated methods.</p>
+</section>
+<section id="full-and-fractional-factorial-sampling">
+<h2>Full and Fractional Factorial Sampling<a class="headerlink" href="#full-and-fractional-factorial-sampling" title="Link to this heading">#</a></h2>
+<p>In full factorial sampling each factor is treated as being discrete by considering two or more levels (or intervals) of its values. The sampling process then generates samples within each possible combination of levels, corresponding to each parameter. This scheme produces a more comprehensive sampling of the factors’ variability space, as it accounts for all candidate combinations of factor levels (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a)). If the number of levels is the same across all factors, the number of generated samples is estimated using <span class="math notranslate nohighlight">\(n^k\)</span>, where <span class="math notranslate nohighlight">\(n\)</span> is the number of levels and <span class="math notranslate nohighlight">\(k\)</span> is the number of factors. For example, <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a) presents a full factorial sampling of three uncertain factors <span class="math notranslate nohighlight">\((x_1,\)</span> <span class="math notranslate nohighlight">\(x_2,\)</span> and <span class="math notranslate nohighlight">\(x_3)\)</span>, each considered as having four discrete levels. The total number of samples necessary for such an experiment is <span class="math notranslate nohighlight">\(4^3=64\)</span>. As the number of factors increases, the number of simulations necessary will also grow exponentially, making full factorial sampling computationally burdensome (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (b)). As a result, it is common in the literature to apply full factorial sampling at only two levels per factor, typically the two extremes <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id32" title="Douglas C Montgomery. Design and analysis of experiments. John wiley &amp; sons, 2017.">60</a>]</span>. This significantly reduces computational burden but is only considered appropriate in cases where factors can indeed only assume two discrete values (e.g., when testing the effects of epistemic uncertainty and comparing between model structure A and model structure B). In the case of physical parameters on continuous distributions (e.g., when considering the effects of measurement uncertainty in a temperature sensor), discretizing the range of a factor to only extreme levels can bias its estimated importance.</p>
+<p>Fractional factorial sampling is a widely used alternative to full factorial sampling that allows the analyst to significantly reduce the number of simulations by focusing on the main effects of a factor and seeking to avoid model runs that yield redundant response information <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. In other words, if one can reasonably assume that higher-order interactions are negligible, information about the most significant effects and lower-order interactions (e.g., effects from pairs of factors) can be obtained using a fraction of the full factorial design. Traditionally, fractional factorial design has also been limited to two levels <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id32" title="Douglas C Montgomery. Design and analysis of experiments. John wiley &amp; sons, 2017.">60</a>]</span>, referred to as Fractional Factorial designs 2k-p <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id34" title="George EP Box and J Stuart Hunter. The 2 k—p fractional factorial designs. Technometrics, 3(3):311–351, 1961.">61</a>]</span>. Recently, Generalized Fractional Factorial designs have also been proposed that allow for the structured generation of samples at more than two levels per factor <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id33" title="Izabella Surowiec, Ludvig Vikstrom, Gustaf Hector, Erik Johansson, Conny Vikstrom, and Johan Trygg. Generalized subset designs in analytical chemistry. Analytical chemistry, 89(12):6491–6497, 2017.">62</a>]</span>. Consider a case where the modeling team dealing with the problem in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a) cannot afford to perform 64 simulations of their model. They can afford 32 runs for their experiment and instead decide to fractionally sample the variability space of their factors. A potential design of such a sampling strategy is presented in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (c).</p>
+<figure class="margin-caption align-center" id="id33">
+<span id="figure-3-3"></span><a class="reference internal image-reference" href="_images/figure3_3_alternative_designs.png"><img alt="Figure 3.3" src="_images/figure3_3_alternative_designs.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Alternative designs of experiments and their computational costs for three uncertain factors <span class="math notranslate nohighlight">\((x_1,\)</span> <span class="math notranslate nohighlight">\(x_2,\)</span> and <span class="math notranslate nohighlight">\(x_3)\)</span>. (a) Full factorial design sampling of three factors at four levels, at a total of 64 samples; (b) exponential growth of necessary number of samples when applying full factorial design at four levels; (c) fractional factorial design of three factors at four levels, at a total of 32 samples; and (d) Latin Hypercube sample of three factors with uniform distributions, at a total of 32 samples.</span><a class="headerlink" href="#id33" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="latin-hypercube-sampling-lhs">
+<h2>Latin Hypercube Sampling (LHS)<a class="headerlink" href="#latin-hypercube-sampling-lhs" title="Link to this heading">#</a></h2>
+<p>Latin hypercube sampling (LHS) <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id91" title="Michael D McKay, Richard J Beckman, and William J Conover. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21(1):239-2451, 1979.">63</a>]</span> is one of the most common methods in space-filling experimental designs. With this sampling technique, for <span class="math notranslate nohighlight">\(N\)</span> uncertain factors, an <span class="math notranslate nohighlight">\(N\)</span>-dimensional hypercube is generated, with each factor divided into an equal number of levels depending on the total number of samples to be generated. Equal numbers of samples are then randomly generated at each level, across all factors. In this manner, latin hypercube design guarantees sampling from every level of the variability space and without any overlaps. When the number of samples generated is much larger than the number of uncertain factors, LHS can be very effective in examining the effects of each factor <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. LHS is an attractive technique, because it guarantees a diverse coverage of the space, through the use of subintervals, without being constrained to discrete levels for each factor - compare <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (c) with <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (d) for the same number of samples.</p>
+<p>LHS is less effective when the number of samples is not much larger than the number of uncertain factors, and the effects of each factor cannot be appropriately distinguished. The samples between factors can also be highly correlated, biasing any subsequent sensitivity analysis results. To address this, the sampling scheme can be modified to control for the correlation in parameters while maximizing the information derived. An example of such modification is through the use of orthogonal arrays <span id="id17">[<a class="reference internal" href="R.Bibliography.html#id21" title="Boxin Tang. Orthogonal array-based latin hypercubes. Journal of the American statistical association, 88(424):1392–1397, 1993.">64</a>]</span>.</p>
+</section>
+<section id="low-discrepancy-sequences">
+<h2>Low-Discrepancy Sequences<a class="headerlink" href="#low-discrepancy-sequences" title="Link to this heading">#</a></h2>
+<p>Low-discrepancy sequences is another sampling technique that employs a pseudo-random generator for Monte Carlo sampling <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>, <a class="reference internal" href="R.Bibliography.html#id92" title="SK Zaremba. The mathematical basis of monte carlo and quasi-monte carlo methods. SIAM review, 10(3):303–314, 1968.">66</a>]</span>. These quasi-Monte Carlo methods eliminate ‘lumpiness’ across samples (i.e, the presence of gaps and clusters) by minimizing discrepancy across the hypercube samples. Discrepancy can be quantitatively measured using the deviations of sampled points from a uniform distribution <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>, <a class="reference internal" href="R.Bibliography.html#id95" title="Sergei Kucherenko, Daniel Albrecht, and Andrea Saltelli. Exploring multi-dimensional spaces: a comparison of latin hypercube and quasi monte carlo sampling techniques. arXiv preprint arXiv:1505.02350, 2015.">67</a>]</span>. Low-discrepancy sequences ensure that the number of samples in any subspace of the variability hypercube is approximately the same. This is not something guaranteed by Latin Hypercube sampling, and even though its design can be improved through optimization with various criteria, such adjustments are limited to small sample sizes and low dimensions <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id95" title="Sergei Kucherenko, Daniel Albrecht, and Andrea Saltelli. Exploring multi-dimensional spaces: a comparison of latin hypercube and quasi monte carlo sampling techniques. arXiv preprint arXiv:1505.02350, 2015.">67</a>, <a class="reference internal" href="R.Bibliography.html#id93" title="Bertrand Iooss, Loïc Boussouf, Vincent Feuillard, and Amandine Marrel. Numerical studies of the metamodel fitting and validation processes. arXiv preprint arXiv:1001.1049, 2010.">68</a>, <a class="reference internal" href="R.Bibliography.html#id94" title="Ruichen Jin, Wei Chen, and Agus Sudjianto. An efficient algorithm for constructing optimal design of computer experiments. In International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, volume 37009, 545–554. 2003.">69</a>, <a class="reference internal" href="R.Bibliography.html#id96" title="Max D Morris and Toby J Mitchell. Exploratory designs for computational experiments. Journal of statistical planning and inference, 43(3):381–402, 1995.">70</a>, <a class="reference internal" href="R.Bibliography.html#id97" title="Jeong-Soo Park. Optimal latin-hypercube designs for computer experiments. Journal of statistical planning and inference, 39(1):95–111, 1994.">71</a>]</span>. In contrast, the Sobol sequence <span id="id21">[<a class="reference internal" href="R.Bibliography.html#id98" title="Ilya M Sobol. Uniformly distributed sequences with an additional uniform property. USSR Computational Mathematics and Mathematical Physics, 16(5):236–242, 1976.">72</a>, <a class="reference internal" href="R.Bibliography.html#id99" title="Il'ya Meerovich Sobol'. On the distribution of points in a cube and the approximate evaluation of integrals. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 7(4):784–802, 1967.">73</a>]</span>, one of the most widely used sampling techniques, utilizes the low-discrepancy approach to uniformly fill the sampled factor space. A core advantage of this style of sampling is that it takes far fewer samples (i.e., simulations) to attain a much lower level of error in estimating model output statistics (e.g., the mean and variance of outputs).</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial which uses Sobol sequence sampling for the purposes of a Sobol sensitivity analysis:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA using SALib Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="other-types-of-sampling">
+<h2>Other types of sampling<a class="headerlink" href="#other-types-of-sampling" title="Link to this heading">#</a></h2>
+<p>The sampling techniques mentioned so far are general sampling methods useful for a variety of applications beyond sensitivity analysis. There are however techniques that have been developed for specific sensitivity analysis methods. Examples of these methods include the Morris One-At-a-Time <span id="id22">[<a class="reference internal" href="R.Bibliography.html#id132" title="Max D Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.">74</a>]</span>, Fourier Amplitude Sensitivity Test (FAST; <span id="id23">[<a class="reference internal" href="R.Bibliography.html#id100" title="RI Cukier, CM Fortuin, Kurt E Shuler, AG Petschek, and JH Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. i theory. The Journal of chemical physics, 59(8):3873–3878, 1973.">75</a>]</span>), Extended FAST <span id="id24">[<a class="reference internal" href="R.Bibliography.html#id101" title="Andrea Saltelli, Stefano Tarantola, and KP-S Chan. A quantitative model-independent method for global sensitivity analysis of model output. Technometrics, 41(1):39–56, 1999.">76</a>]</span>, and Extended Sobol methods <span id="id25">[<a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>]</span>. For example, the Morris sampling strategy builds a number of trajectories (usually referred to as repetitions and denoted by <span class="math notranslate nohighlight">\(r\)</span>) in the input space each composed of <span class="math notranslate nohighlight">\(N+1\)</span> factor points, where <span class="math notranslate nohighlight">\(N\)</span> is the number of uncertain factors. The first point of the trajectory is selected randomly and the subsequent <span class="math notranslate nohighlight">\(N\)</span> points are generated by moving one factor at a time by a fixed amount. Each factor is perturbed once along the trajectory, while the starting points of all of the trajectories are randomly and uniformly distributed. Several variations of this strategy also exist in the literature; for more details on each approach and their differences the reader is directed to <span id="id26">Pianosi <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>.</p>
+</section>
+<section id="synthetic-generation-of-input-time-series">
+<h2>Synthetic generation of input time series<a class="headerlink" href="#synthetic-generation-of-input-time-series" title="Link to this heading">#</a></h2>
+<p>Models often have input time series or processes with strong temporal and/or spatial correlations (e.g., streamflow, energy demand, pricing of commodities, etc.) that, while they might not immediately come to mind as factors to be examined in sensitivity analysis, can be treated as such. Synthetic input time series are used for a variety of reasons, for example, when observations are not available or are limited, or when past observations are not considered sufficiently representative to capture rare or extreme events of interest <span id="id27">[<a class="reference internal" href="R.Bibliography.html#id103" title="Jonathan D Herman, Harrison B Zeff, Jonathan R Lamontagne, Patrick M Reed, and Gregory W Characklis. Synthetic drought scenario generation to support bottom-up water supply vulnerability assessments. Journal of Water Resources Planning and Management, 142(11):04016050, 2016.">78</a>, <a class="reference internal" href="R.Bibliography.html#id104" title="PCD Milly, Julio Betancourt, Malin Falkenmark, Robert M Hirsch, Zbigniew W Kundzewicz, Dennis P Lettenmaier, and Ronald J Stouffer. Stationarity is dead: whither water management? Earth, 4:20, 2008.">79</a>]</span>. Synthetic generation of input time series provides a valuable tool to consider non-stationarity and incorporate potential stressors, such as climate change impacts into input time series <span id="id28">[<a class="reference internal" href="R.Bibliography.html#id105" title="Edoardo Borgomeo, Christopher L Farmer, and Jim W Hall. Numerical rivers: a synthetic streamflow generator for water resources vulnerability assessments. Water Resources Research, 51(7):5382–5405, 2015.">80</a>]</span>. For example, a century of record will be insufficient to capture very high impact rare extreme events (e.g., persistent multi-year droughts). A large body of statistical literature exists focusing on the topics of synthetic weather <span id="id29">[<a class="reference internal" href="R.Bibliography.html#id74" title="Manuel Herrera, Sukumar Natarajan, David A Coley, Tristan Kershaw, Alfonso P Ramallo-González, Matthew Eames, Daniel Fosas, and Michael Wood. A review of current and future weather data for building simulation. Building Services Engineering Research and Technology, 38(5):602–627, 2017.">81</a>, <a class="reference internal" href="R.Bibliography.html#id75" title="Daniel S Wilks and Robert L Wilby. The weather generation game: a review of stochastic weather models. Progress in physical geography, 23(3):329–357, 1999.">82</a>]</span> and streamflow <span id="id30">[<a class="reference internal" href="R.Bibliography.html#id76" title="JR Lamontagne and JR Stedinger. Generating synthetic streamflow forecasts with specified precision. Journal of Water Resources Planning and Management, 144(4):04018007, 2018.">83</a>, <a class="reference internal" href="R.Bibliography.html#id77" title="Sanghamitra Medda and Kalyan Kumar Bhar. Comparison of single-site and multi-site stochastic models for streamflow generation. Applied Water Science, 9(3):67, 2019.">84</a>]</span> generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound (hot, severe drought) extremes. It is beyond the scope of this text to review these methods, but readers are encouraged to explore the studies cited above as well as the following publications for discussions and comparisons of these methods: <span id="id31">[<a class="reference internal" href="R.Bibliography.html#id103" title="Jonathan D Herman, Harrison B Zeff, Jonathan R Lamontagne, Patrick M Reed, and Gregory W Characklis. Synthetic drought scenario generation to support bottom-up water supply vulnerability assessments. Journal of Water Resources Planning and Management, 142(11):04016050, 2016.">78</a>, <a class="reference internal" href="R.Bibliography.html#id105" title="Edoardo Borgomeo, Christopher L Farmer, and Jim W Hall. Numerical rivers: a synthetic streamflow generator for water resources vulnerability assessments. Water Resources Research, 51(7):5382–5405, 2015.">80</a>, <a class="reference internal" href="R.Bibliography.html#id106" title="Brian R Kirsch, Gregory W Characklis, and Harrison B Zeff. Evaluating the impact of alternative hydro-climate scenarios on transfer agreements: practical improvement for generating synthetic streamflows. Journal of Water Resources Planning and Management, 139(4):396–406, 2013.">85</a>, <a class="reference internal" href="R.Bibliography.html#id107" title="Daniel P Loucks and Eelco Van Beek. Water resource systems planning and management: An introduction to methods, models, and applications. Springer, 2017.">86</a>, <a class="reference internal" href="R.Bibliography.html#id108" title="Scott Steinschneider, Sungwook Wi, and Casey Brown. The integrated effects of climate and hydrologic uncertainty on future flood risk assessments. Hydrological Processes, 29(12):2823–2839, 2015.">87</a>, <a class="reference internal" href="R.Bibliography.html#id109" title="Richard M Vogel. Stochastic watershed models for hydrologic risk management. Water Security, 1:28–35, 2017.">88</a>, <a class="reference internal" href="R.Bibliography.html#id110" title="Richard M Vogel and Jery R Stedinger. The value of stochastic streamflow models in overyear reservoir design applications. Water Resources Research, 24(9):1483–1490, 1988.">89</a>]</span>. The use of these methods for the purposes of exploratory modeling, especially in the context of well-characterized versus deep uncertainty, is further discussed in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a>.</p>
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+  <section id="derivative-based-methods">
+<h1>Derivative-based Methods<a class="headerlink" href="#derivative-based-methods" title="Link to this heading">#</a></h1>
+<p>Derivative-based methods explore how model outputs are affected by perturbations in a single model input around a particular input value. These methods are local and are performed using OAT sampling. For simplicity of mathematical notations, let us assume that the model <span class="math notranslate nohighlight">\(g(X)\)</span> only returns one output. Following <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id117" title="Emanuele Borgonovo. Sensitivity analysis of model output with input constraints: a generalized rationale for local methods. Risk Analysis: An International Journal, 28(3):667–680, 2008.">90</a>]</span> and <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>, the sensitivity index, <span class="math notranslate nohighlight">\(S_i\)</span> , of the model’s <em>i</em>-th input factor, <span class="math notranslate nohighlight">\(x_i\)</span> , can be measured using the partial derivative evaluated at a nominal value, <span class="math notranslate nohighlight">\(\bar{x}\)</span>, of the vector of inputs:</p>
+<div class="math notranslate nohighlight">
+\[S_i (\bar{x}) = \frac{\partial g}{\partial x} |_{\bar{x}{^c{_i}}}\]</div>
+<p>where <em>c</em><sub>i</sub> is the scaling factor. In most applications however, the relationship <span class="math notranslate nohighlight">\(g(X)\)</span> is not fully known in its analytical form, and therefore the above partial derivative is usually approximated:</p>
+<div class="math notranslate nohighlight">
+\[S_i (\bar{x}) = \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i\]</div>
+<p>Using this approximation, the <em>i</em>-th input factor is perturbed by a magnitude of <span class="math notranslate nohighlight">\(\Delta_i\)</span>, and its relative importance is calculated. Derivative-based methods are some of the oldest sensitivity analysis methods as they only require <span class="math notranslate nohighlight">\(N+1\)</span> model evaluations to estimate indices for <span class="math notranslate nohighlight">\(N\)</span> uncertain factors. As described above, being computationally very cheap comes at the cost of not being able to explore the entire input space, but only (local) perturbations to the nominal value. Additionally, as these methods examine the effects of each input factor one at a time, they cannot assess parametric interactions or capture the interacting nature of many real systems and the models that abstract them.</p>
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+    <h1>Elementary Effect Methods</h1>
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+  <section id="elementary-effect-methods">
+<h1>Elementary Effect Methods<a class="headerlink" href="#elementary-effect-methods" title="Link to this heading">#</a></h1>
+<p>Elementary effect (EE) SA methods provide a solution to the local nature of the derivative-based methods by exploring the entire parametric range of each input parameter <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>. However, EE methods still use OAT sampling and do not vary all input parameters simultaneously while exploring the parametric space. The OAT nature of EEs methods therefore prevents them from properly capturing the interactions between uncertain factors. EEs methods are computationally efficient compared to their All-At-a-Time (AAT) counterparts, making them more suitable when computational capacity is a limiting factor, while still allowing for some inferences regarding factor interactions.
+The most popular EE method is the Method of Morris <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id132" title="Max D Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.">74</a>]</span>. Following the notation by <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>, this method calculates global sensitivity using the mean of the EEs (finite differences) of each parameter at different locations:</p>
+<div class="math notranslate nohighlight">
+\[S_i = \mu_i^* = \frac{1}{r}\sum_{j=1}^r EE^j_i = \frac{1}{r}\sum_{j=1}^r \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i\]</div>
+<p>with <span class="math notranslate nohighlight">\(r\)</span> representing the number of sample repetitions (also refered to as trajectories) in the input space, usually set between 4 and 10 <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. Each <span class="math notranslate nohighlight">\(x_j\)</span> represents the points of each trajectory, with <span class="math notranslate nohighlight">\(j=1,…, r\)</span>, selected as described in the sampling strategy for this method, found above. This method also produces the standard deviation of the EEs:</p>
+<div class="math notranslate nohighlight">
+\[\sigma_i = \sqrt{\frac{1}{r}\sum_{j=1}^r(EE_i^j-\frac{1}{r}\sum_{j=1}^r EE^j_i)^2}\]</div>
+<p>which is a measure of parametric interactions. Higher values of <span class="math notranslate nohighlight">\(\sigma_i\)</span> suggest model responses at different levels of factor <span class="math notranslate nohighlight">\(x_i\)</span> are significantly different, which indicates considerable interactions between that and other uncertain factors. The values of <span class="math notranslate nohighlight">\(\mu_i^*\)</span> and <span class="math notranslate nohighlight">\(\sigma_i\)</span> for each factor allow us to draw several different conclusions, illustrated in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-4"><span class="std std-numref">Fig. 3.4</span></a>, following the example by <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>. In this example, factors <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> can be said to have an influence on the model outputs, with <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> having some interactive or non-linear effects. Depending on the orders of magnitude of <span class="math notranslate nohighlight">\(\mu_i^*\)</span> and <span class="math notranslate nohighlight">\(\sigma_i\)</span> one can indirectly deduce whether the factors have strong interactive effects, for example if a factor <span class="math notranslate nohighlight">\(\sigma_i &lt;&lt; \mu_i^*\)</span> then the relationship between that factor and the output can be assumed to be largely linear (note that this is still an OAT method and assumptions on factor interactions should be strongly caveated). Extensions of the Method of Morris have also been developed specifically for the purposes of factor fixing and explorations of parametric interactions (e.g., <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id27" title="Emanuele Borgonovo. Sensitivity analysis with finite changes: an application to modified eoq models. European Journal of Operational Research, 200(1):127–138, 2010.">48</a>, <a class="reference internal" href="R.Bibliography.html#id133" title="Francesca Campolongo and Roger Braddock. The use of graph theory in the sensitivity analysis of the model output: a second order screening method. Reliability Engineering &amp; System Safety, 64(1):1–12, 1999.">92</a>, <a class="reference internal" href="R.Bibliography.html#id134" title="Roger A Cropp and Roger D Braddock. The new morris method: an efficient second-order screening method. Reliability Engineering &amp; System Safety, 78(1):77–83, 2002.">93</a>]</span>).</p>
+<figure class="margin-caption align-center" id="id7">
+<span id="figure-3-4"></span><a class="reference internal image-reference" href="_images/figure3_4_morris_method.png"><img alt="Figure 3.4" src="_images/figure3_4_morris_method.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Illustrative results of the Morris Method. Factors <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> have an influence on the model outputs, with <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> having interactive or non-linear effects. Whether or not a factor should be considered influential to the output depends on the output selected and is specific to the research context and purpose of the analysis, as discussed in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#why-sa"><span class="std std-numref">Chapter 3.2</span></a>.</span><a class="headerlink" href="#id7" title="Link to this image">#</a></p>
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+  <section id="regression-based-methods">
+<h1>Regression-based Methods<a class="headerlink" href="#regression-based-methods" title="Link to this heading">#</a></h1>
+<p>Regression analysis is one of the oldest ways of investigating parametric importance and sensitivity <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. Here, we describe some of the most popular regression-based sensitivity indices. One of the main sensitivity indices of this category is the standardized regression coefficient (SRC). To calculate SRC, a linear regression relationship needs to be fitted between the input vector, <span class="math notranslate nohighlight">\(x\)</span>, and the model output of interest by using a least-square minimizing method:</p>
+<div class="math notranslate nohighlight">
+\[y = b_0 + \sum_{i=1}^N b_ix_i\]</div>
+<p>where <span class="math notranslate nohighlight">\(b_0\)</span> and <span class="math notranslate nohighlight">\(b_i\)</span> (corresponding to the <em>i</em>-th model input) are regression coefficients. The following relationship can then be used to calculate the SRCs for different input values:</p>
+<div class="math notranslate nohighlight">
+\[S_i=SRC_i=b_i\frac{\sigma_i}{\sigma_y}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\sigma_i\)</span> and <span class="math notranslate nohighlight">\(\sigma_y\)</span> are standard deviations of <em>i</em>-th model input and output, respectively.</p>
+<p>Several other regression-based indices explore the correlation between input and output parameters as a proxy to model parametric sensitivity <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>, <a class="reference internal" href="R.Bibliography.html#id135" title="Jon C Helton. Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal. Reliability Engineering &amp; System Safety, 42(2-3):327–367, 1993.">94</a>, <a class="reference internal" href="R.Bibliography.html#id136" title="Gemma Manache and Charles S Melching. Identification of reliable regression-and correlation-based sensitivity measures for importance ranking of water-quality model parameters. Environmental Modelling &amp; Software, 23(5):549–562, 2008.">95</a>]</span>. The Pearson correlation coefficient (PCC) can be used when a linear relationship exists between an uncertain factor, <span class="math notranslate nohighlight">\(x_i\)</span>, and the output <span class="math notranslate nohighlight">\(y\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i=PCC=\frac{cov(x_i,y)}{\sigma_i\sigma_y}\]</div>
+<p>In cases when there are outliers in the data or the relationship between the uncertain factors and the output is not linear, rank-based correlation coefficients are preferred, for example, Spearman’s rank correlation coefficient (SRCC):</p>
+<div class="math notranslate nohighlight">
+\[S_i=SRCC=\frac{cov(rx_i,ryi)}{\sigma_{ri}\sigma_{ry}}\]</div>
+<p>where the raw values of <span class="math notranslate nohighlight">\(x_i\)</span> and <span class="math notranslate nohighlight">\(y\)</span> and converted to ranks <span class="math notranslate nohighlight">\(rx_i\)</span> and <span class="math notranslate nohighlight">\(ry\)</span> respectively, which instead represent a measurement of the strength of the monotonic relationship, rather than linear relationship, between the input and output. Other regression-based metrics include the partial correlations coefficient, the partial rank correlations coefficient, and the Nash-Sutcliffe coefficient, more discussion on which can be found in <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>.</p>
+<p>Tree-based regression techniques have also been used for sensitivity analysis in an effort to address the challenges faced with nonlinear models <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id137" title="F Pappenberger and Keith J Beven. Ignorance is bliss: or seven reasons not to use uncertainty analysis. Water resources research, 2006.">96</a>]</span>. Examples of these methods include the Patient Rule Induction Method (PRIM; <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id79" title="Jerome H Friedman and Nicholas I Fisher. Bump hunting in high-dimensional data. Statistics and Computing, 9(2):123–143, 1999.">97</a>]</span>) and Classification And Regression Trees (CART; <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id80" title="Leo Breiman, Jerome Friedman, Charles J Stone, and Richard A Olshen. Classification and regression trees. CRC press, 1984.">98</a>]</span>). CART-based approaches also include boosting and bagging extensions <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id89" title="Yoav Freund, Robert Schapire, and Naoki Abe. A short introduction to boosting. Journal-Japanese Society For Artificial Intelligence, 14(771-780):1612, 1999.">99</a>, <a class="reference internal" href="R.Bibliography.html#id138" title="Leo Breiman. Bagging predictors. Machine learning, 24(2):123–140, 1996.">100</a>]</span>. These methods are particularly useful when sensitivity analysis is used for factor mapping (i.e., when trying to identify which uncertain model factors produce a certain model behavior). <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a> elaborates on the use of these methods. Regression-based sensitivity analysis methods are global by nature and can explore the entire space of variables. However, the true level of comprehensiveness depends on the design of experiments and the number of simulations providing data to establish the regression relationships. Although they are usually computationally efficient, they do not produce significant information about parametric interactions <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>]</span>.</p>
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+  <section id="regional-sensitivity-analysis">
+<h1>Regional Sensitivity Analysis<a class="headerlink" href="#regional-sensitivity-analysis" title="Link to this heading">#</a></h1>
+<p>Another method primarily applied for basic factor mapping applications is Regional Sensitivity Analysis (RSA; <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id157" title="George M Hornberger and Robert C Spear. Approach to the preliminary analysis of environmental systems. J. Environ. Mgmt., 12(1):7–18, 1981.">101</a>]</span>). RSA is a global sensitivity analysis method that is typically implemented using standard sampling methods such as latin hypercube sampling. It is performed by specifying a condition on the output space (e.g., an upper threshold) and classifying outputs that meet the condition as behavioral and the ones that fail it as non-behavioral (illustrated in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (b)). Note that the specified threshold depends on the nature of the problem, model, and the research question. It can reflect model-performance metrics (such as errors) or consequential decision-relevant metrics (such as unacceptable system outcomes). The behavioral and non-behavioral outputs are then traced back to their originating sampled factors, where differences between the distributions of samples can be used to determine their significance in producing each part of the output. The Kolmogorov-Smirnov divergence is commonly used to quantify the difference between the distribution of behavioral and non-behavioral parameters <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i=|F_{x_i|y_b} (y \in Y_b)-F_{x_i|y_{nb}} (y \in Y_{nb})|\]</div>
+<p>where <span class="math notranslate nohighlight">\(Y_b\)</span> represents the set of behavioral outputs, and <span class="math notranslate nohighlight">\(F_{x_i|y_b}\)</span> is the empirical cumulative distribution function of the values of <span class="math notranslate nohighlight">\(x_i\)</span> associated with values of <span class="math notranslate nohighlight">\(y\)</span> that belong in the behavioral set. The <span class="math notranslate nohighlight">\(nb\)</span> notation indicates the equivalent elements related to the non-behavioral set. Large differences between the two distributions indicate stronger effects by the parameters on the respective part of the output space.</p>
+<p>Used in a factor mapping setting, RSA can be applied for scenario discovery <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id71" title="Robert J Lempert, David G Groves, Steven W Popper, and Steve C Bankes. A general, analytic method for generating robust strategies and narrative scenarios. Management science, 52(4):514–528, 2006.">102</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>, the Generalized Likelihood Uncertainty Estimation method (GLUE; <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>, <a class="reference internal" href="R.Bibliography.html#id158" title="Keith Beven and Andrew Binley. Glue: 20 years on. Hydrological processes, 28(24):5897–5918, 2014.">104</a>, <a class="reference internal" href="R.Bibliography.html#id159" title="Roberta-Serena Blasone, Jasper A Vrugt, Henrik Madsen, Dan Rosbjerg, Bruce A Robinson, and George A Zyvoloski. Generalized likelihood uncertainty estimation (glue) using adaptive markov chain monte carlo sampling. Advances in Water Resources, 31(4):630–648, 2008.">105</a>]</span>) and other hybrid sensitivity analysis methods (e.g., <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id160" title="SA Cryer and PL Havens. Regional sensitivity analysis using a fractional factorial method for the usda model gleams. Environmental modelling &amp; software, 14(6):613–624, 1999.">106</a>, <a class="reference internal" href="R.Bibliography.html#id161" title="Pengfei Wei, Zhenzhou Lu, and Xiukai Yuan. Monte carlo simulation for moment-independent sensitivity analysis. Reliability Engineering &amp; System Safety, 110:60–67, 2013.">107</a>]</span>). The fundamental shortcomings of RSA are that, in some cases, it could be hard to interpret the difference between behavioral and non-behavioral sample sets, and that insights about parametric correlations and interactions cannot always be uncovered <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. For more elaborate discussions and illustrations of the RSA method, readers are directed to <span id="id7">Tang <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id20" title="Yong Tang, Patrick Reed, Thibaut Wagener, and K van Werkhoven. Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation. Hydrology and Earth System Sciences, 11(2):793–817, 2007.">42</a>], Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>], Young [<a class="reference internal" href="R.Bibliography.html#id162" title="Peter Young. Data-based mechanistic modelling, generalised sensitivity and dominant mode analysis. Computer Physics Communications, 117(1-2):113–129, 1999.">108</a>]</span> and references therein.</p>
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+  <section id="variance-based-methods">
+<span id="id1"></span><h1>Variance-based Methods<a class="headerlink" href="#variance-based-methods" title="Link to this heading">#</a></h1>
+<p>Variance-based sensitivity analysis methods hypothesize that various specified model factors contribute differently to the variation of model outputs; therefore, decomposition and analysis of output variance can determine a model’s sensitivity to input parameters <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>]</span>. The most popular variance-based method is the Sobol method, which is a global sensitivity analysis method that takes into account complex and nonlinear factor interaction when calculating sensitivity indices, and employs more sophisticated sampling methods (e.g., the Sobol sampling method). The Sobol method is able to calculate three types of sensitivity indices that provide different types of information about model sensitivities. These indices include first-order, higher-order (e.g., second-, third-, etc. orders), and total-order sensitivities.</p>
+<p>The first-order sensitivity index indicates the percent of model output variance contributed by a factor individually (i.e., the effect of varying <span class="math notranslate nohighlight">\(x_i\)</span> alone) and is obtained using the following <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>, <a class="reference internal" href="R.Bibliography.html#id148" title="Andrea Saltelli. Making best use of model evaluations to compute sensitivity indices. Computer physics communications, 145(2):280–297, 2002.">109</a>]</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i^1=\frac{V_{x_i}[E_{x_{\sim i}}(x_i)]}{V(y)}\]</div>
+<p>with <span class="math notranslate nohighlight">\(E\)</span> and <span class="math notranslate nohighlight">\(V\)</span> denoting the expected value and the variance, respectively. <span class="math notranslate nohighlight">\(x_{\sim i}\)</span> denotes all factors expect for <span class="math notranslate nohighlight">\(x_i\)</span>. The first-order sensitivity index (<span class="math notranslate nohighlight">\(S_i^1\)</span>) can therefore also be thought of as the portion of total output variance (<span class="math notranslate nohighlight">\(V_y\)</span>) that can be reduced if the uncertainty in factor <span class="math notranslate nohighlight">\(x_i\)</span> is eliminated <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id149" title="Andrea Saltelli. Sensitivity analysis for importance assessment. Risk analysis, 22(3):579–590, 2002.">110</a>]</span>. First-order sensitivity indices are usually used to understand the independent effect of a factor and to distinguish its individual versus interactive influence. It would be expected for linearly independent factors that they would only have first order indices (no interactions) that should correspond well with sensitivities obtained from simpler methods using OAT sampling.</p>
+<p>Higher-order sensitivity indices explore the interaction between two or more parameters that contribute to model output variations. For example, a second-order index indicates how interactions between a pair of factors can lead to change in model output variance and is calculated using the following relationship:</p>
+<div class="math notranslate nohighlight">
+\[S_{ij}^2=\frac{V_{x_{i,j}}[E_{x_{\sim i,j}}(x_i,x_j)]}{V(y)}\]</div>
+<p>with <span class="math notranslate nohighlight">\(i \ne j\)</span>. Higher order indices can be calculated by similar extensions (i.e., fixing additional operators together), but it is usually computationally expensive in practice.</p>
+<p>The total sensitivity analysis index represents the entire influence of an input factor on model outputs including all of its interactions with other factors <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id150" title="Toshimitsu Homma and Andrea Saltelli. Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering &amp; System Safety, 52(1):1–17, 1996.">111</a>]</span>. In other words, total-order indices include first-order and all higher-order interactions associated with each factor and can be estimated calculated using the following:</p>
+<div class="math notranslate nohighlight">
+\[S_i^T= \frac{E_{x_{\sim i}}[V_{x_i}(x_{\sim i})]}{V(y)} = 1 - \frac{V_{x_{\sim i}}[E_{x_{i}}(x_{\sim i})]}{V(y)}\]</div>
+<p>This index reveals the expected portion of variance that remains if uncertainty is eliminated in all factors but <span class="math notranslate nohighlight">\(x_i\)</span> <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id149" title="Andrea Saltelli. Sensitivity analysis for importance assessment. Risk analysis, 22(3):579–590, 2002.">110</a>]</span>. The total sensitivity index is the overall best measure of sensitivity as it captures the full individual and interactive effects of model factors.</p>
+<p>Besides the Sobol method, there are some other variance-based sensitivity analysis methods, such as the Fourier amplitude sensitivity test (FAST; <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id100" title="RI Cukier, CM Fortuin, Kurt E Shuler, AG Petschek, and JH Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. i theory. The Journal of chemical physics, 59(8):3873–3878, 1973.">75</a>, <a class="reference internal" href="R.Bibliography.html#id151" title="Gregory J McRae, William R Goodin, and John H Seinfeld. Development of a second-generation mathematical model for urban air pollution—i. model formulation. Atmospheric Environment (1967), 16(4):679–696, 1982.">112</a>]</span>) and extended-FAST <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id152" title="Andrea Saltelli and Ricardo Bolado. An alternative way to compute fourier amplitude sensitivity test (fast). Computational Statistics &amp; Data Analysis, 26(4):445–460, 1998.">113</a>, <a class="reference internal" href="R.Bibliography.html#id153" title="MA Vazquez-Cruz, R Guzman-Cruz, IL Lopez-Cruz, O Cornejo-Perez, I Torres-Pacheco, and RG Guevara-Gonzalez. Global sensitivity analysis by means of efast and sobol’methods and calibration of reduced state-variable tomgro model using genetic algorithms. Computers and Electronics in Agriculture, 100:1–12, 2014.">114</a>]</span>, that have been used by the scientific community. However, Sobol remains by far the most common method of this class. Variance-based techniques have been widely used and have proved to be powerful in a variety of applications. Despite their popularity, some authors have expressed concerns about the methods’ appropriateness in some settings. Specifically, the presence of heavy-tailed distributions or outliers, or when model outputs are multimodal can bias the sensitivity indices produced by these methods <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id154" title="Benjamin Auder and Bertrand Iooss. Global sensitivity analysis based on entropy. In Safety, reliability and risk analysis-Proceedings of the ESREL 2008 Conference, 2107–2115. 2008.">115</a>, <a class="reference internal" href="R.Bibliography.html#id155" title="Farkhondeh Khorashadi Zadeh, Jiri Nossent, Fanny Sarrazin, Francesca Pianosi, Ann van Griensven, Thorsten Wagener, and Willy Bauwens. Comparison of variance-based and moment-independent global sensitivity analysis approaches by application to the swat model. Environmental Modelling &amp; Software, 91:210–222, 2017.">116</a>, <a class="reference internal" href="R.Bibliography.html#id156" title="Francesca Pianosi and Thorsten Wagener. A simple and efficient method for global sensitivity analysis based on cumulative distribution functions. Environmental Modelling &amp; Software, 67:1–11, 2015.">117</a>]</span>. Moment-independent measures, discussed below, attempt to overcome these challenges.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial which demonstrates the application of a Sobol sensitivity analysis:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA using SALib Jupyter Notebook</a></p>
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+  <section id="analysis-of-variance-anova">
+<h1>Analysis of Variance (ANOVA)<a class="headerlink" href="#analysis-of-variance-anova" title="Link to this heading">#</a></h1>
+<p>Analysis of Variance (ANOVA) was first introduced by <span id="id1">Fisher and others [<a class="reference internal" href="R.Bibliography.html#id144" title="Ronald Aylmer Fisher and others. Statistical methods for research workers. Statistical methods for research workers., 1934.">118</a>]</span> and has since become a popular factor analysis method in physical experiments. ANOVA can be used as a sensitivity analysis method in computational experiments with a factorial design of experiment (referred to as factorial ANOVA). Note that Sobol can also be categorized as an ANOVA sensitivity analysis method, and that is why Sobol is sometimes referred to as a functional ANOVA <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id145" title="Loıc Brevault, Mathieu Balesdent, Nicolas Bérend, and Rodolphe Le Riche. Comparison of different global sensitivity analysis methods for aerospace vehicle optimal design. In 10th World Congress on Structural and Multidisciplinary Optimization, WCSMO-10. 2013.">119</a>]</span>. Factorial ANOVA methods are particularly suited for models and problems that have discrete input spaces, significantly reducing the computational time. More information about these methods can be found in <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id145" title="Loıc Brevault, Mathieu Balesdent, Nicolas Bérend, and Rodolphe Le Riche. Comparison of different global sensitivity analysis methods for aerospace vehicle optimal design. In 10th World Congress on Structural and Multidisciplinary Optimization, WCSMO-10. 2013.">119</a>, <a class="reference internal" href="R.Bibliography.html#id146" title="GEB Archer, Andrea Saltelli, and IM Sobol. Sensitivity measures, anova-like techniques and the use of bootstrap. Journal of Statistical Computation and Simulation, 58(2):99–120, 1997.">120</a>, <a class="reference internal" href="R.Bibliography.html#id147" title="Art B Owen. Variance components and generalized sobol'indices. SIAM/ASA Journal on Uncertainty Quantification, 1(1):19–41, 2013.">121</a>]</span>.</p>
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+  <section id="moment-independent-density-based-methods">
+<h1>Moment-Independent (Density-Based) Methods<a class="headerlink" href="#moment-independent-density-based-methods" title="Link to this heading">#</a></h1>
+<p>These methods typically compare the entire distribution (i.e., not just the variance) of input and output parameters in order to determine the sensitivity of the output to a particular input variable. Several moment-independent sensitivity analysis methods have been proposed in recent years. The delta (<span class="math notranslate nohighlight">\(\delta\)</span>) moment-independent method calculates the difference between unconditional and conditional cumulative distribution functions of the output. The method was first introduced by <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id142" title="Emanuele Borgonovo. Measuring uncertainty importance: investigation and comparison of alternative approaches. Risk analysis, 26(5):1349–1361, 2006.">122</a>, <a class="reference internal" href="R.Bibliography.html#id141" title="Emanuele Borgonovo. A new uncertainty importance measure. Reliability Engineering &amp; System Safety, 92(6):771–784, 2007.">123</a>]</span> and has become widely used in various disciplines. The <span class="math notranslate nohighlight">\(\delta\)</span> sensitivity index is defined as follows:</p>
+<div class="math notranslate nohighlight">
+\[S_i=\delta_i=\frac{1}{2}E_{x_i}|f_y(y)-f_{y|x_i}(y)|dy\]</div>
+<p>where <span class="math notranslate nohighlight">\(f_y(y)\)</span> is the probability density function of the entire model output <span class="math notranslate nohighlight">\(y\)</span>, and <span class="math notranslate nohighlight">\(f_{y|x_i}(y)\)</span> is the conditional density of <span class="math notranslate nohighlight">\(y\)</span>, given that factor <span class="math notranslate nohighlight">\(x_i\)</span> assumes a fixed value. The <span class="math notranslate nohighlight">\(\delta_i\)</span> sensitivity indicator therefore represents the normalized expected shift in the distribution of <span class="math notranslate nohighlight">\(y\)</span> provoked by <span class="math notranslate nohighlight">\(x_i\)</span>. Moment-independent methods are advantageous in cases where we are concerned about the entire distribution of events, such as when uncertain factors lead to more extreme events in a system <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id47" title="Antonia Hadjimichael, Julianne Quinn, and Patrick Reed. Advancing Diagnostic Model Evaluation to Better Understand Water Shortage Mechanisms in Institutionally Complex River Basins. Water Resources Research, 56(10):e2020WR028079, 2020. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR028079 (visited on 2020-10-16), doi:10.1029/2020WR028079.">13</a>]</span>. Further, they can be used with a pre-existing sample of data, without requiring a specific sampling scheme, unlike the previously reviewed methods <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id143" title="Elmar Plischke, Emanuele Borgonovo, and Curtis L Smith. Global sensitivity measures from given data. European Journal of Operational Research, 226(3):536–550, 2013.">124</a>]</span>. The <span class="math notranslate nohighlight">\(\delta\)</span> sensitivity index does not include interactions between factors and it is therefore akin to the first order index produced by the Sobol method. Interactions between factors can still be estimated using this method, by conditioning the calculation on more than one uncertain factor being fixed <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id141" title="Emanuele Borgonovo. A new uncertainty importance measure. Reliability Engineering &amp; System Safety, 92(6):771–784, 2007.">123</a>]</span>.</p>
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+    <h1>Sensitivity Analysis Methods</h1>
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+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivative-based-methods">Derivative-based Methods</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#elementary-effect-methods">Elementary Effect Methods</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#regression-based-methods">Regression-based Methods</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#regional-sensitivity-analysis">Regional Sensitivity Analysis</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#variance-based-methods">Variance-based Methods</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysis-of-variance-anova">Analysis of Variance (ANOVA)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#moment-independent-density-based-methods">Moment-Independent (Density-Based) Methods</a></li>
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+  <section id="sensitivity-analysis-methods">
+<span id="id1"></span><h1>Sensitivity Analysis Methods<a class="headerlink" href="#sensitivity-analysis-methods" title="Link to this heading">#</a></h1>
+<p>In this section, we describe some of the most widely applied sensitivity analysis methods along with their mathematical definitions. We also provide a detailed discussion on applying each method, as well as a comparison of and their features and limitations.</p>
+<section id="derivative-based-methods">
+<h2>Derivative-based Methods<a class="headerlink" href="#derivative-based-methods" title="Link to this heading">#</a></h2>
+<p>Derivative-based methods explore how model outputs are affected by perturbations in a single model input around a particular input value. These methods are local and are performed using OAT sampling. For simplicity of mathematical notations, let us assume that the model <span class="math notranslate nohighlight">\(g(X)\)</span> only returns one output. Following <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id117" title="Emanuele Borgonovo. Sensitivity analysis of model output with input constraints: a generalized rationale for local methods. Risk Analysis: An International Journal, 28(3):667–680, 2008.">90</a>]</span> and <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>, the sensitivity index, <span class="math notranslate nohighlight">\(S_i\)</span> , of the model’s <em>i</em>-th input factor, <span class="math notranslate nohighlight">\(x_i\)</span> , can be measured using the partial derivative evaluated at a nominal value, <span class="math notranslate nohighlight">\(\bar{x}\)</span>, of the vector of inputs:</p>
+<div class="math notranslate nohighlight">
+\[S_i (\bar{x}) = \frac{\partial g}{\partial x} |_{\bar{x}{^c{_i}}}\]</div>
+<p>where <em>c</em><sub>i</sub> is the scaling factor. In most applications however, the relationship <span class="math notranslate nohighlight">\(g(X)\)</span> is not fully known in its analytical form, and therefore the above partial derivative is usually approximated:</p>
+<div class="math notranslate nohighlight">
+\[S_i (\bar{x}) = \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i\]</div>
+<p>Using this approximation, the <em>i</em>-th input factor is perturbed by a magnitude of <span class="math notranslate nohighlight">\(\Delta_i\)</span>, and its relative importance is calculated. Derivative-based methods are some of the oldest sensitivity analysis methods as they only require <span class="math notranslate nohighlight">\(N+1\)</span> model evaluations to estimate indices for <span class="math notranslate nohighlight">\(N\)</span> uncertain factors. As described above, being computationally very cheap comes at the cost of not being able to explore the entire input space, but only (local) perturbations to the nominal value. Additionally, as these methods examine the effects of each input factor one at a time, they cannot assess parametric interactions or capture the interacting nature of many real systems and the models that abstract them.</p>
+</section>
+<section id="elementary-effect-methods">
+<h2>Elementary Effect Methods<a class="headerlink" href="#elementary-effect-methods" title="Link to this heading">#</a></h2>
+<p>Elementary effect (EE) SA methods provide a solution to the local nature of the derivative-based methods by exploring the entire parametric range of each input parameter <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>. However, EE methods still use OAT sampling and do not vary all input parameters simultaneously while exploring the parametric space. The OAT nature of EEs methods therefore prevents them from properly capturing the interactions between uncertain factors. EEs methods are computationally efficient compared to their All-At-a-Time (AAT) counterparts, making them more suitable when computational capacity is a limiting factor, while still allowing for some inferences regarding factor interactions.
+The most popular EE method is the Method of Morris <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id132" title="Max D Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.">74</a>]</span>. Following the notation by <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>, this method calculates global sensitivity using the mean of the EEs (finite differences) of each parameter at different locations:</p>
+<div class="math notranslate nohighlight">
+\[S_i = \mu_i^* = \frac{1}{r}\sum_{j=1}^r EE^j_i = \frac{1}{r}\sum_{j=1}^r \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i\]</div>
+<p>with <span class="math notranslate nohighlight">\(r\)</span> representing the number of sample repetitions (also refered to as trajectories) in the input space, usually set between 4 and 10 <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. Each <span class="math notranslate nohighlight">\(x_j\)</span> represents the points of each trajectory, with <span class="math notranslate nohighlight">\(j=1,…, r\)</span>, selected as described in the sampling strategy for this method, found above. This method also produces the standard deviation of the EEs:</p>
+<div class="math notranslate nohighlight">
+\[\sigma_i = \sqrt{\frac{1}{r}\sum_{j=1}^r(EE_i^j-\frac{1}{r}\sum_{j=1}^r EE^j_i)^2}\]</div>
+<p>which is a measure of parametric interactions. Higher values of <span class="math notranslate nohighlight">\(\sigma_i\)</span> suggest model responses at different levels of factor <span class="math notranslate nohighlight">\(x_i\)</span> are significantly different, which indicates considerable interactions between that and other uncertain factors. The values of <span class="math notranslate nohighlight">\(\mu_i^*\)</span> and <span class="math notranslate nohighlight">\(\sigma_i\)</span> for each factor allow us to draw several different conclusions, illustrated in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-4"><span class="std std-numref">Fig. 3.4</span></a>, following the example by <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>. In this example, factors <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> can be said to have an influence on the model outputs, with <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> having some interactive or non-linear effects. Depending on the orders of magnitude of <span class="math notranslate nohighlight">\(\mu_i^*\)</span> and <span class="math notranslate nohighlight">\(\sigma_i\)</span> one can indirectly deduce whether the factors have strong interactive effects, for example if a factor <span class="math notranslate nohighlight">\(\sigma_i &lt;&lt; \mu_i^*\)</span> then the relationship between that factor and the output can be assumed to be largely linear (note that this is still an OAT method and assumptions on factor interactions should be strongly caveated). Extensions of the Method of Morris have also been developed specifically for the purposes of factor fixing and explorations of parametric interactions (e.g., <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id27" title="Emanuele Borgonovo. Sensitivity analysis with finite changes: an application to modified eoq models. European Journal of Operational Research, 200(1):127–138, 2010.">48</a>, <a class="reference internal" href="R.Bibliography.html#id133" title="Francesca Campolongo and Roger Braddock. The use of graph theory in the sensitivity analysis of the model output: a second order screening method. Reliability Engineering &amp; System Safety, 64(1):1–12, 1999.">92</a>, <a class="reference internal" href="R.Bibliography.html#id134" title="Roger A Cropp and Roger D Braddock. The new morris method: an efficient second-order screening method. Reliability Engineering &amp; System Safety, 78(1):77–83, 2002.">93</a>]</span>).</p>
+<figure class="margin-caption align-center" id="id41">
+<span id="figure-3-4"></span><a class="reference internal image-reference" href="_images/figure3_4_morris_method.png"><img alt="Figure 3.4" src="_images/figure3_4_morris_method.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Illustrative results of the Morris Method. Factors <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> have an influence on the model outputs, with <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> having interactive or non-linear effects. Whether or not a factor should be considered influential to the output depends on the output selected and is specific to the research context and purpose of the analysis, as discussed in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#why-sa"><span class="std std-numref">Chapter 3.2</span></a>.</span><a class="headerlink" href="#id41" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="regression-based-methods">
+<h2>Regression-based Methods<a class="headerlink" href="#regression-based-methods" title="Link to this heading">#</a></h2>
+<p>Regression analysis is one of the oldest ways of investigating parametric importance and sensitivity <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. Here, we describe some of the most popular regression-based sensitivity indices. One of the main sensitivity indices of this category is the standardized regression coefficient (SRC). To calculate SRC, a linear regression relationship needs to be fitted between the input vector, <span class="math notranslate nohighlight">\(x\)</span>, and the model output of interest by using a least-square minimizing method:</p>
+<div class="math notranslate nohighlight">
+\[y = b_0 + \sum_{i=1}^N b_ix_i\]</div>
+<p>where <span class="math notranslate nohighlight">\(b_0\)</span> and <span class="math notranslate nohighlight">\(b_i\)</span> (corresponding to the <em>i</em>-th model input) are regression coefficients. The following relationship can then be used to calculate the SRCs for different input values:</p>
+<div class="math notranslate nohighlight">
+\[S_i=SRC_i=b_i\frac{\sigma_i}{\sigma_y}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\sigma_i\)</span> and <span class="math notranslate nohighlight">\(\sigma_y\)</span> are standard deviations of <em>i</em>-th model input and output, respectively.</p>
+<p>Several other regression-based indices explore the correlation between input and output parameters as a proxy to model parametric sensitivity <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>, <a class="reference internal" href="R.Bibliography.html#id135" title="Jon C Helton. Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal. Reliability Engineering &amp; System Safety, 42(2-3):327–367, 1993.">94</a>, <a class="reference internal" href="R.Bibliography.html#id136" title="Gemma Manache and Charles S Melching. Identification of reliable regression-and correlation-based sensitivity measures for importance ranking of water-quality model parameters. Environmental Modelling &amp; Software, 23(5):549–562, 2008.">95</a>]</span>. The Pearson correlation coefficient (PCC) can be used when a linear relationship exists between an uncertain factor, <span class="math notranslate nohighlight">\(x_i\)</span>, and the output <span class="math notranslate nohighlight">\(y\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i=PCC=\frac{cov(x_i,y)}{\sigma_i\sigma_y}\]</div>
+<p>In cases when there are outliers in the data or the relationship between the uncertain factors and the output is not linear, rank-based correlation coefficients are preferred, for example, Spearman’s rank correlation coefficient (SRCC):</p>
+<div class="math notranslate nohighlight">
+\[S_i=SRCC=\frac{cov(rx_i,ryi)}{\sigma_{ri}\sigma_{ry}}\]</div>
+<p>where the raw values of <span class="math notranslate nohighlight">\(x_i\)</span> and <span class="math notranslate nohighlight">\(y\)</span> and converted to ranks <span class="math notranslate nohighlight">\(rx_i\)</span> and <span class="math notranslate nohighlight">\(ry\)</span> respectively, which instead represent a measurement of the strength of the monotonic relationship, rather than linear relationship, between the input and output. Other regression-based metrics include the partial correlations coefficient, the partial rank correlations coefficient, and the Nash-Sutcliffe coefficient, more discussion on which can be found in <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>.</p>
+<p>Tree-based regression techniques have also been used for sensitivity analysis in an effort to address the challenges faced with nonlinear models <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id137" title="F Pappenberger and Keith J Beven. Ignorance is bliss: or seven reasons not to use uncertainty analysis. Water resources research, 2006.">96</a>]</span>. Examples of these methods include the Patient Rule Induction Method (PRIM; <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id79" title="Jerome H Friedman and Nicholas I Fisher. Bump hunting in high-dimensional data. Statistics and Computing, 9(2):123–143, 1999.">97</a>]</span>) and Classification And Regression Trees (CART; <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id80" title="Leo Breiman, Jerome Friedman, Charles J Stone, and Richard A Olshen. Classification and regression trees. CRC press, 1984.">98</a>]</span>). CART-based approaches also include boosting and bagging extensions <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id89" title="Yoav Freund, Robert Schapire, and Naoki Abe. A short introduction to boosting. Journal-Japanese Society For Artificial Intelligence, 14(771-780):1612, 1999.">99</a>, <a class="reference internal" href="R.Bibliography.html#id138" title="Leo Breiman. Bagging predictors. Machine learning, 24(2):123–140, 1996.">100</a>]</span>. These methods are particularly useful when sensitivity analysis is used for factor mapping (i.e., when trying to identify which uncertain model factors produce a certain model behavior). <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a> elaborates on the use of these methods. Regression-based sensitivity analysis methods are global by nature and can explore the entire space of variables. However, the true level of comprehensiveness depends on the design of experiments and the number of simulations providing data to establish the regression relationships. Although they are usually computationally efficient, they do not produce significant information about parametric interactions <span id="id17">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>]</span>.</p>
+</section>
+<section id="regional-sensitivity-analysis">
+<h2>Regional Sensitivity Analysis<a class="headerlink" href="#regional-sensitivity-analysis" title="Link to this heading">#</a></h2>
+<p>Another method primarily applied for basic factor mapping applications is Regional Sensitivity Analysis (RSA; <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id157" title="George M Hornberger and Robert C Spear. Approach to the preliminary analysis of environmental systems. J. Environ. Mgmt., 12(1):7–18, 1981.">101</a>]</span>). RSA is a global sensitivity analysis method that is typically implemented using standard sampling methods such as latin hypercube sampling. It is performed by specifying a condition on the output space (e.g., an upper threshold) and classifying outputs that meet the condition as behavioral and the ones that fail it as non-behavioral (illustrated in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (b)). Note that the specified threshold depends on the nature of the problem, model, and the research question. It can reflect model-performance metrics (such as errors) or consequential decision-relevant metrics (such as unacceptable system outcomes). The behavioral and non-behavioral outputs are then traced back to their originating sampled factors, where differences between the distributions of samples can be used to determine their significance in producing each part of the output. The Kolmogorov-Smirnov divergence is commonly used to quantify the difference between the distribution of behavioral and non-behavioral parameters <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i=|F_{x_i|y_b} (y \in Y_b)-F_{x_i|y_{nb}} (y \in Y_{nb})|\]</div>
+<p>where <span class="math notranslate nohighlight">\(Y_b\)</span> represents the set of behavioral outputs, and <span class="math notranslate nohighlight">\(F_{x_i|y_b}\)</span> is the empirical cumulative distribution function of the values of <span class="math notranslate nohighlight">\(x_i\)</span> associated with values of <span class="math notranslate nohighlight">\(y\)</span> that belong in the behavioral set. The <span class="math notranslate nohighlight">\(nb\)</span> notation indicates the equivalent elements related to the non-behavioral set. Large differences between the two distributions indicate stronger effects by the parameters on the respective part of the output space.</p>
+<p>Used in a factor mapping setting, RSA can be applied for scenario discovery <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id71" title="Robert J Lempert, David G Groves, Steven W Popper, and Steve C Bankes. A general, analytic method for generating robust strategies and narrative scenarios. Management science, 52(4):514–528, 2006.">102</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>, the Generalized Likelihood Uncertainty Estimation method (GLUE; <span id="id21">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>, <a class="reference internal" href="R.Bibliography.html#id158" title="Keith Beven and Andrew Binley. Glue: 20 years on. Hydrological processes, 28(24):5897–5918, 2014.">104</a>, <a class="reference internal" href="R.Bibliography.html#id159" title="Roberta-Serena Blasone, Jasper A Vrugt, Henrik Madsen, Dan Rosbjerg, Bruce A Robinson, and George A Zyvoloski. Generalized likelihood uncertainty estimation (glue) using adaptive markov chain monte carlo sampling. Advances in Water Resources, 31(4):630–648, 2008.">105</a>]</span>) and other hybrid sensitivity analysis methods (e.g., <span id="id22">[<a class="reference internal" href="R.Bibliography.html#id160" title="SA Cryer and PL Havens. Regional sensitivity analysis using a fractional factorial method for the usda model gleams. Environmental modelling &amp; software, 14(6):613–624, 1999.">106</a>, <a class="reference internal" href="R.Bibliography.html#id161" title="Pengfei Wei, Zhenzhou Lu, and Xiukai Yuan. Monte carlo simulation for moment-independent sensitivity analysis. Reliability Engineering &amp; System Safety, 110:60–67, 2013.">107</a>]</span>). The fundamental shortcomings of RSA are that, in some cases, it could be hard to interpret the difference between behavioral and non-behavioral sample sets, and that insights about parametric correlations and interactions cannot always be uncovered <span id="id23">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. For more elaborate discussions and illustrations of the RSA method, readers are directed to <span id="id24">Tang <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id20" title="Yong Tang, Patrick Reed, Thibaut Wagener, and K van Werkhoven. Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation. Hydrology and Earth System Sciences, 11(2):793–817, 2007.">42</a>], Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>], Young [<a class="reference internal" href="R.Bibliography.html#id162" title="Peter Young. Data-based mechanistic modelling, generalised sensitivity and dominant mode analysis. Computer Physics Communications, 117(1-2):113–129, 1999.">108</a>]</span> and references therein.</p>
+</section>
+<section id="variance-based-methods">
+<span id="id25"></span><h2>Variance-based Methods<a class="headerlink" href="#variance-based-methods" title="Link to this heading">#</a></h2>
+<p>Variance-based sensitivity analysis methods hypothesize that various specified model factors contribute differently to the variation of model outputs; therefore, decomposition and analysis of output variance can determine a model’s sensitivity to input parameters <span id="id26">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>]</span>. The most popular variance-based method is the Sobol method, which is a global sensitivity analysis method that takes into account complex and nonlinear factor interaction when calculating sensitivity indices, and employs more sophisticated sampling methods (e.g., the Sobol sampling method). The Sobol method is able to calculate three types of sensitivity indices that provide different types of information about model sensitivities. These indices include first-order, higher-order (e.g., second-, third-, etc. orders), and total-order sensitivities.</p>
+<p>The first-order sensitivity index indicates the percent of model output variance contributed by a factor individually (i.e., the effect of varying <span class="math notranslate nohighlight">\(x_i\)</span> alone) and is obtained using the following <span id="id27">[<a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>, <a class="reference internal" href="R.Bibliography.html#id148" title="Andrea Saltelli. Making best use of model evaluations to compute sensitivity indices. Computer physics communications, 145(2):280–297, 2002.">109</a>]</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i^1=\frac{V_{x_i}[E_{x_{\sim i}}(x_i)]}{V(y)}\]</div>
+<p>with <span class="math notranslate nohighlight">\(E\)</span> and <span class="math notranslate nohighlight">\(V\)</span> denoting the expected value and the variance, respectively. <span class="math notranslate nohighlight">\(x_{\sim i}\)</span> denotes all factors expect for <span class="math notranslate nohighlight">\(x_i\)</span>. The first-order sensitivity index (<span class="math notranslate nohighlight">\(S_i^1\)</span>) can therefore also be thought of as the portion of total output variance (<span class="math notranslate nohighlight">\(V_y\)</span>) that can be reduced if the uncertainty in factor <span class="math notranslate nohighlight">\(x_i\)</span> is eliminated <span id="id28">[<a class="reference internal" href="R.Bibliography.html#id149" title="Andrea Saltelli. Sensitivity analysis for importance assessment. Risk analysis, 22(3):579–590, 2002.">110</a>]</span>. First-order sensitivity indices are usually used to understand the independent effect of a factor and to distinguish its individual versus interactive influence. It would be expected for linearly independent factors that they would only have first order indices (no interactions) that should correspond well with sensitivities obtained from simpler methods using OAT sampling.</p>
+<p>Higher-order sensitivity indices explore the interaction between two or more parameters that contribute to model output variations. For example, a second-order index indicates how interactions between a pair of factors can lead to change in model output variance and is calculated using the following relationship:</p>
+<div class="math notranslate nohighlight">
+\[S_{ij}^2=\frac{V_{x_{i,j}}[E_{x_{\sim i,j}}(x_i,x_j)]}{V(y)}\]</div>
+<p>with <span class="math notranslate nohighlight">\(i \ne j\)</span>. Higher order indices can be calculated by similar extensions (i.e., fixing additional operators together), but it is usually computationally expensive in practice.</p>
+<p>The total sensitivity analysis index represents the entire influence of an input factor on model outputs including all of its interactions with other factors <span id="id29">[<a class="reference internal" href="R.Bibliography.html#id150" title="Toshimitsu Homma and Andrea Saltelli. Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering &amp; System Safety, 52(1):1–17, 1996.">111</a>]</span>. In other words, total-order indices include first-order and all higher-order interactions associated with each factor and can be estimated calculated using the following:</p>
+<div class="math notranslate nohighlight">
+\[S_i^T= \frac{E_{x_{\sim i}}[V_{x_i}(x_{\sim i})]}{V(y)} = 1 - \frac{V_{x_{\sim i}}[E_{x_{i}}(x_{\sim i})]}{V(y)}\]</div>
+<p>This index reveals the expected portion of variance that remains if uncertainty is eliminated in all factors but <span class="math notranslate nohighlight">\(x_i\)</span> <span id="id30">[<a class="reference internal" href="R.Bibliography.html#id149" title="Andrea Saltelli. Sensitivity analysis for importance assessment. Risk analysis, 22(3):579–590, 2002.">110</a>]</span>. The total sensitivity index is the overall best measure of sensitivity as it captures the full individual and interactive effects of model factors.</p>
+<p>Besides the Sobol method, there are some other variance-based sensitivity analysis methods, such as the Fourier amplitude sensitivity test (FAST; <span id="id31">[<a class="reference internal" href="R.Bibliography.html#id100" title="RI Cukier, CM Fortuin, Kurt E Shuler, AG Petschek, and JH Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. i theory. The Journal of chemical physics, 59(8):3873–3878, 1973.">75</a>, <a class="reference internal" href="R.Bibliography.html#id151" title="Gregory J McRae, William R Goodin, and John H Seinfeld. Development of a second-generation mathematical model for urban air pollution—i. model formulation. Atmospheric Environment (1967), 16(4):679–696, 1982.">112</a>]</span>) and extended-FAST <span id="id32">[<a class="reference internal" href="R.Bibliography.html#id152" title="Andrea Saltelli and Ricardo Bolado. An alternative way to compute fourier amplitude sensitivity test (fast). Computational Statistics &amp; Data Analysis, 26(4):445–460, 1998.">113</a>, <a class="reference internal" href="R.Bibliography.html#id153" title="MA Vazquez-Cruz, R Guzman-Cruz, IL Lopez-Cruz, O Cornejo-Perez, I Torres-Pacheco, and RG Guevara-Gonzalez. Global sensitivity analysis by means of efast and sobol’methods and calibration of reduced state-variable tomgro model using genetic algorithms. Computers and Electronics in Agriculture, 100:1–12, 2014.">114</a>]</span>, that have been used by the scientific community. However, Sobol remains by far the most common method of this class. Variance-based techniques have been widely used and have proved to be powerful in a variety of applications. Despite their popularity, some authors have expressed concerns about the methods’ appropriateness in some settings. Specifically, the presence of heavy-tailed distributions or outliers, or when model outputs are multimodal can bias the sensitivity indices produced by these methods <span id="id33">[<a class="reference internal" href="R.Bibliography.html#id154" title="Benjamin Auder and Bertrand Iooss. Global sensitivity analysis based on entropy. In Safety, reliability and risk analysis-Proceedings of the ESREL 2008 Conference, 2107–2115. 2008.">115</a>, <a class="reference internal" href="R.Bibliography.html#id155" title="Farkhondeh Khorashadi Zadeh, Jiri Nossent, Fanny Sarrazin, Francesca Pianosi, Ann van Griensven, Thorsten Wagener, and Willy Bauwens. Comparison of variance-based and moment-independent global sensitivity analysis approaches by application to the swat model. Environmental Modelling &amp; Software, 91:210–222, 2017.">116</a>, <a class="reference internal" href="R.Bibliography.html#id156" title="Francesca Pianosi and Thorsten Wagener. A simple and efficient method for global sensitivity analysis based on cumulative distribution functions. Environmental Modelling &amp; Software, 67:1–11, 2015.">117</a>]</span>. Moment-independent measures, discussed below, attempt to overcome these challenges.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial which demonstrates the application of a Sobol sensitivity analysis:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA using SALib Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="analysis-of-variance-anova">
+<h2>Analysis of Variance (ANOVA)<a class="headerlink" href="#analysis-of-variance-anova" title="Link to this heading">#</a></h2>
+<p>Analysis of Variance (ANOVA) was first introduced by <span id="id34">Fisher and others [<a class="reference internal" href="R.Bibliography.html#id144" title="Ronald Aylmer Fisher and others. Statistical methods for research workers. Statistical methods for research workers., 1934.">118</a>]</span> and has since become a popular factor analysis method in physical experiments. ANOVA can be used as a sensitivity analysis method in computational experiments with a factorial design of experiment (referred to as factorial ANOVA). Note that Sobol can also be categorized as an ANOVA sensitivity analysis method, and that is why Sobol is sometimes referred to as a functional ANOVA <span id="id35">[<a class="reference internal" href="R.Bibliography.html#id145" title="Loıc Brevault, Mathieu Balesdent, Nicolas Bérend, and Rodolphe Le Riche. Comparison of different global sensitivity analysis methods for aerospace vehicle optimal design. In 10th World Congress on Structural and Multidisciplinary Optimization, WCSMO-10. 2013.">119</a>]</span>. Factorial ANOVA methods are particularly suited for models and problems that have discrete input spaces, significantly reducing the computational time. More information about these methods can be found in <span id="id36">[<a class="reference internal" href="R.Bibliography.html#id145" title="Loıc Brevault, Mathieu Balesdent, Nicolas Bérend, and Rodolphe Le Riche. Comparison of different global sensitivity analysis methods for aerospace vehicle optimal design. In 10th World Congress on Structural and Multidisciplinary Optimization, WCSMO-10. 2013.">119</a>, <a class="reference internal" href="R.Bibliography.html#id146" title="GEB Archer, Andrea Saltelli, and IM Sobol. Sensitivity measures, anova-like techniques and the use of bootstrap. Journal of Statistical Computation and Simulation, 58(2):99–120, 1997.">120</a>, <a class="reference internal" href="R.Bibliography.html#id147" title="Art B Owen. Variance components and generalized sobol'indices. SIAM/ASA Journal on Uncertainty Quantification, 1(1):19–41, 2013.">121</a>]</span>.</p>
+</section>
+<section id="moment-independent-density-based-methods">
+<h2>Moment-Independent (Density-Based) Methods<a class="headerlink" href="#moment-independent-density-based-methods" title="Link to this heading">#</a></h2>
+<p>These methods typically compare the entire distribution (i.e., not just the variance) of input and output parameters in order to determine the sensitivity of the output to a particular input variable. Several moment-independent sensitivity analysis methods have been proposed in recent years. The delta (<span class="math notranslate nohighlight">\(\delta\)</span>) moment-independent method calculates the difference between unconditional and conditional cumulative distribution functions of the output. The method was first introduced by <span id="id37">[<a class="reference internal" href="R.Bibliography.html#id142" title="Emanuele Borgonovo. Measuring uncertainty importance: investigation and comparison of alternative approaches. Risk analysis, 26(5):1349–1361, 2006.">122</a>, <a class="reference internal" href="R.Bibliography.html#id141" title="Emanuele Borgonovo. A new uncertainty importance measure. Reliability Engineering &amp; System Safety, 92(6):771–784, 2007.">123</a>]</span> and has become widely used in various disciplines. The <span class="math notranslate nohighlight">\(\delta\)</span> sensitivity index is defined as follows:</p>
+<div class="math notranslate nohighlight">
+\[S_i=\delta_i=\frac{1}{2}E_{x_i}|f_y(y)-f_{y|x_i}(y)|dy\]</div>
+<p>where <span class="math notranslate nohighlight">\(f_y(y)\)</span> is the probability density function of the entire model output <span class="math notranslate nohighlight">\(y\)</span>, and <span class="math notranslate nohighlight">\(f_{y|x_i}(y)\)</span> is the conditional density of <span class="math notranslate nohighlight">\(y\)</span>, given that factor <span class="math notranslate nohighlight">\(x_i\)</span> assumes a fixed value. The <span class="math notranslate nohighlight">\(\delta_i\)</span> sensitivity indicator therefore represents the normalized expected shift in the distribution of <span class="math notranslate nohighlight">\(y\)</span> provoked by <span class="math notranslate nohighlight">\(x_i\)</span>. Moment-independent methods are advantageous in cases where we are concerned about the entire distribution of events, such as when uncertain factors lead to more extreme events in a system <span id="id38">[<a class="reference internal" href="R.Bibliography.html#id47" title="Antonia Hadjimichael, Julianne Quinn, and Patrick Reed. Advancing Diagnostic Model Evaluation to Better Understand Water Shortage Mechanisms in Institutionally Complex River Basins. Water Resources Research, 56(10):e2020WR028079, 2020. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR028079 (visited on 2020-10-16), doi:10.1029/2020WR028079.">13</a>]</span>. Further, they can be used with a pre-existing sample of data, without requiring a specific sampling scheme, unlike the previously reviewed methods <span id="id39">[<a class="reference internal" href="R.Bibliography.html#id143" title="Elmar Plischke, Emanuele Borgonovo, and Curtis L Smith. Global sensitivity measures from given data. European Journal of Operational Research, 226(3):536–550, 2013.">124</a>]</span>. The <span class="math notranslate nohighlight">\(\delta\)</span> sensitivity index does not include interactions between factors and it is therefore akin to the first order index produced by the Sobol method. Interactions between factors can still be estimated using this method, by conditioning the calculation on more than one uncertain factor being fixed <span id="id40">[<a class="reference internal" href="R.Bibliography.html#id141" title="Emanuele Borgonovo. A new uncertainty importance measure. Reliability Engineering &amp; System Safety, 92(6):771–784, 2007.">123</a>]</span>.</p>
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+  <section id="how-to-choose-a-sensitivity-analysis-method-model-traits-and-dimensionality">
+<h1>How To Choose A Sensitivity Analysis Method: Model Traits And Dimensionality<a class="headerlink" href="#how-to-choose-a-sensitivity-analysis-method-model-traits-and-dimensionality" title="Link to this heading">#</a></h1>
+<p><a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-5"><span class="std std-numref">Fig. 3.5</span></a>, synthesized from variants found in <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>, presents a graphical synthesis of the methods overviewed in this section, with regards to their appropriateness of application based on the complexity of the model at hand and the computational limits on the number of model evaluations afforded. The bars below each method also indicate the sensitivity analysis purposes they are most appropriate to address, which are in turn a reflection of the motivations and research questions the sensitivity analysis is called to address. Computational intensity is measured as a multiple of the number of model factors that are considered uncertain (<span class="math notranslate nohighlight">\(d\)</span>). Increasing model complexity mandates that more advanced sensitivity analysis methods are applied to address potential nonlinearities, factor interactions, and discontinuities. Such methods can only be performed at increasing computational expense. For example, computationally cheap linear regression should not be used to assess factors’ importance if the model cannot be proven linear and the factors independent, because important relationships will invariably be missed (recall the example in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-5"><span class="std std-numref">Fig. 3.5</span></a>). When computational limits do constrain applications to make simplified assumptions and sensitivity techniques, any conclusions in such cases should be delivered with clear statements of the appropriate caveats.</p>
+<figure class="margin-caption align-center" id="id7">
+<span id="figure-3-5"></span><a class="reference internal image-reference" href="_images/figure3_5classificationofmethods.png"><img alt="Figure 3_5" src="_images/figure3_5classificationofmethods.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Classification of the sensitivity analysis methods overviewed in this section, with regards to their computational cost (horizontal axis), their appropriateness to model complexity (vertical axis), and the purpose they can be used for (colored bars). d: number of uncertain factors considered; ANOVA: Analysis of Variance; FAST: Fourier Amplitude Sensitivity Test; PRIM: Patient Rule Induction Method; CART: Classification and Regression Trees; SRCC: Spearman’s rank correlation coefficient: NSE: Nash–Sutcliffe efficiency; SRC: standardized regression coefficient; PCC: Pearson correlation coefficient. This figure is synthesized from variants found in <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>.</span><a class="headerlink" href="#id7" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The reader should also be aware that the estimates of computational intensity that are given here are indicative of magnitude and would vary depending on the sampling technique, model complexity and the level of information being asked. For example, a Sobol sensitivity analysis typically requires a sample of size <span class="math notranslate nohighlight">\(n * d+2\)</span> to produce first- and total-order indices, where <span class="math notranslate nohighlight">\(d\)</span> is the number of uncertain factors and <span class="math notranslate nohighlight">\(n\)</span> is a scaling factor, selected ad hoc, depending on model complexity <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id25" title="Andrea Saltelli and Stefano Tarantola. On the relative importance of input factors in mathematical models: safety assessment for nuclear waste disposal. Journal of the American Statistical Association, 97(459):702–709, 2002.">46</a>]</span>. The scaling factor <span class="math notranslate nohighlight">\(n\)</span> is typically set to at least 1000, but it should most appropriately be set on the basis of index convergence. In other words, a prudent analyst would perform the analysis several times with increasing <span class="math notranslate nohighlight">\(n\)</span> and observe at what level the indices converge to stable values <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id140" title="Jiri Nossent, Pieter Elsen, and Willy Bauwens. Sobol’ sensitivity analysis of a complex environmental model. Environmental Modelling &amp; Software, 26(12):1515-1525, 2011. URL: https://www.sciencedirect.com/science/article/pii/S1364815211001939, doi:https://doi.org/10.1016/j.envsoft.2011.08.010.">125</a>]</span>. The level should be the minimum sample size used in subsequent sensitivity analyses of the same system. Furthermore, if the analyst would like to better understand the degrees of interaction between factors, requiring second-order indices, the sample size would have to increase to <span class="math notranslate nohighlight">\(n * 2d+2\)</span> <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id25" title="Andrea Saltelli and Stefano Tarantola. On the relative importance of input factors in mathematical models: safety assessment for nuclear waste disposal. Journal of the American Statistical Association, 97(459):702–709, 2002.">46</a>]</span>.</p>
+<p>Another important consideration is that methods that do not require specific sampling schemes can be performed in conjunction with others without requiring additional model evaluations. None of the regression-based methods, for example, require samples of specific structures or sizes, and can be combined with other methods for complementary purposes. For instance, one could complement a Sobol analysis with an application of CART, using the same data, but to address questions relating to factor mapping (e.g., we know factor <span class="math notranslate nohighlight">\(x_i\)</span>  is important for a model output, but we would like to also know which of its values specifically push the output to undesirable states). Lastly, comparing results from different methods performed together can be especially useful in model diagnostic settings. For example, <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id47" title="Antonia Hadjimichael, Julianne Quinn, and Patrick Reed. Advancing Diagnostic Model Evaluation to Better Understand Water Shortage Mechanisms in Institutionally Complex River Basins. Water Resources Research, 56(10):e2020WR028079, 2020. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR028079 (visited on 2020-10-16), doi:10.1029/2020WR028079.">13</a>]</span> used <span class="math notranslate nohighlight">\(\delta\)</span> indices, first-order Sobol indices, and <span class="math notranslate nohighlight">\(R^2\)</span>  values from linear regression, all performed on the same factors, to derive insights about the effects on factors on different moments of the output distribution and about the linearity of their relationship.</p>
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+  <section id="software-toolkits">
+<span id="id1"></span><h1>Software Toolkits<a class="headerlink" href="#software-toolkits" title="Link to this heading">#</a></h1>
+<p>This section presents available open source sensitivity analysis software tools, based on the programming language they use and the methods they support <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-6"><span class="std std-numref">Fig. 3.6</span></a>. Our review covers five widely used programming languages: R, MATLAB, Julia, Python, and C++, as well as one tool that provides a graphical user interface (GUI). Each available SA tool was assessed on the number of SA methods and design of experiments methods it supports. For example, the <em>sensobol</em> package in R only supports the variance-based Sobol method. However, it is the only package we came across that calculates third-order interactions among parameters. On the other side of the spectrum, there are SA software packages that contain several popular SA methods. For example, <em>SALib</em> in Python <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id139" title="Jon Herman and Will Usher. Salib: an open-source python library for sensitivity analysis. Journal of Open Source Software, 2(9):97, 2017.">126</a>]</span> supports seven different SA methods. The <em>DifferentialEquations</em> package is a comprehensive package developed for Julia, and <em>GlobalSensitivityAnalysis</em> is another Julia package that has mostly adapted SALib methods. <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-6"><span class="std std-numref">Fig. 3.6</span></a> also identifies the SA packages that have been updated since 2018, indicating active support and development.</p>
+<figure class="margin-caption align-center" id="id3">
+<span id="figure-3-6"></span><a class="reference internal image-reference" href="_images/figure3_6_softwaretoolkits.png"><img alt="Figure 3_6" src="_images/figure3_6_softwaretoolkits.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Sensitivity analysis packages available in different programming language platforms (R, Python, Julia, MATLAB, and C++), with the number of methods they support. Packages supporting more than five methods are indicated in pink. Packages updated since 2018 are indicated with asterisks.</span><a class="headerlink" href="#id3" title="Link to this image">#</a></p>
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+    <h1>Sensitivity Analysis: The Basics</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#global-versus-local-sensitivity">3.1. Global Versus Local Sensitivity</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#why-perform-sensitivity-analysis">3.2. Why Perform Sensitivity Analysis</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#design-of-experiments">3.3. Design of Experiments</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#one-at-a-time-oat">3.3.1. One-At-a-Time (OAT)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#full-and-fractional-factorial-sampling">3.3.2. Full and Fractional Factorial Sampling</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#latin-hypercube-sampling-lhs">3.3.3. Latin Hypercube Sampling (LHS)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#low-discrepancy-sequences">3.3.4. Low-Discrepancy Sequences</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#other-types-of-sampling">3.3.5. Other types of sampling</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-generation-of-input-time-series">3.3.6. Synthetic generation of input time series</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sensitivity-analysis-methods">3.4. Sensitivity Analysis Methods</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#derivative-based-methods">3.4.1. Derivative-based Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#elementary-effect-methods">3.4.2. Elementary Effect Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#regression-based-methods">3.4.3. Regression-based Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#regional-sensitivity-analysis">3.4.4. Regional Sensitivity Analysis</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#variance-based-methods">3.4.5. Variance-based Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#analysis-of-variance-anova">3.4.6. Analysis of Variance (ANOVA)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#moment-independent-density-based-methods">3.4.7. Moment-Independent (Density-Based) Methods</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#how-to-choose-a-sensitivity-analysis-method-model-traits-and-dimensionality">3.5. How To Choose A Sensitivity Analysis Method: Model Traits And Dimensionality</a></li>
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+  <section id="sensitivity-analysis-the-basics">
+<span id="id1"></span><h1><span class="section-number">3. </span>Sensitivity Analysis: The Basics<a class="headerlink" href="#sensitivity-analysis-the-basics" title="Link to this heading">#</a></h1>
+<section id="global-versus-local-sensitivity">
+<span id="global-vs-local"></span><h2><span class="section-number">3.1. </span>Global Versus Local Sensitivity<a class="headerlink" href="#global-versus-local-sensitivity" title="Link to this heading">#</a></h2>
+<p>Out of the several definitions for sensitivity analysis presented in the literature, the most widely used has been proposed by <span id="id2">Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span> as “the study of how uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input”. In other words, sensitivity analysis explores the relationship between the model’s <span class="math notranslate nohighlight">\(N\)</span> input variables, <span class="math notranslate nohighlight">\(x=[x_1,x_2,...,x_N]\)</span>, and <span class="math notranslate nohighlight">\(M\)</span> output variables, <span class="math notranslate nohighlight">\(y=[y_1,y_2,...,y_M]\)</span> with <span class="math notranslate nohighlight">\(y=g(x)\)</span>, where <span class="math notranslate nohighlight">\(g\)</span> is the model that maps the model inputs to the outputs <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>]</span>.</p>
+<p>Historically, there have been two broad categories of sensitivity analysis techniques: local and global. Local sensitivity analysis is performed by varying model parameters around specific reference values, with the goal of exploring how small input perturbations influence model performance. Due to its ease-of-use and limited computational demands, this approach has been widely used in literature, but has important limitations <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id18" title="O Rakovec, Mary C Hill, MP Clark, AH Weerts, AJ Teuling, and R Uijlenhoet. Distributed evaluation of local sensitivity analysis (delsa), with application to hydrologic models. Water Resources Research, 50(1):409–426, 2014.">40</a>, <a class="reference internal" href="R.Bibliography.html#id19" title="Andrea Saltelli and Paola Annoni. How to avoid a perfunctory sensitivity analysis. Environmental Modelling &amp; Software, 25(12):1508–1517, 2010.">41</a>]</span>. If the model is not linear, the results of local sensitivity analysis can be heavily biased, as they are strongly influenced by independence assumptions and a limited exploration of model inputs (e.g., <span id="id5">Tang <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id20" title="Yong Tang, Patrick Reed, Thibaut Wagener, and K van Werkhoven. Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation. Hydrology and Earth System Sciences, 11(2):793–817, 2007.">42</a>]</span>). If the model’s factors interact, local sensitivity analysis will underestimate their importance, as it does not account for those effects (e.g., <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id23" title="Nicholas AS Hamm, Jim W Hall, and MG Anderson. Variance-based sensitivity analysis of the probability of hydrologically induced slope instability. Computers &amp; geosciences, 32(6):803–817, 2006.">43</a>]</span>). In general, as local sensitivity analysis only partially and locally explores a model’s parametric space, it is not considered a valid approach for nonlinear models <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id24" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">44</a>]</span>. This is illustrated in <a class="reference internal" href="#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (a-b), presenting contour plots of a model response (<span class="math notranslate nohighlight">\(y\)</span>) with an additive linear model (a) and with a nonlinear model (b). In a linear model without interactions between the input terms <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span>, local sensitivity analysis (assuming deviations from some reference values) can produce appropriate sensitivity indices (<a class="reference internal" href="#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (a)). If however, factors <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> interact, the local and partial consideration of the space can not properly account for each factor’s effects on the model response (<a class="reference internal" href="#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (b)), as it is only informative at the reference value where it is applied. In contrast, a global sensitivity analysis varies uncertain factors within the entire feasible space of variable model responses (<a class="reference internal" href="#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (c)). This approach reveals the global effects of each parameter on the model output, including any interactive effects. For models that cannot be proven linear, global sensitivity analysis is preferred and this text is primarily discussing global sensitivity analysis methods. In the text that follows, whenever we use the term sensitivity analysis we are referring to its global application.</p>
+<figure class="margin-caption align-center" id="id94">
+<span id="figure-3-1"></span><a class="reference internal image-reference" href="_images/figure3_1_global_versus_local.png"><img alt="Figure 3.1" src="_images/figure3_1_global_versus_local.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 3.1 </span><span class="caption-text">Treatment of a two-dimensional space of variability by local (panels a-b) and global (panel c) sensitivity analyses. Panels depict contour plots with the value of a model response (<span class="math notranslate nohighlight">\(y\)</span>) changing with changes in the values of input terms <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span>. Local sensitivity analysis is only an appropriate approach to sensitivity in the case of linear models without interactions between terms, for example in panel (a), where <span class="math notranslate nohighlight">\(y=3x_1+5x_2\)</span>. In the case of more complex models, for example in panels (b-c), where <span class="math notranslate nohighlight">\(y={1 \above 1pt e^{x^2_1+x^2_2}} + {50 \above 1pt e^{(0.1x_1)^2+(0.1x_2)^3}}\)</span>, local sensitivity will miscalculate sensitivity indices as the assessed changes in the value <span class="math notranslate nohighlight">\(y\)</span> depend on the assumed base values chose for <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> (panel (b)). In these cases, global sensitivity methods should be used instead (panel (c)). The points in panel (c) are generated using a uniform random sample of <span class="math notranslate nohighlight">\(n=50\)</span>, but many other methods are available.</span><a class="headerlink" href="#id94" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="why-perform-sensitivity-analysis">
+<span id="why-sa"></span><h2><span class="section-number">3.2. </span>Why Perform Sensitivity Analysis<a class="headerlink" href="#why-perform-sensitivity-analysis" title="Link to this heading">#</a></h2>
+<p>It is important to understand the many ways in which a SA might be of use to your modeling effort. Most commonly, one might be motivated to perform sensitivity analysis for the following reasons:</p>
+<p><em>Model evaluation</em>: Sensitivity analysis can be used to gauge model inferences when assumptions about the structure of the model or its parameterization are dubious or have changed. For instance, consider a numerical model that uses a set of calibrated parameter values to produce outputs, which we then use to inform decisions about the real-world system represented. One might like to know if small changes in these parameter values significantly change this model’s output and the decisions it informs or if, instead, our parameter inferences yield stable model behavior regardless of the uncertainty present in the specific parameterized processes or properties. This can either discredit or lend credence to the model at hand, as well as any inferences drawn that are founded on its accurate representation of the system. Sensitivity analysis can identify which uncertain model factors cause this undesirable model behavior.</p>
+<p><em>Model simplification</em>: Sensitivity analysis can also be used to identify factors or components of the model that appear to have limited effects on direct outputs or metrics of interest. Consider a model that has been developed in an organization for the purposes of a specific research question and is later used in the context of a different application. Some processes represented in significant detail might no longer be of the same importance while consuming significant data or computational resources, as different outputs might be pertinent to the new application. Sensitivity analysis can be used to identify unimportant model components and simplify them to nominal values and reduced model forms. Model complexity and computational costs can therefore be reduced.</p>
+<p><em>Model refinement</em>: Alternatively, sensitivity analysis can reveal the factors or processes that are highly influential to the outputs or metrics of interest, by assessing their relative importance. In the context of model evaluation, this can inform which model components warrant additional investigation or measurement so the uncertainty surrounding them and the resulting model outputs or metrics of interest can be reduced.</p>
+<p><em>Exploratory modeling</em>: When sufficient credence has been established in the model, sensitivity analysis can be applied to a host of other inquiries. Inferences about the factors and processes that most (or least) control a model’s outputs of interest can be extrapolated to the real system they represent and be used in a heuristic manner to inform model-based inferences. On this foundation, a model paired with the advanced techniques presented in this text can be used to “discover” decision relevant and highly consequential outcomes (i.e., scenario discovery, discussed in more detail in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a> <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>, <a class="reference internal" href="R.Bibliography.html#id16" title="Benjamin P Bryant and Robert J Lempert. Thinking inside the box: a participatory, computer-assisted approach to scenario discovery. Technological Forecasting and Social Change, 77(1):34–49, 2010.">45</a>]</span>).</p>
+<p>The nature and context of the model shapes the specific objectives of applying a sensitivity analysis, as well as methods and tools most appropriate and defensible for each application setting <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id203" title="Thorsten Wagener and Francesca Pianosi. What has Global Sensitivity Analysis ever done for us? A systematic review to support scientific advancement and to inform policy-making in earth system modelling. Earth-Science Reviews, 194:1–18, July 2019. URL: https://www.sciencedirect.com/science/article/pii/S0012825218300990 (visited on 2021-08-30), doi:10.1016/j.earscirev.2019.04.006.">35</a>, <a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id25" title="Andrea Saltelli and Stefano Tarantola. On the relative importance of input factors in mathematical models: safety assessment for nuclear waste disposal. Journal of the American Statistical Association, 97(459):702–709, 2002.">46</a>]</span>. The three most common sensitivity analysis modes (<em>Factor Prioritization</em>, <em>Factor Fixing</em>, and <em>Factor Mapping</em>) are presented below, but the reader should be aware that other uses have been proposed in the literature (e.g., <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id26" title="Barry Anderson, Emanuele Borgonovo, Marzio Galeotti, and Roberto Roson. Uncertainty in climate change modeling: can global sensitivity analysis be of help? Risk analysis, 34(2):271–293, 2014.">47</a>, <a class="reference internal" href="R.Bibliography.html#id27" title="Emanuele Borgonovo. Sensitivity analysis with finite changes: an application to modified eoq models. European Journal of Operational Research, 200(1):127–138, 2010.">48</a>]</span>).</p>
+<p><em>Factor prioritization</em>: This sensitivity analysis application mode (also referred to as <em>factor ranking</em>) refers to when one would like to identify the uncertain factors that have the greatest impact on the variability of the output, and which, when fixed to their true value (i.e., if there were no uncertainty regarding their value), would lead to the greatest reduction in output variability <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. Information from this type of analysis can be crucial to model improvement as these factors can become the focus of future measurement campaigns or numerical experiments so that uncertainty in the model output can be reduced. The impact of each uncertain input on the variance of the model output is often used as the criterion for factor prioritization. <a class="reference internal" href="#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (a) shows the effects of three uncertain variables (<span class="math notranslate nohighlight">\(X_1\)</span>, <span class="math notranslate nohighlight">\(X_2\)</span>, and <span class="math notranslate nohighlight">\(X_3\)</span>) on the variance of output <span class="math notranslate nohighlight">\(Y\)</span>. <span class="math notranslate nohighlight">\(V(E(Y|X_i))\)</span> indicates the variance in <span class="math notranslate nohighlight">\(Y\)</span> if factor <span class="math notranslate nohighlight">\(X_i\)</span> is left to vary freely while all other factors remain fixed to nominal values. In this case, factor <span class="math notranslate nohighlight">\(X_2\)</span> makes the largest contribution to the variability of output <span class="math notranslate nohighlight">\(Y\)</span> and it should therefore be prioritized. In the context of risk analysis, factor prioritization can be used to reduce output variance to below a given tolerable threshold (also known as variance cutting).</p>
+<p><em>Factor fixing</em>: This mode of sensitivity analysis (also referred to as <em>factor screening</em>) aims to identify the model components that have a negligible effect or make no significant contributions to the variability of the outputs or metrics of interest (usually referred to as non-influential <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>). In the stylized example of <a class="reference internal" href="#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (a), <span class="math notranslate nohighlight">\(X_1\)</span> makes the smallest contribution to the variability of output <span class="math notranslate nohighlight">\(Y\)</span> suggesting that the uncertainty in its value could be negligible and the factor itself fixed in subsequent model executions. Eliminating these factors or processes in the model or fixing them to a nominal value can help reduce model complexity as well as the unnecessary computational burden of subsequent model runs, results processing, or other sensitivity analyses (the fewer uncertain factors considered, the fewer runs are necessary to illuminate their effects on the output). Significance of the outcome can be gauged in a variety of manners, depending on the application. For instance, if applying a variance-based method, a minimum threshold value of contribution to the variance could be considered as a significance ‘cutoff’, and factors with indices below that value can be considered non-influential. Conclusions about factor fixing should be made carefully, considering all of the effects a factor has, individually and in interaction with other factors (explained in more detail in the <a class="reference internal" href="#variance-based-methods"><span class="std std-numref">Chapter 3.4.5</span></a>).</p>
+<p><em>Factor mapping</em>: Finally, factor mapping can be used to pinpoint which values of uncertain factors lead to model outputs within a given range of the output space <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. In the context of model diagnostics, it is possible that the model’s output changes in ways considered impossible based on the represented processes, or other observed evidence. In this situation, factor mapping can be used to identify which uncertain model factors cause this undesirable model behavior by ‘filtering’ model runs that are considered ‘non-behavioral’ <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id29" title="Neil R Edwards, David Cameron, and Jonathan Rougier. Precalibrating an intermediate complexity climate model. Climate dynamics, 37(7):1469–1482, 2011.">50</a>, <a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id31" title="RC Spear and GM Hornberger. Eutrophication in peel inlet—ii. identification of critical uncertainties via generalized sensitivity analysis. Water research, 14(1):43–49, 1980.">52</a>]</span>. In <a class="reference internal" href="#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (b), region <span class="math notranslate nohighlight">\(B\)</span> of the output space <span class="math notranslate nohighlight">\(Y\)</span> denotes the set of behavioral model outcomes and region <span class="math notranslate nohighlight">\(\bar{B}\)</span> denotes the set of non-behavioral outcomes, resulting from the entirety of input space <span class="math notranslate nohighlight">\(X\)</span>. Factor mapping refers to the process of tracing which factor values of input space <span class="math notranslate nohighlight">\(X\)</span> produce the behavioral model outcomes in the output space.</p>
+<figure class="margin-caption align-center" id="id95">
+<span id="figure-3-2"></span><a class="reference internal image-reference" href="_images/figure3_2_factor_mapping.png"><img alt="Figure 3.2" src="_images/figure3_2_factor_mapping.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 3.2 </span><span class="caption-text">Factor prioritization, factor fixing and factor mapping settings of sensitivity analysis.</span><a class="headerlink" href="#id95" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The language used above reflects a use of sensitivity analysis for model fidelity evaluation and refinement. However, as previously mentioned, when a model has been established as a sufficiently accurate representation of the system, sensitivity analysis can produce additional inferences (i.e., exploratory modeling and scenario discovery). For instance, under the factor mapping use, the analyst can now focus on undesirable system states and discover which factors are most responsible for them: for instance, “population growth of above 25% would be responsible for unacceptably high energy demands”. Factor prioritization and factor fixing can be used to make equivalent inferences, such as “growing populations and increasing temperatures are the leading factors for changing energy demands” (prioritizing of factors) or “changing dietary needs are inconsequential to increasing energy demands for this region” (a factor that can be fixed in subsequent model runs). All these inferences hinge on the assumption that the real system’s stakeholders consider the model states faithful enough representations of system states. As elaborated in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>, this view on sensitivity analysis is founded on a relativist perspective on modeling, which tends to place more value on model usefulness rather than strict accuracy of representation in terms of error. As such, sensitivity analysis performed with decision-making relevance in mind will focus on model outputs or metrics that are consequential and decision relevant (e.g., energy demand in the examples above).</p>
+</section>
+<section id="design-of-experiments">
+<span id="id15"></span><h2><span class="section-number">3.3. </span>Design of Experiments<a class="headerlink" href="#design-of-experiments" title="Link to this heading">#</a></h2>
+<p>Before conducting a sensitivity analysis, the first element that needs to be clarified is the uncertainty space of the model <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id111" title="Jon C Helton, Jay Dean Johnson, Cedric J Sallaberry, and Curt B Storlie. Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliability Engineering &amp; System Safety, 91(10-11):1175–1209, 2006.">53</a>]</span>. In other words, how many and which factors making up the mathematical model are considered uncertain and can potentially affect the model output and the inferences drawn from it. Uncertain factors can be model parameters, model structures, inputs, or alternative model resolution levels (scales), all of which can be assessed through the tools presented in this text. Depending on the kind of factor, its variability can be elicited through various means: expert opinion, values reported in the literature, historical observations, its physical meaning (e.g., population values in a city can never be negative), or through the use of more formal UQ methods (<a class="reference internal" href="A1_Uncertainty_Quantification.html#a1-uncertainty-quantification"><span class="std std-numref">Chapter A</span></a>). The model uncertainty space represents the entire space of variability present in each of the uncertain factors of a model. The complexity of most real-world models means that the response function, <span class="math notranslate nohighlight">\(y=g(x)\)</span>, mapping inputs to outputs, is hardly ever available in an analytical form and therefore analytically computing the sensitivity of the output to each uncertain factor becomes impossible. In these cases, sensitivity analysis is only feasible through numerical procedures that employ different strategies to sample the uncertainty space and calculate sensitivity indices.</p>
+<p>A sampling strategy is often referred to as a <em>design of experiments</em> and represents a methodological choice made before conducting any sensitivity analysis. Experimental design was first introduced by <span id="id17">Fisher [<a class="reference internal" href="R.Bibliography.html#id112" title="Ronald Aylmer Fisher. Design of experiments. Br Med J, 1(3923):554–554, 1936.">54</a>]</span> in the context of laboratory or field-based experiments. Its application in sensitivity analysis is similar to setting up a physical experiment in that it is used to discover the behavior of a system under specific conditions. An ideal design of experiments should provide a framework for the extraction of all plausible information about the impact of each factor on the output of the model. The design of experiments is used to set up a simulation platform with the minimum computational cost to answer specific questions that cannot be readily drawn from the data through analytical or common data mining techniques. Models representing coupled human-natural systems usually have a large number of inputs, state variables and parameters, but not all of them exert fundamental control over the numerical process, despite their uncertainty, nor have substantial impacts on the model output, either independently or through their interactions. Each factor influences the model output in different ways that need to be discovered. For example, the influence of a parameter on model output can be linear or non-linear and can be continuous or only be active during specific times or at particular states of the system <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id53" title="JD Herman, PM Reed, and T Wagener. Time-varying sensitivity analysis clarifies the effects of watershed model formulation on model behavior. Water Resources Research, 49(3):1400–1414, 2013.">55</a>, <a class="reference internal" href="R.Bibliography.html#id113" title="Carolina Massmann, Thorsten Wagener, and Hubert Holzmann. A new approach to visualizing time-varying sensitivity indices for environmental model diagnostics across evaluation time-scales. Environmental modelling &amp; software, 51:190–194, 2014.">56</a>]</span>. An effective and efficient design of experiments allows the analyst to explore these complex relationships and evaluate different behaviors of the model for various scientific questions <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id114" title="An Van Schepdael, Aurélie Carlier, and Liesbet Geris. Sensitivity analysis by design of experiments. In Uncertainty in Biology, pages 327–366. Springer, 2016.">57</a>]</span>. The rest of this section overviews some of the most commonly used designs of experiments. Table 1 summarizes the designs discussed.</p>
+<div class="pst-scrollable-table-container"><table class="table" id="id96">
+<caption><span class="caption-number">Table 3.1 </span><span class="caption-text">Summary of designs of experiments overviewed in this section. * Depends on the sample size.</span><a class="headerlink" href="#id96" title="Link to this table">#</a></caption>
+<thead>
+<tr class="row-odd"><th class="head"><p><em>Design of experiments</em></p></th>
+<th class="head"><p><em>Factor interactions considered</em></p></th>
+<th class="head"><p><em>Treatment of factor domains</em></p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>One-At-a-Time (OAT)</p></td>
+<td><p>No - main effects only</p></td>
+<td><p>Continuous (distributions)</p></td>
+</tr>
+<tr class="row-odd"><td><p>Full Factorial Sampling</p></td>
+<td><p>Yes - including total effects</p></td>
+<td><p>Discrete (levels)</p></td>
+</tr>
+<tr class="row-even"><td><p>Fractional Factorial Sampling</p></td>
+<td><p>Yes - only lower-order effects*</p></td>
+<td><p>Discrete (levels)</p></td>
+</tr>
+<tr class="row-odd"><td><p>Latin Hypercube (LH) Sampling</p></td>
+<td><p>Yes - including total effects*</p></td>
+<td><p>Continuous (distributions)</p></td>
+</tr>
+<tr class="row-even"><td><p>Quasi-Random Sampling with Low-Discrepancy Sequences</p></td>
+<td><p>Yes - including total effects*</p></td>
+<td><p>Continuous (distributions)</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>There are a few different approaches to the design of experiments, closely related to the chosen sensitivity analysis approach, which is in turn shaped by the research motivations, scientific questions, and computational constraints at hand (additional discussion of this can be found at the end of <a class="reference internal" href="#sensitivity-analysis-the-basics"><span class="std std-numref">Chapter 3</span></a>). For example, in a sensitivity analysis using perturbation and derivatives methods, the model input parameters vary from their nominal values one at a time, something that the design of experiments needs to reflect. If, instead, one were to perform sensitivity analysis using a multiple-starts perturbation method, the design of experiments needs to consider that multiple points across the factor space are used. The design of experiments specifically defines two key characteristics of samples that are fed to the numerical model: the number of samples and the range of each factor.</p>
+<p>Generally, sampling can be performed randomly or by applying a stratifying approach. In random sampling, such as Monte Carlo <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id115" title="Nicholas Metropolis and Stanislaw Ulam. The monte carlo method. Journal of the American statistical association, 44(247):335–341, 1949.">58</a>]</span>, samples are randomly generated by a pseudo-random number generator with an a-priori assumption about the distribution of parameters and their possible ranges. Random seeds can also be used to ensure consistency and higher control over the random process. However, this method could leave some gaps in the parameter space and cause clustering in some spaces, especially for a large number of parameters <span id="id21">[<a class="reference internal" href="R.Bibliography.html#id116" title="John Norton. An introduction to sensitivity assessment of simulation models. Environmental Modelling &amp; Software, 69:166–174, 2015.">59</a>]</span>. Most sampling strategies use stratified sampling to mitigate these disadvantages. Stratified sampling techniques divide the domain of each factor into subintervals, often of equal lengths. From each subinterval, an equal number of samples is drawn randomly, or based on the specific locations within the subintervals <span id="id22">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>.</p>
+<section id="one-at-a-time-oat">
+<h3><span class="section-number">3.3.1. </span>One-At-a-Time (OAT)<a class="headerlink" href="#one-at-a-time-oat" title="Link to this heading">#</a></h3>
+<p>In this approach, only one model factor is changed at a time while all others are kept fixed across each iteration in a sampling sequence. The OAT method assumes that model factors of focus are linearly independent (i.e., there are no interactions) and can analyze how factors individually influence model outputs or metrics of interest. While popular given its ease of implementation, OAT  is ultimately limited in its exploration of a model’s sensitivities <span id="id23">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. It is primarily used with local sensitivity techniques with similar criticisms: applying this sampling scheme on a system with nonlinear and interactive processes will miss important information on the effect uncertain factors have on the model. OAT samplings can be repeated multiple times in a more sophisticated manner and across different locations of the parameter space to overcome some of these challenges, which would increase computational costs and negate the main reasons for its selection. Given these limitations OAT methods could be used as preliminary, low-cost analyses of the factors’ individual effects, but should ultimately be complemented with more sophisticated methods.</p>
+</section>
+<section id="full-and-fractional-factorial-sampling">
+<h3><span class="section-number">3.3.2. </span>Full and Fractional Factorial Sampling<a class="headerlink" href="#full-and-fractional-factorial-sampling" title="Link to this heading">#</a></h3>
+<p>In full factorial sampling each factor is treated as being discrete by considering two or more levels (or intervals) of its values. The sampling process then generates samples within each possible combination of levels, corresponding to each parameter. This scheme produces a more comprehensive sampling of the factors’ variability space, as it accounts for all candidate combinations of factor levels (<a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a)). If the number of levels is the same across all factors, the number of generated samples is estimated using <span class="math notranslate nohighlight">\(n^k\)</span>, where <span class="math notranslate nohighlight">\(n\)</span> is the number of levels and <span class="math notranslate nohighlight">\(k\)</span> is the number of factors. For example, <a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a) presents a full factorial sampling of three uncertain factors <span class="math notranslate nohighlight">\((x_1,\)</span> <span class="math notranslate nohighlight">\(x_2,\)</span> and <span class="math notranslate nohighlight">\(x_3)\)</span>, each considered as having four discrete levels. The total number of samples necessary for such an experiment is <span class="math notranslate nohighlight">\(4^3=64\)</span>. As the number of factors increases, the number of simulations necessary will also grow exponentially, making full factorial sampling computationally burdensome (<a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (b)). As a result, it is common in the literature to apply full factorial sampling at only two levels per factor, typically the two extremes <span id="id24">[<a class="reference internal" href="R.Bibliography.html#id32" title="Douglas C Montgomery. Design and analysis of experiments. John wiley &amp; sons, 2017.">60</a>]</span>. This significantly reduces computational burden but is only considered appropriate in cases where factors can indeed only assume two discrete values (e.g., when testing the effects of epistemic uncertainty and comparing between model structure A and model structure B). In the case of physical parameters on continuous distributions (e.g., when considering the effects of measurement uncertainty in a temperature sensor), discretizing the range of a factor to only extreme levels can bias its estimated importance.</p>
+<p>Fractional factorial sampling is a widely used alternative to full factorial sampling that allows the analyst to significantly reduce the number of simulations by focusing on the main effects of a factor and seeking to avoid model runs that yield redundant response information <span id="id25">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. In other words, if one can reasonably assume that higher-order interactions are negligible, information about the most significant effects and lower-order interactions (e.g., effects from pairs of factors) can be obtained using a fraction of the full factorial design. Traditionally, fractional factorial design has also been limited to two levels <span id="id26">[<a class="reference internal" href="R.Bibliography.html#id32" title="Douglas C Montgomery. Design and analysis of experiments. John wiley &amp; sons, 2017.">60</a>]</span>, referred to as Fractional Factorial designs 2k-p <span id="id27">[<a class="reference internal" href="R.Bibliography.html#id34" title="George EP Box and J Stuart Hunter. The 2 k—p fractional factorial designs. Technometrics, 3(3):311–351, 1961.">61</a>]</span>. Recently, Generalized Fractional Factorial designs have also been proposed that allow for the structured generation of samples at more than two levels per factor <span id="id28">[<a class="reference internal" href="R.Bibliography.html#id33" title="Izabella Surowiec, Ludvig Vikstrom, Gustaf Hector, Erik Johansson, Conny Vikstrom, and Johan Trygg. Generalized subset designs in analytical chemistry. Analytical chemistry, 89(12):6491–6497, 2017.">62</a>]</span>. Consider a case where the modeling team dealing with the problem in <a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a) cannot afford to perform 64 simulations of their model. They can afford 32 runs for their experiment and instead decide to fractionally sample the variability space of their factors. A potential design of such a sampling strategy is presented in <a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (c).</p>
+<figure class="margin-caption align-center" id="id97">
+<span id="figure-3-3"></span><a class="reference internal image-reference" href="_images/figure3_3_alternative_designs.png"><img alt="Figure 3.3" src="_images/figure3_3_alternative_designs.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 3.3 </span><span class="caption-text">Alternative designs of experiments and their computational costs for three uncertain factors <span class="math notranslate nohighlight">\((x_1,\)</span> <span class="math notranslate nohighlight">\(x_2,\)</span> and <span class="math notranslate nohighlight">\(x_3)\)</span>. (a) Full factorial design sampling of three factors at four levels, at a total of 64 samples; (b) exponential growth of necessary number of samples when applying full factorial design at four levels; (c) fractional factorial design of three factors at four levels, at a total of 32 samples; and (d) Latin Hypercube sample of three factors with uniform distributions, at a total of 32 samples.</span><a class="headerlink" href="#id97" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="latin-hypercube-sampling-lhs">
+<h3><span class="section-number">3.3.3. </span>Latin Hypercube Sampling (LHS)<a class="headerlink" href="#latin-hypercube-sampling-lhs" title="Link to this heading">#</a></h3>
+<p>Latin hypercube sampling (LHS) <span id="id29">[<a class="reference internal" href="R.Bibliography.html#id91" title="Michael D McKay, Richard J Beckman, and William J Conover. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21(1):239-2451, 1979.">63</a>]</span> is one of the most common methods in space-filling experimental designs. With this sampling technique, for <span class="math notranslate nohighlight">\(N\)</span> uncertain factors, an <span class="math notranslate nohighlight">\(N\)</span>-dimensional hypercube is generated, with each factor divided into an equal number of levels depending on the total number of samples to be generated. Equal numbers of samples are then randomly generated at each level, across all factors. In this manner, latin hypercube design guarantees sampling from every level of the variability space and without any overlaps. When the number of samples generated is much larger than the number of uncertain factors, LHS can be very effective in examining the effects of each factor <span id="id30">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. LHS is an attractive technique, because it guarantees a diverse coverage of the space, through the use of subintervals, without being constrained to discrete levels for each factor - compare <a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (c) with <a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (d) for the same number of samples.</p>
+<p>LHS is less effective when the number of samples is not much larger than the number of uncertain factors, and the effects of each factor cannot be appropriately distinguished. The samples between factors can also be highly correlated, biasing any subsequent sensitivity analysis results. To address this, the sampling scheme can be modified to control for the correlation in parameters while maximizing the information derived. An example of such modification is through the use of orthogonal arrays <span id="id31">[<a class="reference internal" href="R.Bibliography.html#id21" title="Boxin Tang. Orthogonal array-based latin hypercubes. Journal of the American statistical association, 88(424):1392–1397, 1993.">64</a>]</span>.</p>
+</section>
+<section id="low-discrepancy-sequences">
+<h3><span class="section-number">3.3.4. </span>Low-Discrepancy Sequences<a class="headerlink" href="#low-discrepancy-sequences" title="Link to this heading">#</a></h3>
+<p>Low-discrepancy sequences is another sampling technique that employs a pseudo-random generator for Monte Carlo sampling <span id="id32">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>, <a class="reference internal" href="R.Bibliography.html#id92" title="SK Zaremba. The mathematical basis of monte carlo and quasi-monte carlo methods. SIAM review, 10(3):303–314, 1968.">66</a>]</span>. These quasi-Monte Carlo methods eliminate ‘lumpiness’ across samples (i.e, the presence of gaps and clusters) by minimizing discrepancy across the hypercube samples. Discrepancy can be quantitatively measured using the deviations of sampled points from a uniform distribution <span id="id33">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>, <a class="reference internal" href="R.Bibliography.html#id95" title="Sergei Kucherenko, Daniel Albrecht, and Andrea Saltelli. Exploring multi-dimensional spaces: a comparison of latin hypercube and quasi monte carlo sampling techniques. arXiv preprint arXiv:1505.02350, 2015.">67</a>]</span>. Low-discrepancy sequences ensure that the number of samples in any subspace of the variability hypercube is approximately the same. This is not something guaranteed by Latin Hypercube sampling, and even though its design can be improved through optimization with various criteria, such adjustments are limited to small sample sizes and low dimensions <span id="id34">[<a class="reference internal" href="R.Bibliography.html#id95" title="Sergei Kucherenko, Daniel Albrecht, and Andrea Saltelli. Exploring multi-dimensional spaces: a comparison of latin hypercube and quasi monte carlo sampling techniques. arXiv preprint arXiv:1505.02350, 2015.">67</a>, <a class="reference internal" href="R.Bibliography.html#id93" title="Bertrand Iooss, Loïc Boussouf, Vincent Feuillard, and Amandine Marrel. Numerical studies of the metamodel fitting and validation processes. arXiv preprint arXiv:1001.1049, 2010.">68</a>, <a class="reference internal" href="R.Bibliography.html#id94" title="Ruichen Jin, Wei Chen, and Agus Sudjianto. An efficient algorithm for constructing optimal design of computer experiments. In International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, volume 37009, 545–554. 2003.">69</a>, <a class="reference internal" href="R.Bibliography.html#id96" title="Max D Morris and Toby J Mitchell. Exploratory designs for computational experiments. Journal of statistical planning and inference, 43(3):381–402, 1995.">70</a>, <a class="reference internal" href="R.Bibliography.html#id97" title="Jeong-Soo Park. Optimal latin-hypercube designs for computer experiments. Journal of statistical planning and inference, 39(1):95–111, 1994.">71</a>]</span>. In contrast, the Sobol sequence <span id="id35">[<a class="reference internal" href="R.Bibliography.html#id98" title="Ilya M Sobol. Uniformly distributed sequences with an additional uniform property. USSR Computational Mathematics and Mathematical Physics, 16(5):236–242, 1976.">72</a>, <a class="reference internal" href="R.Bibliography.html#id99" title="Il'ya Meerovich Sobol'. On the distribution of points in a cube and the approximate evaluation of integrals. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 7(4):784–802, 1967.">73</a>]</span>, one of the most widely used sampling techniques, utilizes the low-discrepancy approach to uniformly fill the sampled factor space. A core advantage of this style of sampling is that it takes far fewer samples (i.e., simulations) to attain a much lower level of error in estimating model output statistics (e.g., the mean and variance of outputs).</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial which uses Sobol sequence sampling for the purposes of a Sobol sensitivity analysis:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA using SALib Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="other-types-of-sampling">
+<h3><span class="section-number">3.3.5. </span>Other types of sampling<a class="headerlink" href="#other-types-of-sampling" title="Link to this heading">#</a></h3>
+<p>The sampling techniques mentioned so far are general sampling methods useful for a variety of applications beyond sensitivity analysis. There are however techniques that have been developed for specific sensitivity analysis methods. Examples of these methods include the Morris One-At-a-Time <span id="id36">[<a class="reference internal" href="R.Bibliography.html#id132" title="Max D Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.">74</a>]</span>, Fourier Amplitude Sensitivity Test (FAST; <span id="id37">[<a class="reference internal" href="R.Bibliography.html#id100" title="RI Cukier, CM Fortuin, Kurt E Shuler, AG Petschek, and JH Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. i theory. The Journal of chemical physics, 59(8):3873–3878, 1973.">75</a>]</span>), Extended FAST <span id="id38">[<a class="reference internal" href="R.Bibliography.html#id101" title="Andrea Saltelli, Stefano Tarantola, and KP-S Chan. A quantitative model-independent method for global sensitivity analysis of model output. Technometrics, 41(1):39–56, 1999.">76</a>]</span>, and Extended Sobol methods <span id="id39">[<a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>]</span>. For example, the Morris sampling strategy builds a number of trajectories (usually referred to as repetitions and denoted by <span class="math notranslate nohighlight">\(r\)</span>) in the input space each composed of <span class="math notranslate nohighlight">\(N+1\)</span> factor points, where <span class="math notranslate nohighlight">\(N\)</span> is the number of uncertain factors. The first point of the trajectory is selected randomly and the subsequent <span class="math notranslate nohighlight">\(N\)</span> points are generated by moving one factor at a time by a fixed amount. Each factor is perturbed once along the trajectory, while the starting points of all of the trajectories are randomly and uniformly distributed. Several variations of this strategy also exist in the literature; for more details on each approach and their differences the reader is directed to <span id="id40">Pianosi <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>.</p>
+</section>
+<section id="synthetic-generation-of-input-time-series">
+<h3><span class="section-number">3.3.6. </span>Synthetic generation of input time series<a class="headerlink" href="#synthetic-generation-of-input-time-series" title="Link to this heading">#</a></h3>
+<p>Models often have input time series or processes with strong temporal and/or spatial correlations (e.g., streamflow, energy demand, pricing of commodities, etc.) that, while they might not immediately come to mind as factors to be examined in sensitivity analysis, can be treated as such. Synthetic input time series are used for a variety of reasons, for example, when observations are not available or are limited, or when past observations are not considered sufficiently representative to capture rare or extreme events of interest <span id="id41">[<a class="reference internal" href="R.Bibliography.html#id103" title="Jonathan D Herman, Harrison B Zeff, Jonathan R Lamontagne, Patrick M Reed, and Gregory W Characklis. Synthetic drought scenario generation to support bottom-up water supply vulnerability assessments. Journal of Water Resources Planning and Management, 142(11):04016050, 2016.">78</a>, <a class="reference internal" href="R.Bibliography.html#id104" title="PCD Milly, Julio Betancourt, Malin Falkenmark, Robert M Hirsch, Zbigniew W Kundzewicz, Dennis P Lettenmaier, and Ronald J Stouffer. Stationarity is dead: whither water management? Earth, 4:20, 2008.">79</a>]</span>. Synthetic generation of input time series provides a valuable tool to consider non-stationarity and incorporate potential stressors, such as climate change impacts into input time series <span id="id42">[<a class="reference internal" href="R.Bibliography.html#id105" title="Edoardo Borgomeo, Christopher L Farmer, and Jim W Hall. Numerical rivers: a synthetic streamflow generator for water resources vulnerability assessments. Water Resources Research, 51(7):5382–5405, 2015.">80</a>]</span>. For example, a century of record will be insufficient to capture very high impact rare extreme events (e.g., persistent multi-year droughts). A large body of statistical literature exists focusing on the topics of synthetic weather <span id="id43">[<a class="reference internal" href="R.Bibliography.html#id74" title="Manuel Herrera, Sukumar Natarajan, David A Coley, Tristan Kershaw, Alfonso P Ramallo-González, Matthew Eames, Daniel Fosas, and Michael Wood. A review of current and future weather data for building simulation. Building Services Engineering Research and Technology, 38(5):602–627, 2017.">81</a>, <a class="reference internal" href="R.Bibliography.html#id75" title="Daniel S Wilks and Robert L Wilby. The weather generation game: a review of stochastic weather models. Progress in physical geography, 23(3):329–357, 1999.">82</a>]</span> and streamflow <span id="id44">[<a class="reference internal" href="R.Bibliography.html#id76" title="JR Lamontagne and JR Stedinger. Generating synthetic streamflow forecasts with specified precision. Journal of Water Resources Planning and Management, 144(4):04018007, 2018.">83</a>, <a class="reference internal" href="R.Bibliography.html#id77" title="Sanghamitra Medda and Kalyan Kumar Bhar. Comparison of single-site and multi-site stochastic models for streamflow generation. Applied Water Science, 9(3):67, 2019.">84</a>]</span> generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound (hot, severe drought) extremes. It is beyond the scope of this text to review these methods, but readers are encouraged to explore the studies cited above as well as the following publications for discussions and comparisons of these methods: <span id="id45">[<a class="reference internal" href="R.Bibliography.html#id103" title="Jonathan D Herman, Harrison B Zeff, Jonathan R Lamontagne, Patrick M Reed, and Gregory W Characklis. Synthetic drought scenario generation to support bottom-up water supply vulnerability assessments. Journal of Water Resources Planning and Management, 142(11):04016050, 2016.">78</a>, <a class="reference internal" href="R.Bibliography.html#id105" title="Edoardo Borgomeo, Christopher L Farmer, and Jim W Hall. Numerical rivers: a synthetic streamflow generator for water resources vulnerability assessments. Water Resources Research, 51(7):5382–5405, 2015.">80</a>, <a class="reference internal" href="R.Bibliography.html#id106" title="Brian R Kirsch, Gregory W Characklis, and Harrison B Zeff. Evaluating the impact of alternative hydro-climate scenarios on transfer agreements: practical improvement for generating synthetic streamflows. Journal of Water Resources Planning and Management, 139(4):396–406, 2013.">85</a>, <a class="reference internal" href="R.Bibliography.html#id107" title="Daniel P Loucks and Eelco Van Beek. Water resource systems planning and management: An introduction to methods, models, and applications. Springer, 2017.">86</a>, <a class="reference internal" href="R.Bibliography.html#id108" title="Scott Steinschneider, Sungwook Wi, and Casey Brown. The integrated effects of climate and hydrologic uncertainty on future flood risk assessments. Hydrological Processes, 29(12):2823–2839, 2015.">87</a>, <a class="reference internal" href="R.Bibliography.html#id109" title="Richard M Vogel. Stochastic watershed models for hydrologic risk management. Water Security, 1:28–35, 2017.">88</a>, <a class="reference internal" href="R.Bibliography.html#id110" title="Richard M Vogel and Jery R Stedinger. The value of stochastic streamflow models in overyear reservoir design applications. Water Resources Research, 24(9):1483–1490, 1988.">89</a>]</span>. The use of these methods for the purposes of exploratory modeling, especially in the context of well-characterized versus deep uncertainty, is further discussed in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a>.</p>
+</section>
+</section>
+<section id="sensitivity-analysis-methods">
+<span id="id46"></span><h2><span class="section-number">3.4. </span>Sensitivity Analysis Methods<a class="headerlink" href="#sensitivity-analysis-methods" title="Link to this heading">#</a></h2>
+<p>In this section, we describe some of the most widely applied sensitivity analysis methods along with their mathematical definitions. We also provide a detailed discussion on applying each method, as well as a comparison of and their features and limitations.</p>
+<section id="derivative-based-methods">
+<h3><span class="section-number">3.4.1. </span>Derivative-based Methods<a class="headerlink" href="#derivative-based-methods" title="Link to this heading">#</a></h3>
+<p>Derivative-based methods explore how model outputs are affected by perturbations in a single model input around a particular input value. These methods are local and are performed using OAT sampling. For simplicity of mathematical notations, let us assume that the model <span class="math notranslate nohighlight">\(g(X)\)</span> only returns one output. Following <span id="id47">[<a class="reference internal" href="R.Bibliography.html#id117" title="Emanuele Borgonovo. Sensitivity analysis of model output with input constraints: a generalized rationale for local methods. Risk Analysis: An International Journal, 28(3):667–680, 2008.">90</a>]</span> and <span id="id48">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>, the sensitivity index, <span class="math notranslate nohighlight">\(S_i\)</span> , of the model’s <em>i</em>-th input factor, <span class="math notranslate nohighlight">\(x_i\)</span> , can be measured using the partial derivative evaluated at a nominal value, <span class="math notranslate nohighlight">\(\bar{x}\)</span>, of the vector of inputs:</p>
+<div class="math notranslate nohighlight">
+\[S_i (\bar{x}) = \frac{\partial g}{\partial x} |_{\bar{x}{^c{_i}}}\]</div>
+<p>where <em>c</em><sub>i</sub> is the scaling factor. In most applications however, the relationship <span class="math notranslate nohighlight">\(g(X)\)</span> is not fully known in its analytical form, and therefore the above partial derivative is usually approximated:</p>
+<div class="math notranslate nohighlight">
+\[S_i (\bar{x}) = \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i\]</div>
+<p>Using this approximation, the <em>i</em>-th input factor is perturbed by a magnitude of <span class="math notranslate nohighlight">\(\Delta_i\)</span>, and its relative importance is calculated. Derivative-based methods are some of the oldest sensitivity analysis methods as they only require <span class="math notranslate nohighlight">\(N+1\)</span> model evaluations to estimate indices for <span class="math notranslate nohighlight">\(N\)</span> uncertain factors. As described above, being computationally very cheap comes at the cost of not being able to explore the entire input space, but only (local) perturbations to the nominal value. Additionally, as these methods examine the effects of each input factor one at a time, they cannot assess parametric interactions or capture the interacting nature of many real systems and the models that abstract them.</p>
+</section>
+<section id="elementary-effect-methods">
+<h3><span class="section-number">3.4.2. </span>Elementary Effect Methods<a class="headerlink" href="#elementary-effect-methods" title="Link to this heading">#</a></h3>
+<p>Elementary effect (EE) SA methods provide a solution to the local nature of the derivative-based methods by exploring the entire parametric range of each input parameter <span id="id49">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>. However, EE methods still use OAT sampling and do not vary all input parameters simultaneously while exploring the parametric space. The OAT nature of EEs methods therefore prevents them from properly capturing the interactions between uncertain factors. EEs methods are computationally efficient compared to their All-At-a-Time (AAT) counterparts, making them more suitable when computational capacity is a limiting factor, while still allowing for some inferences regarding factor interactions.
+The most popular EE method is the Method of Morris <span id="id50">[<a class="reference internal" href="R.Bibliography.html#id132" title="Max D Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.">74</a>]</span>. Following the notation by <span id="id51">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>, this method calculates global sensitivity using the mean of the EEs (finite differences) of each parameter at different locations:</p>
+<div class="math notranslate nohighlight">
+\[S_i = \mu_i^* = \frac{1}{r}\sum_{j=1}^r EE^j_i = \frac{1}{r}\sum_{j=1}^r \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i\]</div>
+<p>with <span class="math notranslate nohighlight">\(r\)</span> representing the number of sample repetitions (also refered to as trajectories) in the input space, usually set between 4 and 10 <span id="id52">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. Each <span class="math notranslate nohighlight">\(x_j\)</span> represents the points of each trajectory, with <span class="math notranslate nohighlight">\(j=1,…, r\)</span>, selected as described in the sampling strategy for this method, found above. This method also produces the standard deviation of the EEs:</p>
+<div class="math notranslate nohighlight">
+\[\sigma_i = \sqrt{\frac{1}{r}\sum_{j=1}^r(EE_i^j-\frac{1}{r}\sum_{j=1}^r EE^j_i)^2}\]</div>
+<p>which is a measure of parametric interactions. Higher values of <span class="math notranslate nohighlight">\(\sigma_i\)</span> suggest model responses at different levels of factor <span class="math notranslate nohighlight">\(x_i\)</span> are significantly different, which indicates considerable interactions between that and other uncertain factors. The values of <span class="math notranslate nohighlight">\(\mu_i^*\)</span> and <span class="math notranslate nohighlight">\(\sigma_i\)</span> for each factor allow us to draw several different conclusions, illustrated in <a class="reference internal" href="#figure-3-4"><span class="std std-numref">Fig. 3.4</span></a>, following the example by <span id="id53">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>. In this example, factors <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> can be said to have an influence on the model outputs, with <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> having some interactive or non-linear effects. Depending on the orders of magnitude of <span class="math notranslate nohighlight">\(\mu_i^*\)</span> and <span class="math notranslate nohighlight">\(\sigma_i\)</span> one can indirectly deduce whether the factors have strong interactive effects, for example if a factor <span class="math notranslate nohighlight">\(\sigma_i &lt;&lt; \mu_i^*\)</span> then the relationship between that factor and the output can be assumed to be largely linear (note that this is still an OAT method and assumptions on factor interactions should be strongly caveated). Extensions of the Method of Morris have also been developed specifically for the purposes of factor fixing and explorations of parametric interactions (e.g., <span id="id54">[<a class="reference internal" href="R.Bibliography.html#id27" title="Emanuele Borgonovo. Sensitivity analysis with finite changes: an application to modified eoq models. European Journal of Operational Research, 200(1):127–138, 2010.">48</a>, <a class="reference internal" href="R.Bibliography.html#id133" title="Francesca Campolongo and Roger Braddock. The use of graph theory in the sensitivity analysis of the model output: a second order screening method. Reliability Engineering &amp; System Safety, 64(1):1–12, 1999.">92</a>, <a class="reference internal" href="R.Bibliography.html#id134" title="Roger A Cropp and Roger D Braddock. The new morris method: an efficient second-order screening method. Reliability Engineering &amp; System Safety, 78(1):77–83, 2002.">93</a>]</span>).</p>
+<figure class="margin-caption align-center" id="id98">
+<span id="figure-3-4"></span><a class="reference internal image-reference" href="_images/figure3_4_morris_method.png"><img alt="Figure 3.4" src="_images/figure3_4_morris_method.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 3.4 </span><span class="caption-text">Illustrative results of the Morris Method. Factors <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> have an influence on the model outputs, with <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> having interactive or non-linear effects. Whether or not a factor should be considered influential to the output depends on the output selected and is specific to the research context and purpose of the analysis, as discussed in <a class="reference internal" href="#why-sa"><span class="std std-numref">Chapter 3.2</span></a>.</span><a class="headerlink" href="#id98" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="regression-based-methods">
+<h3><span class="section-number">3.4.3. </span>Regression-based Methods<a class="headerlink" href="#regression-based-methods" title="Link to this heading">#</a></h3>
+<p>Regression analysis is one of the oldest ways of investigating parametric importance and sensitivity <span id="id55">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. Here, we describe some of the most popular regression-based sensitivity indices. One of the main sensitivity indices of this category is the standardized regression coefficient (SRC). To calculate SRC, a linear regression relationship needs to be fitted between the input vector, <span class="math notranslate nohighlight">\(x\)</span>, and the model output of interest by using a least-square minimizing method:</p>
+<div class="math notranslate nohighlight">
+\[y = b_0 + \sum_{i=1}^N b_ix_i\]</div>
+<p>where <span class="math notranslate nohighlight">\(b_0\)</span> and <span class="math notranslate nohighlight">\(b_i\)</span> (corresponding to the <em>i</em>-th model input) are regression coefficients. The following relationship can then be used to calculate the SRCs for different input values:</p>
+<div class="math notranslate nohighlight">
+\[S_i=SRC_i=b_i\frac{\sigma_i}{\sigma_y}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\sigma_i\)</span> and <span class="math notranslate nohighlight">\(\sigma_y\)</span> are standard deviations of <em>i</em>-th model input and output, respectively.</p>
+<p>Several other regression-based indices explore the correlation between input and output parameters as a proxy to model parametric sensitivity <span id="id56">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>, <a class="reference internal" href="R.Bibliography.html#id135" title="Jon C Helton. Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal. Reliability Engineering &amp; System Safety, 42(2-3):327–367, 1993.">94</a>, <a class="reference internal" href="R.Bibliography.html#id136" title="Gemma Manache and Charles S Melching. Identification of reliable regression-and correlation-based sensitivity measures for importance ranking of water-quality model parameters. Environmental Modelling &amp; Software, 23(5):549–562, 2008.">95</a>]</span>. The Pearson correlation coefficient (PCC) can be used when a linear relationship exists between an uncertain factor, <span class="math notranslate nohighlight">\(x_i\)</span>, and the output <span class="math notranslate nohighlight">\(y\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i=PCC=\frac{cov(x_i,y)}{\sigma_i\sigma_y}\]</div>
+<p>In cases when there are outliers in the data or the relationship between the uncertain factors and the output is not linear, rank-based correlation coefficients are preferred, for example, Spearman’s rank correlation coefficient (SRCC):</p>
+<div class="math notranslate nohighlight">
+\[S_i=SRCC=\frac{cov(rx_i,ryi)}{\sigma_{ri}\sigma_{ry}}\]</div>
+<p>where the raw values of <span class="math notranslate nohighlight">\(x_i\)</span> and <span class="math notranslate nohighlight">\(y\)</span> and converted to ranks <span class="math notranslate nohighlight">\(rx_i\)</span> and <span class="math notranslate nohighlight">\(ry\)</span> respectively, which instead represent a measurement of the strength of the monotonic relationship, rather than linear relationship, between the input and output. Other regression-based metrics include the partial correlations coefficient, the partial rank correlations coefficient, and the Nash-Sutcliffe coefficient, more discussion on which can be found in <span id="id57">[<a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>.</p>
+<p>Tree-based regression techniques have also been used for sensitivity analysis in an effort to address the challenges faced with nonlinear models <span id="id58">[<a class="reference internal" href="R.Bibliography.html#id137" title="F Pappenberger and Keith J Beven. Ignorance is bliss: or seven reasons not to use uncertainty analysis. Water resources research, 2006.">96</a>]</span>. Examples of these methods include the Patient Rule Induction Method (PRIM; <span id="id59">[<a class="reference internal" href="R.Bibliography.html#id79" title="Jerome H Friedman and Nicholas I Fisher. Bump hunting in high-dimensional data. Statistics and Computing, 9(2):123–143, 1999.">97</a>]</span>) and Classification And Regression Trees (CART; <span id="id60">[<a class="reference internal" href="R.Bibliography.html#id80" title="Leo Breiman, Jerome Friedman, Charles J Stone, and Richard A Olshen. Classification and regression trees. CRC press, 1984.">98</a>]</span>). CART-based approaches also include boosting and bagging extensions <span id="id61">[<a class="reference internal" href="R.Bibliography.html#id89" title="Yoav Freund, Robert Schapire, and Naoki Abe. A short introduction to boosting. Journal-Japanese Society For Artificial Intelligence, 14(771-780):1612, 1999.">99</a>, <a class="reference internal" href="R.Bibliography.html#id138" title="Leo Breiman. Bagging predictors. Machine learning, 24(2):123–140, 1996.">100</a>]</span>. These methods are particularly useful when sensitivity analysis is used for factor mapping (i.e., when trying to identify which uncertain model factors produce a certain model behavior). <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a> elaborates on the use of these methods. Regression-based sensitivity analysis methods are global by nature and can explore the entire space of variables. However, the true level of comprehensiveness depends on the design of experiments and the number of simulations providing data to establish the regression relationships. Although they are usually computationally efficient, they do not produce significant information about parametric interactions <span id="id62">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>]</span>.</p>
+</section>
+<section id="regional-sensitivity-analysis">
+<h3><span class="section-number">3.4.4. </span>Regional Sensitivity Analysis<a class="headerlink" href="#regional-sensitivity-analysis" title="Link to this heading">#</a></h3>
+<p>Another method primarily applied for basic factor mapping applications is Regional Sensitivity Analysis (RSA; <span id="id63">[<a class="reference internal" href="R.Bibliography.html#id157" title="George M Hornberger and Robert C Spear. Approach to the preliminary analysis of environmental systems. J. Environ. Mgmt., 12(1):7–18, 1981.">101</a>]</span>). RSA is a global sensitivity analysis method that is typically implemented using standard sampling methods such as latin hypercube sampling. It is performed by specifying a condition on the output space (e.g., an upper threshold) and classifying outputs that meet the condition as behavioral and the ones that fail it as non-behavioral (illustrated in <a class="reference internal" href="#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (b)). Note that the specified threshold depends on the nature of the problem, model, and the research question. It can reflect model-performance metrics (such as errors) or consequential decision-relevant metrics (such as unacceptable system outcomes). The behavioral and non-behavioral outputs are then traced back to their originating sampled factors, where differences between the distributions of samples can be used to determine their significance in producing each part of the output. The Kolmogorov-Smirnov divergence is commonly used to quantify the difference between the distribution of behavioral and non-behavioral parameters <span id="id64">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i=|F_{x_i|y_b} (y \in Y_b)-F_{x_i|y_{nb}} (y \in Y_{nb})|\]</div>
+<p>where <span class="math notranslate nohighlight">\(Y_b\)</span> represents the set of behavioral outputs, and <span class="math notranslate nohighlight">\(F_{x_i|y_b}\)</span> is the empirical cumulative distribution function of the values of <span class="math notranslate nohighlight">\(x_i\)</span> associated with values of <span class="math notranslate nohighlight">\(y\)</span> that belong in the behavioral set. The <span class="math notranslate nohighlight">\(nb\)</span> notation indicates the equivalent elements related to the non-behavioral set. Large differences between the two distributions indicate stronger effects by the parameters on the respective part of the output space.</p>
+<p>Used in a factor mapping setting, RSA can be applied for scenario discovery <span id="id65">[<a class="reference internal" href="R.Bibliography.html#id71" title="Robert J Lempert, David G Groves, Steven W Popper, and Steve C Bankes. A general, analytic method for generating robust strategies and narrative scenarios. Management science, 52(4):514–528, 2006.">102</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>, the Generalized Likelihood Uncertainty Estimation method (GLUE; <span id="id66">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>, <a class="reference internal" href="R.Bibliography.html#id158" title="Keith Beven and Andrew Binley. Glue: 20 years on. Hydrological processes, 28(24):5897–5918, 2014.">104</a>, <a class="reference internal" href="R.Bibliography.html#id159" title="Roberta-Serena Blasone, Jasper A Vrugt, Henrik Madsen, Dan Rosbjerg, Bruce A Robinson, and George A Zyvoloski. Generalized likelihood uncertainty estimation (glue) using adaptive markov chain monte carlo sampling. Advances in Water Resources, 31(4):630–648, 2008.">105</a>]</span>) and other hybrid sensitivity analysis methods (e.g., <span id="id67">[<a class="reference internal" href="R.Bibliography.html#id160" title="SA Cryer and PL Havens. Regional sensitivity analysis using a fractional factorial method for the usda model gleams. Environmental modelling &amp; software, 14(6):613–624, 1999.">106</a>, <a class="reference internal" href="R.Bibliography.html#id161" title="Pengfei Wei, Zhenzhou Lu, and Xiukai Yuan. Monte carlo simulation for moment-independent sensitivity analysis. Reliability Engineering &amp; System Safety, 110:60–67, 2013.">107</a>]</span>). The fundamental shortcomings of RSA are that, in some cases, it could be hard to interpret the difference between behavioral and non-behavioral sample sets, and that insights about parametric correlations and interactions cannot always be uncovered <span id="id68">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. For more elaborate discussions and illustrations of the RSA method, readers are directed to <span id="id69">Tang <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id20" title="Yong Tang, Patrick Reed, Thibaut Wagener, and K van Werkhoven. Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation. Hydrology and Earth System Sciences, 11(2):793–817, 2007.">42</a>], Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>], Young [<a class="reference internal" href="R.Bibliography.html#id162" title="Peter Young. Data-based mechanistic modelling, generalised sensitivity and dominant mode analysis. Computer Physics Communications, 117(1-2):113–129, 1999.">108</a>]</span> and references therein.</p>
+</section>
+<section id="variance-based-methods">
+<span id="id70"></span><h3><span class="section-number">3.4.5. </span>Variance-based Methods<a class="headerlink" href="#variance-based-methods" title="Link to this heading">#</a></h3>
+<p>Variance-based sensitivity analysis methods hypothesize that various specified model factors contribute differently to the variation of model outputs; therefore, decomposition and analysis of output variance can determine a model’s sensitivity to input parameters <span id="id71">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>]</span>. The most popular variance-based method is the Sobol method, which is a global sensitivity analysis method that takes into account complex and nonlinear factor interaction when calculating sensitivity indices, and employs more sophisticated sampling methods (e.g., the Sobol sampling method). The Sobol method is able to calculate three types of sensitivity indices that provide different types of information about model sensitivities. These indices include first-order, higher-order (e.g., second-, third-, etc. orders), and total-order sensitivities.</p>
+<p>The first-order sensitivity index indicates the percent of model output variance contributed by a factor individually (i.e., the effect of varying <span class="math notranslate nohighlight">\(x_i\)</span> alone) and is obtained using the following <span id="id72">[<a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>, <a class="reference internal" href="R.Bibliography.html#id148" title="Andrea Saltelli. Making best use of model evaluations to compute sensitivity indices. Computer physics communications, 145(2):280–297, 2002.">109</a>]</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i^1=\frac{V_{x_i}[E_{x_{\sim i}}(x_i)]}{V(y)}\]</div>
+<p>with <span class="math notranslate nohighlight">\(E\)</span> and <span class="math notranslate nohighlight">\(V\)</span> denoting the expected value and the variance, respectively. <span class="math notranslate nohighlight">\(x_{\sim i}\)</span> denotes all factors expect for <span class="math notranslate nohighlight">\(x_i\)</span>. The first-order sensitivity index (<span class="math notranslate nohighlight">\(S_i^1\)</span>) can therefore also be thought of as the portion of total output variance (<span class="math notranslate nohighlight">\(V_y\)</span>) that can be reduced if the uncertainty in factor <span class="math notranslate nohighlight">\(x_i\)</span> is eliminated <span id="id73">[<a class="reference internal" href="R.Bibliography.html#id149" title="Andrea Saltelli. Sensitivity analysis for importance assessment. Risk analysis, 22(3):579–590, 2002.">110</a>]</span>. First-order sensitivity indices are usually used to understand the independent effect of a factor and to distinguish its individual versus interactive influence. It would be expected for linearly independent factors that they would only have first order indices (no interactions) that should correspond well with sensitivities obtained from simpler methods using OAT sampling.</p>
+<p>Higher-order sensitivity indices explore the interaction between two or more parameters that contribute to model output variations. For example, a second-order index indicates how interactions between a pair of factors can lead to change in model output variance and is calculated using the following relationship:</p>
+<div class="math notranslate nohighlight">
+\[S_{ij}^2=\frac{V_{x_{i,j}}[E_{x_{\sim i,j}}(x_i,x_j)]}{V(y)}\]</div>
+<p>with <span class="math notranslate nohighlight">\(i \ne j\)</span>. Higher order indices can be calculated by similar extensions (i.e., fixing additional operators together), but it is usually computationally expensive in practice.</p>
+<p>The total sensitivity analysis index represents the entire influence of an input factor on model outputs including all of its interactions with other factors <span id="id74">[<a class="reference internal" href="R.Bibliography.html#id150" title="Toshimitsu Homma and Andrea Saltelli. Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering &amp; System Safety, 52(1):1–17, 1996.">111</a>]</span>. In other words, total-order indices include first-order and all higher-order interactions associated with each factor and can be estimated calculated using the following:</p>
+<div class="math notranslate nohighlight">
+\[S_i^T= \frac{E_{x_{\sim i}}[V_{x_i}(x_{\sim i})]}{V(y)} = 1 - \frac{V_{x_{\sim i}}[E_{x_{i}}(x_{\sim i})]}{V(y)}\]</div>
+<p>This index reveals the expected portion of variance that remains if uncertainty is eliminated in all factors but <span class="math notranslate nohighlight">\(x_i\)</span> <span id="id75">[<a class="reference internal" href="R.Bibliography.html#id149" title="Andrea Saltelli. Sensitivity analysis for importance assessment. Risk analysis, 22(3):579–590, 2002.">110</a>]</span>. The total sensitivity index is the overall best measure of sensitivity as it captures the full individual and interactive effects of model factors.</p>
+<p>Besides the Sobol method, there are some other variance-based sensitivity analysis methods, such as the Fourier amplitude sensitivity test (FAST; <span id="id76">[<a class="reference internal" href="R.Bibliography.html#id100" title="RI Cukier, CM Fortuin, Kurt E Shuler, AG Petschek, and JH Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. i theory. The Journal of chemical physics, 59(8):3873–3878, 1973.">75</a>, <a class="reference internal" href="R.Bibliography.html#id151" title="Gregory J McRae, William R Goodin, and John H Seinfeld. Development of a second-generation mathematical model for urban air pollution—i. model formulation. Atmospheric Environment (1967), 16(4):679–696, 1982.">112</a>]</span>) and extended-FAST <span id="id77">[<a class="reference internal" href="R.Bibliography.html#id152" title="Andrea Saltelli and Ricardo Bolado. An alternative way to compute fourier amplitude sensitivity test (fast). Computational Statistics &amp; Data Analysis, 26(4):445–460, 1998.">113</a>, <a class="reference internal" href="R.Bibliography.html#id153" title="MA Vazquez-Cruz, R Guzman-Cruz, IL Lopez-Cruz, O Cornejo-Perez, I Torres-Pacheco, and RG Guevara-Gonzalez. Global sensitivity analysis by means of efast and sobol’methods and calibration of reduced state-variable tomgro model using genetic algorithms. Computers and Electronics in Agriculture, 100:1–12, 2014.">114</a>]</span>, that have been used by the scientific community. However, Sobol remains by far the most common method of this class. Variance-based techniques have been widely used and have proved to be powerful in a variety of applications. Despite their popularity, some authors have expressed concerns about the methods’ appropriateness in some settings. Specifically, the presence of heavy-tailed distributions or outliers, or when model outputs are multimodal can bias the sensitivity indices produced by these methods <span id="id78">[<a class="reference internal" href="R.Bibliography.html#id154" title="Benjamin Auder and Bertrand Iooss. Global sensitivity analysis based on entropy. In Safety, reliability and risk analysis-Proceedings of the ESREL 2008 Conference, 2107–2115. 2008.">115</a>, <a class="reference internal" href="R.Bibliography.html#id155" title="Farkhondeh Khorashadi Zadeh, Jiri Nossent, Fanny Sarrazin, Francesca Pianosi, Ann van Griensven, Thorsten Wagener, and Willy Bauwens. Comparison of variance-based and moment-independent global sensitivity analysis approaches by application to the swat model. Environmental Modelling &amp; Software, 91:210–222, 2017.">116</a>, <a class="reference internal" href="R.Bibliography.html#id156" title="Francesca Pianosi and Thorsten Wagener. A simple and efficient method for global sensitivity analysis based on cumulative distribution functions. Environmental Modelling &amp; Software, 67:1–11, 2015.">117</a>]</span>. Moment-independent measures, discussed below, attempt to overcome these challenges.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial which demonstrates the application of a Sobol sensitivity analysis:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA using SALib Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="analysis-of-variance-anova">
+<h3><span class="section-number">3.4.6. </span>Analysis of Variance (ANOVA)<a class="headerlink" href="#analysis-of-variance-anova" title="Link to this heading">#</a></h3>
+<p>Analysis of Variance (ANOVA) was first introduced by <span id="id79">Fisher and others [<a class="reference internal" href="R.Bibliography.html#id144" title="Ronald Aylmer Fisher and others. Statistical methods for research workers. Statistical methods for research workers., 1934.">118</a>]</span> and has since become a popular factor analysis method in physical experiments. ANOVA can be used as a sensitivity analysis method in computational experiments with a factorial design of experiment (referred to as factorial ANOVA). Note that Sobol can also be categorized as an ANOVA sensitivity analysis method, and that is why Sobol is sometimes referred to as a functional ANOVA <span id="id80">[<a class="reference internal" href="R.Bibliography.html#id145" title="Loıc Brevault, Mathieu Balesdent, Nicolas Bérend, and Rodolphe Le Riche. Comparison of different global sensitivity analysis methods for aerospace vehicle optimal design. In 10th World Congress on Structural and Multidisciplinary Optimization, WCSMO-10. 2013.">119</a>]</span>. Factorial ANOVA methods are particularly suited for models and problems that have discrete input spaces, significantly reducing the computational time. More information about these methods can be found in <span id="id81">[<a class="reference internal" href="R.Bibliography.html#id145" title="Loıc Brevault, Mathieu Balesdent, Nicolas Bérend, and Rodolphe Le Riche. Comparison of different global sensitivity analysis methods for aerospace vehicle optimal design. In 10th World Congress on Structural and Multidisciplinary Optimization, WCSMO-10. 2013.">119</a>, <a class="reference internal" href="R.Bibliography.html#id146" title="GEB Archer, Andrea Saltelli, and IM Sobol. Sensitivity measures, anova-like techniques and the use of bootstrap. Journal of Statistical Computation and Simulation, 58(2):99–120, 1997.">120</a>, <a class="reference internal" href="R.Bibliography.html#id147" title="Art B Owen. Variance components and generalized sobol'indices. SIAM/ASA Journal on Uncertainty Quantification, 1(1):19–41, 2013.">121</a>]</span>.</p>
+</section>
+<section id="moment-independent-density-based-methods">
+<h3><span class="section-number">3.4.7. </span>Moment-Independent (Density-Based) Methods<a class="headerlink" href="#moment-independent-density-based-methods" title="Link to this heading">#</a></h3>
+<p>These methods typically compare the entire distribution (i.e., not just the variance) of input and output parameters in order to determine the sensitivity of the output to a particular input variable. Several moment-independent sensitivity analysis methods have been proposed in recent years. The delta (<span class="math notranslate nohighlight">\(\delta\)</span>) moment-independent method calculates the difference between unconditional and conditional cumulative distribution functions of the output. The method was first introduced by <span id="id82">[<a class="reference internal" href="R.Bibliography.html#id142" title="Emanuele Borgonovo. Measuring uncertainty importance: investigation and comparison of alternative approaches. Risk analysis, 26(5):1349–1361, 2006.">122</a>, <a class="reference internal" href="R.Bibliography.html#id141" title="Emanuele Borgonovo. A new uncertainty importance measure. Reliability Engineering &amp; System Safety, 92(6):771–784, 2007.">123</a>]</span> and has become widely used in various disciplines. The <span class="math notranslate nohighlight">\(\delta\)</span> sensitivity index is defined as follows:</p>
+<div class="math notranslate nohighlight">
+\[S_i=\delta_i=\frac{1}{2}E_{x_i}|f_y(y)-f_{y|x_i}(y)|dy\]</div>
+<p>where <span class="math notranslate nohighlight">\(f_y(y)\)</span> is the probability density function of the entire model output <span class="math notranslate nohighlight">\(y\)</span>, and <span class="math notranslate nohighlight">\(f_{y|x_i}(y)\)</span> is the conditional density of <span class="math notranslate nohighlight">\(y\)</span>, given that factor <span class="math notranslate nohighlight">\(x_i\)</span> assumes a fixed value. The <span class="math notranslate nohighlight">\(\delta_i\)</span> sensitivity indicator therefore represents the normalized expected shift in the distribution of <span class="math notranslate nohighlight">\(y\)</span> provoked by <span class="math notranslate nohighlight">\(x_i\)</span>. Moment-independent methods are advantageous in cases where we are concerned about the entire distribution of events, such as when uncertain factors lead to more extreme events in a system <span id="id83">[<a class="reference internal" href="R.Bibliography.html#id47" title="Antonia Hadjimichael, Julianne Quinn, and Patrick Reed. Advancing Diagnostic Model Evaluation to Better Understand Water Shortage Mechanisms in Institutionally Complex River Basins. Water Resources Research, 56(10):e2020WR028079, 2020. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR028079 (visited on 2020-10-16), doi:10.1029/2020WR028079.">13</a>]</span>. Further, they can be used with a pre-existing sample of data, without requiring a specific sampling scheme, unlike the previously reviewed methods <span id="id84">[<a class="reference internal" href="R.Bibliography.html#id143" title="Elmar Plischke, Emanuele Borgonovo, and Curtis L Smith. Global sensitivity measures from given data. European Journal of Operational Research, 226(3):536–550, 2013.">124</a>]</span>. The <span class="math notranslate nohighlight">\(\delta\)</span> sensitivity index does not include interactions between factors and it is therefore akin to the first order index produced by the Sobol method. Interactions between factors can still be estimated using this method, by conditioning the calculation on more than one uncertain factor being fixed <span id="id85">[<a class="reference internal" href="R.Bibliography.html#id141" title="Emanuele Borgonovo. A new uncertainty importance measure. Reliability Engineering &amp; System Safety, 92(6):771–784, 2007.">123</a>]</span>.</p>
+</section>
+</section>
+<section id="how-to-choose-a-sensitivity-analysis-method-model-traits-and-dimensionality">
+<h2><span class="section-number">3.5. </span>How To Choose A Sensitivity Analysis Method: Model Traits And Dimensionality<a class="headerlink" href="#how-to-choose-a-sensitivity-analysis-method-model-traits-and-dimensionality" title="Link to this heading">#</a></h2>
+<p><a class="reference internal" href="#figure-3-5"><span class="std std-numref">Fig. 3.5</span></a>, synthesized from variants found in <span id="id86">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>, presents a graphical synthesis of the methods overviewed in this section, with regards to their appropriateness of application based on the complexity of the model at hand and the computational limits on the number of model evaluations afforded. The bars below each method also indicate the sensitivity analysis purposes they are most appropriate to address, which are in turn a reflection of the motivations and research questions the sensitivity analysis is called to address. Computational intensity is measured as a multiple of the number of model factors that are considered uncertain (<span class="math notranslate nohighlight">\(d\)</span>). Increasing model complexity mandates that more advanced sensitivity analysis methods are applied to address potential nonlinearities, factor interactions, and discontinuities. Such methods can only be performed at increasing computational expense. For example, computationally cheap linear regression should not be used to assess factors’ importance if the model cannot be proven linear and the factors independent, because important relationships will invariably be missed (recall the example in <a class="reference internal" href="#figure-3-5"><span class="std std-numref">Fig. 3.5</span></a>). When computational limits do constrain applications to make simplified assumptions and sensitivity techniques, any conclusions in such cases should be delivered with clear statements of the appropriate caveats.</p>
+<figure class="margin-caption align-center" id="id99">
+<span id="figure-3-5"></span><a class="reference internal image-reference" href="_images/figure3_5classificationofmethods.png"><img alt="Figure 3_5" src="_images/figure3_5classificationofmethods.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 3.5 </span><span class="caption-text">Classification of the sensitivity analysis methods overviewed in this section, with regards to their computational cost (horizontal axis), their appropriateness to model complexity (vertical axis), and the purpose they can be used for (colored bars). d: number of uncertain factors considered; ANOVA: Analysis of Variance; FAST: Fourier Amplitude Sensitivity Test; PRIM: Patient Rule Induction Method; CART: Classification and Regression Trees; SRCC: Spearman’s rank correlation coefficient: NSE: Nash–Sutcliffe efficiency; SRC: standardized regression coefficient; PCC: Pearson correlation coefficient. This figure is synthesized from variants found in <span id="id87">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>.</span><a class="headerlink" href="#id99" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The reader should also be aware that the estimates of computational intensity that are given here are indicative of magnitude and would vary depending on the sampling technique, model complexity and the level of information being asked. For example, a Sobol sensitivity analysis typically requires a sample of size <span class="math notranslate nohighlight">\(n * d+2\)</span> to produce first- and total-order indices, where <span class="math notranslate nohighlight">\(d\)</span> is the number of uncertain factors and <span class="math notranslate nohighlight">\(n\)</span> is a scaling factor, selected ad hoc, depending on model complexity <span id="id88">[<a class="reference internal" href="R.Bibliography.html#id25" title="Andrea Saltelli and Stefano Tarantola. On the relative importance of input factors in mathematical models: safety assessment for nuclear waste disposal. Journal of the American Statistical Association, 97(459):702–709, 2002.">46</a>]</span>. The scaling factor <span class="math notranslate nohighlight">\(n\)</span> is typically set to at least 1000, but it should most appropriately be set on the basis of index convergence. In other words, a prudent analyst would perform the analysis several times with increasing <span class="math notranslate nohighlight">\(n\)</span> and observe at what level the indices converge to stable values <span id="id89">[<a class="reference internal" href="R.Bibliography.html#id140" title="Jiri Nossent, Pieter Elsen, and Willy Bauwens. Sobol’ sensitivity analysis of a complex environmental model. Environmental Modelling &amp; Software, 26(12):1515-1525, 2011. URL: https://www.sciencedirect.com/science/article/pii/S1364815211001939, doi:https://doi.org/10.1016/j.envsoft.2011.08.010.">125</a>]</span>. The level should be the minimum sample size used in subsequent sensitivity analyses of the same system. Furthermore, if the analyst would like to better understand the degrees of interaction between factors, requiring second-order indices, the sample size would have to increase to <span class="math notranslate nohighlight">\(n * 2d+2\)</span> <span id="id90">[<a class="reference internal" href="R.Bibliography.html#id25" title="Andrea Saltelli and Stefano Tarantola. On the relative importance of input factors in mathematical models: safety assessment for nuclear waste disposal. Journal of the American Statistical Association, 97(459):702–709, 2002.">46</a>]</span>.</p>
+<p>Another important consideration is that methods that do not require specific sampling schemes can be performed in conjunction with others without requiring additional model evaluations. None of the regression-based methods, for example, require samples of specific structures or sizes, and can be combined with other methods for complementary purposes. For instance, one could complement a Sobol analysis with an application of CART, using the same data, but to address questions relating to factor mapping (e.g., we know factor <span class="math notranslate nohighlight">\(x_i\)</span>  is important for a model output, but we would like to also know which of its values specifically push the output to undesirable states). Lastly, comparing results from different methods performed together can be especially useful in model diagnostic settings. For example, <span id="id91">[<a class="reference internal" href="R.Bibliography.html#id47" title="Antonia Hadjimichael, Julianne Quinn, and Patrick Reed. Advancing Diagnostic Model Evaluation to Better Understand Water Shortage Mechanisms in Institutionally Complex River Basins. Water Resources Research, 56(10):e2020WR028079, 2020. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR028079 (visited on 2020-10-16), doi:10.1029/2020WR028079.">13</a>]</span> used <span class="math notranslate nohighlight">\(\delta\)</span> indices, first-order Sobol indices, and <span class="math notranslate nohighlight">\(R^2\)</span>  values from linear regression, all performed on the same factors, to derive insights about the effects on factors on different moments of the output distribution and about the linearity of their relationship.</p>
+</section>
+<section id="software-toolkits">
+<span id="id92"></span><h2><span class="section-number">3.6. </span>Software Toolkits<a class="headerlink" href="#software-toolkits" title="Link to this heading">#</a></h2>
+<p>This section presents available open source sensitivity analysis software tools, based on the programming language they use and the methods they support <a class="reference internal" href="#figure-3-6"><span class="std std-numref">Fig. 3.6</span></a>. Our review covers five widely used programming languages: R, MATLAB, Julia, Python, and C++, as well as one tool that provides a graphical user interface (GUI). Each available SA tool was assessed on the number of SA methods and design of experiments methods it supports. For example, the <em>sensobol</em> package in R only supports the variance-based Sobol method. However, it is the only package we came across that calculates third-order interactions among parameters. On the other side of the spectrum, there are SA software packages that contain several popular SA methods. For example, <em>SALib</em> in Python <span id="id93">[<a class="reference internal" href="R.Bibliography.html#id139" title="Jon Herman and Will Usher. Salib: an open-source python library for sensitivity analysis. Journal of Open Source Software, 2(9):97, 2017.">126</a>]</span> supports seven different SA methods. The <em>DifferentialEquations</em> package is a comprehensive package developed for Julia, and <em>GlobalSensitivityAnalysis</em> is another Julia package that has mostly adapted SALib methods. <a class="reference internal" href="#figure-3-6"><span class="std std-numref">Fig. 3.6</span></a> also identifies the SA packages that have been updated since 2018, indicating active support and development.</p>
+<figure class="margin-caption align-center" id="id100">
+<span id="figure-3-6"></span><a class="reference internal image-reference" href="_images/figure3_6_softwaretoolkits.png"><img alt="Figure 3_6" src="_images/figure3_6_softwaretoolkits.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 3.6 </span><span class="caption-text">Sensitivity analysis packages available in different programming language platforms (R, Python, Julia, MATLAB, and C++), with the number of methods they support. Packages supporting more than five methods are indicated in pink. Packages updated since 2018 are indicated with asterisks.</span><a class="headerlink" href="#id100" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The following articles are suggested as fundamental reading for the information presented in this section:</p>
+<ul class="simple">
+<li><p>Wagener, T., Pianosi, F., 2019. What has Global Sensitivity Analysis ever done for us? A systematic review to support scientific advancement and to inform policy-making in earth system modelling. <em>Earth-Science Reviews</em> 194, 1–18. <a class="reference external" href="https://doi.org/10.1016/j.earscirev.2019.04.006">https://doi.org/10.1016/j.earscirev.2019.04.006</a></p></li>
+<li><p>Pianosi, F., Beven, K., Freer, J., Hall, J.W., Rougier, J., Stephenson, D.B., Wagener, T., 2016. Sensitivity analysis of environmental models: A systematic review with practical workflow. Environmental Modelling &amp; Software 79, 214–232. <a class="reference external" href="https://doi.org/10.1016/j.envsoft.2016.02.008">https://doi.org/10.1016/j.envsoft.2016.02.008</a></p></li>
+</ul>
+<p>The following articles can be used as supplemental reading:</p>
+<ul class="simple">
+<li><p>Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S., 2008. Global Sensitivity Analysis: The Primer, 1st edition. ed. Wiley-Interscience, Chichester, England ; Hoboken, NJ.</p></li>
+<li><p>Montgomery, D.C., 2017. Design and analysis of experiments. John Wiley &amp; Sons.</p></li>
+<li><p>Iooss, B., Lemaître, P., 2015. A Review on Global Sensitivity Analysis Methods, in: Dellino, G., Meloni, C. (Eds.), Uncertainty Management in Simulation-Optimization of Complex Systems: Algorithms and Applications, Operations Research/Computer Science Interfaces Series. Springer US, Boston, MA, pp. 101–122. <a class="reference external" href="https://doi.org/10.1007/978-1-4899-7547-8_5">https://doi.org/10.1007/978-1-4899-7547-8_5</a></p></li>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#global-versus-local-sensitivity">3.1. Global Versus Local Sensitivity</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#why-perform-sensitivity-analysis">3.2. Why Perform Sensitivity Analysis</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#design-of-experiments">3.3. Design of Experiments</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#one-at-a-time-oat">3.3.1. One-At-a-Time (OAT)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#full-and-fractional-factorial-sampling">3.3.2. Full and Fractional Factorial Sampling</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#latin-hypercube-sampling-lhs">3.3.3. Latin Hypercube Sampling (LHS)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#low-discrepancy-sequences">3.3.4. Low-Discrepancy Sequences</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#other-types-of-sampling">3.3.5. Other types of sampling</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-generation-of-input-time-series">3.3.6. Synthetic generation of input time series</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sensitivity-analysis-methods">3.4. Sensitivity Analysis Methods</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#derivative-based-methods">3.4.1. Derivative-based Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#elementary-effect-methods">3.4.2. Elementary Effect Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#regression-based-methods">3.4.3. Regression-based Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#regional-sensitivity-analysis">3.4.4. Regional Sensitivity Analysis</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#variance-based-methods">3.4.5. Variance-based Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#analysis-of-variance-anova">3.4.6. Analysis of Variance (ANOVA)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#moment-independent-density-based-methods">3.4.7. Moment-Independent (Density-Based) Methods</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#how-to-choose-a-sensitivity-analysis-method-model-traits-and-dimensionality">3.5. How To Choose A Sensitivity Analysis Method: Model Traits And Dimensionality</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#software-toolkits">3.6. Software Toolkits</a></li>
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+    <h1>Understanding Errors: What Is Controlling Model Performance?</h1>
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+  <section id="understanding-errors-what-is-controlling-model-performance">
+<span id="understanding-errors"></span><h1>Understanding Errors: What Is Controlling Model Performance?<a class="headerlink" href="#understanding-errors-what-is-controlling-model-performance" title="Link to this heading">#</a></h1>
+<p>Sensitivity analysis is a diagnostic tool when reconciling model outputs with observed data. It is helpful for clarifying how and under what conditions modeling choices (structure, parameterization, data inputs, etc.) propagate through model components and manifest in their effects on model outputs. This exploration is performed through carefully designed sampling of multiple combinations of input parameters and subsequent evaluation of the model structures that are emerging as controlling factors. Model structure and parameterization are two of the most commonly explored aspects of models that have been a central focus when evaluating their performance relative to available observations <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id49" title="Keith Beven. Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in water resources, 16(1):41–51, 1993.">17</a>]</span>. Addressing these issues plays an important role in establishing credibility in model predictions, particularly in the positivist natural sciences literature. Traditional model evaluations compare the model with observed data, and then rely on expert judgements of its acceptability based on the closeness between simulation and observation with one or a small number of selected metrics. This approach can be myopic, as it is often impossible to use one metric to attribute a certain error and to link that with different parts of the model and its parameters <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id50" title="Hoshin V Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008.">127</a>]</span>. This means that, even when the error or fitting measure between the model estimates and observations is very small, it is not guaranteed that all the components in the model accurately represent the conceptual reality that the model is abstracting: propagated errors in different parts of the model might cancel each other out, or multiple parameterized implementations of the model can yield similar performance (i.e., equifinality <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id49" title="Keith Beven. Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in water resources, 16(1):41–51, 1993.">17</a>]</span>).</p>
+<p>The inherent complexity of a system hinders accepting or rejecting a model based on one performance measure and different types of measures can aid in evaluating the various components of a model as essentially a multiobjective problem <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id52" title="Keith J Beven. On hypothesis testing in hydrology: why falsification of models is still a really good idea. Wiley Interdisciplinary Reviews: Water, 5(3):e1278, 2018.">25</a>, <a class="reference internal" href="R.Bibliography.html#id51" title="Hoshin V Gupta, Martyn P Clark, Jasper A Vrugt, Gab Abramowitz, and Ming Ye. Towards a comprehensive assessment of model structural adequacy. Water Resources Research, 2012.">26</a>]</span>. In addition, natural systems mimicked by the models contain various interacting components that might act differently across spatial and temporal domains <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id53" title="JD Herman, PM Reed, and T Wagener. Time-varying sensitivity analysis clarifies the effects of watershed model formulation on model behavior. Water Resources Research, 49(3):1400–1414, 2013.">55</a>]</span>. This heterogeneity is lost when a single performance measure is used, as a highly dimensional and interactive system becomes aggregated through the averaging of spatial or temporal output errors <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id54" title="Keith Beven. Towards a coherent philosophy for modelling the environment. Proceedings of the royal society of London. Series A: mathematical, physical and engineering sciences, 458(2026):2465–2484, 2002.">15</a>]</span>. Therefore, diagnostic error analyses should consider multiple error signatures across different scales and states of concern when seeking to understand how model performance relates to observed data (<a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-1"><span class="std std-numref">Fig. 4.1</span></a>). Diverse error signatures can be used to measure the consistency of underlying processes and behaviors of the model and to evaluate the dynamics of model controls under changing temporal and spatial conditions <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id56" title="Keith Beven and Jim Freer. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the glue methodology. Journal of hydrology, 249(1-4):11–29, 2001.">128</a>]</span>. Within this framework, even minimal information extracted from the data can be beneficial as it helps us unearth structural inadequacies in the model. In this context, proper selection of measures of model performance and the number of measures could play consequential roles in our understanding of the model and its predictions <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id55" title="Cameron McPhail, HR Maier, JH Kwakkel, M Giuliani, A Castelletti, and S Westra. Robustness metrics: how are they calculated, when should they be used and why do they give different results? Earth's Future, 6(2):169–191, 2018.">129</a>]</span>.</p>
+<p>As discussed earlier, instead of the traditional focus on using deterministic prediction that results in a single error measure, many plausible states and spaces could be searched for making different inferences and quantifying uncertainties. This process also requires estimates of prior probability distributions of all the important parameters and quantification of model behavior across input space. One strategy to reduce the search space is filtering of some model alternatives that are not consistent with observations and known system behaviors. Those implausible parts of the search space can be referred to as non-physical or non-behavioral alternatives <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id29" title="Neil R Edwards, David Cameron, and Jonathan Rougier. Precalibrating an intermediate complexity climate model. Climate dynamics, 37(7):1469–1482, 2011.">50</a>, <a class="reference internal" href="R.Bibliography.html#id158" title="Keith Beven and Andrew Binley. Glue: 20 years on. Hydrological processes, 28(24):5897–5918, 2014.">104</a>]</span>. This step is conducted before the Bayesian calibration exercise (see <a class="reference internal" href="A1_Uncertainty_Quantification.html#a1-uncertainty-quantification"><span class="std std-numref">Chapter A</span></a>).</p>
+<p>A comprehensive model diagnostic workflow typically entails the components demonstrated in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-1"><span class="std std-numref">Fig. 4.1</span></a>. The workflow begins with the selection of model input parameters and their plausible ranges. After the parameter selection, we need to specify the design of experiment (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#design-of-experiments"><span class="std std-numref">Chapter 3.3</span></a>) and the sensitivity analysis method (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#sensitivity-analysis-methods"><span class="std std-numref">Chapter 3.4</span></a>) to be used. As previously discussed, these methods require different numbers of model simulations, and each method provides a different insights into the direct effects and interactions of the uncertain factors. In addition, the simulation time of the model and the available computational resources are two of the primary considerations that influence these decisions. After identifying the appropriate methods, we generate a matrix of input parameters, where each set of input parameters will be used to conduct a model simulation. The model can include one or more output variables that fluctuate in time and space. The next step is to analyze model performance by comparing model outputs with observations. As discussed earlier, the positivist model evaluation paradigm focuses on a single model performance metric (error), leading to a loss of information about model parameters and the suitability of the model’s structure. However, a thorough investigation of the temporal and spatial signatures of model outputs using various performance metrics or time- and space-varying sensitivity analyses can shed more light on the fitness of each parameter set and the model’s internal structure. This analysis provides diagnostic feedback on the importance and range of model parameters and can guide further improvement of the model algorithm.</p>
+<figure class="margin-caption align-center" id="id10">
+<span id="figure-4-1"></span><a class="reference internal image-reference" href="_images/figure4_1_diagnostic_workflow.png"><img alt="Figure 4.1" src="_images/figure4_1_diagnostic_workflow.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Diagnostic evaluation of model fidelity using sensitivity analysis methods.</span><a class="headerlink" href="#id10" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following badge to try out an interactive tutorial on implementing a time-varying sensitivity analysis of HYMOD model parameters:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">HYMOD Jupyter Notebook</a></p>
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+    <h1>Consequential Dynamics: What is Controlling Model Behaviors of Interest?</h1>
+    <!-- Table of contents -->
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+  <section id="consequential-dynamics-what-is-controlling-model-behaviors-of-interest">
+<h1>Consequential Dynamics: What is Controlling Model Behaviors of Interest?<a class="headerlink" href="#consequential-dynamics-what-is-controlling-model-behaviors-of-interest" title="Link to this heading">#</a></h1>
+<p>Consequential changes in dynamic systems can take many forms, but most dynamic behavior can be categorized in a few basic patterns. Feedback structures inherent in a system, be they positive or negative, generate these patterns which can be grouped into three groups for the simplest of systems: exponential growth, goal seeking, and oscillation (<a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-2"><span class="std std-numref">Fig. 4.2</span></a>). A positive (or self-reinforcing) feedback gives rise to exponential growth, a negative (or self-correcting) feedback gives rise to a goal seeking mode, and negative feedbacks with time delays give rise to oscillatory behavior. Nonlinear interactions between the system’s feedback structures can give rise to more complex dynamic behavior modes, examples of which are also shown in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-2"><span class="std std-numref">Fig. 4.2</span></a>, adapted from <span id="id1">Sterman [<a class="reference internal" href="R.Bibliography.html#id57" title="John D Sterman. System dynamics modeling: tools for learning in a complex world. California management review, 43(4):8–25, 2001.">130</a>]</span>.</p>
+<figure class="margin-caption align-center" id="id16">
+<span id="figure-4-2"></span><a class="reference internal image-reference" href="_images/figure4_2_behavior_modes.png"><img alt="Figure 4.2" src="_images/figure4_2_behavior_modes.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Common modes of behavior in dynamic systems, occurring based on the presence of positive and negative feedback relationships, and linear and non-linear interactions. Adapted from <span id="id2">Sterman [<a class="reference internal" href="R.Bibliography.html#id57" title="John D Sterman. System dynamics modeling: tools for learning in a complex world. California management review, 43(4):8–25, 2001.">130</a>]</span>.</span><a class="headerlink" href="#id16" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The nature of feedback processes in a dynamic system shapes its fundamental behavior: positive feedbacks generate their own growth, negative feedbacks self-limit, seeking balance and equilibrium. In this manner, feedback processes give rise to different regimes, multiples of which could be present in each mode of behavior. Consider a population of mammals growing exponentially until it reaches the carrying capacity of its environment (referred to as S-shaped growth). When the population is exponentially growing, the regime is dominated by positive feedback relationships that reinforce its growth. As the population approaches its carrying capacity limit, negative feedback structures begin to dominate, counteracting the growth and establishing a stable equilibrium. Shifts between regimes can be thought of as tipping points, mathematically defined as unstable equilibria, where the presence of positive feedbacks amplifies disturbances and moves the system to a new equilibrium point. In the case of stable equilibria, the presence of negative feedbacks dampens any small disturbance and maintains the system at a stable state. As different feedback relationships govern each regime, different factors (those making up each feedback mechanism) are activated and shape the states the system is found in, as well as define the points of equilibria.</p>
+<p>For simple stylized models with a small number of states, system dynamics analysis can analytically derive these equilibria, the conditions for their stability and the factors determining them. The ability for this to be performed is, however, significantly challenged when it comes to systems that attempt to more closely resemble real complex systems. We argue this is the case for several reasons. First, besides generally exhibiting complex nonlinear dynamics, real world systems are also made up from larger numbers of interacting elements, which often makes the analytic derivation of system characteristics intractable <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id59" title="John M Anderies, Jean-Denis Mathias, and Marco A Janssen. Knowledge infrastructure and safe operating spaces in social–ecological systems. Proceedings of the National Academy of Sciences, 116(12):5277–5284, 2019.">131</a>, <a class="reference internal" href="R.Bibliography.html#id58" title="Rachata Muneepeerakul and John M Anderies. The emergence and resilience of self-organized governance in coupled infrastructure systems. Proceedings of the National Academy of Sciences, 117(9):4617–4622, 2020.">132</a>]</span>. Second, human-natural systems temporally evolve and transform when human state-aware action is present. Consider, for instance, humans recreationally hunting the aforementioned population of mammals. Humans act based on the mammal population levels by enforcing hunting quotas or establishing protected territories or eliminating other predators. The mammal population reacts in a response, giving birth to ever changing state-action-consequence feedbacks, the path dependencies of which become difficult to diagnose and understand (e.g., <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id60" title="Antonia Hadjimichael, Patrick M Reed, and Julianne D Quinn. Navigating deeply uncertain tradeoffs in harvested predator-prey systems. Complexity, 2020.">133</a>]</span>). Trying to simulate the combination of these two challenges (large numbers of state-aware agents interacting with a natural resource and with each other) produces intractable models that require advanced heuristics to analyze their properties and establish useful inferences.</p>
+<p>Sensitivity analysis paired with exploratory modeling methods offers a promising set of tools to address these challenges. We present a simple demonstrative application based on <span id="id5">Quinn <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id61" title="Julianne D Quinn, Patrick M Reed, and Klaus Keller. Direct policy search for robust multi-objective management of deeply uncertain socio-ecological tipping points. Environmental modelling &amp; software, 92:125–141, 2017.">134</a>]</span>. This stylistic example was first developed by <span id="id6">Carpenter <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id163" title="S. R. Carpenter, D. Ludwig, and W. A. Brock. Management of Eutrophication for Lakes Subject to Potentially Irreversible Change. Ecological Applications, 9(3):751–771, August 1999. URL: http://onlinelibrary.wiley.com/doi/10.1890/1051-0761(1999)009[0751:MOEFLS]2.0.CO;2/abstract, doi:10.1890/1051-0761(1999)009[0751:MOEFLS]2.0.CO;2.">135</a>]</span> and represents a town that must balance its agricultural and industrial productivity with the pollution it creates in a downstream lake. Increased productivity allows for increased profits, which the town aims to maximize, but it also produces more pollution for the lake. Too much phosphorus pollution can cause irreversible eutrophication, a process known as “tipping” the lake. The model of phosphorus in the lake <span class="math notranslate nohighlight">\(X_t\)</span> at time <span class="math notranslate nohighlight">\(t\)</span> is governed by:</p>
+<div class="math notranslate nohighlight">
+\[X_{t+1}= X_{t}+a_{t}+\frac{X_{t}^q} {1+X_{t}^q}-bX_t+\varepsilon\]</div>
+<p>where <span class="math notranslate nohighlight">\(a_t \in [0,0.1]\)</span> is the town’s pollution release at each timestep, <span class="math notranslate nohighlight">\(b\)</span> is the natural decay rate of phosphorus in the lake, <span class="math notranslate nohighlight">\(q\)</span> defines the lake’s recycling rate (primarily through sediments), and <span class="math notranslate nohighlight">\(\varepsilon\)</span> represents uncontrollable natural inflows of pollution modeled as a log-normal distribution with a given mean, <span class="math notranslate nohighlight">\(\mu\)</span>, and standard deviation <span class="math notranslate nohighlight">\(\sigma\)</span>.</p>
+<p>Panels (a-c) in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> plot the fluxes of phosphorus into the lake versus the mass accumulation of phosphorus in the lake. The red line corresponds to the phosphorus sinks in the lake (natural decay), given by <span class="math notranslate nohighlight">\(bX_t\)</span>. The grey shaded area represents the lake’s phosphorus recycling flux, given by <span class="math notranslate nohighlight">\(\frac{X_{t}^q} {1+X_{t}^q}\)</span>. The points of intersection indicate the system’s equilibria, two of which are stable, and one is unstable (also known as the tipping point). The stable equilibrium in the bottom left of the figure reflects an oligotrophic lake, whereas the stable equilibrium in the top right represents a eutrophic lake. With increasing phosphorus values, the tipping point can be crossed, and the lake will experience irreversible eutrophication, as the recycling rate would exceed the removal rate even if the town’s pollution became zero. In the absence of anthropogenic and natural inflows of pollution in the lake (<span class="math notranslate nohighlight">\(a_t\)</span> and <span class="math notranslate nohighlight">\(\varepsilon\)</span> respectively), the area between the bottom-left black point and the white point in the middle can be considered as the safe operating space, before emission levels cross the tipping point.</p>
+<figure class="margin-caption align-center" id="id17">
+<span id="figure-4-3"></span><a class="reference internal image-reference" href="_images/figure4_3_lake_problem_fluxes.png"><img alt="Figure 4.3" src="_images/figure4_3_lake_problem_fluxes.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Fluxes of phosphorus with regards to mass of phosphorus in the lake and sensitivity analysis results, assuming <span class="math notranslate nohighlight">\(b=0.42\)</span> and <span class="math notranslate nohighlight">\(q=2\)</span>. (a) Fluxes of phosphorus assuming no emmisions policy and no natural inflows. (b-c) Fluxes phosphorus when applying two different emissions policies. The “Best economic policy” and the “Most reliable policy” have been identified by <span id="id7">Quinn <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id61" title="Julianne D Quinn, Patrick M Reed, and Klaus Keller. Direct policy search for robust multi-objective management of deeply uncertain socio-ecological tipping points. Environmental modelling &amp; software, 92:125–141, 2017.">134</a>]</span> and can be found at <span id="id8">Quinn [<a class="reference internal" href="R.Bibliography.html#id164" title="Julianne Quinn. Julianneq/Lake_problem_dps. December 2017. original-date: 2017-02-06T18:33:54Z. URL: https://github.com/julianneq/Lake_Problem_DPS (visited on 2021-06-14).">136</a>]</span>. (d) Results of a sensitivity analysis on the parameters of the model most consequential to the reliability of the “Most reliable policy”. The code to replicate the sensitivity analysis can be found at <span id="id9">Hadka [<a class="reference internal" href="R.Bibliography.html#id165" title="David Hadka. Project-Platypus/Rhodium. 2017. original-date: 2015-10-29T18:08:43Z. URL: https://github.com/Project-Platypus/Rhodium (visited on 2021-06-14).">137</a>]</span>. Panels (a-c) are used courtesy of Julianne Quinn, University of Virginia.</span><a class="headerlink" href="#id17" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The town has identified two potential policies that can be used to manage this lake, one that maximizes its economic profits (“best economic policy”) and one that maximizes the time below the tipping point (“most reliable policy”). Panels (b-c) in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> add the emissions from these policies to the recycling flux and show how the equilibria points shift as a result. In both cases the stable oligotrophic equilibrium increases and the tipping point decreases, narrowing the safe operating space <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id59" title="John M Anderies, Jean-Denis Mathias, and Marco A Janssen. Knowledge infrastructure and safe operating spaces in social–ecological systems. Proceedings of the National Academy of Sciences, 116(12):5277–5284, 2019.">131</a>, <a class="reference internal" href="R.Bibliography.html#id166" title="Stephen R. Carpenter, William A. Brock, Carl Folke, Egbert H. van Nes, and Marten Scheffer. Allowing variance may enlarge the safe operating space for exploited ecosystems. Proceedings of the National Academy of Sciences, 112(46):14384–14389, November 2015. URL: http://www.pnas.org/content/112/46/14384 (visited on 2017-08-18), doi:10.1073/pnas.1511804112.">138</a>]</span>. The best economic policy results in a much narrower space of action, with the tipping point very close to the oligotrophic equilibrium. The performance of both policies depends significantly on the system parameters. For example, a higher value of <span class="math notranslate nohighlight">\(b\)</span>, the natural decay rate, would shift the red line upward, moving the equilibria points and widening the safe operating space. Inversely, a higher value of <span class="math notranslate nohighlight">\(q\)</span>, the lake’s recycling rate, would shift the recycling line upward, moving the tipping point lower and decreasing the safe operating space. The assumptions under which these policies were identified are therefore critical to their performance and any potential uncertainty in the parameter values could be detrimental to the system’s objectives being met.</p>
+<p>Sensitivity analysis can be used to clarify the role these parameters play on policy performance. <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> (d) shows the results of a Sobol sensitivity analysis on the reliability of the “most reliable” policy in a radial convergence diagram. The significance of each parameter is indicated by the size of circles corresponding to it. The size of the interior dark circle indicates the parameter’s first-order effects and the size of the exterior circle indicates the parameter’s total-order effects. The thickness of the lines between two parameters indicated the extent of their interaction (second-order effects). In this case, parameters <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(q\)</span> appear to have the most significant importance on the system, followed by the mean, <span class="math notranslate nohighlight">\(\mu\)</span>, of the natural inflows. All these parameters function in a manner that shifts the location of the three equilibria and therefore policies that are identified ignoring this parametric uncertainty might fail to meet their intended goals.</p>
+<p>It is worth mentioning that current sensitivity analysis methods are somewhat challenged in addressing several system dynamics analysis questions. The fundamental reason is that sensitivity analysis methods and tools have been developed to gauge numerical sensitivity of model output to changes in factor values. This is natural, as most simulation studies (e.g., all aforementioned examples) have been traditionally concerned with this type of sensitivity. In system dynamics modeling, however, a more important and pertinent concern is changes between regimes or between behavior modes (also known as bifurcations) as a result of changes in model factors <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id57" title="John D Sterman. System dynamics modeling: tools for learning in a complex world. California management review, 43(4):8–25, 2001.">130</a>, <a class="reference internal" href="R.Bibliography.html#id63" title="Mustafa Hekimoğlu and Yaman Barlas. Sensitivity analysis for models with multiple behavior modes: a method based on behavior pattern measures. System Dynamics Review, 32(3-4):332–362, 2016.">139</a>]</span>. This poses two new challenges. First, identifying a change in regime depends on several characteristics besides a change in output value, like the rate and direction of change. Second, behavior mode changes are qualitative and discontinuous, as equilibria change in stability but also move in and out of existence.</p>
+<p>Despite these challenges, recent advanced sensitivity analysis methods can help illuminate which factors in a system are most important in shaping boundary conditions (tipping points) between different regimes and determining changes in behavior modes. Reviewing such methods is outside the scope of this text, but the reader is directed to the examples of <span id="id12">Eker <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id64" title="Sibel Eker, Elena Rovenskaya, Michael Obersteiner, and Simon Langan. Practice and perspectives in the validation of resource management models. Nature communications, 9(1):1–10, 2018.">22</a>]</span> and <span id="id13">Hadjimichael <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id60" title="Antonia Hadjimichael, Patrick M Reed, and Julianne D Quinn. Navigating deeply uncertain tradeoffs in harvested predator-prey systems. Complexity, 2020.">133</a>]</span>, who apply parameterised perturbation on the functional relationships of a system to study the effects of model structural uncertainty on model outputs and bifurcations, and <span id="id14">Hekimoğlu and Barlas [<a class="reference internal" href="R.Bibliography.html#id63" title="Mustafa Hekimoğlu and Yaman Barlas. Sensitivity analysis for models with multiple behavior modes: a method based on behavior pattern measures. System Dynamics Review, 32(3-4):332–362, 2016.">139</a>]</span> and <span id="id15">Steinmann <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id65" title="Patrick Steinmann, Willem L Auping, and Jan H Kwakkel. Behavior-based scenario discovery using time series clustering. Technological Forecasting and Social Change, 156:120052, 2020.">140</a>]</span> who, following wide sampling of uncertain inputs, cluster the resulting time series in modes of behavior and identify most important factors for each.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following badge to try out an interactive tutorial on performing a sensitivity analysis to discover consequential dynamics:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/fishery_dynamics.ipynb">Factor Discovery Jupyter Notebook</a></p>
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+    <h1>Consequential Scenarios: What is Controlling Consequential Outcomes?</h1>
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+  <section id="consequential-scenarios-what-is-controlling-consequential-outcomes">
+<span id="consequential-scenarios"></span><h1>Consequential Scenarios: What is Controlling Consequential Outcomes?<a class="headerlink" href="#consequential-scenarios-what-is-controlling-consequential-outcomes" title="Link to this heading">#</a></h1>
+<p>As overviewed in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>, most models are abstractions of systems in the real world. When sufficient confidence has been established in a model, it can then act as a surrogate for the actual system, in that the consequences of potential stressors, proposed actions or other changes can be evaluated by computer model simulations <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>]</span>. A model simulation then represents a computational experiment, which can be used to assess how the modeled system would behave should the various changes come to be. Steven Bankes coined the term exploratory modeling to describe the use of large sets of such computational experiments to investigate their implications on the system. <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a> presents a typical workflow of an exploratory modeling application. Exploratory modeling approaches typically use sampling designs to generate large ensembles of states that represent combinations of changes happening together, spanning the entire range of potential values a factor might take (indicated in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a> by numbers 2-5). This perspective on modeling is particularly relevant to studies making long term projections into the future.</p>
+<figure class="margin-caption align-center" id="id26">
+<span id="figure-4-4"></span><a class="reference internal image-reference" href="_images/figure4_4_exploratory_workflow.png"><img alt="Figure 4_4" src="_images/figure4_4_exploratory_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">A typical exploratory modeling workflow</span><a class="headerlink" href="#id26" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>In the long-term policy analysis literature, exploratory modeling has prominently placed itself as an alternative to traditional narrative scenario or assumptions-based planning approaches, in what can be summarized in the following two-pronged critique <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>, <a class="reference internal" href="R.Bibliography.html#id67" title="Steven C Bankes, Robert J Lempert, and Steven W Popper. Computer-assisted reasoning. Computing in Science &amp; Engineering, 3(2):71–77, 2001.">141</a>, <a class="reference internal" href="R.Bibliography.html#id66" title="Robert J. Lempert, Steven W. Popper, and Steven C. Bankes. Shaping the Next One Hundred Years. RAND Corporation, 2003. URL: https://www.rand.org/pubs/monograph_reports/MR1626.html (visited on 2017-09-14).">142</a>]</span>. The most prevalent criticism sees that the future and how it might evolve is both highly complex and deeply uncertain. Despite its benefits for interpretation and intuitive appeal, a small number of scenarios invariably misses many other potential futures that did not get selected as sufficiently representative of the future. This is especially the case for aggregate, narrative scenarios that describe simultaneous changes in multiple sectors together (e.g., “increased energy demand, combined with high agricultural land use and large economic growth”), such as the emission scenarios produced by the Intergovernmental Panel on Climate Change <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id68" title="Jonathan R Lamontagne, Patrick M Reed, Robert Link, Katherine V Calvin, Leon E Clarke, and James A Edmonds. Large ensemble analytic framework for consequence-driven discovery of climate change scenarios. Earth's Future, 6(3):488–504, 2018.">143</a>]</span>. The bias introduced by this reduced set of potential changes can skew inferences drawn from the model, particularly when the original narrative scenarios are focused on a single or narrow set of measures of system behavior.</p>
+<p>The second main criticism of traditional narrative scenario-based planning methods is that they provide no systematic way to distinguish which of the constituent factors lead to the undesirable consequences produced by a scenario. Narrative scenarios (e.g., the scenario matrix framework of RCPs-SSPs-SPAs; <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id69" title="Brian C O’Neill, Elmar Kriegler, Keywan Riahi, Kristie L Ebi, Stephane Hallegatte, Timothy R Carter, Ritu Mathur, and Detlef P van Vuuren. A new scenario framework for climate change research: the concept of shared socioeconomic pathways. Climatic change, 122(3):387–400, 2014.">144</a>]</span>) encompass multiple changes happening together selected to span the range of potential changes but are not typically generated in a systematic factorial manner that considers the multiple ways the factors can be combined. This has two critical limitations. It obfuscates the role each component factor plays in the system, both in isolation and in combination with others (e.g., “is it the increased energy demand or the high agricultural land use that cause unbearable water stress?”). It also renders the delineation of how much change in a factor is critical near impossible. Consider, for example, narrative scenario A with a 5% increase in energy demand, and scenario B with a 30% increase in energy demand, which would have dire consequences. At which point between 5% and 30% do the dire consequences actually begin to occur? Such questions cannot be answered without a wide exploration of the space of potential changes. It should be noted that for some levels of model complexity and computational demands (e.g., global-scale models) there is little feasible recourse beyond the use of narrative scenarios.</p>
+<p>Exploratory modeling is typically paired with scenario discovery methods (indicated by number 9 in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>) that identify which of the scenarios (also known as states of the world) generated indeed have consequences of interest for stakeholders and policy makers, in an approach referred to as ensemble-based scenario-discovery <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id16" title="Benjamin P Bryant and Robert J Lempert. Thinking inside the box: a participatory, computer-assisted approach to scenario discovery. Technological Forecasting and Social Change, 77(1):34–49, 2010.">45</a>, <a class="reference internal" href="R.Bibliography.html#id71" title="Robert J Lempert, David G Groves, Steven W Popper, and Steve C Bankes. A general, analytic method for generating robust strategies and narrative scenarios. Management science, 52(4):514–528, 2006.">102</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>. This approach therefore flips the planning analysis from one that attempts to predict future system conditions to one that attempts to discover the (un)desirable future conditions. Ensemble-based scenario discovery can thus inform what modeling choices yield the most consequential behavioral changes or outcomes, especially when considering deeply uncertain, scenario-informed projections <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id12" title="Enayat A Moallemi, Jan Kwakkel, Fjalar J de Haan, and Brett A Bryan. Exploratory modeling for analyzing coupled human-natural systems under uncertainty. Global Environmental Change, 65:102186, 2020.">9</a>, <a class="reference internal" href="R.Bibliography.html#id72" title="Warren E Walker, Marjolijn Haasnoot, and Jan H Kwakkel. Adapt or perish: a review of planning approaches for adaptation under deep uncertainty. Sustainability, 5(3):955–979, 2013.">145</a>]</span>. The relative likelihoods and relevance of the discovered scenarios can be subsequently evaluated by the practitioners a posteriori, within a richer context of knowing the wider set of potential consequences <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id73" title="Suraje Dessai, Mike Hulme, Robert Lempert, and Roger Pielke Jr. Climate prediction: a limit to adaptation. Adapting to climate change: thresholds, values, governance, 64:78, 2009.">146</a>]</span>. This can include changing how an analysis is framed (number 10 in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>). For instance, one could initially focus on ensemble modeling of vulnerability using a single uncertain factor that is assumed to be well characterized by historical observations (e.g., streamflow; this step is represented by numbers 2-3 in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>). The analysis can then shift to include projections of more factors treated as deeply uncertain (e.g., urbanization, population demands, temperature, and snow-melt) to yield a far wider space of challenging projected futures. UC experiments contrasting these two framings can be highly valuable for tracing how vulnerability inferences change as the modeled space of futures expands from the historical baseline <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id61" title="Julianne D Quinn, Patrick M Reed, and Klaus Keller. Direct policy search for robust multi-objective management of deeply uncertain socio-ecological tipping points. Environmental modelling &amp; software, 92:125–141, 2017.">134</a>]</span>.</p>
+<p>An important nuance to be clarified here is that the focus or purpose of a modeling exercise plays a major role in whether a given factor of interest is considered well-characterized or deeply uncertain. Take the example context of characterizing temperature or streamflow extremes, where for each state variable of interest for a given location of focus there is a century of historical observations. Clearly, the observation technologies will have evolved over time uniquely for temperature and streamflow measurements and they likely lack replicate experiments (data uncertainty). A century of record will be insufficient to capture very high impact and rare extreme events (i.e., increasingly poor structural/parametric inference for the distributions of specific extreme single or compound events). The mechanistic processes as well as their evolving variability will be interdependent but uniquely different for each of these state variables. A large body of statistical literature exists focusing on the topics of synthetic weather <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id74" title="Manuel Herrera, Sukumar Natarajan, David A Coley, Tristan Kershaw, Alfonso P Ramallo-González, Matthew Eames, Daniel Fosas, and Michael Wood. A review of current and future weather data for building simulation. Building Services Engineering Research and Technology, 38(5):602–627, 2017.">81</a>, <a class="reference internal" href="R.Bibliography.html#id75" title="Daniel S Wilks and Robert L Wilby. The weather generation game: a review of stochastic weather models. Progress in physical geography, 23(3):329–357, 1999.">82</a>]</span> or streamflow <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id76" title="JR Lamontagne and JR Stedinger. Generating synthetic streamflow forecasts with specified precision. Journal of Water Resources Planning and Management, 144(4):04018007, 2018.">83</a>, <a class="reference internal" href="R.Bibliography.html#id77" title="Sanghamitra Medda and Kalyan Kumar Bhar. Comparison of single-site and multi-site stochastic models for streamflow generation. Applied Water Science, 9(3):67, 2019.">84</a>]</span> generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound extremes. These history-focused approaches can be viewed as providing well-characterized quantifications of streamflow or temperature distributions; however, they do not capture how coupled natural-human processes can fundamentally change their dynamics when transitioning to projections of longer-term futures (e.g., streamflow and temperature in 2055). Consequently, changing the focus of the modeling to making long term projections of future streamflow or temperature now makes these processes deeply uncertain.</p>
+<p>Scenario discovery methods (number 9 in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>) can be qualitative or quantitative and they generally attempt to distinguish futures in which a system or proposed policies to manage the system meet or miss their goals <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>. The emphasis placed by exploratory modeling on model outputs that have decision relevant consequences represents a shift toward a broader class of metrics that are reflective of the stakeholders’ concerns, agency and preferences (also discussed in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>). As a result, sensitivity analysis and scenario discovery methods in this context are therefore applied to performance metrics that go beyond model error but are rather focused on broader metrics such as the resilience of a sector, the reliability of a process, or the vulnerability of a population in the face of uncertainty. In exploratory modeling literature, this metric is most typically—but not always—a measure of robustness (number 8 in Fig. 13). Robustness is a property of a system or a design choice capturing its insensitivity to uncertainty and can be measured via a variety of means, most recently reviewed by <span id="id12">Herman <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id78" title="Jonathan D Herman, Patrick M Reed, Harrison B Zeff, and Gregory W Characklis. How should robustness be defined for water systems planning under change? Journal of Water Resources Planning and Management, 141(10):04015012, 2015.">147</a>]</span> and <span id="id13">McPhail <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id55" title="Cameron McPhail, HR Maier, JH Kwakkel, M Giuliani, A Castelletti, and S Westra. Robustness metrics: how are they calculated, when should they be used and why do they give different results? Earth's Future, 6(2):169–191, 2018.">129</a>]</span>.</p>
+<p>Scenario discovery is typically performed through the use of algorithms applied on large databases of model runs, generated through exploratory modeling, with each model run representing the performance of the system in one potential state of the world. The algorithms seek to identify the combinations of factor values (e.g., future conditions) that best distinguish the cases in which the system does or does not meet its objectives. The most widely known classification algorithms are the Patient Rule Induction Method (PRIM; <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id79" title="Jerome H Friedman and Nicholas I Fisher. Bump hunting in high-dimensional data. Statistics and Computing, 9(2):123–143, 1999.">97</a>]</span>) and Classification and Regression Trees (CART; <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id80" title="Leo Breiman, Jerome Friedman, Charles J Stone, and Richard A Olshen. Classification and regression trees. CRC press, 1984.">98</a>]</span>). These factor mapping algorithms create orthogonal boundaries (multi-dimensional hypercubes) between states of the world that are successful or unsuccessful in meeting the system’s goals <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>]</span>. The algorithms attempt to strike a balance between simplicity of classification (and as a result, interpretability) and accuracy <span id="id17">[<a class="reference internal" href="R.Bibliography.html#id16" title="Benjamin P Bryant and Robert J Lempert. Thinking inside the box: a participatory, computer-assisted approach to scenario discovery. Technological Forecasting and Social Change, 77(1):34–49, 2010.">45</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>, <a class="reference internal" href="R.Bibliography.html#id82" title="Robert J Lempert. Robust decision making (rdm). In Decision making under deep uncertainty, pages 23–51. Springer, Cham, 2019.">148</a>]</span>.</p>
+<p>Even though these approaches have been shown to yield interpretable and relevant scenarios <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id83" title="Jan H Kwakkel and Marjolijn Haasnoot. Supporting dmdu: a taxonomy of approaches and tools. In Decision Making under Deep Uncertainty, pages 355–374. Springer, Cham, 2019.">149</a>]</span>, several authors have pointed out the limitations of these methods with regards to their division of space in orthogonal behavioral and non-behavioral regions <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id84" title="BC Trindade, PM Reed, and GW Characklis. Deeply uncertain pathways: integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134:103442, 2019.">150</a>]</span>. Due to their reliance on boundaries orthogonal to the uncertainty axes, PRIM and CART cannot capture interactions between the various uncertain factors considered, which can often be significant <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id85" title="Julianne D Quinn, Patrick M Reed, Matteo Giuliani, Andrea Castelletti, Jared W Oyler, and Robert E Nicholas. Exploring how changing monsoonal dynamics and human pressures challenge multireservoir management for flood protection, hydropower production, and agricultural water supply. Water Resources Research, 54(7):4638–4662, 2018.">151</a>]</span>. More advanced methods have been proposed to address this drawback, with logistic regression being perhaps the most prominent <span id="id21">[<a class="reference internal" href="R.Bibliography.html#id85" title="Julianne D Quinn, Patrick M Reed, Matteo Giuliani, Andrea Castelletti, Jared W Oyler, and Robert E Nicholas. Exploring how changing monsoonal dynamics and human pressures challenge multireservoir management for flood protection, hydropower production, and agricultural water supply. Water Resources Research, 54(7):4638–4662, 2018.">151</a>, <a class="reference internal" href="R.Bibliography.html#id86" title="DF Gold, PM Reed, BC Trindade, and GW Characklis. Identifying actionable compromises: navigating multi-city robustness conflicts to discover cooperative safe operating spaces for regional water supply portfolios. Water Resources Research, 55(11):9024–9050, 2019.">152</a>, <a class="reference internal" href="R.Bibliography.html#id87" title="JR Lamontagne, PM Reed, G Marangoni, K Keller, and GG Garner. Robust abatement pathways to tolerable climate futures require immediate global action. Nature Climate Change, 9(4):290–294, 2019.">153</a>]</span>. Logistic regression can produce boundaries that are not necessarily orthogonal to each uncertainty axis, nor necessarily linear, if interactive terms between two parameters are used to build the regression model. It also describes the probability that a state of the world belongs to the scenarios that lead to failure. This feature allows users to define regions of success based on a gradient of estimated probability of success in those worlds, unlike PRIM which only classifies states of the world in two regions <span id="id22">[<a class="reference internal" href="R.Bibliography.html#id85" title="Julianne D Quinn, Patrick M Reed, Matteo Giuliani, Andrea Castelletti, Jared W Oyler, and Robert E Nicholas. Exploring how changing monsoonal dynamics and human pressures challenge multireservoir management for flood protection, hydropower production, and agricultural water supply. Water Resources Research, 54(7):4638–4662, 2018.">151</a>, <a class="reference internal" href="R.Bibliography.html#id202" title="Antonia Hadjimichael, Julianne Quinn, Erin Wilson, Patrick Reed, Leon Basdekas, David Yates, and Michelle Garrison. Defining Robustness, Vulnerabilities, and Consequential Scenarios for Diverse Stakeholder Interests in Institutionally Complex River Basins. Earth's Future, 8(7):e2020EF001503, 2020. URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020EF001503 (visited on 2020-07-13), doi:10.1029/2020EF001503.">154</a>]</span>.</p>
+<p>Another more advanced factor mapping method is boosted trees <span id="id23">[<a class="reference internal" href="R.Bibliography.html#id89" title="Yoav Freund, Robert Schapire, and Naoki Abe. A short introduction to boosting. Journal-Japanese Society For Artificial Intelligence, 14(771-780):1612, 1999.">99</a>, <a class="reference internal" href="R.Bibliography.html#id88" title="Harris Drucker and Corinna Cortes. Boosting decision trees. Advances in neural information processing systems, pages 479–485, 1996.">155</a>]</span>. Boosted trees can avoid two limitations inherent to the application of logistic regression: i) to build a nonlinear classification model the interactive term between two uncertainties needs to be pre-specified and cannot be discovered (e.g., we need to know a-priori whether factor <span class="math notranslate nohighlight">\(x_1\)</span> interacts with <span class="math notranslate nohighlight">\(x_2\)</span> in a relationship that looks like <span class="math notranslate nohighlight">\(x_1\)</span>·<span class="math notranslate nohighlight">\(x_2\)</span> or <span class="math notranslate nohighlight">\(x_1^{x_2}\)</span>); and ii) the subspaces defined are always convex. The application of such a factor mapping algorithm is limited in the presence of threshold-based rules with discrete actions in a modeled system (e.g., “if network capacity is low, build new infrastructure”), which results in failure regions that are nonlinear and non-convex <span id="id24">[<a class="reference internal" href="R.Bibliography.html#id84" title="BC Trindade, PM Reed, and GW Characklis. Deeply uncertain pathways: integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134:103442, 2019.">150</a>]</span>. Boosting works by creating an ensemble of classifiers and forcing some of them to focus on the hard-to-learn parts of the problem, and others to focus on the easy-to-learn parts. Boosting applied to CART trees can avoid the aforementioned challenges faced by other scenario discovery methods, while resisting overfitting <span id="id25">[<a class="reference internal" href="R.Bibliography.html#id90" title="Kevin P Murphy. Machine learning: a probabilistic perspective. MIT press, 2012.">156</a>]</span>, assuring the identified success and failure regions are still easy to interpret.</p>
+<p>Below we provide an example application of two scenario discovery methods, PRIM and logistic regression, using the lake problem introduced in the previous section. From the sensitivity analysis results presented in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> (d), we can already infer that parameters <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(q\)</span> have important effects on model outputs (i.e., we have performed factor prioritization). Scenario discovery (i.e., factor mapping) complements this analysis by further identifying the specific values of b and q that can lead to consequential and undesirable outcomes. For the purposes of demonstration, we can assume the undesirable outcome in this case is defined as the management policy failing to achieve 90% reliability in a state of the world.</p>
+<figure class="margin-caption align-center" id="id27">
+<span id="figure-4-5"></span><a class="reference internal image-reference" href="_images/figure14lake_problem_SD.png"><img alt="Figure 14." src="_images/figure14lake_problem_SD.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Scenario discovery for the lake problem, using (a) PRIM and (b) logistic regression.</span><a class="headerlink" href="#id27" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-5"><span class="std std-numref">Fig. 4.5</span></a> shows the results of scenario discovery, performed through (a) PRIM and (b) logistic regression. Each point in the two panels indicates a potential state of the world, generated through Latin Hypercube Sampling. Each point is colored by whether the policy meets the above performance criterion, with blue indicating success and red indicating failure. PRIM identifies several orthogonal areas of interest, one of which is shown in panel (a). As discussed above, this necessary orthogonality limits how PRIM identifies areas of success (the area within the box). As factors  <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(q\)</span> interact in this system, the transition boundary between the regions of success and failure is not orthogonal to any of the axes. As a result, a large number of points in the bottom right and the top left of the figure are left outside of the identified region. Logistic regression can overcome this limitation by identifying a diagonal boundary between the two regions, seen in panel (b). This method also produces a gradient of estimated probability of success across these regions.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial on performing factor mapping using logistic regression:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/basin_users_logistic_regression.ipynb">Logistic Regression Jupyter Notebook</a></p>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#understanding-errors-what-is-controlling-model-performance">4.1. Understanding Errors: What Is Controlling Model Performance?</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#consequential-dynamics-what-is-controlling-model-behaviors-of-interest">4.2. Consequential Dynamics: What is Controlling Model Behaviors of Interest?</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#consequential-scenarios-what-is-controlling-consequential-outcomes">4.3. Consequential Scenarios: What is Controlling Consequential Outcomes?</a></li>
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+  <section id="sensitivity-analysis-diagnostic-exploratory-modeling">
+<span id="sensitivity-analysis"></span><h1><span class="section-number">4. </span>Sensitivity Analysis: Diagnostic &amp; Exploratory Modeling<a class="headerlink" href="#sensitivity-analysis-diagnostic-exploratory-modeling" title="Link to this heading">#</a></h1>
+<section id="understanding-errors-what-is-controlling-model-performance">
+<span id="understanding-errors"></span><h2><span class="section-number">4.1. </span>Understanding Errors: What Is Controlling Model Performance?<a class="headerlink" href="#understanding-errors-what-is-controlling-model-performance" title="Link to this heading">#</a></h2>
+<p>Sensitivity analysis is a diagnostic tool when reconciling model outputs with observed data. It is helpful for clarifying how and under what conditions modeling choices (structure, parameterization, data inputs, etc.) propagate through model components and manifest in their effects on model outputs. This exploration is performed through carefully designed sampling of multiple combinations of input parameters and subsequent evaluation of the model structures that are emerging as controlling factors. Model structure and parameterization are two of the most commonly explored aspects of models that have been a central focus when evaluating their performance relative to available observations <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id49" title="Keith Beven. Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in water resources, 16(1):41–51, 1993.">17</a>]</span>. Addressing these issues plays an important role in establishing credibility in model predictions, particularly in the positivist natural sciences literature. Traditional model evaluations compare the model with observed data, and then rely on expert judgements of its acceptability based on the closeness between simulation and observation with one or a small number of selected metrics. This approach can be myopic, as it is often impossible to use one metric to attribute a certain error and to link that with different parts of the model and its parameters <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id50" title="Hoshin V Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008.">127</a>]</span>. This means that, even when the error or fitting measure between the model estimates and observations is very small, it is not guaranteed that all the components in the model accurately represent the conceptual reality that the model is abstracting: propagated errors in different parts of the model might cancel each other out, or multiple parameterized implementations of the model can yield similar performance (i.e., equifinality <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id49" title="Keith Beven. Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in water resources, 16(1):41–51, 1993.">17</a>]</span>).</p>
+<p>The inherent complexity of a system hinders accepting or rejecting a model based on one performance measure and different types of measures can aid in evaluating the various components of a model as essentially a multiobjective problem <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id52" title="Keith J Beven. On hypothesis testing in hydrology: why falsification of models is still a really good idea. Wiley Interdisciplinary Reviews: Water, 5(3):e1278, 2018.">25</a>, <a class="reference internal" href="R.Bibliography.html#id51" title="Hoshin V Gupta, Martyn P Clark, Jasper A Vrugt, Gab Abramowitz, and Ming Ye. Towards a comprehensive assessment of model structural adequacy. Water Resources Research, 2012.">26</a>]</span>. In addition, natural systems mimicked by the models contain various interacting components that might act differently across spatial and temporal domains <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id53" title="JD Herman, PM Reed, and T Wagener. Time-varying sensitivity analysis clarifies the effects of watershed model formulation on model behavior. Water Resources Research, 49(3):1400–1414, 2013.">55</a>]</span>. This heterogeneity is lost when a single performance measure is used, as a highly dimensional and interactive system becomes aggregated through the averaging of spatial or temporal output errors <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id54" title="Keith Beven. Towards a coherent philosophy for modelling the environment. Proceedings of the royal society of London. Series A: mathematical, physical and engineering sciences, 458(2026):2465–2484, 2002.">15</a>]</span>. Therefore, diagnostic error analyses should consider multiple error signatures across different scales and states of concern when seeking to understand how model performance relates to observed data (<a class="reference internal" href="#figure-4-1"><span class="std std-numref">Fig. 4.1</span></a>). Diverse error signatures can be used to measure the consistency of underlying processes and behaviors of the model and to evaluate the dynamics of model controls under changing temporal and spatial conditions <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id56" title="Keith Beven and Jim Freer. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the glue methodology. Journal of hydrology, 249(1-4):11–29, 2001.">128</a>]</span>. Within this framework, even minimal information extracted from the data can be beneficial as it helps us unearth structural inadequacies in the model. In this context, proper selection of measures of model performance and the number of measures could play consequential roles in our understanding of the model and its predictions <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id55" title="Cameron McPhail, HR Maier, JH Kwakkel, M Giuliani, A Castelletti, and S Westra. Robustness metrics: how are they calculated, when should they be used and why do they give different results? Earth's Future, 6(2):169–191, 2018.">129</a>]</span>.</p>
+<p>As discussed earlier, instead of the traditional focus on using deterministic prediction that results in a single error measure, many plausible states and spaces could be searched for making different inferences and quantifying uncertainties. This process also requires estimates of prior probability distributions of all the important parameters and quantification of model behavior across input space. One strategy to reduce the search space is filtering of some model alternatives that are not consistent with observations and known system behaviors. Those implausible parts of the search space can be referred to as non-physical or non-behavioral alternatives <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id29" title="Neil R Edwards, David Cameron, and Jonathan Rougier. Precalibrating an intermediate complexity climate model. Climate dynamics, 37(7):1469–1482, 2011.">50</a>, <a class="reference internal" href="R.Bibliography.html#id158" title="Keith Beven and Andrew Binley. Glue: 20 years on. Hydrological processes, 28(24):5897–5918, 2014.">104</a>]</span>. This step is conducted before the Bayesian calibration exercise (see <a class="reference internal" href="A1_Uncertainty_Quantification.html#a1-uncertainty-quantification"><span class="std std-numref">Chapter A</span></a>).</p>
+<p>A comprehensive model diagnostic workflow typically entails the components demonstrated in <a class="reference internal" href="#figure-4-1"><span class="std std-numref">Fig. 4.1</span></a>. The workflow begins with the selection of model input parameters and their plausible ranges. After the parameter selection, we need to specify the design of experiment (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#design-of-experiments"><span class="std std-numref">Chapter 3.3</span></a>) and the sensitivity analysis method (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#sensitivity-analysis-methods"><span class="std std-numref">Chapter 3.4</span></a>) to be used. As previously discussed, these methods require different numbers of model simulations, and each method provides a different insights into the direct effects and interactions of the uncertain factors. In addition, the simulation time of the model and the available computational resources are two of the primary considerations that influence these decisions. After identifying the appropriate methods, we generate a matrix of input parameters, where each set of input parameters will be used to conduct a model simulation. The model can include one or more output variables that fluctuate in time and space. The next step is to analyze model performance by comparing model outputs with observations. As discussed earlier, the positivist model evaluation paradigm focuses on a single model performance metric (error), leading to a loss of information about model parameters and the suitability of the model’s structure. However, a thorough investigation of the temporal and spatial signatures of model outputs using various performance metrics or time- and space-varying sensitivity analyses can shed more light on the fitness of each parameter set and the model’s internal structure. This analysis provides diagnostic feedback on the importance and range of model parameters and can guide further improvement of the model algorithm.</p>
+<figure class="margin-caption align-center" id="id50">
+<span id="figure-4-1"></span><a class="reference internal image-reference" href="_images/figure4_1_diagnostic_workflow.png"><img alt="Figure 4.1" src="_images/figure4_1_diagnostic_workflow.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 4.1 </span><span class="caption-text">Diagnostic evaluation of model fidelity using sensitivity analysis methods.</span><a class="headerlink" href="#id50" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following badge to try out an interactive tutorial on implementing a time-varying sensitivity analysis of HYMOD model parameters:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">HYMOD Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="consequential-dynamics-what-is-controlling-model-behaviors-of-interest">
+<h2><span class="section-number">4.2. </span>Consequential Dynamics: What is Controlling Model Behaviors of Interest?<a class="headerlink" href="#consequential-dynamics-what-is-controlling-model-behaviors-of-interest" title="Link to this heading">#</a></h2>
+<p>Consequential changes in dynamic systems can take many forms, but most dynamic behavior can be categorized in a few basic patterns. Feedback structures inherent in a system, be they positive or negative, generate these patterns which can be grouped into three groups for the simplest of systems: exponential growth, goal seeking, and oscillation (<a class="reference internal" href="#figure-4-2"><span class="std std-numref">Fig. 4.2</span></a>). A positive (or self-reinforcing) feedback gives rise to exponential growth, a negative (or self-correcting) feedback gives rise to a goal seeking mode, and negative feedbacks with time delays give rise to oscillatory behavior. Nonlinear interactions between the system’s feedback structures can give rise to more complex dynamic behavior modes, examples of which are also shown in <a class="reference internal" href="#figure-4-2"><span class="std std-numref">Fig. 4.2</span></a>, adapted from <span id="id10">Sterman [<a class="reference internal" href="R.Bibliography.html#id57" title="John D Sterman. System dynamics modeling: tools for learning in a complex world. California management review, 43(4):8–25, 2001.">130</a>]</span>.</p>
+<figure class="margin-caption align-center" id="id51">
+<span id="figure-4-2"></span><a class="reference internal image-reference" href="_images/figure4_2_behavior_modes.png"><img alt="Figure 4.2" src="_images/figure4_2_behavior_modes.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 4.2 </span><span class="caption-text">Common modes of behavior in dynamic systems, occurring based on the presence of positive and negative feedback relationships, and linear and non-linear interactions. Adapted from <span id="id11">Sterman [<a class="reference internal" href="R.Bibliography.html#id57" title="John D Sterman. System dynamics modeling: tools for learning in a complex world. California management review, 43(4):8–25, 2001.">130</a>]</span>.</span><a class="headerlink" href="#id51" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The nature of feedback processes in a dynamic system shapes its fundamental behavior: positive feedbacks generate their own growth, negative feedbacks self-limit, seeking balance and equilibrium. In this manner, feedback processes give rise to different regimes, multiples of which could be present in each mode of behavior. Consider a population of mammals growing exponentially until it reaches the carrying capacity of its environment (referred to as S-shaped growth). When the population is exponentially growing, the regime is dominated by positive feedback relationships that reinforce its growth. As the population approaches its carrying capacity limit, negative feedback structures begin to dominate, counteracting the growth and establishing a stable equilibrium. Shifts between regimes can be thought of as tipping points, mathematically defined as unstable equilibria, where the presence of positive feedbacks amplifies disturbances and moves the system to a new equilibrium point. In the case of stable equilibria, the presence of negative feedbacks dampens any small disturbance and maintains the system at a stable state. As different feedback relationships govern each regime, different factors (those making up each feedback mechanism) are activated and shape the states the system is found in, as well as define the points of equilibria.</p>
+<p>For simple stylized models with a small number of states, system dynamics analysis can analytically derive these equilibria, the conditions for their stability and the factors determining them. The ability for this to be performed is, however, significantly challenged when it comes to systems that attempt to more closely resemble real complex systems. We argue this is the case for several reasons. First, besides generally exhibiting complex nonlinear dynamics, real world systems are also made up from larger numbers of interacting elements, which often makes the analytic derivation of system characteristics intractable <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id59" title="John M Anderies, Jean-Denis Mathias, and Marco A Janssen. Knowledge infrastructure and safe operating spaces in social–ecological systems. Proceedings of the National Academy of Sciences, 116(12):5277–5284, 2019.">131</a>, <a class="reference internal" href="R.Bibliography.html#id58" title="Rachata Muneepeerakul and John M Anderies. The emergence and resilience of self-organized governance in coupled infrastructure systems. Proceedings of the National Academy of Sciences, 117(9):4617–4622, 2020.">132</a>]</span>. Second, human-natural systems temporally evolve and transform when human state-aware action is present. Consider, for instance, humans recreationally hunting the aforementioned population of mammals. Humans act based on the mammal population levels by enforcing hunting quotas or establishing protected territories or eliminating other predators. The mammal population reacts in a response, giving birth to ever changing state-action-consequence feedbacks, the path dependencies of which become difficult to diagnose and understand (e.g., <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id60" title="Antonia Hadjimichael, Patrick M Reed, and Julianne D Quinn. Navigating deeply uncertain tradeoffs in harvested predator-prey systems. Complexity, 2020.">133</a>]</span>). Trying to simulate the combination of these two challenges (large numbers of state-aware agents interacting with a natural resource and with each other) produces intractable models that require advanced heuristics to analyze their properties and establish useful inferences.</p>
+<p>Sensitivity analysis paired with exploratory modeling methods offers a promising set of tools to address these challenges. We present a simple demonstrative application based on <span id="id14">Quinn <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id61" title="Julianne D Quinn, Patrick M Reed, and Klaus Keller. Direct policy search for robust multi-objective management of deeply uncertain socio-ecological tipping points. Environmental modelling &amp; software, 92:125–141, 2017.">134</a>]</span>. This stylistic example was first developed by <span id="id15">Carpenter <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id163" title="S. R. Carpenter, D. Ludwig, and W. A. Brock. Management of Eutrophication for Lakes Subject to Potentially Irreversible Change. Ecological Applications, 9(3):751–771, August 1999. URL: http://onlinelibrary.wiley.com/doi/10.1890/1051-0761(1999)009[0751:MOEFLS]2.0.CO;2/abstract, doi:10.1890/1051-0761(1999)009[0751:MOEFLS]2.0.CO;2.">135</a>]</span> and represents a town that must balance its agricultural and industrial productivity with the pollution it creates in a downstream lake. Increased productivity allows for increased profits, which the town aims to maximize, but it also produces more pollution for the lake. Too much phosphorus pollution can cause irreversible eutrophication, a process known as “tipping” the lake. The model of phosphorus in the lake <span class="math notranslate nohighlight">\(X_t\)</span> at time <span class="math notranslate nohighlight">\(t\)</span> is governed by:</p>
+<div class="math notranslate nohighlight">
+\[X_{t+1}= X_{t}+a_{t}+\frac{X_{t}^q} {1+X_{t}^q}-bX_t+\varepsilon\]</div>
+<p>where <span class="math notranslate nohighlight">\(a_t \in [0,0.1]\)</span> is the town’s pollution release at each timestep, <span class="math notranslate nohighlight">\(b\)</span> is the natural decay rate of phosphorus in the lake, <span class="math notranslate nohighlight">\(q\)</span> defines the lake’s recycling rate (primarily through sediments), and <span class="math notranslate nohighlight">\(\varepsilon\)</span> represents uncontrollable natural inflows of pollution modeled as a log-normal distribution with a given mean, <span class="math notranslate nohighlight">\(\mu\)</span>, and standard deviation <span class="math notranslate nohighlight">\(\sigma\)</span>.</p>
+<p>Panels (a-c) in <a class="reference internal" href="#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> plot the fluxes of phosphorus into the lake versus the mass accumulation of phosphorus in the lake. The red line corresponds to the phosphorus sinks in the lake (natural decay), given by <span class="math notranslate nohighlight">\(bX_t\)</span>. The grey shaded area represents the lake’s phosphorus recycling flux, given by <span class="math notranslate nohighlight">\(\frac{X_{t}^q} {1+X_{t}^q}\)</span>. The points of intersection indicate the system’s equilibria, two of which are stable, and one is unstable (also known as the tipping point). The stable equilibrium in the bottom left of the figure reflects an oligotrophic lake, whereas the stable equilibrium in the top right represents a eutrophic lake. With increasing phosphorus values, the tipping point can be crossed, and the lake will experience irreversible eutrophication, as the recycling rate would exceed the removal rate even if the town’s pollution became zero. In the absence of anthropogenic and natural inflows of pollution in the lake (<span class="math notranslate nohighlight">\(a_t\)</span> and <span class="math notranslate nohighlight">\(\varepsilon\)</span> respectively), the area between the bottom-left black point and the white point in the middle can be considered as the safe operating space, before emission levels cross the tipping point.</p>
+<figure class="margin-caption align-center" id="id52">
+<span id="figure-4-3"></span><a class="reference internal image-reference" href="_images/figure4_3_lake_problem_fluxes.png"><img alt="Figure 4.3" src="_images/figure4_3_lake_problem_fluxes.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 4.3 </span><span class="caption-text">Fluxes of phosphorus with regards to mass of phosphorus in the lake and sensitivity analysis results, assuming <span class="math notranslate nohighlight">\(b=0.42\)</span> and <span class="math notranslate nohighlight">\(q=2\)</span>. (a) Fluxes of phosphorus assuming no emmisions policy and no natural inflows. (b-c) Fluxes phosphorus when applying two different emissions policies. The “Best economic policy” and the “Most reliable policy” have been identified by <span id="id16">Quinn <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id61" title="Julianne D Quinn, Patrick M Reed, and Klaus Keller. Direct policy search for robust multi-objective management of deeply uncertain socio-ecological tipping points. Environmental modelling &amp; software, 92:125–141, 2017.">134</a>]</span> and can be found at <span id="id17">Quinn [<a class="reference internal" href="R.Bibliography.html#id164" title="Julianne Quinn. Julianneq/Lake_problem_dps. December 2017. original-date: 2017-02-06T18:33:54Z. URL: https://github.com/julianneq/Lake_Problem_DPS (visited on 2021-06-14).">136</a>]</span>. (d) Results of a sensitivity analysis on the parameters of the model most consequential to the reliability of the “Most reliable policy”. The code to replicate the sensitivity analysis can be found at <span id="id18">Hadka [<a class="reference internal" href="R.Bibliography.html#id165" title="David Hadka. Project-Platypus/Rhodium. 2017. original-date: 2015-10-29T18:08:43Z. URL: https://github.com/Project-Platypus/Rhodium (visited on 2021-06-14).">137</a>]</span>. Panels (a-c) are used courtesy of Julianne Quinn, University of Virginia.</span><a class="headerlink" href="#id52" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The town has identified two potential policies that can be used to manage this lake, one that maximizes its economic profits (“best economic policy”) and one that maximizes the time below the tipping point (“most reliable policy”). Panels (b-c) in <a class="reference internal" href="#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> add the emissions from these policies to the recycling flux and show how the equilibria points shift as a result. In both cases the stable oligotrophic equilibrium increases and the tipping point decreases, narrowing the safe operating space <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id59" title="John M Anderies, Jean-Denis Mathias, and Marco A Janssen. Knowledge infrastructure and safe operating spaces in social–ecological systems. Proceedings of the National Academy of Sciences, 116(12):5277–5284, 2019.">131</a>, <a class="reference internal" href="R.Bibliography.html#id166" title="Stephen R. Carpenter, William A. Brock, Carl Folke, Egbert H. van Nes, and Marten Scheffer. Allowing variance may enlarge the safe operating space for exploited ecosystems. Proceedings of the National Academy of Sciences, 112(46):14384–14389, November 2015. URL: http://www.pnas.org/content/112/46/14384 (visited on 2017-08-18), doi:10.1073/pnas.1511804112.">138</a>]</span>. The best economic policy results in a much narrower space of action, with the tipping point very close to the oligotrophic equilibrium. The performance of both policies depends significantly on the system parameters. For example, a higher value of <span class="math notranslate nohighlight">\(b\)</span>, the natural decay rate, would shift the red line upward, moving the equilibria points and widening the safe operating space. Inversely, a higher value of <span class="math notranslate nohighlight">\(q\)</span>, the lake’s recycling rate, would shift the recycling line upward, moving the tipping point lower and decreasing the safe operating space. The assumptions under which these policies were identified are therefore critical to their performance and any potential uncertainty in the parameter values could be detrimental to the system’s objectives being met.</p>
+<p>Sensitivity analysis can be used to clarify the role these parameters play on policy performance. <a class="reference internal" href="#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> (d) shows the results of a Sobol sensitivity analysis on the reliability of the “most reliable” policy in a radial convergence diagram. The significance of each parameter is indicated by the size of circles corresponding to it. The size of the interior dark circle indicates the parameter’s first-order effects and the size of the exterior circle indicates the parameter’s total-order effects. The thickness of the lines between two parameters indicated the extent of their interaction (second-order effects). In this case, parameters <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(q\)</span> appear to have the most significant importance on the system, followed by the mean, <span class="math notranslate nohighlight">\(\mu\)</span>, of the natural inflows. All these parameters function in a manner that shifts the location of the three equilibria and therefore policies that are identified ignoring this parametric uncertainty might fail to meet their intended goals.</p>
+<p>It is worth mentioning that current sensitivity analysis methods are somewhat challenged in addressing several system dynamics analysis questions. The fundamental reason is that sensitivity analysis methods and tools have been developed to gauge numerical sensitivity of model output to changes in factor values. This is natural, as most simulation studies (e.g., all aforementioned examples) have been traditionally concerned with this type of sensitivity. In system dynamics modeling, however, a more important and pertinent concern is changes between regimes or between behavior modes (also known as bifurcations) as a result of changes in model factors <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id57" title="John D Sterman. System dynamics modeling: tools for learning in a complex world. California management review, 43(4):8–25, 2001.">130</a>, <a class="reference internal" href="R.Bibliography.html#id63" title="Mustafa Hekimoğlu and Yaman Barlas. Sensitivity analysis for models with multiple behavior modes: a method based on behavior pattern measures. System Dynamics Review, 32(3-4):332–362, 2016.">139</a>]</span>. This poses two new challenges. First, identifying a change in regime depends on several characteristics besides a change in output value, like the rate and direction of change. Second, behavior mode changes are qualitative and discontinuous, as equilibria change in stability but also move in and out of existence.</p>
+<p>Despite these challenges, recent advanced sensitivity analysis methods can help illuminate which factors in a system are most important in shaping boundary conditions (tipping points) between different regimes and determining changes in behavior modes. Reviewing such methods is outside the scope of this text, but the reader is directed to the examples of <span id="id21">Eker <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id64" title="Sibel Eker, Elena Rovenskaya, Michael Obersteiner, and Simon Langan. Practice and perspectives in the validation of resource management models. Nature communications, 9(1):1–10, 2018.">22</a>]</span> and <span id="id22">Hadjimichael <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id60" title="Antonia Hadjimichael, Patrick M Reed, and Julianne D Quinn. Navigating deeply uncertain tradeoffs in harvested predator-prey systems. Complexity, 2020.">133</a>]</span>, who apply parameterised perturbation on the functional relationships of a system to study the effects of model structural uncertainty on model outputs and bifurcations, and <span id="id23">Hekimoğlu and Barlas [<a class="reference internal" href="R.Bibliography.html#id63" title="Mustafa Hekimoğlu and Yaman Barlas. Sensitivity analysis for models with multiple behavior modes: a method based on behavior pattern measures. System Dynamics Review, 32(3-4):332–362, 2016.">139</a>]</span> and <span id="id24">Steinmann <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id65" title="Patrick Steinmann, Willem L Auping, and Jan H Kwakkel. Behavior-based scenario discovery using time series clustering. Technological Forecasting and Social Change, 156:120052, 2020.">140</a>]</span> who, following wide sampling of uncertain inputs, cluster the resulting time series in modes of behavior and identify most important factors for each.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following badge to try out an interactive tutorial on performing a sensitivity analysis to discover consequential dynamics:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/fishery_dynamics.ipynb">Factor Discovery Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="consequential-scenarios-what-is-controlling-consequential-outcomes">
+<span id="consequential-scenarios"></span><h2><span class="section-number">4.3. </span>Consequential Scenarios: What is Controlling Consequential Outcomes?<a class="headerlink" href="#consequential-scenarios-what-is-controlling-consequential-outcomes" title="Link to this heading">#</a></h2>
+<p>As overviewed in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>, most models are abstractions of systems in the real world. When sufficient confidence has been established in a model, it can then act as a surrogate for the actual system, in that the consequences of potential stressors, proposed actions or other changes can be evaluated by computer model simulations <span id="id25">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>]</span>. A model simulation then represents a computational experiment, which can be used to assess how the modeled system would behave should the various changes come to be. Steven Bankes coined the term exploratory modeling to describe the use of large sets of such computational experiments to investigate their implications on the system. <a class="reference internal" href="#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a> presents a typical workflow of an exploratory modeling application. Exploratory modeling approaches typically use sampling designs to generate large ensembles of states that represent combinations of changes happening together, spanning the entire range of potential values a factor might take (indicated in <a class="reference internal" href="#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a> by numbers 2-5). This perspective on modeling is particularly relevant to studies making long term projections into the future.</p>
+<figure class="margin-caption align-center" id="id53">
+<span id="figure-4-4"></span><a class="reference internal image-reference" href="_images/figure4_4_exploratory_workflow.png"><img alt="Figure 4_4" src="_images/figure4_4_exploratory_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 4.4 </span><span class="caption-text">A typical exploratory modeling workflow</span><a class="headerlink" href="#id53" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>In the long-term policy analysis literature, exploratory modeling has prominently placed itself as an alternative to traditional narrative scenario or assumptions-based planning approaches, in what can be summarized in the following two-pronged critique <span id="id26">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>, <a class="reference internal" href="R.Bibliography.html#id67" title="Steven C Bankes, Robert J Lempert, and Steven W Popper. Computer-assisted reasoning. Computing in Science &amp; Engineering, 3(2):71–77, 2001.">141</a>, <a class="reference internal" href="R.Bibliography.html#id66" title="Robert J. Lempert, Steven W. Popper, and Steven C. Bankes. Shaping the Next One Hundred Years. RAND Corporation, 2003. URL: https://www.rand.org/pubs/monograph_reports/MR1626.html (visited on 2017-09-14).">142</a>]</span>. The most prevalent criticism sees that the future and how it might evolve is both highly complex and deeply uncertain. Despite its benefits for interpretation and intuitive appeal, a small number of scenarios invariably misses many other potential futures that did not get selected as sufficiently representative of the future. This is especially the case for aggregate, narrative scenarios that describe simultaneous changes in multiple sectors together (e.g., “increased energy demand, combined with high agricultural land use and large economic growth”), such as the emission scenarios produced by the Intergovernmental Panel on Climate Change <span id="id27">[<a class="reference internal" href="R.Bibliography.html#id68" title="Jonathan R Lamontagne, Patrick M Reed, Robert Link, Katherine V Calvin, Leon E Clarke, and James A Edmonds. Large ensemble analytic framework for consequence-driven discovery of climate change scenarios. Earth's Future, 6(3):488–504, 2018.">143</a>]</span>. The bias introduced by this reduced set of potential changes can skew inferences drawn from the model, particularly when the original narrative scenarios are focused on a single or narrow set of measures of system behavior.</p>
+<p>The second main criticism of traditional narrative scenario-based planning methods is that they provide no systematic way to distinguish which of the constituent factors lead to the undesirable consequences produced by a scenario. Narrative scenarios (e.g., the scenario matrix framework of RCPs-SSPs-SPAs; <span id="id28">[<a class="reference internal" href="R.Bibliography.html#id69" title="Brian C O’Neill, Elmar Kriegler, Keywan Riahi, Kristie L Ebi, Stephane Hallegatte, Timothy R Carter, Ritu Mathur, and Detlef P van Vuuren. A new scenario framework for climate change research: the concept of shared socioeconomic pathways. Climatic change, 122(3):387–400, 2014.">144</a>]</span>) encompass multiple changes happening together selected to span the range of potential changes but are not typically generated in a systematic factorial manner that considers the multiple ways the factors can be combined. This has two critical limitations. It obfuscates the role each component factor plays in the system, both in isolation and in combination with others (e.g., “is it the increased energy demand or the high agricultural land use that cause unbearable water stress?”). It also renders the delineation of how much change in a factor is critical near impossible. Consider, for example, narrative scenario A with a 5% increase in energy demand, and scenario B with a 30% increase in energy demand, which would have dire consequences. At which point between 5% and 30% do the dire consequences actually begin to occur? Such questions cannot be answered without a wide exploration of the space of potential changes. It should be noted that for some levels of model complexity and computational demands (e.g., global-scale models) there is little feasible recourse beyond the use of narrative scenarios.</p>
+<p>Exploratory modeling is typically paired with scenario discovery methods (indicated by number 9 in <a class="reference internal" href="#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>) that identify which of the scenarios (also known as states of the world) generated indeed have consequences of interest for stakeholders and policy makers, in an approach referred to as ensemble-based scenario-discovery <span id="id29">[<a class="reference internal" href="R.Bibliography.html#id16" title="Benjamin P Bryant and Robert J Lempert. Thinking inside the box: a participatory, computer-assisted approach to scenario discovery. Technological Forecasting and Social Change, 77(1):34–49, 2010.">45</a>, <a class="reference internal" href="R.Bibliography.html#id71" title="Robert J Lempert, David G Groves, Steven W Popper, and Steve C Bankes. A general, analytic method for generating robust strategies and narrative scenarios. Management science, 52(4):514–528, 2006.">102</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>. This approach therefore flips the planning analysis from one that attempts to predict future system conditions to one that attempts to discover the (un)desirable future conditions. Ensemble-based scenario discovery can thus inform what modeling choices yield the most consequential behavioral changes or outcomes, especially when considering deeply uncertain, scenario-informed projections <span id="id30">[<a class="reference internal" href="R.Bibliography.html#id12" title="Enayat A Moallemi, Jan Kwakkel, Fjalar J de Haan, and Brett A Bryan. Exploratory modeling for analyzing coupled human-natural systems under uncertainty. Global Environmental Change, 65:102186, 2020.">9</a>, <a class="reference internal" href="R.Bibliography.html#id72" title="Warren E Walker, Marjolijn Haasnoot, and Jan H Kwakkel. Adapt or perish: a review of planning approaches for adaptation under deep uncertainty. Sustainability, 5(3):955–979, 2013.">145</a>]</span>. The relative likelihoods and relevance of the discovered scenarios can be subsequently evaluated by the practitioners a posteriori, within a richer context of knowing the wider set of potential consequences <span id="id31">[<a class="reference internal" href="R.Bibliography.html#id73" title="Suraje Dessai, Mike Hulme, Robert Lempert, and Roger Pielke Jr. Climate prediction: a limit to adaptation. Adapting to climate change: thresholds, values, governance, 64:78, 2009.">146</a>]</span>. This can include changing how an analysis is framed (number 10 in <a class="reference internal" href="#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>). For instance, one could initially focus on ensemble modeling of vulnerability using a single uncertain factor that is assumed to be well characterized by historical observations (e.g., streamflow; this step is represented by numbers 2-3 in <a class="reference internal" href="#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>). The analysis can then shift to include projections of more factors treated as deeply uncertain (e.g., urbanization, population demands, temperature, and snow-melt) to yield a far wider space of challenging projected futures. UC experiments contrasting these two framings can be highly valuable for tracing how vulnerability inferences change as the modeled space of futures expands from the historical baseline <span id="id32">[<a class="reference internal" href="R.Bibliography.html#id61" title="Julianne D Quinn, Patrick M Reed, and Klaus Keller. Direct policy search for robust multi-objective management of deeply uncertain socio-ecological tipping points. Environmental modelling &amp; software, 92:125–141, 2017.">134</a>]</span>.</p>
+<p>An important nuance to be clarified here is that the focus or purpose of a modeling exercise plays a major role in whether a given factor of interest is considered well-characterized or deeply uncertain. Take the example context of characterizing temperature or streamflow extremes, where for each state variable of interest for a given location of focus there is a century of historical observations. Clearly, the observation technologies will have evolved over time uniquely for temperature and streamflow measurements and they likely lack replicate experiments (data uncertainty). A century of record will be insufficient to capture very high impact and rare extreme events (i.e., increasingly poor structural/parametric inference for the distributions of specific extreme single or compound events). The mechanistic processes as well as their evolving variability will be interdependent but uniquely different for each of these state variables. A large body of statistical literature exists focusing on the topics of synthetic weather <span id="id33">[<a class="reference internal" href="R.Bibliography.html#id74" title="Manuel Herrera, Sukumar Natarajan, David A Coley, Tristan Kershaw, Alfonso P Ramallo-González, Matthew Eames, Daniel Fosas, and Michael Wood. A review of current and future weather data for building simulation. Building Services Engineering Research and Technology, 38(5):602–627, 2017.">81</a>, <a class="reference internal" href="R.Bibliography.html#id75" title="Daniel S Wilks and Robert L Wilby. The weather generation game: a review of stochastic weather models. Progress in physical geography, 23(3):329–357, 1999.">82</a>]</span> or streamflow <span id="id34">[<a class="reference internal" href="R.Bibliography.html#id76" title="JR Lamontagne and JR Stedinger. Generating synthetic streamflow forecasts with specified precision. Journal of Water Resources Planning and Management, 144(4):04018007, 2018.">83</a>, <a class="reference internal" href="R.Bibliography.html#id77" title="Sanghamitra Medda and Kalyan Kumar Bhar. Comparison of single-site and multi-site stochastic models for streamflow generation. Applied Water Science, 9(3):67, 2019.">84</a>]</span> generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound extremes. These history-focused approaches can be viewed as providing well-characterized quantifications of streamflow or temperature distributions; however, they do not capture how coupled natural-human processes can fundamentally change their dynamics when transitioning to projections of longer-term futures (e.g., streamflow and temperature in 2055). Consequently, changing the focus of the modeling to making long term projections of future streamflow or temperature now makes these processes deeply uncertain.</p>
+<p>Scenario discovery methods (number 9 in <a class="reference internal" href="#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>) can be qualitative or quantitative and they generally attempt to distinguish futures in which a system or proposed policies to manage the system meet or miss their goals <span id="id35">[<a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>. The emphasis placed by exploratory modeling on model outputs that have decision relevant consequences represents a shift toward a broader class of metrics that are reflective of the stakeholders’ concerns, agency and preferences (also discussed in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>). As a result, sensitivity analysis and scenario discovery methods in this context are therefore applied to performance metrics that go beyond model error but are rather focused on broader metrics such as the resilience of a sector, the reliability of a process, or the vulnerability of a population in the face of uncertainty. In exploratory modeling literature, this metric is most typically—but not always—a measure of robustness (number 8 in Fig. 13). Robustness is a property of a system or a design choice capturing its insensitivity to uncertainty and can be measured via a variety of means, most recently reviewed by <span id="id36">Herman <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id78" title="Jonathan D Herman, Patrick M Reed, Harrison B Zeff, and Gregory W Characklis. How should robustness be defined for water systems planning under change? Journal of Water Resources Planning and Management, 141(10):04015012, 2015.">147</a>]</span> and <span id="id37">McPhail <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id55" title="Cameron McPhail, HR Maier, JH Kwakkel, M Giuliani, A Castelletti, and S Westra. Robustness metrics: how are they calculated, when should they be used and why do they give different results? Earth's Future, 6(2):169–191, 2018.">129</a>]</span>.</p>
+<p>Scenario discovery is typically performed through the use of algorithms applied on large databases of model runs, generated through exploratory modeling, with each model run representing the performance of the system in one potential state of the world. The algorithms seek to identify the combinations of factor values (e.g., future conditions) that best distinguish the cases in which the system does or does not meet its objectives. The most widely known classification algorithms are the Patient Rule Induction Method (PRIM; <span id="id38">[<a class="reference internal" href="R.Bibliography.html#id79" title="Jerome H Friedman and Nicholas I Fisher. Bump hunting in high-dimensional data. Statistics and Computing, 9(2):123–143, 1999.">97</a>]</span>) and Classification and Regression Trees (CART; <span id="id39">[<a class="reference internal" href="R.Bibliography.html#id80" title="Leo Breiman, Jerome Friedman, Charles J Stone, and Richard A Olshen. Classification and regression trees. CRC press, 1984.">98</a>]</span>). These factor mapping algorithms create orthogonal boundaries (multi-dimensional hypercubes) between states of the world that are successful or unsuccessful in meeting the system’s goals <span id="id40">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>]</span>. The algorithms attempt to strike a balance between simplicity of classification (and as a result, interpretability) and accuracy <span id="id41">[<a class="reference internal" href="R.Bibliography.html#id16" title="Benjamin P Bryant and Robert J Lempert. Thinking inside the box: a participatory, computer-assisted approach to scenario discovery. Technological Forecasting and Social Change, 77(1):34–49, 2010.">45</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>, <a class="reference internal" href="R.Bibliography.html#id82" title="Robert J Lempert. Robust decision making (rdm). In Decision making under deep uncertainty, pages 23–51. Springer, Cham, 2019.">148</a>]</span>.</p>
+<p>Even though these approaches have been shown to yield interpretable and relevant scenarios <span id="id42">[<a class="reference internal" href="R.Bibliography.html#id83" title="Jan H Kwakkel and Marjolijn Haasnoot. Supporting dmdu: a taxonomy of approaches and tools. In Decision Making under Deep Uncertainty, pages 355–374. Springer, Cham, 2019.">149</a>]</span>, several authors have pointed out the limitations of these methods with regards to their division of space in orthogonal behavioral and non-behavioral regions <span id="id43">[<a class="reference internal" href="R.Bibliography.html#id84" title="BC Trindade, PM Reed, and GW Characklis. Deeply uncertain pathways: integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134:103442, 2019.">150</a>]</span>. Due to their reliance on boundaries orthogonal to the uncertainty axes, PRIM and CART cannot capture interactions between the various uncertain factors considered, which can often be significant <span id="id44">[<a class="reference internal" href="R.Bibliography.html#id85" title="Julianne D Quinn, Patrick M Reed, Matteo Giuliani, Andrea Castelletti, Jared W Oyler, and Robert E Nicholas. Exploring how changing monsoonal dynamics and human pressures challenge multireservoir management for flood protection, hydropower production, and agricultural water supply. Water Resources Research, 54(7):4638–4662, 2018.">151</a>]</span>. More advanced methods have been proposed to address this drawback, with logistic regression being perhaps the most prominent <span id="id45">[<a class="reference internal" href="R.Bibliography.html#id85" title="Julianne D Quinn, Patrick M Reed, Matteo Giuliani, Andrea Castelletti, Jared W Oyler, and Robert E Nicholas. Exploring how changing monsoonal dynamics and human pressures challenge multireservoir management for flood protection, hydropower production, and agricultural water supply. Water Resources Research, 54(7):4638–4662, 2018.">151</a>, <a class="reference internal" href="R.Bibliography.html#id86" title="DF Gold, PM Reed, BC Trindade, and GW Characklis. Identifying actionable compromises: navigating multi-city robustness conflicts to discover cooperative safe operating spaces for regional water supply portfolios. Water Resources Research, 55(11):9024–9050, 2019.">152</a>, <a class="reference internal" href="R.Bibliography.html#id87" title="JR Lamontagne, PM Reed, G Marangoni, K Keller, and GG Garner. Robust abatement pathways to tolerable climate futures require immediate global action. Nature Climate Change, 9(4):290–294, 2019.">153</a>]</span>. Logistic regression can produce boundaries that are not necessarily orthogonal to each uncertainty axis, nor necessarily linear, if interactive terms between two parameters are used to build the regression model. It also describes the probability that a state of the world belongs to the scenarios that lead to failure. This feature allows users to define regions of success based on a gradient of estimated probability of success in those worlds, unlike PRIM which only classifies states of the world in two regions <span id="id46">[<a class="reference internal" href="R.Bibliography.html#id85" title="Julianne D Quinn, Patrick M Reed, Matteo Giuliani, Andrea Castelletti, Jared W Oyler, and Robert E Nicholas. Exploring how changing monsoonal dynamics and human pressures challenge multireservoir management for flood protection, hydropower production, and agricultural water supply. Water Resources Research, 54(7):4638–4662, 2018.">151</a>, <a class="reference internal" href="R.Bibliography.html#id202" title="Antonia Hadjimichael, Julianne Quinn, Erin Wilson, Patrick Reed, Leon Basdekas, David Yates, and Michelle Garrison. Defining Robustness, Vulnerabilities, and Consequential Scenarios for Diverse Stakeholder Interests in Institutionally Complex River Basins. Earth's Future, 8(7):e2020EF001503, 2020. URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020EF001503 (visited on 2020-07-13), doi:10.1029/2020EF001503.">154</a>]</span>.</p>
+<p>Another more advanced factor mapping method is boosted trees <span id="id47">[<a class="reference internal" href="R.Bibliography.html#id89" title="Yoav Freund, Robert Schapire, and Naoki Abe. A short introduction to boosting. Journal-Japanese Society For Artificial Intelligence, 14(771-780):1612, 1999.">99</a>, <a class="reference internal" href="R.Bibliography.html#id88" title="Harris Drucker and Corinna Cortes. Boosting decision trees. Advances in neural information processing systems, pages 479–485, 1996.">155</a>]</span>. Boosted trees can avoid two limitations inherent to the application of logistic regression: i) to build a nonlinear classification model the interactive term between two uncertainties needs to be pre-specified and cannot be discovered (e.g., we need to know a-priori whether factor <span class="math notranslate nohighlight">\(x_1\)</span> interacts with <span class="math notranslate nohighlight">\(x_2\)</span> in a relationship that looks like <span class="math notranslate nohighlight">\(x_1\)</span>·<span class="math notranslate nohighlight">\(x_2\)</span> or <span class="math notranslate nohighlight">\(x_1^{x_2}\)</span>); and ii) the subspaces defined are always convex. The application of such a factor mapping algorithm is limited in the presence of threshold-based rules with discrete actions in a modeled system (e.g., “if network capacity is low, build new infrastructure”), which results in failure regions that are nonlinear and non-convex <span id="id48">[<a class="reference internal" href="R.Bibliography.html#id84" title="BC Trindade, PM Reed, and GW Characklis. Deeply uncertain pathways: integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134:103442, 2019.">150</a>]</span>. Boosting works by creating an ensemble of classifiers and forcing some of them to focus on the hard-to-learn parts of the problem, and others to focus on the easy-to-learn parts. Boosting applied to CART trees can avoid the aforementioned challenges faced by other scenario discovery methods, while resisting overfitting <span id="id49">[<a class="reference internal" href="R.Bibliography.html#id90" title="Kevin P Murphy. Machine learning: a probabilistic perspective. MIT press, 2012.">156</a>]</span>, assuring the identified success and failure regions are still easy to interpret.</p>
+<p>Below we provide an example application of two scenario discovery methods, PRIM and logistic regression, using the lake problem introduced in the previous section. From the sensitivity analysis results presented in <a class="reference internal" href="#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> (d), we can already infer that parameters <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(q\)</span> have important effects on model outputs (i.e., we have performed factor prioritization). Scenario discovery (i.e., factor mapping) complements this analysis by further identifying the specific values of b and q that can lead to consequential and undesirable outcomes. For the purposes of demonstration, we can assume the undesirable outcome in this case is defined as the management policy failing to achieve 90% reliability in a state of the world.</p>
+<figure class="margin-caption align-center" id="id54">
+<span id="figure-4-5"></span><a class="reference internal image-reference" href="_images/figure14lake_problem_SD.png"><img alt="Figure 14." src="_images/figure14lake_problem_SD.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 4.5 </span><span class="caption-text">Scenario discovery for the lake problem, using (a) PRIM and (b) logistic regression.</span><a class="headerlink" href="#id54" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p><a class="reference internal" href="#figure-4-5"><span class="std std-numref">Fig. 4.5</span></a> shows the results of scenario discovery, performed through (a) PRIM and (b) logistic regression. Each point in the two panels indicates a potential state of the world, generated through Latin Hypercube Sampling. Each point is colored by whether the policy meets the above performance criterion, with blue indicating success and red indicating failure. PRIM identifies several orthogonal areas of interest, one of which is shown in panel (a). As discussed above, this necessary orthogonality limits how PRIM identifies areas of success (the area within the box). As factors  <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(q\)</span> interact in this system, the transition boundary between the regions of success and failure is not orthogonal to any of the axes. As a result, a large number of points in the bottom right and the top left of the figure are left outside of the identified region. Logistic regression can overcome this limitation by identifying a diagonal boundary between the two regions, seen in panel (b). This method also produces a gradient of estimated probability of success across these regions.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial on performing factor mapping using logistic regression:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/basin_users_logistic_regression.ipynb">Logistic Regression Jupyter Notebook</a></p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The following articles are suggested as fundamental reading for the information presented in this section:</p>
+<ul class="simple">
+<li><p>Gupta, H.V., Wagener, T., Liu, Y., 2008. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. <em>Hydrological Processes: An International Journal</em> 22, 3802–3813.</p></li>
+<li><p>Bankes, S., 1993. Exploratory Modeling for Policy Analysis. <em>Operations Research</em> 41, 435–449. <a class="reference external" href="https://doi.org/10.1287/opre.41.3.435">https://doi.org/10.1287/opre.41.3.435</a></p></li>
+<li><p>Groves, D.G., Lempert, R.J., 2007. A new analytic method for finding policy-relevant scenarios. <em>Global Environmental Change</em> 17, 73–85. <a class="reference external" href="https://doi.org/10.1016/j.gloenvcha.2006.11.006">https://doi.org/10.1016/j.gloenvcha.2006.11.006</a></p></li>
+</ul>
+<p>The following articles can be used as supplemental reading:</p>
+<ul class="simple">
+<li><p>Marchau, V.A.W.J., Walker, W.E., Bloemen, P.J.T.M., Popper, S.W. (Eds.), 2019. Decision Making under Deep Uncertainty: From Theory to Practice. Springer International Publishing. <a class="reference external" href="https://doi.org/10.1007/978-3-030-05252-2">https://doi.org/10.1007/978-3-030-05252-2</a></p></li>
+<li><p>Sterman, J.D., 1994. Learning in and about complex systems. System Dynamics Review 10, 291–330. <a class="reference external" href="https://doi.org/10.1002/sdr.4260100214">https://doi.org/10.1002/sdr.4260100214</a></p></li>
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+<span id="id1"></span><h1><span class="section-number">5. </span>Conclusion<a class="headerlink" href="#conclusion" title="Link to this heading">#</a></h1>
+<p>As noted in the Introduction (<a class="reference internal" href="1_introduction.html#introduction"><span class="std std-numref">Chapter 1</span></a>), the computational and conceptual challenges of the multi-model, transdisciplinary workflows that characterize ambitious projects such as IM3 have limited UC and UQ analyses. Moreover, the very nature and purpose of modeling and diagnostic model evaluation can have very diverse philosophical framings depending on the disciplines involved (see <code class="xref std std-numref docutils literal notranslate"><span class="pre">Figure_1_1</span></code> and <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>). The guidance provided in this text can be used to frame consistent and rigorous experimental designs for better understanding the consequences and insights from our modeling choices when seeking to capture complex human-natural systems. The progression of sections of this text provide a thorough introduction of the concepts and definitions of diagnostic model evaluation, sensitivity analysis and UC. In addition, we comprehensively discuss how specific modeling objectives and applications should guide the selection of appropriate techniques; broadly, these can include model diagnostics, in-depth analysis of the behavior of the abstracted system, and projections under conditions of deep uncertainty. This text also contains a detailed presentation of the main sensitivity analysis methods and a discussion of their features and main limitations. Readers are also provided with an overview of computer tools and platforms that have been developed and could be considered in addressing IM3 scientific questions. The appendices of this text include an overview of UQ methods, a terminology glossary of the key concepts as well as example test cases and scripts to showcase various UC related capabilities.</p>
+<p>Although we distinguish the UC and UQ model diagnostics, the reader should note that we suggest an overall consistent approach to both in this text by emphasizing “exploratory modeling” (see review by <span id="id2">Moallemi <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id12" title="Enayat A Moallemi, Jan Kwakkel, Fjalar J de Haan, and Brett A Bryan. Exploratory modeling for analyzing coupled human-natural systems under uncertainty. Global Environmental Change, 65:102186, 2020.">9</a>]</span>). Although data support, model complexity, and computational limits strongly distinguish the feasibility and appropriateness of various UC diagnostic tools (e.g., see <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-5"><span class="std std-numref">Fig. 3.5</span></a>), we overall recommend that modelers view their work through the lens of cycles of learning. Iterative and deliberative exploration of model-based hypotheses and inferences for transdisciplinary teams is non-trivial and ultimately critical for mapping where innovations or insights are most consequential. Overall, we recommend approaching modeling with an openness to the diverse disciplinary perspectives such as those mirrored by the IM3 family of models in a progression from evaluating models relative to observed history to advanced formalized analyses to make inferences on multi-sector, multi-scale vulnerabilities and resilience. Exploratory modeling approaches can help fashion experiments with large numbers of alternative hypotheses on the co-evolutionary dynamics of influences, stressors, as well as path-dependent changes in the form and function of coupled human-natural systems <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id130" title="Christopher P. Weaver, Robert J. Lempert, Casey Brown, John A. Hall, David Revell, and Daniel Sarewitz. Improving the contribution of climate model information to decision making: the value and demands of robust decision frameworks. Wiley Interdisciplinary Reviews: Climate Change, 4(1):39–60, 2013. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/wcc.202 (visited on 2019-10-01), doi:10.1002/wcc.202.">37</a>]</span>. This text guides the reader through the use of sensitivity analysis and uncertainty methods across the diverse perspectives that have shaped modern diagnostic and exploratory modeling.</p>
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+<span id="id1"></span><h1>Glossary<a class="headerlink" href="#glossary" title="Link to this heading">#</a></h1>
+<p><strong>Design of experiment</strong>: Provides a framework for the extraction of all plausible information about the impact of each factor on the output of the numerical model</p>
+<p><strong>Exploratory modeling</strong>: Use of large ensembles of uncertain conditions to discover decision-relevant combinations of uncertain factors</p>
+<p><strong>Factor</strong>: Any model component that can affect model outputs: inputs, resolution levels, coupling relationships, model relationships and parameters. In models with acceptable model fidelity these factors may represent elements of the real-world system under study.</p>
+<p><strong>Factor mapping</strong>: A technique to identify which uncertain model factors lead to certain model behavior</p>
+<p><strong>Factor prioritization</strong>: A technique to identify the uncertain factors which, when fixed to their true value, would lead to the greatest reduction in output variability</p>
+<p><strong>Factor screening</strong>: A technique to identify model components that have a negligible effect or make no significant contributions to the variability of the outputs or metrics of interest</p>
+<p><strong>First-, second-, total-order effects</strong>: First-order effects indicate the percent of model output variance contributed by a factor individually. Second-order effects capture how interactions between a pair of parameter input variables can lead to change in model output. Total-order effects consider all the effects a factor has, individually and in interaction with other factors.</p>
+<p><strong>Hindcasting</strong>: A type of predictive check that uses the model to estimate output for past events to see how well the output matches the known results.</p>
+<p><strong>Pre-calibration</strong>: A hybrid uncertainty assessment method that involves identifying a plausible set of parameters using some prespecified screening criterion, such as the distance from the model results to the observations.</p>
+<p><strong>Prior</strong>: The best assessment of the probability of an event based on existing knowledge before a new experiment is conducted</p>
+<p><strong>Posterior</strong>: The revised or updated probability of an event after taking into account new information</p>
+<p><strong>Probabilistic inversion</strong>: Uses additional information, for instance, a probabilistic expert assessment or survey result, to update an existing prior distribution</p>
+<p><strong>Return level</strong>: A value that is expected to be equaled or exceeded on average once every interval of time (T) (with a probability of 1/T)</p>
+<p><strong>Return period</strong>: The estimated time interval between events of a similar size or intensity/</p>
+<p><strong>Sampling</strong>: The process of selecting model parameters or inputs that characterize the model uncertainty space.</p>
+<p><strong>Scenario discovery</strong>: Use of large ensembles of uncertain conditions to discover decision-relevant combinations of uncertain factors</p>
+<p><strong>Sensitivity analysis</strong>: Conducted to understand the factors and processes that most (or least) control a model’s outputs</p>
+<blockquote>
+<div><p><em>Local sensitivity analysis</em>: Model evaluation performed by varying uncertain factors around specific reference values</p>
+<p><em>Global sensitivity analysis</em>: Model evaluation performed by varying uncertain factors throughout their entire feasible value space</p>
+</div></blockquote>
+<p><strong>Uncertainty</strong></p>
+<p><em>Deep uncertainty</em>: Refers to situations where expert opinions consulted on a decision do not know or cannot agree on system boundaries, or the outcomes of interest and their relative importance, or the prior probability Distribution for the various uncertain factors present</p>
+<p><em>Epistemic uncertainty</em>: Systematic uncertainty that comes about due to the lack of knowledge or data to choose the best model</p>
+<p><em>Ontological uncertainty</em>: Uncertainties due to processes, interactions, or futures, that are not contained within current conceptual models</p>
+<p><em>Aleatory uncertainty</em>: Uncertainty due to natural randomness in processes</p>
+<p><em>Uncertainty characterization</em>: Model evaluation under alternative factor hypotheses to explore their implications for model output uncertainty</p>
+<p><em>Uncertainty quantification</em>: Representation of model output uncertainty using probability distributions</p>
+<p><strong>Variance decomposition</strong>: A technique to partition how much of the variability in a model’s output is due to different explanatory variables.</p>
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+<h1>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h1>
+<p>As defined in <a class="reference internal" href="1_introduction.html#introduction"><span class="std std-numref">Chapter 1</span></a>, uncertainty quantification (UQ) refers to the formal focus on the full specification of likelihoods as well as distributional forms necessary to infer the joint  probabilistic response across all modeled factors of interest <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id11" title="Roger Cooke and others. Experts in uncertainty: opinion and subjective probability in science. Oxford University Press on Demand, 1991.">8</a>]</span>. This is in contrast to UC (the primary focus of the main document of this book), which is instead aimed at identifying which modeling choices yield the most consequential changes or outcomes and exploring alternative hypotheses related to the form and function of modeled systems <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id12" title="Enayat A Moallemi, Jan Kwakkel, Fjalar J de Haan, and Brett A Bryan. Exploratory modeling for analyzing coupled human-natural systems under uncertainty. Global Environmental Change, 65:102186, 2020.">9</a>, <a class="reference internal" href="R.Bibliography.html#id13" title="Warren E Walker, Poul Harremoës, Jan Rotmans, Jeroen P Van Der Sluijs, Marjolein BA Van Asselt, Peter Janssen, and Martin P Krayer von Krauss. Defining uncertainty: a conceptual basis for uncertainty management in model-based decision support. Integrated assessment, 4(1):5–17, 2003.">10</a>]</span>.</p>
+<p>UQ is important for quantifying the relative merits of hypotheses for at least three main reasons. First, identifying model parameters that are consistent with observations is an important part of model development. Due to several effects, including correlations between parameters, simplified or incomplete model structures (relative to the full real-world dynamics), and uncertainty in the observations, many different combinations of parameter values can be consistent with the model structure and the observations to varying extents. Accounting for this uncertainty is conceptually preferable to selecting a single “best fit” parameter vector, particularly as consistency with historical or present observations does not necessarily  guarantee skillful future projections.</p>
+<p>The act of quantification requires specific assumptions about distributional forms and likelihoods, which may be more or less justified depending on prior information about the system or model behavior. As a result, UQ is well-suited for studies accounting for or addressing hypotheses related to systems with a relatively large amount of available data and models which are computationally inexpensive, particularly when the emphasis is on prediction. As shown in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-1"><span class="std std-numref">Fig. A.1</span></a>, there is a fundamental tradeoff between the available number of model evaluations (for a fixed computational budget) and the number of parameters treated as uncertain. Sensitivity analyses are therefore part of a typical UQ workflow to identify which factors can be fixed and which ought to be prioritized in the UQ.</p>
+<figure class="margin-caption align-center" id="id4">
+<span id="figure-a1-1"></span><a class="reference internal image-reference" href="_images/figureA1_1_UQ_approaches.png"><img alt="Figure A1.1" src="_images/figureA1_1_UQ_approaches.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Overview of selected existing approaches for uncertainty quantification and their appropriateness given the number of uncertain model parameters and the number of available model simulations. Green shading denotes regions suitable for uncertainty quantification and red shading indicates regions more appropriate for uncertainty characterization.</span><a class="headerlink" href="#id4" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The choice of a particular UQ method depends on both the desired level of quantification and the ability to navigate the tradeoff between computational expense and the number of uncertain parameters (<a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-1"><span class="std std-numref">Fig. A.1</span></a>). For example, Markov chain Monte Carlo with a full system model can provide an improved representation of uncertainty compared to the coarser pre-calibration approach <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id37" title="Kelsey L. Ruckert, Gary Shaffer, David Pollard, Yawen Guan, Tony E. Wong, Chris E. Forest, and Klaus Keller. Assessing the Impact of Retreat Mechanisms in a Simple Antarctic Ice Sheet Model Using Bayesian Calibration. PLOS ONE, 12(1):e0170052, January 2017. doi:10.1371/journal.pone.0170052.">157</a>]</span>, but requires many more model evaluations. The use of a surrogate model to approximate the full system model can reduce the number of needed model evaluations by several orders of magnitude, but the uncertainty quantification can only accommodate a limited number of parameters.</p>
+<p>The remainder of this appendix will focus on introducing workflows for particular UQ methods, including a brief discussion of advantages and limitations.</p>
+</section>
+
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+  <section id="parametric-bootstrap">
+<h1>Parametric Bootstrap<a class="headerlink" href="#parametric-bootstrap" title="Link to this heading">#</a></h1>
+<p>The parametric bootstrap <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id14" title="B. Efron and R. Tibshirani. Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy. Statistical Science, 1(1):54–75, February 1986. doi:10.1214/ss/1177013815.">158</a>]</span> refers to a process of model recalibration to alternate realizations of the data. The bootstrap was originally developed to estimate standard errors and confidence intervals without ascertaining key assumptions that might not hold given the available data. In a setting where observations can be viewed as independent realizations of an underlying stochastic process, a sufficiently rich dataset can be treated as a population representing the data distribution. New datasets are then generated by resampling from the data with replacement, and the model can be refit to each new dataset using maximum-likelihood estimation. The resulting distribution of estimates can then be viewed as a representation of parametric uncertainty.</p>
+<p>A typical workflow for the parametric bootstrap is shown in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-2"><span class="std std-numref">Fig. A.2</span></a>. After identifying outputs of interest and preparing the data, the parametric model is fit by some procedure such as minimizing root-mean-square-error or maximizing the likelihood. Alternate datasets are constructed by resampling from the population or by generating new samples from the fitted data-generating process. It is important at this step that the resampled quantities are independent of one another. For example, in the context of temporally- or spatially-correlated data, such as time series, the raw observations cannot be treated as independent realizations. However, the residuals resulting from fitting the model to the data could be (depending on their structure). For example, if the residuals are treated as independent, they can then be resampled with replacement, and these residuals added to the original model fit to create new realizations. If the residuals are assumed to be the result of an autoregressive process, this process could be fit to the original residual series and new residuals be created using this model <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id35" title="Ryan L. Sriver, Robert J. Lempert, Per Wikman-Svahn, and Klaus Keller. Characterizing uncertain sea-level rise projections to support investment decisions. PLOS ONE, 13(2):e0190641, February 2018. URL: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0190641 (visited on 2021-06-09), doi:10.1371/journal.pone.0190641.">159</a>]</span>. The model is then refit to each new realization.</p>
+<figure class="margin-caption align-center" id="id4">
+<span id="figure-a1-2"></span><a class="reference internal image-reference" href="_images/figureA1_2_bootstrap_workflow.png"><img alt="Figure A1.2" src="_images/figureA1_2_bootstrap_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Workflow for the parametric bootstrap.</span><a class="headerlink" href="#id4" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The bootstrap is computationally convenient, particularly as the process of fitting the model to each realization can be easily parallelized. This approach also requires minimal prior assumptions. However, due to the assumption that the available data are representative of the underlying data distribution, the bootstrap can neglect key uncertainties which might influence the results. For example, when using an autoregressive process to generate new residuals, uncertainty in the autocorrelation parameter and innovation variance is neglected, which may bias estimates of, for example, low-probability but high-impact events <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id36" title="Kelsey L. Ruckert, Yawen Guan, Alexander M. R. Bakker, Chris E. Forest, and Klaus Keller. The effects of time-varying observation errors on semi-empirical sea-level projections. Climatic Change, 140(3):349–360, February 2017. URL: https://doi.org/10.1007/s10584-016-1858-z (visited on 2021-06-09), doi:10.1007/s10584-016-1858-z.">160</a>]</span>.</p>
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+  <section id="pre-calibration">
+<h1>Pre-Calibration<a class="headerlink" href="#pre-calibration" title="Link to this heading">#</a></h1>
+<p>Pre-calibration <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>, <a class="reference internal" href="R.Bibliography.html#id39" title="Neil R Edwards, David Cameron, and Jonathan Rougier. Precalibrating an intermediate complexity climate model. Clim. Dyn., 37(7-8):1469–1482, 2011. URL: http://dx.doi.org/10.1007/s00382-010-0921-0, doi:10.1007/s00382-010-0921-0.">161</a>, <a class="reference internal" href="R.Bibliography.html#id40" title="Alexis Boukouvalas, Pete Sykes, Dan Cornford, and Hugo Maruri-Aguilar. Bayesian Precalibration of a Large Stochastic Microsimulation Model. IEEE Transactions on Intelligent Transportation Systems, 15(3):1337–1347, June 2014. doi:10.1109/TITS.2014.2304394.">162</a>]</span> involves the identification of a plausible set of parameters using some prespecified screening criterion, such as the distance from the model results to the observations (based on an appropriate metric for the desired matching features, such as root-mean-squared error). A typical workflow is shown in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-3"><span class="std std-numref">Fig. A.3</span></a>. Parameter values are obtained by systematically sampling the input space (see <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#design-of-experiments"><span class="std std-numref">Chapter 3.3</span></a>). After the model is evaluated at the samples, only those passing the distance criterion are retained. This selects a subset of the parameter space as “plausible” based on the screening criterion, though there is no assignment of probabilities within this plausible region.</p>
+<figure class="margin-caption align-center" id="id7">
+<span id="figure-a1-3"></span><a class="reference internal image-reference" href="_images/figureA1_3_precal_workflow.png"><img alt="Figure A1.3" src="_images/figureA1_3_precal_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Workflow for pre-calibration.</span><a class="headerlink" href="#id7" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>Pre-calibration can be useful for models which are inexpensive enough that a reasonable
+number of samples can be used to represent the parameter space, but which are too expensive to facilitate full uncertainty quantification. High-dimensional parameter spaces, which can be problematic for the uncertainty quantification methods below, may also be explored using pre-calibration. One key prerequisite to using this method is the ability to place a meaningful distance metric on the output space.</p>
+<p>However, pre-calibration results in a very coarse characterization of uncertainty, especially when considering a large number of parameters, as more samples are needed to fully characterize the parameter space. Due to the inability to evaluate the relative probability of regions of the parameter space beyond the binary plausible-and-implausible characterization, pre-calibration can also result in degraded hindcast and projection skills and parameter estimates <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id37" title="Kelsey L. Ruckert, Gary Shaffer, David Pollard, Yawen Guan, Tony E. Wong, Chris E. Forest, and Klaus Keller. Assessing the Impact of Retreat Mechanisms in a Simple Antarctic Ice Sheet Model Using Bayesian Calibration. PLOS ONE, 12(1):e0170052, January 2017. doi:10.1371/journal.pone.0170052.">157</a>, <a class="reference internal" href="R.Bibliography.html#id41" title="David Makowski, Daniel Wallach, and Marie Tremblay. Using a Bayesian approach to parameter estimation; comparison of the GLUE and MCMC methods. Agronomie, 22(2):191–203, 2002. Publisher: EDP Sciences.">163</a>, <a class="reference internal" href="R.Bibliography.html#id42" title="Mahyar Shafii, Bryan Tolson, and Loren Shawn Matott. Uncertainty-based multi-criteria calibration of rainfall-runoff models: a comparative study. Stochastic Environmental Research and Risk Assessment, 28(6):1493–1510, August 2014. doi:10.1007/s00477-014-0855-x.">164</a>]</span>.</p>
+<p>A related method, widely used in hydrological studies, is generalized likelihood uncertainty estimation, or GLUE <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>]</span>. Unlike pre-calibration, the underlying argument for GLUE relies on the concept of equifinality <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id43" title="Keith Beven and Jim Freer. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology. Journal of hydrology, 249(1-4):11–29, 2001. Publisher: Elsevier.">165</a>]</span>, which posits that it is impossible to find a uniquely well-performing parameter vector for models of abstract environmental systems <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id43" title="Keith Beven and Jim Freer. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology. Journal of hydrology, 249(1-4):11–29, 2001. Publisher: Elsevier.">165</a>, <a class="reference internal" href="R.Bibliography.html#id44" title="Jasper A. Vrugt and Keith J. Beven. Embracing equifinality with efficiency: Limits of Acceptability sampling using the DREAM(LOA) algorithm. Journal of Hydrology, 559:954–971, April 2018. doi:10.1016/j.jhydrol.2018.02.026.">166</a>]</span>. In other words, there exist multiple parameter vectors which perform equally or similarly well. As with pre-calibration, GLUE uses a goodness-of-fit measure (though this is called a “likelihood” in the GLUE literature, as opposed to a statistical likelihood function <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id45" title="Jery R. Stedinger, Richard M. Vogel, Seung Uk Lee, and Rebecca Batchelder. Appraisal of the generalized likelihood uncertainty estimation (GLUE) method. Water Resources Research, 2008. doi:10.1029/2008WR006822.">167</a>]</span>) to evaluate samples. After setting a threshold of acceptable performance with respect to that measure, samples are evaluated and classified into “behavioral” or “non-behavioral” according to the threshold.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following badge to try out an interactive tutorial on utilizing Pre-Calibration and GLUE for HYMOD model calibration:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">Pre-Calibration Jupyter Notebook</a></p>
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+    <h1>Markov Chain Monte Carlo</h1>
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+  <section id="markov-chain-monte-carlo">
+<h1>Markov Chain Monte Carlo<a class="headerlink" href="#markov-chain-monte-carlo" title="Link to this heading">#</a></h1>
+<p>Markov chain Monte Carlo (MCMC) is a “gold standard” approach to full uncertainty quantification. MCMC refers to a category of algorithms which systematically sample from a target distribution (in this case, the posterior distribution) by constructing a Markov chain. A Markov chain is a probabilistic structure consisting of a state space, an initial probability distribution over the states, and a transition distribution between states. If a Markov chain satisfies certain properties <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id170" title="Christian Robert and George Casella. Monte Carlo Statistical Methods. Springer Science &amp; Business Media, March 2013. ISBN 978-1-4757-3071-5.">168</a>, <a class="reference internal" href="R.Bibliography.html#id169" title="Christian P. Robert. The Metropolis–Hastings Algorithm. In Wiley StatsRef: Statistics Reference Online, pages 1–15. American Cancer Society, 2015. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat07834 (visited on 2021-06-14), doi:10.1002/9781118445112.stat07834.">169</a>]</span>, the probability of being in each state will eventually converge to a stable, or stationary, distribution, regardless of the initial probabilities.</p>
+<p>MCMC algorithms construct a Markov chain of samples from a parameter space (the combination of model and statistical parameters). This Markov chain is constructed so that the stationary distribution is a target distribution, in this case the (Bayesian) posterior distribution. As a result, after the transient period, the resulting samples can be viewed as a set of dependent samples from the posterior (the dependence is due to the autocorrelation between samples resulting from the Markov chain transitions). Expected values can be computed from these samples (for example, using batch-means estimators <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id171" title="James M. Flegal, Murali Haran, and Galin L. Jones. Markov Chain Monte Carlo: Can We Trust the Third Significant Figure? Statistical Science, 23(2):250–260, May 2008. Publisher: Institute of Mathematical Statistics. URL: https://projecteuclid.org/journals/statistical-science/volume-23/issue-2/Markov-Chain-Monte-Carlo--Can-We-Trust-the-Third/10.1214/08-STS257.full (visited on 2021-06-14), doi:10.1214/08-STS257.">170</a>]</span>), or the chain can be sub-sampled or thinned and the resulting samples used as independent Monte Carlo samples due to the reduced or eliminated autocorrelation.</p>
+<p>A general workflow for MCMC is shown in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-4"><span class="std std-numref">Fig. A.4</span></a>. The first decision is whether to use the full model or a surrogate model (or emulator). Typical surrogates include Gaussian process emulation <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id172" title="Carla Currin, Toby Mitchell, Max Morris, and Don Ylvisaker. Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments. Journal of the American Statistical Association, 86(416):953–963, December 1991. Publisher: Taylor &amp; Francis _eprint: https://www.tandfonline.com/doi/pdf/10.1080/01621459.1991.10475138. URL: https://www.tandfonline.com/doi/abs/10.1080/01621459.1991.10475138 (visited on 2021-06-14), doi:10.1080/01621459.1991.10475138.">171</a>, <a class="reference internal" href="R.Bibliography.html#id173" title="Jerome Sacks, William J. Welch, Toby J. Mitchell, and Henry P. Wynn. Design and Analysis of Computer Experiments. Statistical Science, 4(4):409–423, 1989. Publisher: Institute of Mathematical Statistics. URL: https://www.jstor.org/stable/2245858 (visited on 2021-06-14).">172</a>]</span>, polynomial chaos expansions <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id174" title="Roger G. Ghanem and Pol D. Spanos. Spectral Stochastic Finite‐Element Formulation for Reliability Analysis. Journal of Engineering Mechanics, 117(10):2351–2372, October 1991. Publisher: American Society of Civil Engineers. URL: https://ascelibrary.org/doi/abs/10.1061/%28ASCE%290733-9399%281991%29117%3A10%282351%29 (visited on 2021-06-14), doi:10.1061/(ASCE)0733-9399(1991)117:10(2351).">173</a>, <a class="reference internal" href="R.Bibliography.html#id175" title="Dongbin Xiu and George Em Karniadakis. The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations. SIAM Journal on Scientific Computing, 24(2):619–644, January 2002. Publisher: Society for Industrial and Applied Mathematics. URL: https://epubs.siam.org/doi/abs/10.1137/S1064827501387826 (visited on 2021-06-14), doi:10.1137/S1064827501387826.">174</a>]</span>, support vector machines <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id176" title="Angelo Ciccazzo, Gianni Di Pillo, and Vittorio Latorre. A SVM Surrogate Model-Based Method for Parametric Yield Optimization. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 35(7):1224–1228, July 2016. Conference Name: IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. doi:10.1109/TCAD.2015.2501307.">175</a>, <a class="reference internal" href="R.Bibliography.html#id177" title="W. Andrew Pruett and Robert L. Hester. The Creation of Surrogate Models for Fast Estimation of Complex Model Outcomes. PLOS ONE, 11(6):e0156574, June 2016. Publisher: Public Library of Science. URL: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0156574 (visited on 2021-06-14), doi:10.1371/journal.pone.0156574.">176</a>]</span>, and neural networks <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id178" title="John Eason and Selen Cremaschi. Adaptive sequential sampling for surrogate model generation with artificial neural networks. Computers &amp; Chemical Engineering, 68:220–232, September 2014. URL: https://www.sciencedirect.com/science/article/pii/S0098135414001719 (visited on 2021-06-14), doi:10.1016/j.compchemeng.2014.05.021.">177</a>, <a class="reference internal" href="R.Bibliography.html#id179" title="Dirk Gorissen, Luciano De Tommasi, Karel Crombecq, and Tom Dhaene. Sequential modeling of a low noise amplifier with neural networks and active learning. Neural Computing and Applications, 18(5):485–494, June 2009. URL: https://doi.org/10.1007/s00521-008-0223-1 (visited on 2021-06-14), doi:10.1007/s00521-008-0223-1.">178</a>]</span>. Surrogate modeling can be faster, but requires a sufficient number of model evaluations for the surrogate to accurately represent the model’s response surface, and this typically limits the number of parameters which can be included in the analysis.</p>
+<figure class="margin-caption align-center" id="id16">
+<span id="figure-a1-4"></span><a class="reference internal image-reference" href="_images/figureA1_4_mcmc_workflow.png"><img alt="Figure A1.4" src="_images/figureA1_4_mcmc_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Workflow for Markov chain Monte Carlo.</span><a class="headerlink" href="#id16" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>After selecting the variables which will be treated as uncertain, the next step is to specify the likelihood based on the selected surrogate model or the structure of the data-model residuals. For example, it may not always be appropriate to treat the residuals as independent and identically distributed (as is commonly done in linear regression). A mis-specification of the residual structure can result in biases and over- or under-confident inferences and projections <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id180" title="Jenný Brynjarsdóttir and Anthony O'Hagan. Learning about physical parameters: the importance of model discrepancy. Inverse Problems, 30(11):114007, October 2014. Publisher: IOP Publishing. URL: https://doi.org/10.1088/0266-5611/30/11/114007 (visited on 2021-06-14), doi:10.1088/0266-5611/30/11/114007.">179</a>]</span>.</p>
+<p>After specifying the prior distributions (see <a class="reference internal" href="A1_Uncertainty_Quantification.html#critical-first-step"><span class="std std-numref">Chapter A.6</span></a>), the selected MCMC algorithm should be used to draw samples from the posterior distribution. There are many MCMC algorithms, all of which have advantages and disadvantages for a particular problem. These include the Metropolis-Hastings algorithm <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id169" title="Christian P. Robert. The Metropolis–Hastings Algorithm. In Wiley StatsRef: Statistics Reference Online, pages 1–15. American Cancer Society, 2015. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat07834 (visited on 2021-06-14), doi:10.1002/9781118445112.stat07834.">169</a>]</span> and Hamiltonian Monte Carlo <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id181" title="Michael Betancourt. A conceptual introduction to Hamiltonian Monte Carlo. arXiv preprint arXiv:1701.02434, 2017.">180</a>, <a class="reference internal" href="R.Bibliography.html#id182" title="Radford M. Neal. MCMC using Hamiltonian dynamics. Handbook of markov chain monte carlo, 2(11):2, 2011.">181</a>]</span>. Software packages typically implement one MCMC method, sometimes designed for a particular problem setting or likelihood specification. For example, R’s <em>adaptMCMC</em> implements an adaptive Metropolis-Hastings algorithm <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id183" title="Matti Vihola. Robust adaptive Metropolis algorithm with coerced acceptance rate. Statistics and Computing, 22(5):997–1008, September 2012. URL: https://doi.org/10.1007/s11222-011-9269-5 (visited on 2021-06-14), doi:10.1007/s11222-011-9269-5.">182</a>]</span>, while <em>NIMBLE</em> <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id184" title="Perry de Valpine, Daniel Turek, Christopher J. Paciorek, Clifford Anderson-Bergman, Duncan Temple Lang, and Rastislav Bodik. Programming With Models: Writing Statistical Algorithms for General Model Structures With NIMBLE. Journal of Computational and Graphical Statistics, 26(2):403–413, April 2017. Publisher: Taylor &amp; Francis _eprint: https://doi.org/10.1080/10618600.2016.1172487. URL: https://doi.org/10.1080/10618600.2016.1172487 (visited on 2021-06-14), doi:10.1080/10618600.2016.1172487.">183</a>, <a class="reference internal" href="R.Bibliography.html#id185" title="NIMBLE Development Team. NIMBLE: MCMC, Particle Filtering, and Programmable Hierarchical Modeling. May 2021. URL: https://zenodo.org/record/4829693 (visited on 2021-06-14), doi:10.5281/zenodo.4829693.">184</a>]</span> uses a user-customizable Metropolis-Hastings implementation, as well as functionality for Gibbs sampling (which is a special case of Metropolis-Hastings where the prior distribution has a convenient mathematical form). Some recent implementations, such as <em>Stan</em> <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id186" title="Stan Development Team. Stan Modeling Language Users Guide and Reference Manual. 2021. URL: https://mc-stan.org/docs/2_27/stan-users-guide/index.html (visited on 2021-06-14).">185</a>]</span>, <em>pyMC3</em> <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id187" title="John Salvatier, Thomas V. Wiecki, and Christopher Fonnesbeck. Probabilistic programming in Python using PyMC3. PeerJ Computer Science, 2:e55, April 2016. Publisher: PeerJ Inc. URL: https://peerj.com/articles/cs-55 (visited on 2021-06-14), doi:10.7717/peerj-cs.55.">186</a>]</span>, and <em>Turing</em> <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id188" title="Hong Ge, Kai Xu, and Zoubin Ghahramani. Turing: A Language for Flexible Probabilistic Inference. In International Conference on Artificial Intelligence and Statistics, 1682–1690. PMLR, March 2018. ISSN: 2640-3498. URL: http://proceedings.mlr.press/v84/ge18b.html (visited on 2021-06-14).">187</a>]</span> allow different algorithms to be used.</p>
+<p>A main consideration when using MCMC algorithms is testing for convergence to the target distribution. As convergence is guaranteed only for a sufficiently large number of transitions, it is impossible to conclude for certain that a chain has converged for a fixed number of iterations. However, several heuristics have been developed <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id171" title="James M. Flegal, Murali Haran, and Galin L. Jones. Markov Chain Monte Carlo: Can We Trust the Third Significant Figure? Statistical Science, 23(2):250–260, May 2008. Publisher: Institute of Mathematical Statistics. URL: https://projecteuclid.org/journals/statistical-science/volume-23/issue-2/Markov-Chain-Monte-Carlo--Can-We-Trust-the-Third/10.1214/08-STS257.full (visited on 2021-06-14), doi:10.1214/08-STS257.">170</a>, <a class="reference internal" href="R.Bibliography.html#id189" title="Andrew Gelman and Donald B. Rubin. Inference from Iterative Simulation Using Multiple Sequences. Statistical Science, 7(4):457–472, November 1992. Publisher: Institute of Mathematical Statistics. URL: https://projecteuclid.org/journals/statistical-science/volume-7/issue-4/Inference-from-Iterative-Simulation-Using-Multiple-Sequences/10.1214/ss/1177011136.full (visited on 2021-06-14), doi:10.1214/ss/1177011136.">188</a>]</span> to increase evidence that convergence has occurred.</p>
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+<h1>Other Methods<a class="headerlink" href="#other-methods" title="Link to this heading">#</a></h1>
+<p>Other common methods for UQ exist. These include sequential Monte Carlo, otherwise known as particle filtering <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id190" title="Pierre Del Moral, Arnaud Doucet, and Ajay Jasra. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(3):411–436, 2006. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9868.2006.00553.x. URL: http://rss.onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9868.2006.00553.x (visited on 2021-06-14), doi:10.1111/j.1467-9868.2006.00553.x.">189</a>, <a class="reference internal" href="R.Bibliography.html#id191" title="Arnaud Doucet, Simon Godsill, and Christophe Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 10(3):197–208, July 2000. URL: https://doi.org/10.1023/A:1008935410038 (visited on 2021-06-14), doi:10.1023/A:1008935410038.">190</a>, <a class="reference internal" href="R.Bibliography.html#id192" title="Jane Liu and Mike West. Combined Parameter and State Estimation in Simulation-Based Filtering. In Arnaud Doucet, Nando de Freitas, and Neil Gordon, editors, Sequential Monte Carlo Methods in Practice, Statistics for Engineering and Information Science, pages 197–223. Springer, New York, NY, 2001. URL: https://doi.org/10.1007/978-1-4757-3437-9_10 (visited on 2021-06-14), doi:10.1007/978-1-4757-3437-9_10.">191</a>]</span>, where a number of particles are used to evaluate samples. An advantage of sequential Monte Carlo is that the vast majority of the computation can be parallelized, unlike with standard MCMC. A major weakness is the potential for degeneracy <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id191" title="Arnaud Doucet, Simon Godsill, and Christophe Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 10(3):197–208, July 2000. URL: https://doi.org/10.1023/A:1008935410038 (visited on 2021-06-14), doi:10.1023/A:1008935410038.">190</a>]</span>, where many particles have extremely small weights, resulting in the effective use of only a few samples.</p>
+<p>Another method is approximate Bayesian computation (ABC) <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id193" title="Stefano Cabras, Maria Eugenia Castellanos Nueda, and Erlis Ruli. Approximate Bayesian Computation by Modelling Summary Statistics in a Quasi-likelihood Framework. Bayesian Analysis, 10(2):411–439, June 2015. Publisher: International Society for Bayesian Analysis. URL: https://projecteuclid.org/journals/bayesian-analysis/volume-10/issue-2/Approximate-Bayesian-Computation-by-Modelling-Summary-Statistics-in-a-Quasi/10.1214/14-BA921.full (visited on 2021-06-14), doi:10.1214/14-BA921.">192</a>, <a class="reference internal" href="R.Bibliography.html#id194" title="Jarno Lintusaari, Michael U. Gutmann, Ritabrata Dutta, Samuel Kaski, and Jukka Corander. Fundamentals and Recent Developments in Approximate Bayesian Computation. Systematic Biology, 66(1):e66–e82, January 2017. URL: https://doi.org/10.1093/sysbio/syw077 (visited on 2021-06-14), doi:10.1093/sysbio/syw077.">193</a>, <a class="reference internal" href="R.Bibliography.html#id195" title="Mikael Sunnåker, Alberto Giovanni Busetto, Elina Numminen, Jukka Corander, Matthieu Foll, and Christophe Dessimoz. Approximate Bayesian Computation. PLOS Computational Biology, 9(1):e1002803, January 2013. Publisher: Public Library of Science. URL: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002803 (visited on 2021-06-14), doi:10.1371/journal.pcbi.1002803.">194</a>]</span>. ABC is a likelihood-free approach that compares model output to a set of summary statistics. ABC is therefore well-suited for models and residual structures which do not lend themselves to a computationally-tractable likelihood, but the resulting inferences are known to be biased if the set of summary statistics is not sufficient, which can be difficult to know a-priori.</p>
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+    <h1>The Critical First Step: How to Choose a Prior Distribution</h1>
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+<span id="critical-first-step"></span><h1>The Critical First Step: How to Choose a Prior Distribution<a class="headerlink" href="#the-critical-first-step-how-to-choose-a-prior-distribution" title="Link to this heading">#</a></h1>
+<p>Prior distributions play an important role in Bayesian uncertainty quantification, particularly when data is limited relative to the dimension of the model. Bayesian updating can be thought of as an information filter, where each additional datum is added to the information contained in the prior; eventually, the prior makes relatively little impact. In real world problems, it can be extremely difficult to assess how much data is required for the choice of prior to become less relevant. The choice of prior can also be influential when conducting SA prior to or without UQ. This is because a prior distribution for a sensitivity analysis for a given parameter which is much wider than the region where the model response surface is sensitive to the parameter value might cause the sensitivity calculation to underestimate the response in that potentially critical region. Similarly, a prior which is too narrow may miss regions where the model responds to the parameter altogether.</p>
+<p>Ideally, prior distributions are constructed independently of any analysis of the new data considered. This is because using data to inform the prior as well as to compute the likelihood reuses information in a potentially inappropriate way, which can lead to overconfident inferences. Following <span id="id1">Jaynes [<a class="reference internal" href="R.Bibliography.html#id196" title="Edwin T. Jaynes. Probability theory: the logic of science. Washington University St. Louis, MO, 1996.">195</a>], Gelman <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id197" title="Andrew Gelman, Daniel Simpson, and Michael Betancourt. The Prior Can Often Only Be Understood in the Context of the Likelihood. Entropy, 19(10):555, October 2017. Number: 10 Publisher: Multidisciplinary Digital Publishing Institute. URL: https://www.mdpi.com/1099-4300/19/10/555 (visited on 2021-06-14), doi:10.3390/e19100555.">196</a>]</span> refers to the ideal prior as one which encodes all available information about the model. For practical reasons (difficulty of construction or computational inconvenience), most priors fail to achieve this ideal. These compromises mean that priors should be transparently articulated and justified, so that the impact of the choice of prior can be fully understood. When there is ambiguity about an appropriate prior, such as how fat the tails should be, an analyst should examine how sensitive the UQ results are to the choice of prior.</p>
+<p>Priors can also be classified in terms of the information encoded by them, demonstrated in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a>. Non-informative priors (illustrated in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a> (a)) allegedly correspond to (and are frequently justified by) a position of ignorance. A classic example is the use of a uniform distribution. A uniform prior can, however, be problematic, as it can lead to improper inferences by giving extremely large values the same prior probability as values which may seem more likely <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id197" title="Andrew Gelman, Daniel Simpson, and Michael Betancourt. The Prior Can Often Only Be Understood in the Context of the Likelihood. Entropy, 19(10):555, October 2017. Number: 10 Publisher: Multidisciplinary Digital Publishing Institute. URL: https://www.mdpi.com/1099-4300/19/10/555 (visited on 2021-06-14), doi:10.3390/e19100555.">196</a>, <a class="reference internal" href="R.Bibliography.html#id198" title="Christian Robert. The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer Science &amp; Business Media, 2007.">197</a>]</span>, and therefore does not really reflect a state of complete ignorance. In the extreme case of a uniform prior over the entire real line, every particular region has effectively a prior weight of zero, even though not all regions are a priori unlikely <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id198" title="Christian Robert. The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer Science &amp; Business Media, 2007.">197</a>]</span>. Moreover, a uniform prior which excludes possible parameter values is not actually noninformative, as it assigns zero probability to those values while jumping to a nonzero probability as soon as the boundary is crossed. While a uniform prior can be problematic for the task of uncertainty quantification, it may be useful for an initial sensitivity analysis to identify the boundary of any regions where the model is sensitive to the parameter.</p>
+<figure class="margin-caption align-center" id="id5">
+<span id="figure-a1-5"></span><a class="reference internal image-reference" href="_images/figureA1_5_priors_posteriors.png"><img alt="Figure A1.5" src="_images/figureA1_5_priors_posteriors.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Impact of priors on posterior inferences. These plots show the results of inference for a linear regression model with 15 data points. The true value of the parameter is equal to -3. All priors have mean 0. In panel (a), a non-informative prior allows the tails of the posterior to extend freely, which may result in unreasonably large parameter values. In panel (b), a weakly informative prior constrains the tails more, but allows them to extend without too much restriction. In panel (c), an informative prior strongly constrains the tails of the posterior and biases the inference closer towards the prior mean (the posterior mean is -0.89 in this case, and closer to -3 in the other two cases).</span><a class="headerlink" href="#id5" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>Informative priors strongly bound the range of probable values (illustrated in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a> (c)). One example is a Gaussian distribution with a relatively small standard deviation, so that large values are assigned a close to null prior probability. Another example is the jump from zero to non-zero probability occurring at the truncation point of a truncated Gaussian, which could be justified based on information that the parameter cannot take on values beyond this point. Without this type of justification, however, priors may be too informative, failing to allow the information contained in the available data to update them.</p>
+<p>Finally, weakly informative priors (illustrated in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a> (b)) fall in between <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id197" title="Andrew Gelman, Daniel Simpson, and Michael Betancourt. The Prior Can Often Only Be Understood in the Context of the Likelihood. Entropy, 19(10):555, October 2017. Number: 10 Publisher: Multidisciplinary Digital Publishing Institute. URL: https://www.mdpi.com/1099-4300/19/10/555 (visited on 2021-06-14), doi:10.3390/e19100555.">196</a>]</span>. They regularize better than non-informative priors, but allow for more inference flexibility than fully informative priors. An example might be a Gaussian distribution with a moderate standard deviation, which still assigns negligible probability for values far away from the mean, but is less constrained than a narrow Gaussian for a reasonably large area. A key note is that it is not necessarily better to be more informative if this cannot be justified by the available information.</p>
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+    <h1>The Critical Final Step: Predictive Checks</h1>
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+  <section id="the-critical-final-step-predictive-checks">
+<h1>The Critical Final Step: Predictive Checks<a class="headerlink" href="#the-critical-final-step-predictive-checks" title="Link to this heading">#</a></h1>
+<p>Every UQ workflow requires a number of choices, potentially including selecting prior distributions, the likelihood specification, and any used numerical models. Checking the appropriateness of these choices is an essential step for sound inferences, as misspecification can produce biased results <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id180" title="Jenný Brynjarsdóttir and Anthony O'Hagan. Learning about physical parameters: the importance of model discrepancy. Inverse Problems, 30(11):114007, October 2014. Publisher: IOP Publishing. URL: https://doi.org/10.1088/0266-5611/30/11/114007 (visited on 2021-06-14), doi:10.1088/0266-5611/30/11/114007.">179</a>]</span>. Model checking in this fashion is part of an iterative UQ process, as the results can reveal adjustments to the statistical model or the need to select a different numerical model <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id200" title="Andrew Gelman, Xiao-Li Meng, and Hal Stern. Posterior Predictive Assessment of Model Fitness via Realized Discrepancies. Statistica Sinica, 6(4):733–760, 1996. Publisher: Institute of Statistical Science, Academia Sinica. URL: https://www.jstor.org/stable/24306036 (visited on 2021-06-14).">198</a>, <a class="reference internal" href="R.Bibliography.html#id199" title="Andrew Gelman, Aki Vehtari, Daniel Simpson, Charles C. Margossian, Bob Carpenter, Yuling Yao, Lauren Kennedy, Jonah Gabry, Paul-Christian Bürkner, and Martin Modrák. Bayesian Workflow. arXiv:2011.01808 [stat], November 2020. arXiv: 2011.01808. URL: http://arxiv.org/abs/2011.01808 (visited on 2021-06-14).">199</a>, <a class="reference internal" href="R.Bibliography.html#id201" title="Andrew Gelman and Cosma Rohilla Shalizi. Philosophy and the practice of Bayesian statistics. British Journal of Mathematical and Statistical Psychology, 66(1):8–38, 2013. _eprint: https://bpspsychub.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2044-8317.2011.02037.x. URL: https://bpspsychub.onlinelibrary.wiley.com/doi/abs/10.1111/j.2044-8317.2011.02037.x (visited on 2021-06-14), doi:10.1111/j.2044-8317.2011.02037.x.">200</a>]</span>.</p>
+<p>A classic example is the need to check the structure of residuals for correlations. Many standard statistical models, such as linear regression, assume that the residuals are independent and identically distributed from the error distribution. The presence of correlations, including temporal autocorrelations and spatial correlations, indicates a structural mismatch between the likelihood and the data. In these cases, the likelihood should be adjusted to account for these correlations.</p>
+<p>Checking residuals in this fashion is one example of a predictive check (or a posterior predictive check in the Bayesian setting). One way to view UQ is as a means to recover data-generating processes (associated with each parameter vector) consistent with the observations. Predictive checks compare the inferred data-generating process to the observations to determine whether the model is capable of appropriately capturing uncertainty. After conducting the UQ analysis, alternatively realized datasets are simulated from sampled parameters. These alternative datasets, or their summary statistics, can be tested against the observations to determine adequacy of the fit. Predictive checks are therefore a way of probing various model components to identify shortcomings that might result in biased inferences or poor projections, depending on the goal of the analysis.</p>
+<p>One example of a graphical predictive check for time series models is hindcasting, where predictive intervals are constructed from the alternative datasets and plotted along with the data. Hindcasts demonstrate how well the model is capable of capturing the broader dynamics of the data, as well as whether the parameter distributions produce appropriate levels of output uncertainty. A related quantitative check is the surprise index, which calculates the percentage of data points located within a fixed predictive interval. For example, the 90% predictive interval should contain approximately 90% of the data. More uncertainty than this reflects underconfidence, while less uncertainty reflects overconfidence. This could be the result of priors that are not appropriately informative, or a likelihood that does not account for correlations between data points appropriately. It could also be the result of a numerical model that isn’t sufficiently sensitive to the parameters that are treated as uncertain.</p>
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+<h1>Key Take-Home Points<a class="headerlink" href="#key-take-home-points" title="Link to this heading">#</a></h1>
+<p>When appropriate, UQ is an important component of the exploratory modeling workflow. While a number of parameter sets could be consistent with observations, they may result in divergent model outputs when exposed to different future conditions. This can result in identifying risks which are not visible when selecting a “best fit” parameterization. Quantifying uncertainties also allows us to quantify the support for hypotheses, which is an essential part of the scientific process.</p>
+<p>Due to the scale and complexity of the experiments taking place in IM3, UQ has not been extensively used. The tradeoff between the available number of function evaluations and the number of uncertain parameters illustrated in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-1"><span class="std std-numref">Fig. A.1</span></a> is particularly challenging due to the increasing complexity of state-of-the-art models and the movement towards coupled, multisector models. This tradeoff can be addressed somewhat through the use of emulators and parallelizable methods. In particular, when attempting to navigate this tradeoff by limiting the number of uncertain parameters, it is important to carefully iterate with sensitivity analyses to ensure that critical parameters are identified.</p>
+<p>Specifying prior distributions and likelihoods is an ongoing challenge. Prior distributions, in particular, should be treated as deeply uncertain when appropriate. One key advantage of the methods described in this chapter is that they have the potential for increased transparency. When it is not possible to conduct a sensitivity analysis on a number of critical priors due to limited computational budgets, fully specifying and providing a justification for the utilized distributions allows other researchers to identify key assumptions and build on existing work. The same is true for the specification of likelihoods—while likelihood-free methods avoid the need to specify a likelihood function, they require other assumptions or choices, which should be described and justified as transparently as possible.</p>
+<p>We conclude this appendix with some key recommendations:
+1. UQ analysis does not require full confidence in priors and likelihoods. Rather, UQ should be treated as part of an exploratory modeling workflow, where hypotheses related to model structures, prior distributions, and likelihoods can be tested.
+2. For complex multisectoral models, UQ will typically require the use of a reduced set of parameters, either through emulation or by fixing the others to their best-fit values. These parameters should be selected through a careful sensitivity analysis.
+3. Avoid the use of supposedly “non-informative” priors, such as uniform priors, whenever possible. In the absence of strong information about parameter values, the use of weakly informative priors, such as diffuse normals, is preferable.
+4. Be cognizant of the limitations of conclusions that can be drawn by using each method. The bootstrap, for example, may result in overconfidence if the dataset is limited and is not truly representative of the underlying stochastic process.
+5. When using MCMC, Markov chains can not be shown to have converged to the target distribution, but rather evidence can be collected to demonstrate that it is likely that they have.
+6. Conduct predictive checks based on the assumptions underlying the choices made in the analysis, and iteratively update those choices if the assumptions prove to be ill-suited for the problem at hand.</p>
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+    <h1>Uncertainty Quantification</h1>
+    <!-- Table of contents -->
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+                <h2> Contents </h2>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">A.1. Introduction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#parametric-bootstrap">A.2. Parametric Bootstrap</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pre-calibration">A.3. Pre-Calibration</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#markov-chain-monte-carlo">A.4. Markov Chain Monte Carlo</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-methods">A.5. Other Methods</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-critical-first-step-how-to-choose-a-prior-distribution">A.6. The Critical First Step: How to Choose a Prior Distribution</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-critical-final-step-predictive-checks">A.7. The Critical Final Step: Predictive Checks</a></li>
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+  <section id="uncertainty-quantification">
+<span id="a1-uncertainty-quantification"></span><h1><span class="section-number">A. </span>Uncertainty Quantification<a class="headerlink" href="#uncertainty-quantification" title="Link to this heading">#</a></h1>
+<section id="introduction">
+<h2><span class="section-number">A.1. </span>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h2>
+<p>As defined in <a class="reference internal" href="1_introduction.html#introduction"><span class="std std-numref">Chapter 1</span></a>, uncertainty quantification (UQ) refers to the formal focus on the full specification of likelihoods as well as distributional forms necessary to infer the joint  probabilistic response across all modeled factors of interest <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id11" title="Roger Cooke and others. Experts in uncertainty: opinion and subjective probability in science. Oxford University Press on Demand, 1991.">8</a>]</span>. This is in contrast to UC (the primary focus of the main document of this book), which is instead aimed at identifying which modeling choices yield the most consequential changes or outcomes and exploring alternative hypotheses related to the form and function of modeled systems <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id12" title="Enayat A Moallemi, Jan Kwakkel, Fjalar J de Haan, and Brett A Bryan. Exploratory modeling for analyzing coupled human-natural systems under uncertainty. Global Environmental Change, 65:102186, 2020.">9</a>, <a class="reference internal" href="R.Bibliography.html#id13" title="Warren E Walker, Poul Harremoës, Jan Rotmans, Jeroen P Van Der Sluijs, Marjolein BA Van Asselt, Peter Janssen, and Martin P Krayer von Krauss. Defining uncertainty: a conceptual basis for uncertainty management in model-based decision support. Integrated assessment, 4(1):5–17, 2003.">10</a>]</span>.</p>
+<p>UQ is important for quantifying the relative merits of hypotheses for at least three main reasons. First, identifying model parameters that are consistent with observations is an important part of model development. Due to several effects, including correlations between parameters, simplified or incomplete model structures (relative to the full real-world dynamics), and uncertainty in the observations, many different combinations of parameter values can be consistent with the model structure and the observations to varying extents. Accounting for this uncertainty is conceptually preferable to selecting a single “best fit” parameter vector, particularly as consistency with historical or present observations does not necessarily  guarantee skillful future projections.</p>
+<p>The act of quantification requires specific assumptions about distributional forms and likelihoods, which may be more or less justified depending on prior information about the system or model behavior. As a result, UQ is well-suited for studies accounting for or addressing hypotheses related to systems with a relatively large amount of available data and models which are computationally inexpensive, particularly when the emphasis is on prediction. As shown in <a class="reference internal" href="#figure-a1-1"><span class="std std-numref">Fig. A.1</span></a>, there is a fundamental tradeoff between the available number of model evaluations (for a fixed computational budget) and the number of parameters treated as uncertain. Sensitivity analyses are therefore part of a typical UQ workflow to identify which factors can be fixed and which ought to be prioritized in the UQ.</p>
+<figure class="margin-caption align-center" id="id37">
+<span id="figure-a1-1"></span><a class="reference internal image-reference" href="_images/figureA1_1_UQ_approaches.png"><img alt="Figure A1.1" src="_images/figureA1_1_UQ_approaches.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. A.1 </span><span class="caption-text">Overview of selected existing approaches for uncertainty quantification and their appropriateness given the number of uncertain model parameters and the number of available model simulations. Green shading denotes regions suitable for uncertainty quantification and red shading indicates regions more appropriate for uncertainty characterization.</span><a class="headerlink" href="#id37" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The choice of a particular UQ method depends on both the desired level of quantification and the ability to navigate the tradeoff between computational expense and the number of uncertain parameters (<a class="reference internal" href="#figure-a1-1"><span class="std std-numref">Fig. A.1</span></a>). For example, Markov chain Monte Carlo with a full system model can provide an improved representation of uncertainty compared to the coarser pre-calibration approach <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id37" title="Kelsey L. Ruckert, Gary Shaffer, David Pollard, Yawen Guan, Tony E. Wong, Chris E. Forest, and Klaus Keller. Assessing the Impact of Retreat Mechanisms in a Simple Antarctic Ice Sheet Model Using Bayesian Calibration. PLOS ONE, 12(1):e0170052, January 2017. doi:10.1371/journal.pone.0170052.">157</a>]</span>, but requires many more model evaluations. The use of a surrogate model to approximate the full system model can reduce the number of needed model evaluations by several orders of magnitude, but the uncertainty quantification can only accommodate a limited number of parameters.</p>
+<p>The remainder of this appendix will focus on introducing workflows for particular UQ methods, including a brief discussion of advantages and limitations.</p>
+</section>
+<section id="parametric-bootstrap">
+<h2><span class="section-number">A.2. </span>Parametric Bootstrap<a class="headerlink" href="#parametric-bootstrap" title="Link to this heading">#</a></h2>
+<p>The parametric bootstrap <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id14" title="B. Efron and R. Tibshirani. Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy. Statistical Science, 1(1):54–75, February 1986. doi:10.1214/ss/1177013815.">158</a>]</span> refers to a process of model recalibration to alternate realizations of the data. The bootstrap was originally developed to estimate standard errors and confidence intervals without ascertaining key assumptions that might not hold given the available data. In a setting where observations can be viewed as independent realizations of an underlying stochastic process, a sufficiently rich dataset can be treated as a population representing the data distribution. New datasets are then generated by resampling from the data with replacement, and the model can be refit to each new dataset using maximum-likelihood estimation. The resulting distribution of estimates can then be viewed as a representation of parametric uncertainty.</p>
+<p>A typical workflow for the parametric bootstrap is shown in <a class="reference internal" href="#figure-a1-2"><span class="std std-numref">Fig. A.2</span></a>. After identifying outputs of interest and preparing the data, the parametric model is fit by some procedure such as minimizing root-mean-square-error or maximizing the likelihood. Alternate datasets are constructed by resampling from the population or by generating new samples from the fitted data-generating process. It is important at this step that the resampled quantities are independent of one another. For example, in the context of temporally- or spatially-correlated data, such as time series, the raw observations cannot be treated as independent realizations. However, the residuals resulting from fitting the model to the data could be (depending on their structure). For example, if the residuals are treated as independent, they can then be resampled with replacement, and these residuals added to the original model fit to create new realizations. If the residuals are assumed to be the result of an autoregressive process, this process could be fit to the original residual series and new residuals be created using this model <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id35" title="Ryan L. Sriver, Robert J. Lempert, Per Wikman-Svahn, and Klaus Keller. Characterizing uncertain sea-level rise projections to support investment decisions. PLOS ONE, 13(2):e0190641, February 2018. URL: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0190641 (visited on 2021-06-09), doi:10.1371/journal.pone.0190641.">159</a>]</span>. The model is then refit to each new realization.</p>
+<figure class="margin-caption align-center" id="id38">
+<span id="figure-a1-2"></span><a class="reference internal image-reference" href="_images/figureA1_2_bootstrap_workflow.png"><img alt="Figure A1.2" src="_images/figureA1_2_bootstrap_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. A.2 </span><span class="caption-text">Workflow for the parametric bootstrap.</span><a class="headerlink" href="#id38" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The bootstrap is computationally convenient, particularly as the process of fitting the model to each realization can be easily parallelized. This approach also requires minimal prior assumptions. However, due to the assumption that the available data are representative of the underlying data distribution, the bootstrap can neglect key uncertainties which might influence the results. For example, when using an autoregressive process to generate new residuals, uncertainty in the autocorrelation parameter and innovation variance is neglected, which may bias estimates of, for example, low-probability but high-impact events <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id36" title="Kelsey L. Ruckert, Yawen Guan, Alexander M. R. Bakker, Chris E. Forest, and Klaus Keller. The effects of time-varying observation errors on semi-empirical sea-level projections. Climatic Change, 140(3):349–360, February 2017. URL: https://doi.org/10.1007/s10584-016-1858-z (visited on 2021-06-09), doi:10.1007/s10584-016-1858-z.">160</a>]</span>.</p>
+</section>
+<section id="pre-calibration">
+<h2><span class="section-number">A.3. </span>Pre-Calibration<a class="headerlink" href="#pre-calibration" title="Link to this heading">#</a></h2>
+<p>Pre-calibration <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>, <a class="reference internal" href="R.Bibliography.html#id39" title="Neil R Edwards, David Cameron, and Jonathan Rougier. Precalibrating an intermediate complexity climate model. Clim. Dyn., 37(7-8):1469–1482, 2011. URL: http://dx.doi.org/10.1007/s00382-010-0921-0, doi:10.1007/s00382-010-0921-0.">161</a>, <a class="reference internal" href="R.Bibliography.html#id40" title="Alexis Boukouvalas, Pete Sykes, Dan Cornford, and Hugo Maruri-Aguilar. Bayesian Precalibration of a Large Stochastic Microsimulation Model. IEEE Transactions on Intelligent Transportation Systems, 15(3):1337–1347, June 2014. doi:10.1109/TITS.2014.2304394.">162</a>]</span> involves the identification of a plausible set of parameters using some prespecified screening criterion, such as the distance from the model results to the observations (based on an appropriate metric for the desired matching features, such as root-mean-squared error). A typical workflow is shown in <a class="reference internal" href="#figure-a1-3"><span class="std std-numref">Fig. A.3</span></a>. Parameter values are obtained by systematically sampling the input space (see <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#design-of-experiments"><span class="std std-numref">Chapter 3.3</span></a>). After the model is evaluated at the samples, only those passing the distance criterion are retained. This selects a subset of the parameter space as “plausible” based on the screening criterion, though there is no assignment of probabilities within this plausible region.</p>
+<figure class="margin-caption align-center" id="id39">
+<span id="figure-a1-3"></span><a class="reference internal image-reference" href="_images/figureA1_3_precal_workflow.png"><img alt="Figure A1.3" src="_images/figureA1_3_precal_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. A.3 </span><span class="caption-text">Workflow for pre-calibration.</span><a class="headerlink" href="#id39" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>Pre-calibration can be useful for models which are inexpensive enough that a reasonable
+number of samples can be used to represent the parameter space, but which are too expensive to facilitate full uncertainty quantification. High-dimensional parameter spaces, which can be problematic for the uncertainty quantification methods below, may also be explored using pre-calibration. One key prerequisite to using this method is the ability to place a meaningful distance metric on the output space.</p>
+<p>However, pre-calibration results in a very coarse characterization of uncertainty, especially when considering a large number of parameters, as more samples are needed to fully characterize the parameter space. Due to the inability to evaluate the relative probability of regions of the parameter space beyond the binary plausible-and-implausible characterization, pre-calibration can also result in degraded hindcast and projection skills and parameter estimates <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id37" title="Kelsey L. Ruckert, Gary Shaffer, David Pollard, Yawen Guan, Tony E. Wong, Chris E. Forest, and Klaus Keller. Assessing the Impact of Retreat Mechanisms in a Simple Antarctic Ice Sheet Model Using Bayesian Calibration. PLOS ONE, 12(1):e0170052, January 2017. doi:10.1371/journal.pone.0170052.">157</a>, <a class="reference internal" href="R.Bibliography.html#id41" title="David Makowski, Daniel Wallach, and Marie Tremblay. Using a Bayesian approach to parameter estimation; comparison of the GLUE and MCMC methods. Agronomie, 22(2):191–203, 2002. Publisher: EDP Sciences.">163</a>, <a class="reference internal" href="R.Bibliography.html#id42" title="Mahyar Shafii, Bryan Tolson, and Loren Shawn Matott. Uncertainty-based multi-criteria calibration of rainfall-runoff models: a comparative study. Stochastic Environmental Research and Risk Assessment, 28(6):1493–1510, August 2014. doi:10.1007/s00477-014-0855-x.">164</a>]</span>.</p>
+<p>A related method, widely used in hydrological studies, is generalized likelihood uncertainty estimation, or GLUE <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>]</span>. Unlike pre-calibration, the underlying argument for GLUE relies on the concept of equifinality <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id43" title="Keith Beven and Jim Freer. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology. Journal of hydrology, 249(1-4):11–29, 2001. Publisher: Elsevier.">165</a>]</span>, which posits that it is impossible to find a uniquely well-performing parameter vector for models of abstract environmental systems <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id43" title="Keith Beven and Jim Freer. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology. Journal of hydrology, 249(1-4):11–29, 2001. Publisher: Elsevier.">165</a>, <a class="reference internal" href="R.Bibliography.html#id44" title="Jasper A. Vrugt and Keith J. Beven. Embracing equifinality with efficiency: Limits of Acceptability sampling using the DREAM(LOA) algorithm. Journal of Hydrology, 559:954–971, April 2018. doi:10.1016/j.jhydrol.2018.02.026.">166</a>]</span>. In other words, there exist multiple parameter vectors which perform equally or similarly well. As with pre-calibration, GLUE uses a goodness-of-fit measure (though this is called a “likelihood” in the GLUE literature, as opposed to a statistical likelihood function <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id45" title="Jery R. Stedinger, Richard M. Vogel, Seung Uk Lee, and Rebecca Batchelder. Appraisal of the generalized likelihood uncertainty estimation (GLUE) method. Water Resources Research, 2008. doi:10.1029/2008WR006822.">167</a>]</span>) to evaluate samples. After setting a threshold of acceptable performance with respect to that measure, samples are evaluated and classified into “behavioral” or “non-behavioral” according to the threshold.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following badge to try out an interactive tutorial on utilizing Pre-Calibration and GLUE for HYMOD model calibration:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">Pre-Calibration Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="markov-chain-monte-carlo">
+<h2><span class="section-number">A.4. </span>Markov Chain Monte Carlo<a class="headerlink" href="#markov-chain-monte-carlo" title="Link to this heading">#</a></h2>
+<p>Markov chain Monte Carlo (MCMC) is a “gold standard” approach to full uncertainty quantification. MCMC refers to a category of algorithms which systematically sample from a target distribution (in this case, the posterior distribution) by constructing a Markov chain. A Markov chain is a probabilistic structure consisting of a state space, an initial probability distribution over the states, and a transition distribution between states. If a Markov chain satisfies certain properties <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id170" title="Christian Robert and George Casella. Monte Carlo Statistical Methods. Springer Science &amp; Business Media, March 2013. ISBN 978-1-4757-3071-5.">168</a>, <a class="reference internal" href="R.Bibliography.html#id169" title="Christian P. Robert. The Metropolis–Hastings Algorithm. In Wiley StatsRef: Statistics Reference Online, pages 1–15. American Cancer Society, 2015. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat07834 (visited on 2021-06-14), doi:10.1002/9781118445112.stat07834.">169</a>]</span>, the probability of being in each state will eventually converge to a stable, or stationary, distribution, regardless of the initial probabilities.</p>
+<p>MCMC algorithms construct a Markov chain of samples from a parameter space (the combination of model and statistical parameters). This Markov chain is constructed so that the stationary distribution is a target distribution, in this case the (Bayesian) posterior distribution. As a result, after the transient period, the resulting samples can be viewed as a set of dependent samples from the posterior (the dependence is due to the autocorrelation between samples resulting from the Markov chain transitions). Expected values can be computed from these samples (for example, using batch-means estimators <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id171" title="James M. Flegal, Murali Haran, and Galin L. Jones. Markov Chain Monte Carlo: Can We Trust the Third Significant Figure? Statistical Science, 23(2):250–260, May 2008. Publisher: Institute of Mathematical Statistics. URL: https://projecteuclid.org/journals/statistical-science/volume-23/issue-2/Markov-Chain-Monte-Carlo--Can-We-Trust-the-Third/10.1214/08-STS257.full (visited on 2021-06-14), doi:10.1214/08-STS257.">170</a>]</span>), or the chain can be sub-sampled or thinned and the resulting samples used as independent Monte Carlo samples due to the reduced or eliminated autocorrelation.</p>
+<p>A general workflow for MCMC is shown in <a class="reference internal" href="#figure-a1-4"><span class="std std-numref">Fig. A.4</span></a>. The first decision is whether to use the full model or a surrogate model (or emulator). Typical surrogates include Gaussian process emulation <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id172" title="Carla Currin, Toby Mitchell, Max Morris, and Don Ylvisaker. Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments. Journal of the American Statistical Association, 86(416):953–963, December 1991. Publisher: Taylor &amp; Francis _eprint: https://www.tandfonline.com/doi/pdf/10.1080/01621459.1991.10475138. URL: https://www.tandfonline.com/doi/abs/10.1080/01621459.1991.10475138 (visited on 2021-06-14), doi:10.1080/01621459.1991.10475138.">171</a>, <a class="reference internal" href="R.Bibliography.html#id173" title="Jerome Sacks, William J. Welch, Toby J. Mitchell, and Henry P. Wynn. Design and Analysis of Computer Experiments. Statistical Science, 4(4):409–423, 1989. Publisher: Institute of Mathematical Statistics. URL: https://www.jstor.org/stable/2245858 (visited on 2021-06-14).">172</a>]</span>, polynomial chaos expansions <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id174" title="Roger G. Ghanem and Pol D. Spanos. Spectral Stochastic Finite‐Element Formulation for Reliability Analysis. Journal of Engineering Mechanics, 117(10):2351–2372, October 1991. Publisher: American Society of Civil Engineers. URL: https://ascelibrary.org/doi/abs/10.1061/%28ASCE%290733-9399%281991%29117%3A10%282351%29 (visited on 2021-06-14), doi:10.1061/(ASCE)0733-9399(1991)117:10(2351).">173</a>, <a class="reference internal" href="R.Bibliography.html#id175" title="Dongbin Xiu and George Em Karniadakis. The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations. SIAM Journal on Scientific Computing, 24(2):619–644, January 2002. Publisher: Society for Industrial and Applied Mathematics. URL: https://epubs.siam.org/doi/abs/10.1137/S1064827501387826 (visited on 2021-06-14), doi:10.1137/S1064827501387826.">174</a>]</span>, support vector machines <span id="id17">[<a class="reference internal" href="R.Bibliography.html#id176" title="Angelo Ciccazzo, Gianni Di Pillo, and Vittorio Latorre. A SVM Surrogate Model-Based Method for Parametric Yield Optimization. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 35(7):1224–1228, July 2016. Conference Name: IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. doi:10.1109/TCAD.2015.2501307.">175</a>, <a class="reference internal" href="R.Bibliography.html#id177" title="W. Andrew Pruett and Robert L. Hester. The Creation of Surrogate Models for Fast Estimation of Complex Model Outcomes. PLOS ONE, 11(6):e0156574, June 2016. Publisher: Public Library of Science. URL: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0156574 (visited on 2021-06-14), doi:10.1371/journal.pone.0156574.">176</a>]</span>, and neural networks <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id178" title="John Eason and Selen Cremaschi. Adaptive sequential sampling for surrogate model generation with artificial neural networks. Computers &amp; Chemical Engineering, 68:220–232, September 2014. URL: https://www.sciencedirect.com/science/article/pii/S0098135414001719 (visited on 2021-06-14), doi:10.1016/j.compchemeng.2014.05.021.">177</a>, <a class="reference internal" href="R.Bibliography.html#id179" title="Dirk Gorissen, Luciano De Tommasi, Karel Crombecq, and Tom Dhaene. Sequential modeling of a low noise amplifier with neural networks and active learning. Neural Computing and Applications, 18(5):485–494, June 2009. URL: https://doi.org/10.1007/s00521-008-0223-1 (visited on 2021-06-14), doi:10.1007/s00521-008-0223-1.">178</a>]</span>. Surrogate modeling can be faster, but requires a sufficient number of model evaluations for the surrogate to accurately represent the model’s response surface, and this typically limits the number of parameters which can be included in the analysis.</p>
+<figure class="margin-caption align-center" id="id40">
+<span id="figure-a1-4"></span><a class="reference internal image-reference" href="_images/figureA1_4_mcmc_workflow.png"><img alt="Figure A1.4" src="_images/figureA1_4_mcmc_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. A.4 </span><span class="caption-text">Workflow for Markov chain Monte Carlo.</span><a class="headerlink" href="#id40" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>After selecting the variables which will be treated as uncertain, the next step is to specify the likelihood based on the selected surrogate model or the structure of the data-model residuals. For example, it may not always be appropriate to treat the residuals as independent and identically distributed (as is commonly done in linear regression). A mis-specification of the residual structure can result in biases and over- or under-confident inferences and projections <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id180" title="Jenný Brynjarsdóttir and Anthony O'Hagan. Learning about physical parameters: the importance of model discrepancy. Inverse Problems, 30(11):114007, October 2014. Publisher: IOP Publishing. URL: https://doi.org/10.1088/0266-5611/30/11/114007 (visited on 2021-06-14), doi:10.1088/0266-5611/30/11/114007.">179</a>]</span>.</p>
+<p>After specifying the prior distributions (see <a class="reference internal" href="#critical-first-step"><span class="std std-numref">Chapter A.6</span></a>), the selected MCMC algorithm should be used to draw samples from the posterior distribution. There are many MCMC algorithms, all of which have advantages and disadvantages for a particular problem. These include the Metropolis-Hastings algorithm <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id169" title="Christian P. Robert. The Metropolis–Hastings Algorithm. In Wiley StatsRef: Statistics Reference Online, pages 1–15. American Cancer Society, 2015. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat07834 (visited on 2021-06-14), doi:10.1002/9781118445112.stat07834.">169</a>]</span> and Hamiltonian Monte Carlo <span id="id21">[<a class="reference internal" href="R.Bibliography.html#id181" title="Michael Betancourt. A conceptual introduction to Hamiltonian Monte Carlo. arXiv preprint arXiv:1701.02434, 2017.">180</a>, <a class="reference internal" href="R.Bibliography.html#id182" title="Radford M. Neal. MCMC using Hamiltonian dynamics. Handbook of markov chain monte carlo, 2(11):2, 2011.">181</a>]</span>. Software packages typically implement one MCMC method, sometimes designed for a particular problem setting or likelihood specification. For example, R’s <em>adaptMCMC</em> implements an adaptive Metropolis-Hastings algorithm <span id="id22">[<a class="reference internal" href="R.Bibliography.html#id183" title="Matti Vihola. Robust adaptive Metropolis algorithm with coerced acceptance rate. Statistics and Computing, 22(5):997–1008, September 2012. URL: https://doi.org/10.1007/s11222-011-9269-5 (visited on 2021-06-14), doi:10.1007/s11222-011-9269-5.">182</a>]</span>, while <em>NIMBLE</em> <span id="id23">[<a class="reference internal" href="R.Bibliography.html#id184" title="Perry de Valpine, Daniel Turek, Christopher J. Paciorek, Clifford Anderson-Bergman, Duncan Temple Lang, and Rastislav Bodik. Programming With Models: Writing Statistical Algorithms for General Model Structures With NIMBLE. Journal of Computational and Graphical Statistics, 26(2):403–413, April 2017. Publisher: Taylor &amp; Francis _eprint: https://doi.org/10.1080/10618600.2016.1172487. URL: https://doi.org/10.1080/10618600.2016.1172487 (visited on 2021-06-14), doi:10.1080/10618600.2016.1172487.">183</a>, <a class="reference internal" href="R.Bibliography.html#id185" title="NIMBLE Development Team. NIMBLE: MCMC, Particle Filtering, and Programmable Hierarchical Modeling. May 2021. URL: https://zenodo.org/record/4829693 (visited on 2021-06-14), doi:10.5281/zenodo.4829693.">184</a>]</span> uses a user-customizable Metropolis-Hastings implementation, as well as functionality for Gibbs sampling (which is a special case of Metropolis-Hastings where the prior distribution has a convenient mathematical form). Some recent implementations, such as <em>Stan</em> <span id="id24">[<a class="reference internal" href="R.Bibliography.html#id186" title="Stan Development Team. Stan Modeling Language Users Guide and Reference Manual. 2021. URL: https://mc-stan.org/docs/2_27/stan-users-guide/index.html (visited on 2021-06-14).">185</a>]</span>, <em>pyMC3</em> <span id="id25">[<a class="reference internal" href="R.Bibliography.html#id187" title="John Salvatier, Thomas V. Wiecki, and Christopher Fonnesbeck. Probabilistic programming in Python using PyMC3. PeerJ Computer Science, 2:e55, April 2016. Publisher: PeerJ Inc. URL: https://peerj.com/articles/cs-55 (visited on 2021-06-14), doi:10.7717/peerj-cs.55.">186</a>]</span>, and <em>Turing</em> <span id="id26">[<a class="reference internal" href="R.Bibliography.html#id188" title="Hong Ge, Kai Xu, and Zoubin Ghahramani. Turing: A Language for Flexible Probabilistic Inference. In International Conference on Artificial Intelligence and Statistics, 1682–1690. PMLR, March 2018. ISSN: 2640-3498. URL: http://proceedings.mlr.press/v84/ge18b.html (visited on 2021-06-14).">187</a>]</span> allow different algorithms to be used.</p>
+<p>A main consideration when using MCMC algorithms is testing for convergence to the target distribution. As convergence is guaranteed only for a sufficiently large number of transitions, it is impossible to conclude for certain that a chain has converged for a fixed number of iterations. However, several heuristics have been developed <span id="id27">[<a class="reference internal" href="R.Bibliography.html#id171" title="James M. Flegal, Murali Haran, and Galin L. Jones. Markov Chain Monte Carlo: Can We Trust the Third Significant Figure? Statistical Science, 23(2):250–260, May 2008. Publisher: Institute of Mathematical Statistics. URL: https://projecteuclid.org/journals/statistical-science/volume-23/issue-2/Markov-Chain-Monte-Carlo--Can-We-Trust-the-Third/10.1214/08-STS257.full (visited on 2021-06-14), doi:10.1214/08-STS257.">170</a>, <a class="reference internal" href="R.Bibliography.html#id189" title="Andrew Gelman and Donald B. Rubin. Inference from Iterative Simulation Using Multiple Sequences. Statistical Science, 7(4):457–472, November 1992. Publisher: Institute of Mathematical Statistics. URL: https://projecteuclid.org/journals/statistical-science/volume-7/issue-4/Inference-from-Iterative-Simulation-Using-Multiple-Sequences/10.1214/ss/1177011136.full (visited on 2021-06-14), doi:10.1214/ss/1177011136.">188</a>]</span> to increase evidence that convergence has occurred.</p>
+</section>
+<section id="other-methods">
+<h2><span class="section-number">A.5. </span>Other Methods<a class="headerlink" href="#other-methods" title="Link to this heading">#</a></h2>
+<p>Other common methods for UQ exist. These include sequential Monte Carlo, otherwise known as particle filtering <span id="id28">[<a class="reference internal" href="R.Bibliography.html#id190" title="Pierre Del Moral, Arnaud Doucet, and Ajay Jasra. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(3):411–436, 2006. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9868.2006.00553.x. URL: http://rss.onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9868.2006.00553.x (visited on 2021-06-14), doi:10.1111/j.1467-9868.2006.00553.x.">189</a>, <a class="reference internal" href="R.Bibliography.html#id191" title="Arnaud Doucet, Simon Godsill, and Christophe Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 10(3):197–208, July 2000. URL: https://doi.org/10.1023/A:1008935410038 (visited on 2021-06-14), doi:10.1023/A:1008935410038.">190</a>, <a class="reference internal" href="R.Bibliography.html#id192" title="Jane Liu and Mike West. Combined Parameter and State Estimation in Simulation-Based Filtering. In Arnaud Doucet, Nando de Freitas, and Neil Gordon, editors, Sequential Monte Carlo Methods in Practice, Statistics for Engineering and Information Science, pages 197–223. Springer, New York, NY, 2001. URL: https://doi.org/10.1007/978-1-4757-3437-9_10 (visited on 2021-06-14), doi:10.1007/978-1-4757-3437-9_10.">191</a>]</span>, where a number of particles are used to evaluate samples. An advantage of sequential Monte Carlo is that the vast majority of the computation can be parallelized, unlike with standard MCMC. A major weakness is the potential for degeneracy <span id="id29">[<a class="reference internal" href="R.Bibliography.html#id191" title="Arnaud Doucet, Simon Godsill, and Christophe Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 10(3):197–208, July 2000. URL: https://doi.org/10.1023/A:1008935410038 (visited on 2021-06-14), doi:10.1023/A:1008935410038.">190</a>]</span>, where many particles have extremely small weights, resulting in the effective use of only a few samples.</p>
+<p>Another method is approximate Bayesian computation (ABC) <span id="id30">[<a class="reference internal" href="R.Bibliography.html#id193" title="Stefano Cabras, Maria Eugenia Castellanos Nueda, and Erlis Ruli. Approximate Bayesian Computation by Modelling Summary Statistics in a Quasi-likelihood Framework. Bayesian Analysis, 10(2):411–439, June 2015. Publisher: International Society for Bayesian Analysis. URL: https://projecteuclid.org/journals/bayesian-analysis/volume-10/issue-2/Approximate-Bayesian-Computation-by-Modelling-Summary-Statistics-in-a-Quasi/10.1214/14-BA921.full (visited on 2021-06-14), doi:10.1214/14-BA921.">192</a>, <a class="reference internal" href="R.Bibliography.html#id194" title="Jarno Lintusaari, Michael U. Gutmann, Ritabrata Dutta, Samuel Kaski, and Jukka Corander. Fundamentals and Recent Developments in Approximate Bayesian Computation. Systematic Biology, 66(1):e66–e82, January 2017. URL: https://doi.org/10.1093/sysbio/syw077 (visited on 2021-06-14), doi:10.1093/sysbio/syw077.">193</a>, <a class="reference internal" href="R.Bibliography.html#id195" title="Mikael Sunnåker, Alberto Giovanni Busetto, Elina Numminen, Jukka Corander, Matthieu Foll, and Christophe Dessimoz. Approximate Bayesian Computation. PLOS Computational Biology, 9(1):e1002803, January 2013. Publisher: Public Library of Science. URL: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002803 (visited on 2021-06-14), doi:10.1371/journal.pcbi.1002803.">194</a>]</span>. ABC is a likelihood-free approach that compares model output to a set of summary statistics. ABC is therefore well-suited for models and residual structures which do not lend themselves to a computationally-tractable likelihood, but the resulting inferences are known to be biased if the set of summary statistics is not sufficient, which can be difficult to know a-priori.</p>
+</section>
+<section id="the-critical-first-step-how-to-choose-a-prior-distribution">
+<span id="critical-first-step"></span><h2><span class="section-number">A.6. </span>The Critical First Step: How to Choose a Prior Distribution<a class="headerlink" href="#the-critical-first-step-how-to-choose-a-prior-distribution" title="Link to this heading">#</a></h2>
+<p>Prior distributions play an important role in Bayesian uncertainty quantification, particularly when data is limited relative to the dimension of the model. Bayesian updating can be thought of as an information filter, where each additional datum is added to the information contained in the prior; eventually, the prior makes relatively little impact. In real world problems, it can be extremely difficult to assess how much data is required for the choice of prior to become less relevant. The choice of prior can also be influential when conducting SA prior to or without UQ. This is because a prior distribution for a sensitivity analysis for a given parameter which is much wider than the region where the model response surface is sensitive to the parameter value might cause the sensitivity calculation to underestimate the response in that potentially critical region. Similarly, a prior which is too narrow may miss regions where the model responds to the parameter altogether.</p>
+<p>Ideally, prior distributions are constructed independently of any analysis of the new data considered. This is because using data to inform the prior as well as to compute the likelihood reuses information in a potentially inappropriate way, which can lead to overconfident inferences. Following <span id="id31">Jaynes [<a class="reference internal" href="R.Bibliography.html#id196" title="Edwin T. Jaynes. Probability theory: the logic of science. Washington University St. Louis, MO, 1996.">195</a>], Gelman <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id197" title="Andrew Gelman, Daniel Simpson, and Michael Betancourt. The Prior Can Often Only Be Understood in the Context of the Likelihood. Entropy, 19(10):555, October 2017. Number: 10 Publisher: Multidisciplinary Digital Publishing Institute. URL: https://www.mdpi.com/1099-4300/19/10/555 (visited on 2021-06-14), doi:10.3390/e19100555.">196</a>]</span> refers to the ideal prior as one which encodes all available information about the model. For practical reasons (difficulty of construction or computational inconvenience), most priors fail to achieve this ideal. These compromises mean that priors should be transparently articulated and justified, so that the impact of the choice of prior can be fully understood. When there is ambiguity about an appropriate prior, such as how fat the tails should be, an analyst should examine how sensitive the UQ results are to the choice of prior.</p>
+<p>Priors can also be classified in terms of the information encoded by them, demonstrated in <a class="reference internal" href="#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a>. Non-informative priors (illustrated in <a class="reference internal" href="#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a> (a)) allegedly correspond to (and are frequently justified by) a position of ignorance. A classic example is the use of a uniform distribution. A uniform prior can, however, be problematic, as it can lead to improper inferences by giving extremely large values the same prior probability as values which may seem more likely <span id="id32">[<a class="reference internal" href="R.Bibliography.html#id197" title="Andrew Gelman, Daniel Simpson, and Michael Betancourt. The Prior Can Often Only Be Understood in the Context of the Likelihood. Entropy, 19(10):555, October 2017. Number: 10 Publisher: Multidisciplinary Digital Publishing Institute. URL: https://www.mdpi.com/1099-4300/19/10/555 (visited on 2021-06-14), doi:10.3390/e19100555.">196</a>, <a class="reference internal" href="R.Bibliography.html#id198" title="Christian Robert. The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer Science &amp; Business Media, 2007.">197</a>]</span>, and therefore does not really reflect a state of complete ignorance. In the extreme case of a uniform prior over the entire real line, every particular region has effectively a prior weight of zero, even though not all regions are a priori unlikely <span id="id33">[<a class="reference internal" href="R.Bibliography.html#id198" title="Christian Robert. The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer Science &amp; Business Media, 2007.">197</a>]</span>. Moreover, a uniform prior which excludes possible parameter values is not actually noninformative, as it assigns zero probability to those values while jumping to a nonzero probability as soon as the boundary is crossed. While a uniform prior can be problematic for the task of uncertainty quantification, it may be useful for an initial sensitivity analysis to identify the boundary of any regions where the model is sensitive to the parameter.</p>
+<figure class="margin-caption align-center" id="id41">
+<span id="figure-a1-5"></span><a class="reference internal image-reference" href="_images/figureA1_5_priors_posteriors.png"><img alt="Figure A1.5" src="_images/figureA1_5_priors_posteriors.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. A.5 </span><span class="caption-text">Impact of priors on posterior inferences. These plots show the results of inference for a linear regression model with 15 data points. The true value of the parameter is equal to -3. All priors have mean 0. In panel (a), a non-informative prior allows the tails of the posterior to extend freely, which may result in unreasonably large parameter values. In panel (b), a weakly informative prior constrains the tails more, but allows them to extend without too much restriction. In panel (c), an informative prior strongly constrains the tails of the posterior and biases the inference closer towards the prior mean (the posterior mean is -0.89 in this case, and closer to -3 in the other two cases).</span><a class="headerlink" href="#id41" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>Informative priors strongly bound the range of probable values (illustrated in <a class="reference internal" href="#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a> (c)). One example is a Gaussian distribution with a relatively small standard deviation, so that large values are assigned a close to null prior probability. Another example is the jump from zero to non-zero probability occurring at the truncation point of a truncated Gaussian, which could be justified based on information that the parameter cannot take on values beyond this point. Without this type of justification, however, priors may be too informative, failing to allow the information contained in the available data to update them.</p>
+<p>Finally, weakly informative priors (illustrated in <a class="reference internal" href="#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a> (b)) fall in between <span id="id34">[<a class="reference internal" href="R.Bibliography.html#id197" title="Andrew Gelman, Daniel Simpson, and Michael Betancourt. The Prior Can Often Only Be Understood in the Context of the Likelihood. Entropy, 19(10):555, October 2017. Number: 10 Publisher: Multidisciplinary Digital Publishing Institute. URL: https://www.mdpi.com/1099-4300/19/10/555 (visited on 2021-06-14), doi:10.3390/e19100555.">196</a>]</span>. They regularize better than non-informative priors, but allow for more inference flexibility than fully informative priors. An example might be a Gaussian distribution with a moderate standard deviation, which still assigns negligible probability for values far away from the mean, but is less constrained than a narrow Gaussian for a reasonably large area. A key note is that it is not necessarily better to be more informative if this cannot be justified by the available information.</p>
+</section>
+<section id="the-critical-final-step-predictive-checks">
+<h2><span class="section-number">A.7. </span>The Critical Final Step: Predictive Checks<a class="headerlink" href="#the-critical-final-step-predictive-checks" title="Link to this heading">#</a></h2>
+<p>Every UQ workflow requires a number of choices, potentially including selecting prior distributions, the likelihood specification, and any used numerical models. Checking the appropriateness of these choices is an essential step for sound inferences, as misspecification can produce biased results <span id="id35">[<a class="reference internal" href="R.Bibliography.html#id180" title="Jenný Brynjarsdóttir and Anthony O'Hagan. Learning about physical parameters: the importance of model discrepancy. Inverse Problems, 30(11):114007, October 2014. Publisher: IOP Publishing. URL: https://doi.org/10.1088/0266-5611/30/11/114007 (visited on 2021-06-14), doi:10.1088/0266-5611/30/11/114007.">179</a>]</span>. Model checking in this fashion is part of an iterative UQ process, as the results can reveal adjustments to the statistical model or the need to select a different numerical model <span id="id36">[<a class="reference internal" href="R.Bibliography.html#id200" title="Andrew Gelman, Xiao-Li Meng, and Hal Stern. Posterior Predictive Assessment of Model Fitness via Realized Discrepancies. Statistica Sinica, 6(4):733–760, 1996. Publisher: Institute of Statistical Science, Academia Sinica. URL: https://www.jstor.org/stable/24306036 (visited on 2021-06-14).">198</a>, <a class="reference internal" href="R.Bibliography.html#id199" title="Andrew Gelman, Aki Vehtari, Daniel Simpson, Charles C. Margossian, Bob Carpenter, Yuling Yao, Lauren Kennedy, Jonah Gabry, Paul-Christian Bürkner, and Martin Modrák. Bayesian Workflow. arXiv:2011.01808 [stat], November 2020. arXiv: 2011.01808. URL: http://arxiv.org/abs/2011.01808 (visited on 2021-06-14).">199</a>, <a class="reference internal" href="R.Bibliography.html#id201" title="Andrew Gelman and Cosma Rohilla Shalizi. Philosophy and the practice of Bayesian statistics. British Journal of Mathematical and Statistical Psychology, 66(1):8–38, 2013. _eprint: https://bpspsychub.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2044-8317.2011.02037.x. URL: https://bpspsychub.onlinelibrary.wiley.com/doi/abs/10.1111/j.2044-8317.2011.02037.x (visited on 2021-06-14), doi:10.1111/j.2044-8317.2011.02037.x.">200</a>]</span>.</p>
+<p>A classic example is the need to check the structure of residuals for correlations. Many standard statistical models, such as linear regression, assume that the residuals are independent and identically distributed from the error distribution. The presence of correlations, including temporal autocorrelations and spatial correlations, indicates a structural mismatch between the likelihood and the data. In these cases, the likelihood should be adjusted to account for these correlations.</p>
+<p>Checking residuals in this fashion is one example of a predictive check (or a posterior predictive check in the Bayesian setting). One way to view UQ is as a means to recover data-generating processes (associated with each parameter vector) consistent with the observations. Predictive checks compare the inferred data-generating process to the observations to determine whether the model is capable of appropriately capturing uncertainty. After conducting the UQ analysis, alternatively realized datasets are simulated from sampled parameters. These alternative datasets, or their summary statistics, can be tested against the observations to determine adequacy of the fit. Predictive checks are therefore a way of probing various model components to identify shortcomings that might result in biased inferences or poor projections, depending on the goal of the analysis.</p>
+<p>One example of a graphical predictive check for time series models is hindcasting, where predictive intervals are constructed from the alternative datasets and plotted along with the data. Hindcasts demonstrate how well the model is capable of capturing the broader dynamics of the data, as well as whether the parameter distributions produce appropriate levels of output uncertainty. A related quantitative check is the surprise index, which calculates the percentage of data points located within a fixed predictive interval. For example, the 90% predictive interval should contain approximately 90% of the data. More uncertainty than this reflects underconfidence, while less uncertainty reflects overconfidence. This could be the result of priors that are not appropriately informative, or a likelihood that does not account for correlations between data points appropriately. It could also be the result of a numerical model that isn’t sufficiently sensitive to the parameters that are treated as uncertain.</p>
+</section>
+<section id="key-take-home-points">
+<h2><span class="section-number">A.8. </span>Key Take-Home Points<a class="headerlink" href="#key-take-home-points" title="Link to this heading">#</a></h2>
+<p>When appropriate, UQ is an important component of the exploratory modeling workflow. While a number of parameter sets could be consistent with observations, they may result in divergent model outputs when exposed to different future conditions. This can result in identifying risks which are not visible when selecting a “best fit” parameterization. Quantifying uncertainties also allows us to quantify the support for hypotheses, which is an essential part of the scientific process.</p>
+<p>Due to the scale and complexity of the experiments taking place in IM3, UQ has not been extensively used. The tradeoff between the available number of function evaluations and the number of uncertain parameters illustrated in <a class="reference internal" href="#figure-a1-1"><span class="std std-numref">Fig. A.1</span></a> is particularly challenging due to the increasing complexity of state-of-the-art models and the movement towards coupled, multisector models. This tradeoff can be addressed somewhat through the use of emulators and parallelizable methods. In particular, when attempting to navigate this tradeoff by limiting the number of uncertain parameters, it is important to carefully iterate with sensitivity analyses to ensure that critical parameters are identified.</p>
+<p>Specifying prior distributions and likelihoods is an ongoing challenge. Prior distributions, in particular, should be treated as deeply uncertain when appropriate. One key advantage of the methods described in this chapter is that they have the potential for increased transparency. When it is not possible to conduct a sensitivity analysis on a number of critical priors due to limited computational budgets, fully specifying and providing a justification for the utilized distributions allows other researchers to identify key assumptions and build on existing work. The same is true for the specification of likelihoods—while likelihood-free methods avoid the need to specify a likelihood function, they require other assumptions or choices, which should be described and justified as transparently as possible.</p>
+<p>We conclude this appendix with some key recommendations:
+1. UQ analysis does not require full confidence in priors and likelihoods. Rather, UQ should be treated as part of an exploratory modeling workflow, where hypotheses related to model structures, prior distributions, and likelihoods can be tested.
+2. For complex multisectoral models, UQ will typically require the use of a reduced set of parameters, either through emulation or by fixing the others to their best-fit values. These parameters should be selected through a careful sensitivity analysis.
+3. Avoid the use of supposedly “non-informative” priors, such as uniform priors, whenever possible. In the absence of strong information about parameter values, the use of weakly informative priors, such as diffuse normals, is preferable.
+4. Be cognizant of the limitations of conclusions that can be drawn by using each method. The bootstrap, for example, may result in overconfidence if the dataset is limited and is not truly representative of the underlying stochastic process.
+5. When using MCMC, Markov chains can not be shown to have converged to the target distribution, but rather evidence can be collected to demonstrate that it is likely that they have.
+6. Conduct predictive checks based on the assumptions underlying the choices made in the analysis, and iteratively update those choices if the assumptions prove to be ill-suited for the problem at hand.</p>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">A.1. Introduction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#parametric-bootstrap">A.2. Parametric Bootstrap</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pre-calibration">A.3. Pre-Calibration</a></li>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-critical-first-step-how-to-choose-a-prior-distribution">A.6. The Critical First Step: How to Choose a Prior Distribution</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-critical-final-step-predictive-checks">A.7. The Critical Final Step: Predictive Checks</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#key-take-home-points">A.8. Key Take-Home Points</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="3_sensitivity_analysis_the_basics.html">3. Sensitivity Analysis: The Basics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html">4. Sensitivity Analysis: Diagnostic &amp; Exploratory Modeling</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="A1_Uncertainty_Quantification.html">A. Uncertainty Quantification</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="A3_plotting_code.html">C. Plotting Code Samples</a></li>
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+    <h1>Fishery Dynamics Tutorial</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics">Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s get started!</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-identified-solutions-and-explore-performance">Step 1: Load identified solutions and explore performance</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis">Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-evaluate-the-system-over-all-generated-states-of-the-world">Step 3: Evaluate the system over all generated states of the world</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-calculate-sensitivity-indices">Step 4: Calculate sensitivity indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-explore-relationship-between-uncertain-factors-and-performance">Step 5: Explore relationship between uncertain factors and performance</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem">Tips to Apply Sobol SA and Scenario Discovery to your Problem</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
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+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="fishery-dynamics-tutorial">
+<h1>Fishery Dynamics Tutorial<a class="headerlink" href="#fishery-dynamics-tutorial" title="Link to this heading">#</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/fishery_dynamics.ipynb">Fishery Dynamics Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics">
+<h2>Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics<a class="headerlink" href="#tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics" title="Link to this heading">#</a></h2>
+<p>This notebook demonstrates the application of sensitivity analysis to
+discover factors that shape the behavior modes of a socio-ecological
+system with dynamic human action.</p>
+<p>The system of differential equations below represent a system of prey
+(defined in the equation below as x) and predator (defined as y) fish,
+with a human actor harvesting the prey fish. You can read more about
+this system at <a class="reference external" href="https://doi.org/10.1155/2020/4170453">Hadjimichael et
+al. (2020)</a>.</p>
+<img alt="_images/eqn2.png" src="_images/eqn2.png" />
+<p>The table below defines the parameters in the system and also denotes
+the baseline and ranges associated with each uncertain parameter.</p>
+<img alt="_images/table1.png" src="_images/table1.png" />
+<p>The system is simple but very rich in the dynamic behaviors it exhibits.
+This complexity is accompanied by the presence of several equilibria
+that come in and out of existence with different parameter values. The
+equilibria also change in their stability according to different
+parameter values, giving rise to different behavior modes as shown by
+the diverse predator and prey abundace trajectories in the figure below.</p>
+<img alt="_images/Figure_1.png" src="_images/Figure_1.png" />
+<p>In the unharvested system (without the human actor) the stability of
+several of these equilibria can be derived analytically. The task
+becomes significantly more difficult when the adaptive human actor is
+introduced, deciding to harvest the system at different rates according
+to their objectives and preferences.</p>
+<p>Sensitivity analysis methods can help us identify the factors that most
+control these dynamics by exploring the space of parameter values and
+seeing how system outputs change as a result.</p>
+<p>Through previously conducted optimization, there already exists a set of
+potential harvesting strategies that were identified in pursuit of five
+objectives:</p>
+<ul class="simple">
+<li><p>Maximize Harvesting Discounted Profits (Net Present Value)</p></li>
+<li><p>Minimize Prey Population Deficit</p></li>
+<li><p>Minimize Longest Duration of Consecutive Low Harvest</p></li>
+<li><p>Maximize Worst Harvest Instance</p></li>
+<li><p>Minimize Harvest Variance</p></li>
+</ul>
+<p>The identified harvesting strategies also meet the necessary constraint
+of not causing inadvertent predator collapse.</p>
+<p>We will be examining the effects of parametric uncertainty on these
+identified strategies, particularly focusing on two strategies: one
+selected to maximize harvesting profits and one identified through
+previous analysis to perform ‘well enough’ for all objectives across a
+wide range of states of the world (referred to as the ‘robust’
+harvesting policy).</p>
+<section id="lets-get-started">
+<h3>Let’s get started!<a class="headerlink" href="#lets-get-started" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we will be loading in data that has been produced in
+Hadjimichael et al. (2020). Before we start our analysis, we’ll load the
+relevant Python libraries. <strong>NOTE</strong>: To step through the notebook,
+execute each gray (code) box by typing “Shift+Enter”.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Import necessary libraries</span>
+
+<span class="kn">import</span> <span class="nn">msdbook</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">saltelli</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">sobol</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">patheffects</span> <span class="k">as</span> <span class="n">pe</span>
+
+<span class="c1"># load example data</span>
+<span class="n">msdbook</span><span class="o">.</span><span class="n">install_package_data</span><span class="p">()</span>
+
+<span class="o">%</span><span class="k">matplotlib</span> inline
+<span class="o">%</span><span class="k">config</span> InlineBackend.print_figure_kwargs = {&#39;bbox_inches&#39;:None}
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Downloading</span> <span class="n">example</span> <span class="n">data</span> <span class="k">for</span> <span class="n">msdbook</span> <span class="n">version</span> <span class="mf">0.1.5</span><span class="o">...</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">uncertain_params_bounds</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_metric_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">solutions</span><span class="o">.</span><span class="n">resultfile</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LHsamples_original_1000</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">param_values</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_params_256samples</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LeafCatch</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_simulations_256samples</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">Robustness</span><span class="o">.</span><span class="n">txt</span>
+</pre></div>
+</div>
+</section>
+<section id="step-1-load-identified-solutions-and-explore-performance">
+<h3>Step 1: Load identified solutions and explore performance<a class="headerlink" href="#step-1-load-identified-solutions-and-explore-performance" title="Link to this heading">#</a></h3>
+<p>Here we load in the solution set obtained in Hadjimichael et al. (2020).
+The solution set contains the decision variables and objectives
+associated with a variety of harvesting policies. For this tutorial, we
+focus on comparing two policies: harvesting profits and one that
+performs robustly across all objectives. Below, we are reading in the
+decision variables and objectives from an external file that can be
+found within the msdbook package data.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">robustness</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_robustness_data</span><span class="p">()</span>
+<span class="n">results</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_profit_maximization_data</span><span class="p">()</span>
+
+<span class="n">robust_solution</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">robustness</span><span class="p">[:,</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="c1">#pick robust solution</span>
+<span class="n">profit_solution</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">results</span><span class="p">[:,</span><span class="mi">6</span><span class="p">])</span> <span class="c1">#pick profitable solution</span>
+<span class="n">objective_performance</span> <span class="o">=</span> <span class="o">-</span><span class="n">results</span><span class="p">[:,</span><span class="mi">6</span><span class="p">:]</span> <span class="c1">#Retain objective values</span>
+
+<span class="c1"># Get decision variables for each of the policies</span>
+<span class="n">highprofitpolicy</span> <span class="o">=</span> <span class="n">results</span><span class="p">[</span><span class="n">profit_solution</span><span class="p">,</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">]</span>
+<span class="n">mostrobustpolicy</span> <span class="o">=</span> <span class="n">results</span><span class="p">[</span><span class="n">robust_solution</span><span class="p">,</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">]</span>
+</pre></div>
+</div>
+<p>Next we plot the identified solutions with regards to their objective
+performance in a parallel axis plot</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-objective-performance">plot_objective_performance</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ax</span><span class="p">,</span> <span class="n">ax1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_objective_performance</span><span class="p">(</span><span class="n">objective_performance</span><span class="p">,</span> <span class="n">profit_solution</span><span class="p">,</span> <span class="n">robust_solution</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/fishery_output_6_0.png" src="_images/fishery_output_6_0.png" />
+<p>The solution set from the optimization in Hadjimichael et al. (2020) are
+presented in a parallel axis plot where each of the five objectives (and
+one constraint) are represented as an axis. Each solution on the Pareto
+front is represented as a line where the color of the line indicates the
+value of the NPV objective. The preference for objective values is in
+the upward direction. Therefore, the ideal solution would be a line
+straight across the top of the plot that satisfies every objective.
+However, no such line exists because there are tradeoffs when sets of
+objectives are prioritized over the others. When lines cross in between
+axes, this indicates a tradeoff between objectives (as seen in the first
+two axes).The solution that is most robust in the NPV objective has the
+highest value on the first axis and is outlined in dark gold. The
+solution that is most robust across all objectives is outlined in a
+brighter yellow. A parallel axis is an effective visual to characterize
+high-dimensional tradeoffs in the system and visualize differences in
+performance across policies.</p>
+</section>
+<section id="step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis">
+<h3>Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis<a class="headerlink" href="#step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis" title="Link to this heading">#</a></h3>
+<p>In Step 1, we showed how the optimized harvesting policies performed in
+the objective space, which utilized the baseline parameters outlined in
+the table above. Now, we are interested in understanding how sensitive
+our two policies are to alternative states of the world that may be
+characterized by different parameter values. To do so, we first need to
+define the problem dictionary that allows us to generate these
+alternative states of the world.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set up SALib problem</span>
+<span class="n">problem</span> <span class="o">=</span> <span class="p">{</span>
+  <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">9</span><span class="p">,</span>
+  <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;a&#39;</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="s1">&#39;c&#39;</span><span class="p">,</span> <span class="s1">&#39;d&#39;</span><span class="p">,</span> <span class="s1">&#39;h&#39;</span><span class="p">,</span> <span class="s1">&#39;K&#39;</span><span class="p">,</span> <span class="s1">&#39;m&#39;</span><span class="p">,</span> <span class="s1">&#39;sigmaX&#39;</span><span class="p">,</span> <span class="s1">&#39;sigmaY&#39;</span><span class="p">],</span>
+  <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="mf">0.002</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.005</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+             <span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">5000</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">]]</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+<p>Then we use the following command to generate a Saltelli sample from
+these defined ranges:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">param_values</span> <span class="o">=</span> <span class="n">saltelli</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="mi">1024</span><span class="p">,</span> <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Generally, it is a good idea to save the result of the sample since it
+is often reused and regenerating it produces a different sample set. For
+this reason, we will load one from file that was previously generated.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously generated Saltelli sample from our msdbook package data</span>
+<span class="n">param_values</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_saltelli_param_values</span><span class="p">()</span>
+</pre></div>
+</div>
+</section>
+<section id="step-3-evaluate-the-system-over-all-generated-states-of-the-world">
+<h3>Step 3: Evaluate the system over all generated states of the world<a class="headerlink" href="#step-3-evaluate-the-system-over-all-generated-states-of-the-world" title="Link to this heading">#</a></h3>
+<p>Now we re-evaluate how well the policies do in the new states of the
+world. In order to characterize failure of a policy, we identify the
+states where the predator population collapses, as an inadvertent
+consequence of applying the harvesting strategy under a state of the
+world different from the one originally assumed. Due to how long this
+step takes to execute within the tutorial, we will read in the solutions
+from an external file. However, the block of code below shows how
+evaluation can be implemented.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># create array to store collapse values under both policies</span>
+<span class="n">collapse_days</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="nb">len</span><span class="p">(</span><span class="n">param_values</span><span class="p">),</span> <span class="mi">2</span><span class="p">])</span>
+
+<span class="c1"># evaluate performance under every state</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">param_values</span><span class="p">)):</span>
+
+    <span class="n">additional_inputs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="s1">&#39;Previous_Prey&#39;</span><span class="p">],</span>
+                                  <span class="p">[</span><span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">6</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">7</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span>
+
+    <span class="n">collapse_days</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span><span class="n">fish_game</span><span class="p">(</span><span class="n">highprofitpolicy</span><span class="p">,</span> <span class="n">additional_inputs</span><span class="p">)[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+    <span class="n">collapse_days</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">=</span><span class="n">fish_game</span><span class="p">(</span><span class="n">mostrobustpolicy</span><span class="p">,</span> <span class="n">additional_inputs</span><span class="p">)[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load the simulation data from our msdbook package data</span>
+<span class="n">collapse_days</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_collapse_data</span><span class="p">()</span>
+</pre></div>
+</div>
+</section>
+<section id="step-4-calculate-sensitivity-indices">
+<h3>Step 4: Calculate sensitivity indices<a class="headerlink" href="#step-4-calculate-sensitivity-indices" title="Link to this heading">#</a></h3>
+<p>Now we use a Sobol sensitivity analysis to calculate first-order,
+second-order, and total-order sensitivity indices for each parameter and
+for each of the two policies. These indicies help determine which
+factors explain the most variability in the number of days of predator
+population collapse.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Perform the Sobol SA for the profit-maximizing solution</span>
+<span class="n">Si_profit</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="n">collapse_days</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
+                          <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+                          <span class="n">conf_level</span><span class="o">=</span><span class="mf">0.95</span><span class="p">,</span>
+                          <span class="n">print_to_console</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Perform the Sobol SA for the robust solution</span>
+<span class="n">Si_robustness</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span>
+                              <span class="n">collapse_days</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
+                              <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+                              <span class="n">conf_level</span><span class="o">=</span><span class="mf">0.95</span><span class="p">,</span>
+                              <span class="n">print_to_console</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>              <span class="n">ST</span>   <span class="n">ST_conf</span>
+<span class="n">a</span>       <span class="mf">0.226402</span>  <span class="mf">0.036146</span>
+<span class="n">b</span>       <span class="mf">0.066819</span>  <span class="mf">0.013347</span>
+<span class="n">c</span>       <span class="mf">0.004395</span>  <span class="mf">0.004023</span>
+<span class="n">d</span>       <span class="mf">0.024509</span>  <span class="mf">0.006993</span>
+<span class="n">h</span>       <span class="mf">0.009765</span>  <span class="mf">0.005488</span>
+<span class="n">K</span>       <span class="mf">0.020625</span>  <span class="mf">0.009494</span>
+<span class="n">m</span>       <span class="mf">0.897971</span>  <span class="mf">0.066470</span>
+<span class="n">sigmaX</span>  <span class="mf">0.000136</span>  <span class="mf">0.000149</span>
+<span class="n">sigmaY</span>  <span class="mf">0.000739</span>  <span class="mf">0.001040</span>
+              <span class="n">S1</span>   <span class="n">S1_conf</span>
+<span class="n">a</span>       <span class="mf">0.087936</span>  <span class="mf">0.044236</span>
+<span class="n">b</span>       <span class="mf">0.000554</span>  <span class="mf">0.021474</span>
+<span class="n">c</span>      <span class="o">-</span><span class="mf">0.002970</span>  <span class="mf">0.004590</span>
+<span class="n">d</span>       <span class="mf">0.001206</span>  <span class="mf">0.015881</span>
+<span class="n">h</span>       <span class="mf">0.004554</span>  <span class="mf">0.007998</span>
+<span class="n">K</span>       <span class="mf">0.003843</span>  <span class="mf">0.012661</span>
+<span class="n">m</span>       <span class="mf">0.751301</span>  <span class="mf">0.071862</span>
+<span class="n">sigmaX</span> <span class="o">-</span><span class="mf">0.000325</span>  <span class="mf">0.001245</span>
+<span class="n">sigmaY</span> <span class="o">-</span><span class="mf">0.001887</span>  <span class="mf">0.002768</span>
+</pre></div>
+</div>
+<p>Looking at the total-order indices, (ST) factors <span class="math notranslate nohighlight">\(m\)</span>, <span class="math notranslate nohighlight">\(a\)</span>,
+<span class="math notranslate nohighlight">\(b\)</span>, <span class="math notranslate nohighlight">\(d\)</span> and <span class="math notranslate nohighlight">\(K\)</span> explain a non-negligible amount of
+variance therefore have an effect on the stability of this system.
+Looking at the first-order indices (S1), we also see that besides
+factors <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(a\)</span>, all other factors are important in this
+system through their interactions, which make up the difference between
+their S1 and ST indices. This shows the danger of limiting sensitivity
+analyses to first order effects, as factor importance might be
+significantly misjudged.</p>
+<p>These findings are supported by the analytical condition of equilibrium
+stability in this system:</p>
+<img alt="_images/eqn4.png" src="_images/eqn4.png" />
+<p>In an unharvested system, this condition is both necessary and
+sufficient for the equilibrium of the two species coexisting to be
+stable.</p>
+<p>When adaptive human action is introduced however, this condition is
+still necessary, but no longer sufficient, as harvesting reduces the
+numbers of prey fish and as a result reduces the resources for the
+predator fish. Since this harvesting value is not constant, but can
+dynamically adapt according to the harvester’s objectives, it cannot be
+introduced into this simple equation.</p>
+</section>
+<section id="step-5-explore-relationship-between-uncertain-factors-and-performance">
+<h3>Step 5: Explore relationship between uncertain factors and performance<a class="headerlink" href="#step-5-explore-relationship-between-uncertain-factors-and-performance" title="Link to this heading">#</a></h3>
+<p>In the following steps, we will use the results of our sensitivity
+analysis to investigate the relationships between parametric
+uncertainty, equilibrium stability and the performance of the two
+policies.</p>
+<p>We can use the top three factors identified (<span class="math notranslate nohighlight">\(m\)</span>, <span class="math notranslate nohighlight">\(a\)</span>, and
+<span class="math notranslate nohighlight">\(b\)</span>) to visualize the performance of our policies in this
+three-dimensional parametric space.</p>
+<p>We first define the stability condition, as a function of <span class="math notranslate nohighlight">\(b\)</span> and
+<span class="math notranslate nohighlight">\(m\)</span>, and calculate the corresponding values of <span class="math notranslate nohighlight">\(a\)</span>.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">inequality</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
+    <span class="k">return</span> <span class="p">((</span><span class="n">b</span><span class="o">**</span><span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">h</span><span class="o">*</span><span class="n">K</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">m</span><span class="p">))</span>
+
+<span class="c1"># boundary interval that separates successful and failed states of the world</span>
+<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mf">0.005</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">m</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">K</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">2000</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">b</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">inequality</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">clip</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-factor-performance">plot_factor_performance</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># generate plot</span>
+<span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_factor_performance</span><span class="p">(</span><span class="n">param_values</span><span class="p">,</span> <span class="n">collapse_days</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/fishery_output_22_0.png" src="_images/fishery_output_22_0.png" />
+<p>These figures show the combinations of factors that lead to success or
+failure in different states of the world for the profit-maximizing and
+robust policies. Each point is a state of the world, characterized by
+specific values of the parameters, and ideally, we would like the color
+of the point to be blue, to represent that there are a low number of
+days with a predator collapse in that world. The gray curve denotes the
+highly non-linear nature of the boundary, defined by the stability
+condition, that separates successful and failed states of the world. The
+figures demonstrate the following key points:</p>
+<p>First, as asserted above, the policies interact with the system in
+different and complex ways. In the presence of human action, the
+stability condition is not sufficient in determining whether the policy
+will succeed, even though it clearly shapes the system in a fundamental
+manner.</p>
+<p>Secondly, the robust policy manages to avoid collapse in many more of the sampled states of the world, indicated by the number of blue points. The robust policy avoids collapse in 31% of worlds versus 14% in the profit-maximizing policy.This presents a clear tradeoff between profit-maximizing performance androbustness against uncertainty.</p>
+</section>
+<section id="tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem">
+<h3>Tips to Apply Sobol SA and Scenario Discovery to your Problem<a class="headerlink" href="#tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we demonstrated a Sobol SA to identify the most
+important factors driving the behavior of a system (i.e. the number of
+the collapse days). In order to apply this methodology to your problem,
+you will need to have a set of optimized policies for your system that
+you are interested in analyzing. The general workflow is as follows:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters and set up the problem
+dictionary as in Step 2 above.</p></li>
+<li><p>Generate samples, or alternative states of the world using the
+<code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function.</p></li>
+<li><p>Evaluate your policies on the alternative states of the world. For
+your application, you will also need to develop a rule for
+determining success or failure of your policy in a new SOW. In this
+tutorial, success was denoted by a small number of collapse days.
+Ultimately, the rule will be specific to your application and can
+include various satisficing criteria.</p></li>
+<li><p>Calculate the Sobol indices and discover the most important
+parameters driving success and failure.</p></li>
+<li><p>Finally, use a similar plotting procedure as in step 5 to identify
+the combination of parameter values that lead to success and failure
+in the system.</p></li>
+</ol>
+</section>
+</section>
+</section>
+
+
+                </article>
+              
+
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics">Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s get started!</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-identified-solutions-and-explore-performance">Step 1: Load identified solutions and explore performance</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis">Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-evaluate-the-system-over-all-generated-states-of-the-world">Step 3: Evaluate the system over all generated states of the world</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-calculate-sensitivity-indices">Step 4: Calculate sensitivity indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-explore-relationship-between-uncertain-factors-and-performance">Step 5: Explore relationship between uncertain factors and performance</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem">Tips to Apply Sobol SA and Scenario Discovery to your Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html">2. Diagnostic Modeling Overview and Perspectives</a></li>
+<li class="toctree-l1"><a class="reference internal" href="3_sensitivity_analysis_the_basics.html">3. Sensitivity Analysis: The Basics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html">4. Sensitivity Analysis: Diagnostic &amp; Exploratory Modeling</a></li>
+<li class="toctree-l1"><a class="reference internal" href="5_conclusion.html">5. Conclusion</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="A1_Uncertainty_Quantification.html">A. Uncertainty Quantification</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="A3_plotting_code.html">C. Plotting Code Samples</a></li>
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+    <h1>Sobol SA Tutorial</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa">Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s get started!</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-generate-the-problem-dictionary">Step 1: Generate the problem dictionary</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-generate-samples-using-the-saltelli-sampling-scheme">Step 2: Generate samples using the Saltelli sampling scheme</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-execute-the-ishigami-function-over-our-sample-set">Step 3: Execute the Ishigami function over our sample set</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method">Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-interpret-our-results">Step 5: Interpret our results</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-to-your-own-problem">Tips to Apply Sobol SA to Your Own Problem</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="sobol-sa-tutorial">
+<h1>Sobol SA Tutorial<a class="headerlink" href="#sobol-sa-tutorial" title="Link to this heading">#</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA Tutorial</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa">
+<h2>Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA<a class="headerlink" href="#tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa" title="Link to this heading">#</a></h2>
+<p>In this tutorial, we will set up a workflow to investigate how sensitive
+the output of a function is to its inputs. Why might you want to do
+this? Imagine that this function represents a complex system, such as
+the rainfall-runoff process of a watershed model, and that you, the
+researcher, want to investigate how your choice of input parameter
+values are affecting the model’s characterization of runoff in the
+watershed. Your parameter values are likely uncertain and can take on
+any value in a pre-defined range. Using a Sobol SA will allow you to
+sample different values of your parameters and calculate how sensitive
+your output of interest is to certain parameters. Below, we demonstrate
+Sobol SA for a simple function to illustrate the method, but the
+workflow can be applied to your own problem of interest!</p>
+<p>In order to conduct this analysis, we will use the popular Python
+Sensitivity Analysis Library
+(<a class="reference external" href="https://salib.readthedocs.io/en/latest/index.html">SALib</a>) to:</p>
+<ol class="arabic simple">
+<li><p>Generate a problem set as a dictionary for our Ishigami function that has three inputs</p></li>
+<li><p>Generate 2048 samples for our problem set using the Saltelli <a class="footnote-reference brackets" href="#id7" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id8" id="id2" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a> sampling scheme</p></li>
+<li><p>Execute the Ishigami function for each of our samples and gather the outputs</p></li>
+<li><p>Compute the sensitivity analysis to generate first-order and total-order sensitivity indices using the Sobol <a class="footnote-reference brackets" href="#id9" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a> method</p></li>
+<li><p>Interpret the meaning of our results</p></li>
+</ol>
+<section id="lets-get-started">
+<h3>Let’s get started!<a class="headerlink" href="#lets-get-started" title="Link to this heading">#</a></h3>
+<p><strong>NOTE</strong>: Content from this tutorial is taken directly from the SALib
+<a class="reference external" href="https://salib.readthedocs.io/en/latest/basics.html">“Basics”</a>
+walkthrough. To step through the notebook, execute each gray (code) box
+by typing “Shift+Enter”.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Import relevant libraries</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">from</span> <span class="nn">mpl_toolkits</span> <span class="kn">import</span> <span class="n">mplot3d</span>
+
+<span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">saltelli</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">sobol</span>
+<span class="kn">from</span> <span class="nn">SALib.test_functions</span> <span class="kn">import</span> <span class="n">Ishigami</span>
+</pre></div>
+</div>
+</section>
+<section id="step-1-generate-the-problem-dictionary">
+<h3>Step 1: Generate the problem dictionary<a class="headerlink" href="#step-1-generate-the-problem-dictionary" title="Link to this heading">#</a></h3>
+<p>The Ishigami function is of the form:</p>
+<div class="math notranslate nohighlight">
+\[f(x_1,x_2,x_3) = sin(x_1)+asin^2(x_2)+bx_3^4sin(x_1)\]</div>
+<p>The function has three inputs, 𝑥1, 𝑥2, 𝑥3 where 𝑥𝑖 ∈ [−𝜋, 𝜋]. The
+constants <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are defined as 7.0 and 0.1
+respectively.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Create a problem dictionary. Here we supply the number of variables, the names of each variable, and the bounds of the variables.</span>
+<span class="n">problem</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span>
+    <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;x1&#39;</span><span class="p">,</span> <span class="s1">&#39;x2&#39;</span><span class="p">,</span> <span class="s1">&#39;x3&#39;</span><span class="p">],</span>
+    <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="o">-</span><span class="mf">3.14159265359</span><span class="p">,</span> <span class="mf">3.14159265359</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">3.14159265359</span><span class="p">,</span> <span class="mf">3.14159265359</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">3.14159265359</span><span class="p">,</span> <span class="mf">3.14159265359</span><span class="p">]]</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+</section>
+<section id="step-2-generate-samples-using-the-saltelli-sampling-scheme">
+<h3>Step 2: Generate samples using the Saltelli sampling scheme<a class="headerlink" href="#step-2-generate-samples-using-the-saltelli-sampling-scheme" title="Link to this heading">#</a></h3>
+<p>Sobol SA requires the use of the Saltelli sampling scheme. The output of
+the <code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function is a NumPy array that is of shape 2048
+by 3. The sampler generates 𝑁∗(2𝐷+2) samples, where in this example, N
+is 256 (the argument we supplied) and D is 3 (the number of model
+inputs), yielding 2048 samples. The keyword argument
+<code class="docutils literal notranslate"><span class="pre">calc_second_order=False</span></code> will exclude second-order indices, resulting
+in a smaller sample matrix with 𝑁∗(𝐷+2) rows instead. Below, we plot the
+resulting Saltelli sample.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Generate parmeter values using the saltelli.sample function</span>
+<span class="n">param_values</span> <span class="o">=</span> <span class="n">saltelli</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="mi">256</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;`param_values` shape:  </span><span class="si">{</span><span class="n">param_values</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<pre class="literal-block"><cite>param_values</cite> shape:  (2048, 3)</pre>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Plot the 2048 samples of the parameters</span>
+
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
+<span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">axes</span><span class="p">(</span><span class="n">projection</span> <span class="o">=</span><span class="s2">&quot;3d&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter3D</span><span class="p">(</span><span class="n">param_values</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">param_values</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">param_values</span><span class="p">[:,</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;X1 Parameter&#39;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;X2 Parameter&#39;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s1">&#39;X3 Parameter&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Saltelli Sample of Parameter Values&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/output_7_0.png" src="_images/output_7_0.png" />
+</section>
+<section id="step-3-execute-the-ishigami-function-over-our-sample-set">
+<h3>Step 3: Execute the Ishigami function over our sample set<a class="headerlink" href="#step-3-execute-the-ishigami-function-over-our-sample-set" title="Link to this heading">#</a></h3>
+<p>SALib provides a nice wrapper to the Ishigami function that allows the
+user to directly pass the <code class="docutils literal notranslate"><span class="pre">param_values</span></code> array we just generated into
+the function directly.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Y</span> <span class="o">=</span> <span class="n">Ishigami</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">param_values</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method">
+<h3>Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method<a class="headerlink" href="#step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method" title="Link to this heading">#</a></h3>
+<p>The <code class="docutils literal notranslate"><span class="pre">sobol.analyze</span></code> function will use our problem dictionary and the
+result of the Ishigami runs (<code class="docutils literal notranslate"><span class="pre">Y</span></code>) to compute first-, second-, and
+total-order indicies.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Si</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><code class="docutils literal notranslate"><span class="pre">Si</span></code> is a Python dict with the keys “S1”, “S2”, “ST”, “S1_conf”,
+“S2_conf”, and “ST_conf”. The <code class="docutils literal notranslate"><span class="pre">_conf</span></code> keys store the corresponding
+confidence intervals, typically with a confidence level of 95%. Use the
+keyword argument <code class="docutils literal notranslate"><span class="pre">print_to_console=True</span></code> to print all indices. Or, we
+can print the individual values from <code class="docutils literal notranslate"><span class="pre">Si</span></code> as shown in the next step.</p>
+</section>
+<section id="step-5-interpret-our-results">
+<h3>Step 5: Interpret our results<a class="headerlink" href="#step-5-interpret-our-results" title="Link to this heading">#</a></h3>
+<p>We execute the following code and take a look at our first-order indices
+(<code class="docutils literal notranslate"><span class="pre">S1</span></code>) for each of our three inputs. These indicies can be interpreted
+as the fraction of variance in the output that is explained by each
+input individually.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">first_order</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">]</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;First-order:&#39;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1: </span><span class="si">{</span><span class="n">first_order</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">, x2: </span><span class="si">{</span><span class="n">first_order</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">, x3: </span><span class="si">{</span><span class="n">first_order</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">First</span><span class="o">-</span><span class="n">order</span><span class="p">:</span>
+<span class="n">x1</span><span class="p">:</span> <span class="mf">0.3184242969763115</span><span class="p">,</span> <span class="n">x2</span><span class="p">:</span> <span class="mf">0.4303808201623416</span><span class="p">,</span> <span class="n">x3</span><span class="p">:</span> <span class="mf">0.022687722804980225</span>
+</pre></div>
+</div>
+<p>If we were to rank the importance of the inputs in how much they
+individually explain the variance in the output, we would rank them from
+greatest to least importance as follows: 𝑥2, 𝑥1 and then 𝑥3. Since 𝑥3
+only explains 2% of the output variance, it does not explain output
+variability meaningfully. Thus, this indicates that there is
+contribution to the output variance by 𝑥2 and 𝑥1 independently, whereas
+𝑥3 does not contribute to the output variance. Determining what inputs
+are most important or what index value is meaningful is a common
+question, but one for which there is no general rule or threshold. This
+question is problem and context-dependent, but procedures have been
+identified to rank order influential inputs and which can be used to
+identify the least influential factors. These factors can be fixed to
+simplify the model <a class="footnote-reference brackets" href="#id10" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id12" id="id5" role="doc-noteref"><span class="fn-bracket">[</span>5<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id13" id="id6" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a>.</p>
+<p>Next, we evaluate the total-order indices, which measure the
+contribution to the output variance caused by varying the model input,
+including both its first-order effects (the input varying alone) and all
+higher-order interactions across the input parameters.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">total_order</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">]</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Total-order:&#39;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1: </span><span class="si">{</span><span class="n">total_order</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">, x2: </span><span class="si">{</span><span class="n">total_order</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">, x3: </span><span class="si">{</span><span class="n">total_order</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Total</span><span class="o">-</span><span class="n">order</span><span class="p">:</span>
+<span class="n">x1</span><span class="p">:</span> <span class="mf">0.5184119098161343</span><span class="p">,</span> <span class="n">x2</span><span class="p">:</span> <span class="mf">0.41021260250026054</span><span class="p">,</span> <span class="n">x3</span><span class="p">:</span> <span class="mf">0.2299058431439953</span>
+</pre></div>
+</div>
+<p>The magnitude of the total order indices are substantially larger than
+the first-order indices, which reveals that higher-order interactions
+are occurring, i.e. that the interactions across inputs are also
+explaining some of the total variance in the output. Note that 𝑥3 has
+non-negligible total-order indices, which indicates that it is not a
+consequential parameter when considered in isolation, but becomes
+consequential and explains 25% of variance in the output through its
+interactions with 𝑥1 and 𝑥2.</p>
+<p>Finally, we can investigate these higher order interactions by viewing
+the second-order indices. The second-order indicies measure the
+contribution to the output variance caused by the interaction between
+any two model inputs. Some computing error can appear in these
+sensitivity indices, such as negative values. Typically, these computing
+errors shrink as the number of samples increases.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">second_order</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S2&#39;</span><span class="p">]</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Second-order:&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1-x2:  </span><span class="si">{</span><span class="n">second_order</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1-x3:  </span><span class="si">{</span><span class="n">second_order</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x2-x3:  </span><span class="si">{</span><span class="n">second_order</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Second</span><span class="o">-</span><span class="n">order</span><span class="p">:</span>
+<span class="n">x1</span><span class="o">-</span><span class="n">x2</span><span class="p">:</span>  <span class="o">-</span><span class="mf">0.043237389723234154</span>
+<span class="n">x1</span><span class="o">-</span><span class="n">x3</span><span class="p">:</span>  <span class="mf">0.17506452088709862</span>
+<span class="n">x2</span><span class="o">-</span><span class="n">x3</span><span class="p">:</span>  <span class="o">-</span><span class="mf">0.03430682392607577</span>
+</pre></div>
+</div>
+<p>We can see that there are strong interactions between 𝑥1 and 𝑥3. Note
+that in the Ishigami function, these two variables are multiplied in the
+last term of the function, which leads to interactive effects. If we
+were considering first order indices alone, we would erroneously assume
+that 𝑥3 explains no variance in the output, but the second-order and
+total order indices reveal that this is not the case. It’s easy to
+understand where we might see interactive effects in the case of the
+simple Ishigami function. However, it’s important to remember that in
+more complex systems, there may be many higher-order interactions that
+are not apparent, but could be extremely consequential in explaining the
+variance of the output.</p>
+</section>
+<section id="tips-to-apply-sobol-sa-to-your-own-problem">
+<h3>Tips to Apply Sobol SA to Your Own Problem<a class="headerlink" href="#tips-to-apply-sobol-sa-to-your-own-problem" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we demonstrated how to apply an SA analysis to a
+simple mathematical test function. In order to apply a Sobol SA to your
+own problem, you will follow the same general workflow that we defined
+above. You will need to:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters and set up the problem
+dictionary as in Step 1 above.</p></li>
+<li><p>Generate samples using the <code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function. This step is
+problem-dependent and note that the Sobol method can be
+computationally intensive depending on the model being analyzed. For
+example, for a simple rainfall-runoff model such as HYMOD, it has
+been recommended to run a sample size of at least N = 10,000 (which
+translates to 60,000 model runs). More complex models will be slower
+to run and will also require more samples to calculate accurate
+estimates of Sobol indices. Once you complete this process, pay
+attention to the confidence bounds on your sensitivity indices to see
+whether you need to run more samples.</p></li>
+<li><p>Run the parameter sets through your model. In the example above, the
+Ishigami function could be evaluated through SALib since it is a
+built in function. For your application, you will need to run these
+parameter sets through the problem externally and save the output.
+The output file should contain one row of output values for each
+model run.</p></li>
+<li><p>Calculate the Sobol indices. Now, the Y will be a numpy array with
+your external model output and you will need to include the parameter
+samples as an additional argument.</p></li>
+<li><p>Finally, we interpet the results. If the confidence intervals of your
+dominant indices are larger than roughly 10% of the value itself, you
+may want to consider increasing your sample size as computation
+permits. You should additionally read the references noted in Step 5
+above to understand more about identifying important factors.</p></li>
+</ol>
+<p><strong>References</strong></p>
+<aside class="footnote-list brackets">
+<aside class="footnote brackets" id="id7" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">1</a><span class="fn-bracket">]</span></span>
+<p>Saltelli, A. (2002). “Making best use of model evaluations to compute sensitivity indices.” Computer Physics Communications, 145(2):280-297, doi:<a class="reference external" href="https://doi.org/10.1016/S0010-4655(02)00280-1">10.1016/S0010-4655(02)00280-1</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id8" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">2</a><span class="fn-bracket">]</span></span>
+<p>Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and S. Tarantola (2010). “Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index.” Computer Physics Communications, 181(2):259-270, doi:<a class="reference external" href="https://doi.org/10.1016/j.cpc.2009.09.018">10.1016/j.cpc.2009.09.018</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id9" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">3</a><span class="fn-bracket">]</span></span>
+<p>Sobol, I. M. (2001). “Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates.” Mathematics and Computers in Simulation, 55(1-3):271-280, doi:<a class="reference external" href="https://doi.org/10.1016/S0378-4754(00)00270-6">10.1016/S0378-4754(00)00270-6</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id10" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">4</a><span class="fn-bracket">]</span></span>
+<p>T. H. Andres, “Sampling methods and sensitivity analysis for large parameter sets,” Journal of Statistical Computation and Simulation, vol. 57, no. 1–4, pp. 77–110, Apr. 1997, doi:<a class="reference external" href="https://doi.org/10.1080/00949659708811804">10.1080/00949659708811804</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id12" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id5">5</a><span class="fn-bracket">]</span></span>
+<p>Y. Tang, P. Reed, T. Wagener, and K. van Werkhoven, “Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation,” Hydrology and Earth System Sciences, vol. 11, no. 2, pp. 793–817, Feb. 2007, doi:<a class="reference external" href="https://doi.org/10.5194/hess-11-793-2007">10.5194/hess-11-793-2007</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id13" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id6">6</a><span class="fn-bracket">]</span></span>
+<p>J. Nossent, P. Elsen, and W. Bauwens, “Sobol’ sensitivity analysis of a complex environmental model,” Environmental Modelling &amp; Software, vol. 26, no. 12, pp. 1515–1525, Dec. 2011, doi:<a class="reference external" href="https://doi.org/10.1016/j.envsoft.2011.08.010">10.1016/j.envsoft.2011.08.010</a>.</p>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa">Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s get started!</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-generate-the-problem-dictionary">Step 1: Generate the problem dictionary</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-generate-samples-using-the-saltelli-sampling-scheme">Step 2: Generate samples using the Saltelli sampling scheme</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-execute-the-ishigami-function-over-our-sample-set">Step 3: Execute the Ishigami function over our sample set</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method">Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-interpret-our-results">Step 5: Interpret our results</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-to-your-own-problem">Tips to Apply Sobol SA to Your Own Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="3_sensitivity_analysis_the_basics.html">3. Sensitivity Analysis: The Basics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html">4. Sensitivity Analysis: Diagnostic &amp; Exploratory Modeling</a></li>
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+    <h1>Logistic Regression Tutorial</h1>
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+                <h2> Contents </h2>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-logistic-regression-for-factor-mapping">Tutorial: Logistic Regression for Factor Mapping</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#background">Background</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s get started!</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-latin-hypercube-sample-and-set-up-problem">Step 1: Load Latin hypercube sample and set up problem</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-define-decision-relevant-metrics-for-illustration">Step 2: Define decision-relevant metrics for illustration</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-run-the-logistic-regression">Step 3: Run the logistic regression</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-factor-ranking">Step 4: Factor ranking</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-draw-factor-maps">Step 5: Draw factor maps</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-scenario-discovery-to-your-own-problem">Tips to Apply Scenario Discovery to Your Own Problem</a></li>
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+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/basin_users_logistic_regression.ipynb">Logistic Regression Tutorial</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-logistic-regression-for-factor-mapping">
+<h2>Tutorial: Logistic Regression for Factor Mapping<a class="headerlink" href="#tutorial-logistic-regression-for-factor-mapping" title="Link to this heading">#</a></h2>
+<p>This tutorial replicates a scenario discovery analysis performed in
+<a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020EF001503">Hadjimichael et
+al. (2020)</a>.</p>
+<section id="background">
+<h3>Background<a class="headerlink" href="#background" title="Link to this heading">#</a></h3>
+<p>Planners in the the Upper Colorado River Basin (UCRB, shown in the
+figure below) are seeking to understand the vulnerability of water users
+to uncertainties stemming from climate change, population growth and
+water policy changes. The UCRB spans 25,682 km2 in western Colorado and
+is home to approximately 300,000 residents and 1,012 km2 of irrigated
+land. Several thousand irrigation ditches divert water from the main
+river and its tributaties for irrigation (shown as small black dots in
+the figure). Transmountain diversions of approximately 567,400,000 m3
+per year are exported for irrigation, industrial and municipal uses in
+northern and eastern Colorado, serving the major population centers of
+Denver and Colorado Springs. These diversions are carried through
+tunnels, shown as large black dots in the figure.</p>
+<img alt="_images/basin_map.png" src="_images/basin_map.png" />
+<p>An important planning consideration is the water rights of each user,
+defined by seniority across all water uses (irrigation diversions,
+transboundary diversions, power plants etc.) in the basin. To assess the
+vulnerability of users with varying degrees of water rights seniority,
+planners simulate the system across an ensemble of scenarios using the
+state of Colorado’s StateMod platform. The model simulates streamflow,
+diversions, instream demands, and reservoir operations.</p>
+<p>Hadjimichael et al. (2020) employ an exploratory analysis by simulating
+a large ensemble of plausible scenarios using StateMod and then
+identifying consequential decision-relevant combinations of uncertain
+factors, termed scenario discovery. Focusing on decision-relevant
+metrics (metrics that are important to the user, the scenario discovery
+is applied to the water shortages experienced by each individual user
+(i.e., not on a single basin-wide or sector-wide metric). For this
+training example, we’ll be performing scenario discovery for three
+different water users: two irrigation users and one municipal user.</p>
+</section>
+<section id="lets-get-started">
+<h3>Let’s get started!<a class="headerlink" href="#lets-get-started" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we will be loading in data that has been produced in
+Hadjimichael et al. (2020). Before we start our analysis, we’ll load the
+relevant Python libraries, example data, and information for the three
+users.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#import libraries</span>
+<span class="kn">import</span> <span class="nn">msdbook</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="nn">mpl</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="c1"># load example data from Hadjimichael et al. (2020)</span>
+<span class="n">msdbook</span><span class="o">.</span><span class="n">install_package_data</span><span class="p">()</span>
+
+<span class="c1"># Select the IDs for the three users that we will perform the analysis for</span>
+<span class="n">all_IDs</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;7000550&#39;</span><span class="p">,</span><span class="s1">&#39;7200799&#39;</span><span class="p">,</span><span class="s1">&#39;3704614&#39;</span><span class="p">]</span>
+<span class="n">usernames</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Medium seniority irrigation&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Low seniority irrigation&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Transbasin municipal diversion&#39;</span><span class="p">]</span>
+<span class="n">nStructures</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Downloading</span> <span class="n">example</span> <span class="n">data</span> <span class="k">for</span> <span class="n">msdbook</span> <span class="n">version</span> <span class="mf">0.1.5</span><span class="o">...</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">uncertain_params_bounds</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_metric_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">solutions</span><span class="o">.</span><span class="n">resultfile</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LHsamples_original_1000</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">param_values</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_params_256samples</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LeafCatch</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_simulations_256samples</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">Robustness</span><span class="o">.</span><span class="n">txt</span>
+</pre></div>
+</div>
+</section>
+<section id="step-1-load-latin-hypercube-sample-and-set-up-problem">
+<h3>Step 1: Load Latin hypercube sample and set up problem<a class="headerlink" href="#step-1-load-latin-hypercube-sample-and-set-up-problem" title="Link to this heading">#</a></h3>
+<p>To examine regional vulnerability, we generate an ensemble of plausible
+future states of the world (SOWs) using Latin Hypercube Sampling. For
+this tutorial, we’ll load a file containing 1,000 samples across 14
+parameters. The sampled parameters encompass plausible changes to the
+future state of the basin, including changes to hydrology, water demands
+(irrigation, municipal &amp; industry, transbasin), and institutional and
+environmental factors (environmental flows, reservoir storage, operation
+of the Shoshone Power Plant). These samples are taken from ranges
+identified in <code class="docutils literal notranslate"><span class="pre">param_bounds</span></code>. Below we load in the 1000 samples, the
+range of values that the samples can take for each parameter, and the
+parameter names. More information on what each parameter constitutes can
+be found in Table 1 of Hadjimichael et al., 2020.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Identify the bounds for each of the 14 parameters</span>
+<span class="n">param_bounds</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_basin_param_bounds</span><span class="p">()</span>
+
+<span class="c1">#Load in the parameter samples</span>
+<span class="n">LHsamples</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_lhs_basin_sample</span><span class="p">()</span>
+
+<span class="c1">#Create an array of the parameter names</span>
+<span class="n">param_names</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Irrigation demand multiplier&#39;</span><span class="p">,</span><span class="s1">&#39;Reservoir loss&#39;</span><span class="p">,</span><span class="s1">&#39;Transbasin demand multiplier&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Municipal &amp; industrial multiplier&#39;</span><span class="p">,</span> <span class="s1">&#39;Shoshone&#39;</span><span class="p">,</span><span class="s1">&#39;Environmental flows&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Evaporation change&#39;</span><span class="p">,</span><span class="s1">&#39;Mean dry flow&#39;</span><span class="p">,</span><span class="s1">&#39;Dry flow variance&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Mean wet flow&#39;</span><span class="p">,</span><span class="s1">&#39;Wet flow variance&#39;</span><span class="p">,</span><span class="s1">&#39;Dry-dry probability&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Wet-wet probability&#39;</span><span class="p">,</span> <span class="s1">&#39;Snowmelt shift&#39;</span><span class="p">]</span>
+</pre></div>
+</div>
+</section>
+<section id="step-2-define-decision-relevant-metrics-for-illustration">
+<h3>Step 2: Define decision-relevant metrics for illustration<a class="headerlink" href="#step-2-define-decision-relevant-metrics-for-illustration" title="Link to this heading">#</a></h3>
+<p>Scenario discovery attempts to identify parametric regions that lead to
+‘success’ and ‘failure’. For this demonstration we’ll be defining
+‘success’ as states of the world where a shortage level doesn’t exceed
+its historical frequency.</p>
+</section>
+<section id="step-3-run-the-logistic-regression">
+<h3>Step 3: Run the logistic regression<a class="headerlink" href="#step-3-run-the-logistic-regression" title="Link to this heading">#</a></h3>
+<p>Logistic regression estimates the probability that a future SOW will be
+classified as a success or failure given a set of performance criteria.
+A logistic regression model is defined by:</p>
+<div class="math notranslate nohighlight">
+\[ln \bigg (\frac{p_i}{1-p_i} \bigg ) = X^T_i \beta\]</div>
+<p>where <span class="math notranslate nohighlight">\(p_i\)</span> is the probability the performance in the
+<span class="math notranslate nohighlight">\(i^{th}\)</span> SOW will be classified as a success, <span class="math notranslate nohighlight">\(X_i\)</span> is the
+vector of covariates describing the <span class="math notranslate nohighlight">\(i^{th}\)</span> SOW, and
+<span class="math notranslate nohighlight">\(\beta\)</span> is the vector of coefficients describing the relationship
+between the covariates and the response, which here will be estimated
+using maximum likelihood estimation.</p>
+<p>A logistic regression model was fit to the ensemble of SOWs using the
+performance criteria defined in step 2. Logistic regression modeling was
+conducted using the <a class="reference external" href="https://www.statsmodels.org/stable/index.html">Statsmodel
+Python</a> package. The
+data required for the full analysis is too large to include in this
+tutorial, but results can be found in the data file loaded below.</p>
+<p>The results files contain the occurence of different shortage frequency
+and magnitude combinations under the experiment, in increments of 10,
+between 0 and 100. These combinations (100 for each user) are
+alternative decision-relevant metrics that can be used for scenario
+discovery.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set arrays for shortage frequencies and magnitudes</span>
+<span class="n">frequencies</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">110</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+<span class="n">magnitudes</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">110</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+<span class="n">realizations</span> <span class="o">=</span> <span class="mi">10</span>
+
+<span class="c1"># Load performance and pseudo r scores for each of the users</span>
+<span class="n">results</span> <span class="o">=</span> <span class="p">[</span><span class="n">msdbook</span><span class="o">.</span><span class="n">load_user_heatmap_array</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">/</span> <span class="mi">100</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">))]</span>
+</pre></div>
+</div>
+</section>
+<section id="step-4-factor-ranking">
+<h3>Step 4: Factor ranking<a class="headerlink" href="#step-4-factor-ranking" title="Link to this heading">#</a></h3>
+<p>To rank the importance of each uncertain factor, we utilize McFadden’s
+psuedo-R2, a measure that quantifies the improvement of the model when
+utilizing each given predictor as compared to prediction using the mean
+of the data set:</p>
+<div class="math notranslate nohighlight">
+\[R^2_{McFadden}=1-\frac{ln \hat{L}(M_{full})}{ln \hat{L}(M_{intercept})}\]</div>
+<p>Here <span class="math notranslate nohighlight">\(ln \hat{L}(M_{full})\)</span> is the log likelihood of the full
+model (including the predictor) and <span class="math notranslate nohighlight">\(ln \hat{L}(M_{intercept})\)</span> is
+the log likelihood of the intercept model (which predicts the mean
+probability of success across all SOWs).</p>
+<p>Higher values of McFadden’s psuedo-R2 indicate higher factor importance
+(when the likelihood of the full model approaches one, the ratio of the
+likelihood of the full model compared to the intercept model will get
+very small).</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Load the pseudo-R^2 scores</span>
+<span class="n">scores</span> <span class="o">=</span> <span class="p">[</span><span class="n">msdbook</span><span class="o">.</span><span class="n">load_user_pseudo_scores</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">))]</span>
+
+<span class="c1"># Select indices of frequency and magnitudes that will be used for the visualization</span>
+<span class="n">freq</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
+<span class="n">mag</span> <span class="o">=</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">7</span><span class="p">]</span>
+</pre></div>
+</div>
+</section>
+<section id="step-5-draw-factor-maps">
+<h3>Step 5: Draw factor maps<a class="headerlink" href="#step-5-draw-factor-maps" title="Link to this heading">#</a></h3>
+<p>The McFadden’s psuedo-R2 scores files contain preliminary logistic
+regression results on parameter importance for each of these
+combinations. Using these psuedo-R2 scores we will identify the two most
+important factors for each metric which we’ll use to generate the final
+scenario discovery maps (note: there may be more than two important
+metrics for each user, but here we will demonstrate by mapping two).</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># setup figure</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">18</span><span class="p">),</span> <span class="n">tight_layout</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">patch</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">axes</span><span class="o">.</span><span class="n">flat</span><span class="p">)):</span>
+
+    <span class="n">ax</span> <span class="o">=</span> <span class="n">axes</span><span class="o">.</span><span class="n">flat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+    <span class="n">allSOWsperformance</span> <span class="o">=</span> <span class="n">results</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+    <span class="n">all_pseudo_r_scores</span> <span class="o">=</span> <span class="n">scores</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+    <span class="c1"># construct dataframe</span>
+    <span class="n">dta</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">LHsamples</span><span class="p">,</span> <span class="n">realizations</span><span class="p">,</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">),</span> <span class="n">columns</span><span class="o">=</span><span class="n">param_names</span><span class="p">)</span>
+    <span class="n">dta</span><span class="p">[</span><span class="s1">&#39;Success&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">allSOWsperformance</span><span class="p">[</span><span class="n">freq</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">mag</span><span class="p">[</span><span class="n">i</span><span class="p">],:]</span>
+
+    <span class="n">pseudo_r_scores</span> <span class="o">=</span> <span class="n">all_pseudo_r_scores</span><span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">frequencies</span><span class="p">[</span><span class="n">freq</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span><span class="o">+</span><span class="s1">&#39;yrs_&#39;</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">magnitudes</span><span class="p">[</span><span class="n">mag</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span><span class="o">+</span><span class="s1">&#39;prc&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>
+    <span class="n">top_predictors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">pseudo_r_scores</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">][:</span><span class="mi">2</span><span class="p">]</span> <span class="c1">#Sort scores and pick top 2 predictors</span>
+
+    <span class="c1"># define color map for dots representing SOWs in which the policy</span>
+    <span class="c1"># succeeds (light blue) and fails (dark red)</span>
+    <span class="n">dot_cmap</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">ListedColormap</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">227</span><span class="p">,</span><span class="mi">26</span><span class="p">,</span><span class="mi">28</span><span class="p">],[</span><span class="mi">166</span><span class="p">,</span><span class="mi">206</span><span class="p">,</span><span class="mi">227</span><span class="p">]])</span><span class="o">/</span><span class="mf">255.0</span><span class="p">)</span>
+
+    <span class="c1"># define color map for probability contours</span>
+    <span class="n">contour_cmap</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">&#39;RdBu&#39;</span><span class="p">)</span>
+
+    <span class="c1"># define probability contours</span>
+    <span class="n">contour_levels</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.05</span><span class="p">,</span><span class="mf">0.1</span><span class="p">)</span>
+
+    <span class="c1"># define base values of the predictors</span>
+    <span class="n">SOW_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span> <span class="c1"># default parameter values for base SOW</span>
+    <span class="n">base</span> <span class="o">=</span> <span class="n">SOW_values</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">]</span>
+    <span class="n">ranges</span> <span class="o">=</span> <span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">]</span>
+
+    <span class="c1"># define grid of x (1st predictor), and y (2nd predictor) dimensions</span>
+    <span class="c1"># to plot contour map over</span>
+    <span class="n">xgrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="mi">0</span><span class="p">],</span>
+                      <span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="mi">1</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">around</span><span class="p">((</span><span class="n">ranges</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">ranges</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span><span class="o">/</span><span class="mi">500</span><span class="p">,</span><span class="n">decimals</span><span class="o">=</span><span class="mi">4</span><span class="p">))</span>
+    <span class="n">ygrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">]][</span><span class="mi">0</span><span class="p">],</span>
+                      <span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">]][</span><span class="mi">1</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">around</span><span class="p">((</span><span class="n">ranges</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">ranges</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span><span class="o">/</span><span class="mi">500</span><span class="p">,</span><span class="n">decimals</span><span class="o">=</span><span class="mi">4</span><span class="p">))</span>
+    <span class="n">all_predictors</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dta</span><span class="o">.</span><span class="n">columns</span><span class="o">.</span><span class="n">tolist</span><span class="p">()[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">top_predictors</span><span class="p">]</span>
+    <span class="n">dta</span><span class="p">[</span><span class="s1">&#39;Interaction&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">dta</span><span class="p">[</span><span class="n">all_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span><span class="o">*</span><span class="n">dta</span><span class="p">[</span><span class="n">all_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
+
+    <span class="c1"># logistic regression here</span>
+    <span class="n">predictor_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">all_predictors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]]</span>
+    <span class="n">result</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">fit_logit</span><span class="p">(</span><span class="n">dta</span><span class="p">,</span> <span class="n">predictor_list</span><span class="p">)</span>
+
+    <span class="c1"># plot contour map</span>
+    <span class="n">contourset</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_contour_map</span><span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="n">dta</span><span class="p">,</span> <span class="n">contour_cmap</span><span class="p">,</span>
+                                          <span class="n">dot_cmap</span><span class="p">,</span> <span class="n">contour_levels</span><span class="p">,</span> <span class="n">xgrid</span><span class="p">,</span>
+                                          <span class="n">ygrid</span><span class="p">,</span> <span class="n">all_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">all_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">base</span><span class="p">)</span>
+
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">usernames</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+
+<span class="c1"># set up colorbar</span>
+<span class="n">cbar_ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_axes</span><span class="p">([</span><span class="mf">0.98</span><span class="p">,</span> <span class="mf">0.15</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">])</span>
+<span class="n">cbar</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">contourset</span><span class="p">,</span> <span class="n">cax</span><span class="o">=</span><span class="n">cbar_ax</span><span class="p">)</span>
+<span class="n">cbar_ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Probability of Success&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">cbar_ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;y&#39;</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">&#39;major&#39;</span><span class="p">,</span> <span class="n">labelsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">statsmodels</span><span class="o">/</span><span class="n">base</span><span class="o">/</span><span class="n">model</span><span class="o">.</span><span class="n">py</span><span class="p">:</span><span class="mi">127</span><span class="p">:</span> <span class="n">ValueWarning</span><span class="p">:</span> <span class="n">unknown</span> <span class="n">kwargs</span> <span class="p">[</span><span class="s1">&#39;disp&#39;</span><span class="p">]</span>
+  <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="n">msg</span><span class="p">,</span> <span class="n">ValueWarning</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Optimization</span> <span class="n">terminated</span> <span class="n">successfully</span><span class="o">.</span>
+         <span class="n">Current</span> <span class="n">function</span> <span class="n">value</span><span class="p">:</span> <span class="mf">0.378619</span>
+         <span class="n">Iterations</span> <span class="mi">8</span>
+<span class="n">Optimization</span> <span class="n">terminated</span> <span class="n">successfully</span><span class="o">.</span>
+         <span class="n">Current</span> <span class="n">function</span> <span class="n">value</span><span class="p">:</span> <span class="mf">0.397285</span>
+         <span class="n">Iterations</span> <span class="mi">8</span>
+<span class="n">Optimization</span> <span class="n">terminated</span> <span class="n">successfully</span><span class="o">.</span>
+         <span class="n">Current</span> <span class="n">function</span> <span class="n">value</span><span class="p">:</span> <span class="mf">0.377323</span>
+         <span class="n">Iterations</span> <span class="mi">8</span>
+</pre></div>
+</div>
+<img alt="_images/notebook_logistic_output_11_1.png" src="_images/notebook_logistic_output_11_1.png" />
+<p>The figure above demonstrates how different combinations of the
+uncertain factors lead to success or failure in different states of the
+world, which are denoted by the blue and red dots respectively. The
+probability of success and failure are further denoted by the contours
+in the figure. Several insights can be drawn from this figure.</p>
+<p>First, using metrics chosen to be decision-relevant (specific to each
+user) causes different factors to be identified as most important by
+this scenario-discovery exercise (the x- and y-axes for each of the
+subplots). In other words, depending on what the decision makers of this
+system want to prioritize they might choose to monitor different
+uncertain factors to track performance.</p>
+<p>Second, in the top panel, the two identified factors appear to also have
+an interactive effect on the metric used (shortages of a certain level
+and frequency in this example). In terms of scenario discovery, the
+Patient Rule Induction Method (PRIM) or Classification And Regression
+Trees (CART) would not be able to delineate this non-linear space and
+would therefore misclassify parameter combinations as ‘desirable’ when
+they were in fact undesirable, and vice versa.</p>
+<p>Lastly, logistic regression also produces contours of probability of
+success, i.e. different factor-value combinations are assigned different
+probabilities that a shortage level will be exceeded. This allows the
+decision makers to evaluate these insights while considering their risk
+aversion.</p>
+</section>
+<section id="tips-to-apply-scenario-discovery-to-your-own-problem">
+<h3>Tips to Apply Scenario Discovery to Your Own Problem<a class="headerlink" href="#tips-to-apply-scenario-discovery-to-your-own-problem" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we demonstrated how to perform a scenario discovery
+analysis for three different users in the UCRB. The analysis allowed us
+to determine which parameters the users would be most affected by and to
+visualize how different ranges of these parameters lead to success and
+failure for different users. This framework can be applicable to any
+other application where it is of interest to characterize success and
+failure based on uncertain parameter ranges. In order to apply the same
+framework to your own problem:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters of interest, which will
+represent uncertainties that characterize your system.</p></li>
+<li><p>Generate samples for these parameters (this can be done using the
+<code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function or externally).</p></li>
+<li><p>Define what constitutes success and failure in your problem. In this
+tutorial, success was defined based on not surpassing the historical
+drought frequency. Choose a metric that is relevant to your problem
+and decision-makers that might be involved. If your model involves an
+optimization, you can also define metrics based on meeting certain
+values of these objectives.</p></li>
+<li><p>Run the parameter sets through your model and calculate success and
+failure based on your metrics and across different users if
+applicable. This step will allow you to create the scatter plot part
+of the final figure.</p></li>
+<li><p>If it is of interest, the contours on the figure can be created by
+fitting the logistic regression model in a similiar manner as denoted
+in Steps 3 and 5 of the tutorial.</p></li>
+</ol>
+</section>
+</section>
+</section>
+
+
+                </article>
+              
+
+              
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+    <i class="fa-solid fa-list"></i> Contents
+  </div>
+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-logistic-regression-for-factor-mapping">Tutorial: Logistic Regression for Factor Mapping</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#background">Background</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s get started!</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-latin-hypercube-sample-and-set-up-problem">Step 1: Load Latin hypercube sample and set up problem</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-define-decision-relevant-metrics-for-illustration">Step 2: Define decision-relevant metrics for illustration</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-run-the-logistic-regression">Step 3: Run the logistic regression</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-factor-ranking">Step 4: Factor ranking</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-draw-factor-maps">Step 5: Draw factor maps</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-scenario-discovery-to-your-own-problem">Tips to Apply Scenario Discovery to Your Own Problem</a></li>
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+              
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+
+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>HYMOD Dynamics Tutorial</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-of-the-hymod-model">Tutorial: Sensitivity Analysis of the HYMOD Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction-to-hymod">1. Introduction to HYMOD</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#global-sensitivity-analysis">2. Global Sensitivity Analysis</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">1 - Introduction to HYMOD</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">2- Global Sensitivity Analysis</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#running-hymod-model-warm-up">Running HYMOD - Model Warm-up</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#sensitivity-analysis-indices-for-each-month">2.4.1 Sensitivity analysis indices for each month</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#first-order-indices">First-order Indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#total-order-indices">Total-order indices</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#annual-sensitivity-analysis-indices">2.4.2 Annual sensitivity analysis indices</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">First-order indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id4">Total-order indices</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#monthly-time-varying-sensitivity-analysis">2.4.3 Monthly time-varying sensitivity analysis</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id5">First-order indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#total-order-time-varying-sensitivity-analysis">Total order - time varying sensitivity analysis</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-this-methodology-to-your-own-problem">Tips to Apply this methodology to your own problem</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="hymod-dynamics-tutorial">
+<h1>HYMOD Dynamics Tutorial<a class="headerlink" href="#hymod-dynamics-tutorial" title="Link to this heading">#</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">HYMOD Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-sensitivity-analysis-of-the-hymod-model">
+<h2>Tutorial: Sensitivity Analysis of the HYMOD Model<a class="headerlink" href="#tutorial-sensitivity-analysis-of-the-hymod-model" title="Link to this heading">#</a></h2>
+<p>The purpose of this tutorial is to demonstrate the global sensitivity
+analysis concepts and tools established in the Section 3.1 of the main
+text of this eBook. This demonstration will highlight the central role
+of design of experiments (Section 3.3), when implementing global
+sensitivity analysis tools described in Section 3.4.</p>
+<p>We’ll explore these tools and concepts using the HYdrological MODel
+(HYMOD), a rainfall-runoff model developed and used for river flow
+forecasting. HYMOD was chosen for demonstration because its purpose is
+to abstract highly complex and non-linear systems. The methods
+demonstrated in thistutorial can be applied to numerical models that
+simulate other complex non-linear systems.</p>
+<p>This tutorial will first introduce HYMOD and use it to simulate
+streamflows in a river basin. Next, we’ll employ sensitivity analysis
+concepets described in Section 3 of the main text to examine how values
+of HYMOD’s parameters impact streamflow predictions. We’ll then explore
+how the effects of these parameters may change over time using
+time-varying sensitivtiy analysis. Finally, we’ll demonstrate concepts
+presented in Chapter 7 through two ensemble-based methods of uncertainty
+quantification - Generalized Likelihood Uncertainty Estimation (GLUE)
+and Pre-Calibration.</p>
+<p>The tutorial includes the following steps:</p>
+<section id="introduction-to-hymod">
+<h3>1. Introduction to HYMOD<a class="headerlink" href="#introduction-to-hymod" title="Link to this heading">#</a></h3>
+<p>1.1 - Introduction to a simple hydrologic model (HYMOD)</p>
+<p>1.2 - Input Data</p>
+<p>1.3 - Running a basic simulation</p>
+<p>1.4 - Model outputs</p>
+</section>
+<section id="global-sensitivity-analysis">
+<h3>2. Global Sensitivity Analysis<a class="headerlink" href="#global-sensitivity-analysis" title="Link to this heading">#</a></h3>
+<p>2.1 - Design of Experiments</p>
+<p>2.2 - Sensitivity analysis for one output</p>
+<p>2.3 - Sensitivity analysis across multiple outputs</p>
+<p>2.4 - Time-varying sensitivity analysis</p>
+</section>
+</section>
+<section id="id1">
+<h2>1 - Introduction to HYMOD<a class="headerlink" href="#id1" title="Link to this heading">#</a></h2>
+<p id="hymod">1.1 Overview</p>
+<p>HYMOD is a hydrologic model (rainfall-runoff model) that simulates key
+hydrologic fluxes such as infiltration, streamflow and
+evapotranspiration. The model was originally developed and used for
+river flow forecasting, but it has also been been used to explore
+different sensitivity analysis (e.g., <a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/wrcr.20124">Herman et al.,
+2013</a>),
+uncertainty quantification (e.g., <a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2006WR005205">Smith et al.,
+2008</a>),
+and optimization (e.g., <a class="reference external" href="https://www.sciencedirect.com/science/article/pii/S0022169414006362?casa_token=IRqE19Hkfa8AAAAA:_fXOqfwpzxMpchvu8_0njCe0Ok9H29Gyw2F46l9PzG9UVODDTUg6wIOSiyp6uybGevNVnZ7N">Ye et al.,
+2014</a>)
+concepts.</p>
+<p>HYMOD accepts two inputs - daily precepitation and daily potential
+evapotranspiration (PET)- and generates predicitons of daily streamflow.
+HYMOD abstracts the highly non-linear process of runoff routing by
+dividing the flow into two components: quick flow, representing
+precipitation that quickly runs off the surface of the watershed into
+the stream, and slow flow, that moves through the soil and takes much
+longer to arrive at the stream.</p>
+<p>To generate streamflow predictions, HYMOD first models vertical
+processes within the watershed to determine how much water infiltrates
+and evaporates from the soil at a given time step. It then determines
+how much water should be partitioned into quick flow and slow flow
+processes. Within each process it abstracts residence time (the time it
+takes a unit volume of water to move through the watershed and into the
+stream) using a series of “reservoirs” each with a calibrated residence
+time.</p>
+<p>HYMOD’s representation of hydrologic processes are shown Figure 1 below
+and controlled by the following parameters:</p>
+<p><span class="math notranslate nohighlight">\(H_{uz}\)</span>: the maximum water storage capacity of the soil (mm)</p>
+<p><span class="math notranslate nohighlight">\(B_{exp}\)</span>: parameters describing the degree of spatial variability
+within the basin between 0 and Huz</p>
+<p><span class="math notranslate nohighlight">\(Alp\)</span>: Fraction of runoff contributing to quick flow</p>
+<p><span class="math notranslate nohighlight">\(K_q\)</span>: Quick flow residence time of linear infinite reservoir (the
+Kq values of all three linear reservoirs are the same)</p>
+<p><span class="math notranslate nohighlight">\(K_s\)</span>: Slow flow residence time of linear infinite reservoir</p>
+<img alt="_images/hymod_schematic-DAVE.png" src="_images/hymod_schematic-DAVE.png" />
+<p>HYMOD models the fraction of water that is stored in the soil
+<span class="math notranslate nohighlight">\((F(XH_{uz}))\)</span> using the following relationship:</p>
+<div class="math notranslate nohighlight">
+\[F(XH_{uz}) = 1 - (1 - \frac{XH_{uz}}{H_{uz}})^{B}\]</div>
+<p>where <span class="math notranslate nohighlight">\(XH_{uz}\)</span> is the water storage capacity of the soil;
+<span class="math notranslate nohighlight">\(H_{uz}\)</span> is the parameter describing basin maximum water
+storage capacity (mm); and <span class="math notranslate nohighlight">\(B\)</span> is the parameter describing the
+degree of spatial variability within the basin.</p>
+<p>The portion of precipitation that exceeds the water storage capacity is
+treated as runoff.</p>
+<p>To route runoff to streamflow, the excess runoff from the vertical
+processes is split into quick flow and slow flow. The proportion of
+runoff partitioned into quick flow and slow flow is determined by a
+parameter <span class="math notranslate nohighlight">\(Alp\)</span>, which ranges between 0 and 1. Quick flow is
+routed through <span class="math notranslate nohighlight">\(N\)</span> identical quick flow tanks <span class="math notranslate nohighlight">\(Q1, Q2... QN\)</span>
+(shown above as <span class="math notranslate nohighlight">\(N=3\)</span>). The rate at which runoff moves through the
+quick flow system is described by the residence time of the quick frlow
+tanks, <span class="math notranslate nohighlight">\(Kq\)</span> (day). Slow flow is routed through a parallel slow
+flow tank and the rate at which slow flow is routed is described by the
+slow flow residences time, <span class="math notranslate nohighlight">\(Ks\)</span> (day).</p>
+<p>Citation: Wagener, T., Boyle, D. P., Lees, M. J., Wheater, H. S., Gupta,
+H. V., &amp; Sorooshian, S. (2001). A framework for development and
+application of hydrological models. Hydrology and Earth System Sciences,
+5(1), 13-26.</p>
+<p id="inputs">1.2 Input data</p>
+<p>The HYMOD model only requires precipitation and potential
+evapotranspiration as inputs. For this example, we’ll run HYMOD using
+data from the Leaf River, a humid catchment located north of Collins
+Mississippi that has been widely used to explore HYMOD. The dataset also
+includes daily observed runoff that we later use to evaluate the
+performace of each sensitvity analysis sample set.</p>
+<p>In the following section of code, we’ll load the necessary python
+libraries and read in the input file. For this exercise we’ll only use
+the first eleven years of data. The first five rows of the input dataset
+are printed to show what they look like:</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">msdbook</span>
+
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+
+<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">metrics</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
+
+<span class="c1"># load example data</span>
+<span class="n">msdbook</span><span class="o">.</span><span class="n">install_package_data</span><span class="p">()</span>
+
+<span class="c1"># load the Leaf River HYMOD input file</span>
+<span class="n">leaf_data</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_input_file</span><span class="p">()</span>
+
+<span class="c1"># extract the first eleven years of data</span>
+<span class="n">leaf_data</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">4015</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Leaf River Data structure:&#39;</span><span class="p">)</span>
+
+<span class="c1"># There are only three columns in the file including precipitation, potential evapotranspiration and  streamflow</span>
+<span class="n">leaf_data</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Downloading</span> <span class="n">example</span> <span class="n">data</span> <span class="k">for</span> <span class="n">msdbook</span> <span class="n">version</span> <span class="mf">0.1.5</span><span class="o">...</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">uncertain_params_bounds</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_metric_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">solutions</span><span class="o">.</span><span class="n">resultfile</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LHsamples_original_1000</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">param_values</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_params_256samples</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LeafCatch</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_simulations_256samples</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">Robustness</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Leaf</span> <span class="n">River</span> <span class="n">Data</span> <span class="n">structure</span><span class="p">:</span>
+</pre></div>
+</div>
+<div>
+<style scoped>
+    .dataframe tbody tr th:only-of-type {
+        vertical-align: middle;
+    }
+
+    .dataframe tbody tr th {
+        vertical-align: top;
+    }
+
+    .dataframe thead th {
+        text-align: right;
+    }
+</style>
+<table border="1" class="dataframe">
+  <thead>
+    <tr style="text-align: right;">
+      <th></th>
+      <th>Precip</th>
+      <th>Pot_ET</th>
+      <th>Strmflw</th>
+    </tr>
+  </thead>
+  <tbody>
+    <tr>
+      <th>0</th>
+      <td>0.0</td>
+      <td>4.60</td>
+      <td>0.29</td>
+    </tr>
+    <tr>
+      <th>1</th>
+      <td>0.0</td>
+      <td>4.31</td>
+      <td>0.24</td>
+    </tr>
+    <tr>
+      <th>2</th>
+      <td>0.0</td>
+      <td>4.33</td>
+      <td>0.21</td>
+    </tr>
+    <tr>
+      <th>3</th>
+      <td>0.0</td>
+      <td>4.78</td>
+      <td>0.19</td>
+    </tr>
+    <tr>
+      <th>4</th>
+      <td>0.0</td>
+      <td>2.91</td>
+      <td>0.18</td>
+    </tr>
+  </tbody>
+</table>
+</div><p>To visualize catchment hydrology, streamflow and precipitation data are
+usually plotted together as a combined hydrograph (streamflow ) and
+hyetograph (rainfall, from Greek.hyetos, “rain”). Streamflow is plotted
+as a time series, while rainfall is shown as an inverted bar plot along
+the top of the graph. Streamflow labels are shown on the left y-axis,
+while rainfall labels are shown on the right y-axis.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># make an axis for the hydrograph</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">strmflw_ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">12</span><span class="p">,</span><span class="mi">6</span><span class="p">])</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">50</span><span class="p">])</span>
+
+<span class="c1">#make a second y-axis for the hyetograph</span>
+<span class="n">precip_ax</span> <span class="o">=</span> <span class="n">strmflw_ax</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">200</span><span class="p">])</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">invert_yaxis</span><span class="p">()</span>
+
+<span class="n">precip</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">]</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">])),</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;lightcoral&#39;</span><span class="p">)</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Streamflow (mm/day)&#39;</span><span class="p">)</span>
+
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">])),</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">],</span> <span class="n">width</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Rainfall (mm/day)&#39;</span><span class="p">)</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">&#39;Precip&#39;</span><span class="p">],</span> <span class="n">loc</span><span class="o">=</span><span class="s1">&#39;center right&#39;</span><span class="p">)</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">&#39;Streamflow&#39;</span><span class="p">],</span><span class="n">bbox_to_anchor</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.48</span><span class="p">))</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">&lt;</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">legend</span><span class="o">.</span><span class="n">Legend</span> <span class="n">at</span> <span class="mh">0x7f53b95c6850</span><span class="o">&gt;</span>
+</pre></div>
+</div>
+<img alt="_images/hymod1.png" src="_images/hymod1.png" />
+<p id="baseline">1.3 Running a Baseline Model Simulation</p>
+<p>We’ll start our experiment by running HYMOD using its default
+parameters.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># assign input parameters to generate a baseline simulated streamflow</span>
+<span class="n">Nq</span> <span class="o">=</span> <span class="mi">3</span>  <span class="c1"># Number of quickflow routing tanks</span>
+<span class="n">Kq</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="c1"># Quickflow routing tanks&#39; rate parameter</span>
+<span class="n">Ks</span> <span class="o">=</span>  <span class="mf">0.001</span> <span class="c1"># Slowflow routing tank&#39;s rate parameter</span>
+<span class="n">Alp</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="c1"># Quick/slow split parameter</span>
+<span class="n">Huz</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1"># Maximum height of soil moisture accounting tank</span>
+<span class="n">B</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="c1"># Scaled distribution function shape parameter</span>
+
+<span class="c1"># Note that the number of years is 11. One year of model warm-up and ten years are used for actual simulation</span>
+<span class="n">model</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">hymod</span><span class="p">(</span><span class="n">Nq</span><span class="p">,</span> <span class="n">Kq</span><span class="p">,</span> <span class="n">Ks</span><span class="p">,</span> <span class="n">Alp</span><span class="p">,</span> <span class="n">Huz</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">leaf_data</span><span class="p">,</span> <span class="n">ndays</span><span class="o">=</span><span class="mi">4015</span><span class="p">)</span>
+</pre></div>
+</div>
+<p id="outputs">1.4 Model Outputs</p>
+<p>Model outputs include actual evapotranspiration, quick and fast
+streamflow, and combined runoff. In this tutorial we focus on the total
+daily runoff, QQ (<span class="math notranslate nohighlight">\(m^3/s\)</span>). We can use the following script to
+plot simulated streamflow against observed streamflow.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-observed-vs-simulated-streamflow">plot_observed_vs_simulated_streamflow</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ax</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_observed_vs_simulated_streamflow</span><span class="p">(</span><span class="n">df</span><span class="o">=</span><span class="n">leaf_data</span><span class="p">,</span> <span class="n">hymod_dict</span><span class="o">=</span><span class="n">model</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod2.png" src="_images/hymod2.png" />
+<p>So how does our model perform? We can investigate model performance
+across several metrics:</p>
+<p>1: Mean Absolute Error (MAE); MAE conveys how the model performs on
+average across the 10 year simulation period, with smaller values
+indicating better performance. The absolute value is taken so that
+positive and negative errors do not cancel each other out.</p>
+<div class="math notranslate nohighlight">
+\[MAE = \frac{1}{N}\sum_{t=0}^N\left\lvert Q_{sim,t}-Q_{obs,t}\right\rvert\]</div>
+<p>2: Root Mean Square Error (RMSE); RMSE is sum of square errors across
+the 10 year simulation period. RMSE is sensitive to large errors between
+the historical record and the simulated flows, and thus is useful for
+highlighting the model’s ability to capture of extreme flood events.</p>
+<div class="math notranslate nohighlight">
+\[RMSE = \sqrt{\frac{1}{N}\sum_{t=1}^{N}(Q_{sim,t}-Q_{obs,t})^2}\]</div>
+<p>3: Log-Root Mean Square Error (Log(RMSE)) LOG(RMSE) focuses on model
+performance during low-flow events.</p>
+<div class="math notranslate nohighlight">
+\[LOG(RMSE) = log(RMSE)\]</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">mae</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">]</span> <span class="o">-</span> <span class="n">model</span><span class="p">[</span><span class="s1">&#39;Q&#39;</span><span class="p">]))</span>
+<span class="n">mse</span> <span class="o">=</span> <span class="n">metrics</span><span class="o">.</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">model</span><span class="p">[</span><span class="s1">&#39;Q&#39;</span><span class="p">],</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">])</span>
+<span class="n">rmse</span> <span class="o">=</span> <span class="n">mse</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;MAE: &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">mae</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">RMSE: &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">mse</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">LOG(RMSE): &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">rmse</span><span class="p">))</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">MAE</span><span class="p">:</span> <span class="mf">1.0787471470460999</span>
+<span class="n">RMSE</span><span class="p">:</span> <span class="mf">4.375695937555197</span>
+<span class="n">LOG</span><span class="p">(</span><span class="n">RMSE</span><span class="p">):</span> <span class="mf">2.0918164206151544</span>
+</pre></div>
+</div>
+<p>The error metrics show that HYMOD performs reasonably well, the MAE is
+around 1 <span class="math notranslate nohighlight">\(m^3/s\)</span>, the RMSE is on the order of 10% of the largest
+observed streamflow and the LOG(RMSE) is fairly low.</p>
+</section>
+<section id="id2">
+<h2>2- Global Sensitivity Analysis<a class="headerlink" href="#id2" title="Link to this heading">#</a></h2>
+<p id="sensitivity">2.1 Experimental Design and Setup</p>
+<p>Now we’ll examine how sensitive streamflow simulations generated by
+HYMOD are to the model’s input parameters. We’ll perform global
+sensitivity analysis (see Section 3.1 of the main text) using the SALib
+Python library.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">saltelli</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">sobol</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">delta</span>
+</pre></div>
+</div>
+<p>A first and critical step when conducting sensitivity analysis is
+determining the experimental design (see Design of Experiments, Section
+3.4 of the main text). Our experimental design involves defining the
+uncertainties that we’ll be examining, the output of interest, the
+ranges of each uncertainty that will be explored and the strategy for
+sampling the uncertainty space.</p>
+<p>For this experiment we’ll explore the five parameters highlighted in
+Figure 1. We’ll draw their ranges from existing literature on the model
+(note Jon H. paper). We’ll use a Sobol sampling an a quasi-random
+sampling with low sequences approach to sample the uncertainty space
+(Section 3.3.4).</p>
+<p>In this demonstration we’ll utilize Sobol Sensitivity Analysis, a
+variance based method (Section 3.4.5).</p>
+<p>To explore HYMOD’s behavoir, we’ll examine the sensitivity of four model
+ouputs to input parameters: 1) predicted flow, 2) Mean Absolute Error
+(compared with the calibaration data set), 3) Root Mean Square Error and
+4) Log Root Mean Square Error.</p>
+<p>This analysis will employ SALib, a Python implementation also utilized
+in the other SA tutorial (make this more formal).</p>
+<p>To start our analysis, we’ll create a dictionary that describes our
+model uncertainties and their ranges, this dictionary is named
+“problem_hymod” (SALib refers to these dictionaries as “problems”).</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">problem_hymod</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">5</span><span class="p">,</span>
+    <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">],</span>
+    <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>  <span class="c1"># Kq</span>
+               <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">],</span>  <span class="c1"># Ks</span>
+               <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>    <span class="c1"># Alp</span>
+               <span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">500</span><span class="p">],</span>  <span class="c1"># Huz</span>
+               <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">1.9</span><span class="p">]]</span>  <span class="c1"># B</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+<p>After defining our uncertainites and ranges, we’ll use SALib to sample
+the uncertainty space and run the model for each of the sample sets. We
+will load a sample that has already been created <code class="docutils literal notranslate"><span class="pre">param_values_hymod</span></code>
+for demonstration purposes. For HYMOD, literature recommends running at
+least N = 10,000 samples, to keep this demonstration easy to run
+however, we utilize only 256 sobol samples of uncertainties. To generate
+accurate approximations of second order sensitivity indicies SALib
+generates N*(2k+2) sets of samples, where N=256 and k=5 (number of
+uncertainties). For the math behind why this is needed, see (Saltelli,
+A., 2002. Making best use of model evaluations to compute sensitivity
+indices. Computer Physics Communications 145, 280–297.
+<a class="reference external" href="https://doi.org/10.1016/S0010-4655(02)00280-1">https://doi.org/10.1016/S0010-4655(02)00280-1</a>).</p>
+<p>The actual model simulation takes an extended period, so we also load
+the simulation data from a previous run. The following demonstrates how
+to conduct this analysis:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># generate 256 samples.</span>
+<span class="n">param_values_hymod</span> <span class="o">=</span> <span class="n">saltelli</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="mi">256</span><span class="p">)</span>
+
+<span class="c1"># dictionary to store outputs in</span>
+<span class="n">d_outputs</span> <span class="o">=</span> <span class="p">{}</span>
+
+<span class="c1"># run simulation for each parameter sample</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">param_values_hymod</span><span class="p">)):</span>
+
+    <span class="c1"># run model for each sensitivity analysis parameter sets</span>
+    <span class="n">hymod_output</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">hymod</span><span class="p">(</span><span class="n">Nq</span><span class="p">,</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>
+                                 <span class="n">leaf_data</span><span class="p">,</span>
+                                 <span class="n">ndays</span><span class="o">=</span><span class="mi">4015</span><span class="p">)</span>
+
+    <span class="c1"># store the simulated total flow discharge</span>
+    <span class="n">d_outputs</span><span class="p">[</span><span class="sa">f</span><span class="s2">&quot;Q</span><span class="si">{</span><span class="n">i</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">hymod_output</span><span class="p">[</span><span class="s2">&quot;Q&quot;</span><span class="p">]</span>
+
+
+<span class="n">Q_df_bw</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d_outputs</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously generated parameter values</span>
+<span class="n">param_values_hymod</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_params</span><span class="p">()</span>
+
+<span class="c1"># number of samples</span>
+<span class="n">n_samples</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">param_values_hymod</span><span class="p">)</span>
+
+<span class="c1"># load previously generated hymod simulated outputs</span>
+<span class="n">Q_df_bw</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_simulation</span><span class="p">()</span>
+
+<span class="c1"># column names of each sample simulation number</span>
+<span class="n">sample_column_names</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">Q_df_bw</span><span class="o">.</span><span class="n">columns</span> <span class="k">if</span> <span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s1">&#39;Q&#39;</span><span class="p">]</span>
+</pre></div>
+</div>
+<section id="running-hymod-model-warm-up">
+<h3>Running HYMOD - Model Warm-up<a class="headerlink" href="#running-hymod-model-warm-up" title="Link to this heading">#</a></h3>
+<p>A hydrological model such as HYMOD usually includes ordinary
+differential equations that are sensitive to their initial condition.
+They also have components in their underlying formulation that have long
+memory such that prior time steps can affect their current simulations.
+For example, soil moisture or groundwater can hold water for a long time
+and therefore they are often considered to exhibit a long memory. This
+can affect the partitioning of water to runoff and infiltration, while
+also controlling the generation of base flow. Therefore, it is important
+to have a reasonable initial value for them. To achieve this,
+hydrologists usually extend their simulation period and after the
+simulations, they remove that extended time period that has unreasonable
+groundwater or surface water values. This time period is called the
+warm-up time period.</p>
+<p>Here we extended our simulation for one year (from 10 years to 11 years)
+and we removed the first year of simulation, therefore our warm-up
+period is one year.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># exclude the first year of simulation from the simulations and reset the index</span>
+<span class="n">Q_df</span> <span class="o">=</span> <span class="n">Q_df_bw</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">365</span><span class="p">:</span><span class="mi">4015</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">reset_index</span><span class="p">(</span><span class="n">drop</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="c1"># exclude the first year of the input data and reset the index</span>
+<span class="n">leaf_data</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">365</span><span class="p">:</span><span class="mi">4015</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">reset_index</span><span class="p">(</span><span class="n">drop</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now that HYMOD has been warmed up, we’ll examine how HYMOD’s streamflow
+outputs vary under different sample sets, and compare them with the
+observed streamflow.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># add date columns to our simulation data frame; for this data our start date is 1/1/2000</span>
+<span class="n">date_ts</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">date_range</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="s1">&#39;1/1/2000&#39;</span><span class="p">,</span> <span class="n">periods</span><span class="o">=</span><span class="mi">3650</span><span class="p">,</span> <span class="n">freq</span><span class="o">=</span><span class="s1">&#39;D&#39;</span><span class="p">)</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;date&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">year</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;month&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">month</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;day&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">day</span>
+
+<span class="c1"># aggregate the simulated observed streamflow to monthly mean</span>
+<span class="n">df_sim_mth_mean</span> <span class="o">=</span> <span class="n">Q_df</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">,</span> <span class="s1">&#39;month&#39;</span><span class="p">])[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+
+<span class="c1"># do the same for the observed data</span>
+<span class="n">date_ts</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">date_range</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="s1">&#39;1/1/2000&#39;</span><span class="p">,</span> <span class="n">periods</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">),</span> <span class="n">freq</span><span class="o">=</span><span class="s1">&#39;D&#39;</span><span class="p">)</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;date&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">year</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;month&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">month</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;day&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">day</span>
+
+<span class="c1"># aggregate the daily observed streamflow to monthly mean</span>
+<span class="n">df_obs_mth_mean</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">,</span> <span class="s1">&#39;month&#39;</span><span class="p">])</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-observed-vs-sensitivity-streamflow">plot_observed_vs_sensitivity_streamflow</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ax</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_observed_vs_sensitivity_streamflow</span><span class="p">(</span><span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_mth_mean</span><span class="p">,</span>
+                                                     <span class="n">df_sim</span><span class="o">=</span><span class="n">df_sim_mth_mean</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod3.png" src="_images/hymod3.png" />
+<p id="sametrics">2.2 Sensitivity of streamflows to model parameters</p>
+<p>Now we’ll examine how each of HYMOD’s parameters impact the variance of
+simulated streamflows. Using SALib we’ll calculate the first order and
+total order sensitivity indicies of each model parameter. The first
+order sensitivity index measure’s the individual impact that a given
+parameter has on the variance of the simulated streamflows. The total
+order index measures the impact of a given parameter along with all
+interactions that other parameters have with the given parameter on
+simulated streamflows.</p>
+<p>We’ll start with an matrix, Y, which contains our simulated streamflows
+for every uncertainty sample. We’ll then use the sobol.analyze function
+from SALib to calculate the sensitivity indicies (Si). The arguments for
+this function are the problem dictionary defined in part 2.2 of this
+tutorial, and the matrix of simulated streamflows, Y.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># overall aggregated indices</span>
+<span class="n">Y</span> <span class="o">=</span> <span class="n">Q_df</span><span class="p">[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+<span class="c1"># Perform analysis</span>
+<span class="n">Si</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now we can examine our results, we’ll print the first order and total
+order Si’s for each parameter, then visualize the results with bar plots</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;First order indices = &#39;</span><span class="p">,</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">])</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Total order indicies = &#39;</span><span class="p">,</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">])</span>
+
+<span class="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+<span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s1">&#39;&#39;</span><span class="p">,</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;First order Si&#39;</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s1">&#39;&#39;</span><span class="p">,</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Total order Si&#39;</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">First</span> <span class="n">order</span> <span class="n">indices</span> <span class="o">=</span>  <span class="p">[</span><span class="mf">9.55550001e-05</span> <span class="mf">7.49249463e-04</span> <span class="mf">5.62386413e-04</span> <span class="mf">7.03327551e-01</span>
+ <span class="mf">2.53701895e-01</span><span class="p">]</span>
+<span class="n">Total</span> <span class="n">order</span> <span class="n">indicies</span> <span class="o">=</span>  <span class="p">[</span><span class="mf">1.76174200e-06</span> <span class="mf">1.63288175e-03</span> <span class="mf">3.41378460e-04</span> <span class="mf">6.88983864e-01</span>
+ <span class="mf">2.53922146e-01</span><span class="p">]</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">ipykernel_launcher</span><span class="o">.</span><span class="n">py</span><span class="p">:</span><span class="mi">9</span><span class="p">:</span> <span class="ne">UserWarning</span><span class="p">:</span> <span class="n">FixedFormatter</span> <span class="n">should</span> <span class="n">only</span> <span class="n">be</span> <span class="n">used</span> <span class="n">together</span> <span class="k">with</span> <span class="n">FixedLocator</span>
+  <span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">&#39;__main__&#39;</span><span class="p">:</span>
+<span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">ipykernel_launcher</span><span class="o">.</span><span class="n">py</span><span class="p">:</span><span class="mi">15</span><span class="p">:</span> <span class="ne">UserWarning</span><span class="p">:</span> <span class="n">FixedFormatter</span> <span class="n">should</span> <span class="n">only</span> <span class="n">be</span> <span class="n">used</span> <span class="n">together</span> <span class="k">with</span> <span class="n">FixedLocator</span>
+  <span class="kn">from</span> <span class="nn">ipykernel</span> <span class="kn">import</span> <span class="n">kernelapp</span> <span class="k">as</span> <span class="n">app</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod4.png" src="_images/hymod4.png" />
+<p>Our findings indicate that in this instance, the streamflow estimate
+from HYMOD is highly sensitive to soil moisture parameters Huz and B and
+hardly affected by the routing parameters. Notably, there is very little
+interactions between parameters causing the total order indicies to be
+nearly identical to the first order indicies.</p>
+<p id="diffperformance">2.3 How do different performance metrics affect the results of our
+sensitivity analysis?</p>
+<p>Streamflow has many different properties. In this section, we discuss
+how the selection of metrics can lead to fundamentally different
+sensitivity analysis results. For example, one can only focus on
+aggregated streamflow metrics such as mean (what has been presented so
+far), or only on extreme events such as drought or floods.</p>
+<p>Here we compare three different metrics: 1- Mean error (ME) 2- Root Mean
+Square Error (RMSE) 3- Log-Root Mean Square Error (Log(RMSE))</p>
+<p>Each of these metrics focuses on a specific attribute of streamflow. For
+example, RMSE highlights the impacts of extreme flood events, while
+LOG(RMSE) focuses on model performance during low-flow events.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># calculate error metrics</span>
+<span class="n">mae</span> <span class="o">=</span> <span class="n">Q_df</span><span class="p">[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">leaf_data</span><span class="p">[</span><span class="s2">&quot;Strmflw&quot;</span><span class="p">]),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="n">mse</span> <span class="o">=</span> <span class="n">Q_df</span><span class="p">[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">metrics</span><span class="o">.</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s2">&quot;Strmflw&quot;</span><span class="p">]),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="n">rmse</span> <span class="o">=</span> <span class="n">mse</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="c1"># add error metrics to a dictionary</span>
+<span class="n">d_metrics</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;MAE&#39;</span><span class="p">:</span> <span class="n">mae</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">.</span><span class="n">values</span><span class="p">,</span>
+             <span class="s1">&#39;RMSE&#39;</span><span class="p">:</span> <span class="n">rmse</span><span class="o">.</span><span class="n">values</span><span class="p">,</span>
+             <span class="s1">&#39;LOG[RMSE]&#39;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">rmse</span><span class="o">.</span><span class="n">values</span><span class="p">)}</span>
+
+<span class="c1"># convert to a dataframe</span>
+<span class="n">df_metrics_SA</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d_metrics</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>We can use the following to calculate the SA indices for each metric and
+visualize it.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">df_metric_s1_result</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">df_metric_sT_result</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+
+<span class="c1"># conduct sensitivity analysis for each metric</span>
+<span class="k">for</span> <span class="n">index</span><span class="p">,</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">d_metrics</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+
+    <span class="c1"># get the data as a numpy array for the target metric</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">d_metrics</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+    <span class="c1"># use the metric to conduct SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add the sensitivity indices to the output data frame</span>
+    <span class="n">df_metric_s1_result</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">index</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">]</span>
+    <span class="n">df_metric_sT_result</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">index</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">]</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create seaborn heatmap with required labels</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+<span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="c1"># labels for y-axis</span>
+<span class="n">y_axis_labels</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Mean Absoulte Error&#39;</span><span class="p">,</span> <span class="s1">&#39;RSME&#39;</span><span class="p">,</span> <span class="s1">&#39;Log(RMSE)&#39;</span><span class="p">]</span>
+
+<span class="c1"># plot heatmap</span>
+<span class="n">ax1</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">df_metric_s1_result</span><span class="p">,</span> <span class="n">yticklabels</span><span class="o">=</span><span class="n">y_axis_labels</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>  <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;inferno_r&#39;</span><span class="p">,</span> <span class="n">cbar_kws</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;label&#39;</span><span class="p">:</span> <span class="s1">&#39;Si&#39;</span><span class="p">},</span> <span class="n">cbar</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">figure</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">label</span><span class="o">.</span><span class="n">set_size</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;First Order Sensitivity&#39;</span><span class="p">)</span>
+
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">df_metric_sT_result</span><span class="p">,</span> <span class="n">yticklabels</span><span class="o">=</span><span class="n">y_axis_labels</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>  <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;inferno_r&#39;</span><span class="p">,</span> <span class="n">cbar_kws</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;label&#39;</span><span class="p">:</span> <span class="s1">&#39;Si&#39;</span><span class="p">})</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">figure</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">label</span><span class="o">.</span><span class="n">set_size</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Total Order Sensitivity&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Text</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="s1">&#39;Total Order Sensitivity&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod5.png" src="_images/hymod5.png" />
+<p>The first order sensitivity indicies indicate that HYMOD’s sensitivity
+to its parameters is different depending on how its output is measured.
+Unsurprisingly, the mean absolute error is highly sensitive to the soil
+moisture accounting parameters Huz and B, just like the overall
+streamflow predictions above. However, when we examine RMSE and
+log(RMSE), the routing parameters Alp become sensitive, and the
+sensitivity to parameter B is reduced. As described above, RMSE and
+LOG(RMSE) respond to model performance in high-flow and low flow periods
+respectively. Our results indicate that for these flow regimes Alp, the
+parameter that governs the split between quick and slow flow is an
+important factor. While still the parameter with the highest most effect
+on all three measures, Huz is much less influential for RMSE and
+LOG(RMSE) than it is for MAE.</p>
+<p>The total order sensitivity indicies review a different, more complex
+story. While the MAE sensitivity is relatively governed by first order
+effects (like the streamflow predictions above), the RMSE and LOG(RMSE)
+error metrics show significant interactions. Alp has the highest total
+order sensitivity for RMSE and is eqal to Huz for Log(RMSE). Kq, which
+has a relatively low first order sensitivity index, shows strong
+contribution to variance when interactions are taken into account.</p>
+<p>Radial convergence plots are a helpful way to visualize the interactions
+between parameters. These plots array the model parameters in a circle
+and plot the first order, total order and second order Sobol sensitivity
+indices for each parameter. The first order sensitivity is shown as the
+size of a closed circle, the total order as the size of a larger open
+circle and the second order as the thickness of a line connecting two
+parameters. Below is an example of a radial convergence plot for the
+LOG(RMSE) measure. The plot indicates strong interactions between the
+Huz and Alp parameters, as well as Alp and Kq. There is also an
+interaction between Alp and Ks.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">itertools</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+<span class="kn">import</span> <span class="nn">math</span>
+<span class="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s1">&#39;whitegrid&#39;</span><span class="p">,</span> <span class="p">{</span><span class="s1">&#39;axes_linewidth&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="s1">&#39;axes.edgecolor&#39;</span><span class="p">:</span> <span class="s1">&#39;white&#39;</span><span class="p">})</span>
+
+<span class="k">def</span> <span class="nf">is_significant</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">confidence_interval</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="s2">&quot;conf&quot;</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">threshold</span> <span class="o">==</span> <span class="s2">&quot;conf&quot;</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">value</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">(</span><span class="n">confidence_interval</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">value</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">threshold</span><span class="p">))</span> <span class="o">&gt;</span> <span class="mi">0</span>
+
+<span class="k">def</span> <span class="nf">grouped_radial</span><span class="p">(</span><span class="n">SAresults</span><span class="p">,</span> <span class="n">parameters</span><span class="p">,</span> <span class="n">radSc</span><span class="o">=</span><span class="mf">2.0</span><span class="p">,</span> <span class="n">scaling</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">widthSc</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">STthick</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">varNameMult</span><span class="o">=</span><span class="mf">1.3</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">groups</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">gpNameMult</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="s2">&quot;conf&quot;</span><span class="p">):</span>
+    <span class="c1"># Derived from https://github.com/calvinwhealton/SensitivityAnalysisPlots</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+    <span class="n">color_map</span> <span class="o">=</span> <span class="p">{}</span>
+
+    <span class="c1"># initialize parameters and colors</span>
+    <span class="k">if</span> <span class="n">groups</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+
+        <span class="k">if</span> <span class="n">colors</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">colors</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;k&quot;</span><span class="p">]</span>
+
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">parameter</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+            <span class="n">color_map</span><span class="p">[</span><span class="n">parameter</span><span class="p">]</span> <span class="o">=</span> <span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)]</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">if</span> <span class="n">colors</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">colors</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">color_palette</span><span class="p">(</span><span class="s2">&quot;deep&quot;</span><span class="p">,</span> <span class="nb">max</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">groups</span><span class="p">)))</span>
+
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">groups</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+            <span class="c1">#parameters.extend(groups[key])</span>
+
+            <span class="k">for</span> <span class="n">parameter</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">[</span><span class="n">key</span><span class="p">]:</span>
+                <span class="n">color_map</span><span class="p">[</span><span class="n">parameter</span><span class="p">]</span> <span class="o">=</span> <span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)]</span>
+
+    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">parameters</span><span class="p">)</span>
+    <span class="n">angles</span> <span class="o">=</span> <span class="n">radSc</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">radSc</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angles</span><span class="p">)</span>
+    <span class="n">y</span> <span class="o">=</span> <span class="n">radSc</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angles</span><span class="p">)</span>
+
+    <span class="c1"># plot second-order indices</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">combinations</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="mi">2</span><span class="p">):</span>
+        <span class="c1">#key1 = parameters[i]</span>
+        <span class="c1">#key2 = parameters[j]</span>
+
+        <span class="k">if</span> <span class="n">is_significant</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S2&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S2_conf&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">threshold</span><span class="p">):</span>
+            <span class="n">angle</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">atan</span><span class="p">((</span><span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">-</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
+
+            <span class="k">if</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">-</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
+                <span class="n">angle</span> <span class="o">+=</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span>
+
+            <span class="n">line_hw</span> <span class="o">=</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S2&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">])</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
+
+            <span class="n">coords</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">((</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Polygon</span><span class="p">(</span><span class="n">coords</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;0.75&quot;</span><span class="p">))</span>
+
+    <span class="c1"># plot total order indices</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">is_significant</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST_conf&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">threshold</span><span class="p">):</span>
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Circle</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;w&#39;</span><span class="p">))</span>
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Circle</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="n">STthick</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;0.4&#39;</span><span class="p">,</span> <span class="n">fill</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
+
+    <span class="c1"># plot first-order indices</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">is_significant</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S1&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S1_conf&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">threshold</span><span class="p">):</span>
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Circle</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S1&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;0.4&#39;</span><span class="p">))</span>
+
+    <span class="c1"># add labels</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+        <span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">varNameMult</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">varNameMult</span><span class="o">*</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">key</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span>
+                <span class="n">rotation</span><span class="o">=</span><span class="n">angles</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="mi">360</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">-</span> <span class="mi">90</span><span class="p">,</span>
+                <span class="n">color</span><span class="o">=</span><span class="n">color_map</span><span class="p">[</span><span class="n">key</span><span class="p">])</span>
+
+    <span class="k">if</span> <span class="n">groups</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">group</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">groups</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+            <span class="nb">print</span><span class="p">(</span><span class="n">group</span><span class="p">)</span>
+            <span class="n">group_angle</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">angles</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">if</span> <span class="n">parameters</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">[</span><span class="n">group</span><span class="p">]])</span>
+
+            <span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">gpNameMult</span><span class="o">*</span><span class="n">radSc</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">group_angle</span><span class="p">),</span> <span class="n">gpNameMult</span><span class="o">*</span><span class="n">radSc</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">group_angle</span><span class="p">),</span> <span class="n">group</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span>
+                <span class="n">rotation</span><span class="o">=</span><span class="n">group_angle</span><span class="o">*</span><span class="mi">360</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">-</span> <span class="mi">90</span><span class="p">,</span>
+                <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)])</span>
+
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">([])</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;equal&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">])</span>
+    <span class="c1">#plt.show()</span>
+
+
+    <span class="k">return</span> <span class="n">fig</span>
+
+<span class="c1"># define groups for parameter uncertainties</span>
+<span class="n">groups</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;Soil Moisture&quot;</span> <span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Huz&quot;</span><span class="p">,</span> <span class="s2">&quot;B&quot;</span><span class="p">],</span>
+        <span class="s2">&quot;Routing&quot;</span> <span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Alp&quot;</span><span class="p">,</span> <span class="s2">&quot;Kq&quot;</span><span class="p">,</span> <span class="s2">&quot;Ks&quot;</span><span class="p">]}</span>
+
+
+<span class="n">fig</span> <span class="o">=</span> <span class="n">grouped_radial</span><span class="p">(</span><span class="n">Si</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">],</span> <span class="n">groups</span><span class="o">=</span><span class="n">groups</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="mf">0.025</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Soil</span> <span class="n">Moisture</span>
+<span class="n">Routing</span>
+</pre></div>
+</div>
+<img alt="_images/hymod6.png" src="_images/hymod6.png" />
+<p>2.4 Time-Varying Sensitivity Analysis</p>
+<p id="time">In section 2.5 we saw how performing sensitivity analysis on different
+measurements of model output can yeild in different results on the
+importance of each uncertain input. In this section we’ll examine how
+performing this analysis over time can yeild additional insight into the
+performance of HYMOD. We’ll first examine how model sensitivities vary
+by month, then examine how they change across each year of the
+simulation.</p>
+<p>For this demonstration, we’ll focus only on the monthly streamflow
+predictions generated by HYMOD.</p>
+<section id="sensitivity-analysis-indices-for-each-month">
+<h4>2.4.1 Sensitivity analysis indices for each month<a class="headerlink" href="#sensitivity-analysis-indices-for-each-month" title="Link to this heading">#</a></h4>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># aggregate simulated streamflow data to monthly time series</span>
+<span class="n">df_sim_by_mth_mean</span> <span class="o">=</span> <span class="n">Q_df</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s1">&#39;month&#39;</span><span class="p">)[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+
+<span class="c1"># aggregate observed streamflow data to monthly time series</span>
+<span class="n">df_obs_by_mth_mean</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s1">&#39;month&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+<p>We can use the following to calculate the SA indices for each month and
+visualize it. Results are pre-loaded for efficiency.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># set up dataframes to store outputs</span>
+<span class="n">df_mth_s1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">12</span><span class="p">,</span><span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">df_mth_delta</span> <span class="o">=</span> <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="c1"># iterate through each month</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">12</span><span class="p">):</span>
+
+    <span class="c1"># generate the simulation data</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">df_sim_by_mth_mean</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+    <span class="c1"># run SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">delta</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">param_values_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add to output dataframes</span>
+    <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="n">df_mth_delta</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;delta&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+
+<span class="c1"># convert to arrays</span>
+<span class="n">arr_mth_s1</span> <span class="o">=</span> <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">values</span>
+<span class="n">arr_mth_delta</span> <span class="o">=</span> <span class="n">df_mth_delta</span><span class="o">.</span><span class="n">values</span>
+</pre></div>
+</div>
+</section>
+</section>
+<section id="first-order-indices">
+<h3>First-order Indices<a class="headerlink" href="#first-order-indices" title="Link to this heading">#</a></h3>
+<p>The following can be used to visualize the time-varying first-order
+indices. The first order represents the direct impacts of a specific
+parameter on model outputs.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-monthly-heatmap">plot_monthly_heatmap</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously ran data</span>
+<span class="n">arr_mth_delta</span><span class="p">,</span> <span class="n">arr_mth_s1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_monthly_simulations</span><span class="p">()</span>
+
+<span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_monthly_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_mth_s1</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                       <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_mth_mean</span><span class="p">,</span>
+                                       <span class="n">title</span><span class="o">=</span><span class="s1">&#39;First Order - Mean Monthly SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod7.png" src="_images/hymod7.png" />
+<p>This figure demonstrates the first order sensitivity indices when the
+streamflow data are aggregated by month. The purple line represents the
+observed monthly discharge. The figure indicates that the first order
+indices are highest for B and Huz across all months and lowest for Alp,
+Ks, and Kq. Note that in the months with the highest flow, Ks becomes an
+influential parameter.</p>
+</section>
+<section id="total-order-indices">
+<h3>Total-order indices<a class="headerlink" href="#total-order-indices" title="Link to this heading">#</a></h3>
+<p>We can also focus on the total order sensitivity index that includes
+first-order SA indices and interactions between parameters</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_monthly_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_mth_delta</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                       <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_mth_mean</span><span class="p">,</span>
+                                       <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Total Order - Mean monthly SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod8.png" src="_images/hymod8.png" />
+<p>Notably, the total order sensitivity results are different than the
+first order sensitivity results, which indicates that interactions
+between the parameters (particularly in regards to routing parameters
+<span class="math notranslate nohighlight">\(Kq\)</span>, <span class="math notranslate nohighlight">\(Ks\)</span>, and <span class="math notranslate nohighlight">\(Alp\)</span>) contribute to changes in HYMOD
+output.</p>
+<section id="annual-sensitivity-analysis-indices">
+<h4>2.4.2 Annual sensitivity analysis indices<a class="headerlink" href="#annual-sensitivity-analysis-indices" title="Link to this heading">#</a></h4>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># group by year and get mean</span>
+<span class="n">df_sim_by_yr_mean</span> <span class="o">=</span> <span class="n">Q_df</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">])[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+
+<span class="c1"># group input data and get mean</span>
+<span class="n">df_obs_by_yr_mean</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">])</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+<p>We can also calculate the sensitivity analysis indices for each
+individual year. This will allow us to understand if model control
+changes during different years. The following code first aggregates the
+outputs to annual time steps, and then calculates the SA indices.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># set up dataframes to store outputs</span>
+<span class="n">df_yr_s1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">df_yr_delta</span> <span class="o">=</span> <span class="n">df_yr_s1</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="c1"># iterate through each year</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">):</span>
+
+    <span class="c1"># generate the simulation data</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">df_sim_by_yr_mean</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+    <span class="c1"># run SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">delta</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">param_values_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add to output dataframes</span>
+    <span class="n">df_yr_s1</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="n">df_yr_delta</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;delta&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+
+<span class="c1"># convert to arrays</span>
+<span class="n">arr_yr_s1</span> <span class="o">=</span> <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">values</span>
+<span class="n">arr_yr_delta</span> <span class="o">=</span> <span class="n">df_mth_delta</span><span class="o">.</span><span class="n">values</span>
+</pre></div>
+</div>
+</section>
+</section>
+<section id="id3">
+<h3>First-order indices<a class="headerlink" href="#id3" title="Link to this heading">#</a></h3>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-annual-heatmap">plot_annual_heatmap</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously ran data</span>
+<span class="n">arr_yr_delta</span><span class="p">,</span> <span class="n">arr_yr_s1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_annual_simulations</span><span class="p">()</span>
+
+<span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_annual_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_yr_s1</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_yr_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;First Order - Mean Annual SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod9.png" src="_images/hymod9.png" />
+<p>The first order sensitivities at the annual scale are not unlike the
+first order monthly sensitivities. Once again, sensitivities vary across
+year and Huz and B are the most consequential parameters.</p>
+</section>
+<section id="id4">
+<h3>Total-order indices<a class="headerlink" href="#id4" title="Link to this heading">#</a></h3>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_annual_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_yr_delta</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_yr_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Total Order - Mean Annual SA and Observed flow&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod10.png" src="_images/hymod10.png" />
+<p>Our results indicate that sensitivity analysis indices vary in different
+years and now that interactions are included, the Kq, Ks, and Alp
+variables impact the sensitivity of the streamflow output.</p>
+<section id="monthly-time-varying-sensitivity-analysis">
+<h4>2.4.3 Monthly time-varying sensitivity analysis<a class="headerlink" href="#monthly-time-varying-sensitivity-analysis" title="Link to this heading">#</a></h4>
+<p>Although time-varying sensitivity analysis (TVSA) at average monthly and
+average annual temporal resolutions is informative, TVSA is susceptible
+to the aggregation issue that we discussed earlier in section 3-2. To
+avoid that we can further discretize our time domain to zoom into
+individual months. This will provide us with even more information about
+model behavior and the sensitivity of different parameters in different
+states of the system. The block of code demonstrates how to implement
+the monthly TVSA.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># set up dataframes to store outputs</span>
+<span class="n">df_vary_s1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">df_obs_mth_mean</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">5</span><span class="p">)),</span>
+                          <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+
+<span class="n">df_vary_delta</span> <span class="o">=</span> <span class="n">df_vary_s1</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="c1"># iterate through each month</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">df_obs_mth_mean</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+
+    <span class="c1"># generate the simulation data</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">df_sim_mth_mean</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+    <span class="c1"># run SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">delta</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">param_values_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add to output dataframes</span>
+    <span class="n">df_vary_s1</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="n">df_vary_delta</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;delta&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+
+<span class="c1"># convert to arrays</span>
+<span class="n">arr_vary_s1</span> <span class="o">=</span> <span class="n">df_vary_s1</span><span class="o">.</span><span class="n">values</span>
+<span class="n">arr_vary_delta</span> <span class="o">=</span> <span class="n">df_vary_delta</span><span class="o">.</span><span class="n">values</span>
+</pre></div>
+</div>
+</section>
+</section>
+<section id="id5">
+<h3>First-order indices<a class="headerlink" href="#id5" title="Link to this heading">#</a></h3>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-varying-heatmap">plot_varying_heatmap</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load in previously ran data</span>
+<span class="n">arr_vary_delta</span><span class="p">,</span> <span class="n">arr_vary_s1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_varying_simulations</span><span class="p">()</span>
+
+<span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_varying_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_vary_s1</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_mth_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;First Order - Time-Varying SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod11.png" src="_images/hymod11.png" />
+<p>Compared to the TVSA when streamflow was aggregated, this figure
+suggests that Kq is indeed a relevant parameter for influencing
+streamflow output when individual months are considered.</p>
+</section>
+<section id="total-order-time-varying-sensitivity-analysis">
+<h3>Total order - time varying sensitivity analysis<a class="headerlink" href="#total-order-time-varying-sensitivity-analysis" title="Link to this heading">#</a></h3>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_varying_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_vary_delta</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_mth_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Total Order - Time-Varying SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod12.png" src="_images/hymod12.png" />
+<p>As above, the total order sensitivities further indicate the importance
+of Kq that is not apparent if aggregation is utilized.</p>
+</section>
+</section>
+<section id="tips-to-apply-this-methodology-to-your-own-problem">
+<h2>Tips to Apply this methodology to your own problem<a class="headerlink" href="#tips-to-apply-this-methodology-to-your-own-problem" title="Link to this heading">#</a></h2>
+<p>In this tutorial, we demonstrated how to use global sensitivtiy analysis
+to explore a complex, non-linear model. We showed how measuring
+sensitivity across multiple measures of model performance and temporal
+aggregations yeilding differing results about model
+sensitivity/behavoir. While these results may seem contraditory, they
+provide useful insight into the behavoir of HYMOD. Would we expect the
+same parameters to control high flow and low flow regimes within the
+model? Maybe, depending on the system, but also, maybe not. This
+analysis can provide insight into how the model responds to its input
+parameters, allowing us to compare the results to our expectaions. This
+may allow us to find problems with our intial assumptions, or call
+attention to model features that can be improved or expanded upon.
+Depending on the model and context, it may also yield insight into the
+workings of the underlying system.</p>
+<p>To run this tutorial on your own model you will need to:</p>
+<ol class="arabic simple">
+<li><p>Design your experiment by choosing sampling bounds for your
+parameters and setting up the problem dictionary as in step 2-2</p></li>
+<li><p>Choose the parameters of interest</p></li>
+<li><p>Generate samples using the saltelli.sample function. This step is
+problem-dependent and note that the Sobol method can be
+computationally intensive depending on the model being analyzed. More
+complex models will be slower to run and will also require more
+samples to calculate accurate estimates of Sobol indices. Once you
+complete this process, pay attention to the confidence bounds on your
+sensitivity indices to see whether you need to run more samples.</p></li>
+<li><p>Run the parameter sets through your model and record each of the
+desired model outputs.</p></li>
+<li><p>Calculate the Sobol indices for each performance criteria. Now, the Y
+will be a numpy array with your external model output and you will
+need to include the parameter samples as an additional argument.</p></li>
+<li><p>Follow the procedure in step 2.6 to disaggregate performance across
+time</p></li>
+</ol>
+</section>
+</section>
+
+
+                </article>
+              
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+    <i class="fa-solid fa-list"></i> Contents
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+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-of-the-hymod-model">Tutorial: Sensitivity Analysis of the HYMOD Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction-to-hymod">1. Introduction to HYMOD</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#global-sensitivity-analysis">2. Global Sensitivity Analysis</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">1 - Introduction to HYMOD</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">2- Global Sensitivity Analysis</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#running-hymod-model-warm-up">Running HYMOD - Model Warm-up</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#sensitivity-analysis-indices-for-each-month">2.4.1 Sensitivity analysis indices for each month</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#first-order-indices">First-order Indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#total-order-indices">Total-order indices</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#annual-sensitivity-analysis-indices">2.4.2 Annual sensitivity analysis indices</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">First-order indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id4">Total-order indices</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#monthly-time-varying-sensitivity-analysis">2.4.3 Monthly time-varying sensitivity analysis</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id5">First-order indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#total-order-time-varying-sensitivity-analysis">Total order - time varying sensitivity analysis</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-this-methodology-to-your-own-problem">Tips to Apply this methodology to your own problem</a></li>
+</ul>
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+    <h1>Time-evolving scenario discovery for infrastructure pathways</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Time-evolving scenario discovery for infrastructure pathways</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#background">Background</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluating-robustness-over-time">Evaluating Robustness over time</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exploring-performance-evolution">Exploring performance evolution</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#how-do-deep-uncertainties-generate-vulnerability">How do deep uncertainties generate vulnerability</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#which-uncertainties-have-the-most-influence-on-time-evolving-performance">Which uncertainties have the most influence on time-evolving performance?</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#open-exploration">Open exploration</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-this-methodology-to-your-own-problem">Tips to apply this methodology to your own problem</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#references">References</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="time-evolving-scenario-discovery-for-infrastructure-pathways">
+<h1>Time-evolving scenario discovery for infrastructure pathways<a class="headerlink" href="#time-evolving-scenario-discovery-for-infrastructure-pathways" title="Link to this heading">#</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/Bedford_Greene_SD_tutorial.ipynb">Scenario Discovery Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="id1">
+<h2>Time-evolving scenario discovery for infrastructure pathways<a class="headerlink" href="#id1" title="Link to this heading">#</a></h2>
+<p>The purpose of this tutorial is to explore time-evolving vulnerability for systems that dynamically adapt to changing conditions. Using an example from water supply planning, we’ll first examine how performance of a dynamic infrastructure pathway policy changes over time, then use factor mapping (main text Chapter 4.3) to understand which combinations of uncertaities generate vulnerability for two water utilities. Next, we’ll perform factor prioritization (main text Chapter 4.3) to determine which uncertainties have the most influence on water supply performance. Finally, we’ll provide an open platform to explore vulnerability across multiple measures of performance and different combinations of uncertainties.</p>
+<section id="background">
+<h3>Background<a class="headerlink" href="#background" title="Link to this heading">#</a></h3>
+<p>The Bedford-Greene metropolitan area (Figure 1) is a stylized water resources test case where two urban water utilities seek to develop an infrastructure and investment and management strategy to confront growing demands and changing climate. The utilities have agreed to jointly construct a new water treatment plant on Lake Classon, a large regional resource. Both utilities have also identified a set of individual infrastructure options to construct if necessary.</p>
+<figure class="align-default" id="id2">
+<img alt="Figure 1" src="_images/Map_small.png" />
+<figcaption>
+<p><span class="caption-text">Figure 1</span><a class="headerlink" href="#id2" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The utilities are formulating a cooperative and adaptive regional management strategy that uses a risk-of-failure (ROF) metric to trigger both short term drought mitigation actions (water use restrictions and treated transfers between utilities) and long-term infrastructure investment decisions (shown in Figure 2a). Both utilities have specified a set of risk triggers and developed a construction order for available infrastructure options.</p>
+<p>The utilities have run a Monte Carlo simulation to evaluate how these policies respond to a wide array of future States Of the World (SOWs). Each SOW represents a different combinations of thirteen uncertain system inputs including demand growth rates, changes to streamflows and financial variables. In this context, a fully specified SOW is composed of one sample of uncertain model inputs (e.g. one projection of demand growth rate coupled with one future streamflow scenario and one projection of future financial conditions). The water utilities used Latin Hypercube sampling (Chapter 3.3 of the main text) to develop an ensemble of 1,000 plausible future SOWs. The Monte Carlo simulation evaluates each candidate water supply infrastructure investment and management policy across all 1,000 SOWs, as shown in Figure 2b. For more details on the Monte Carlo sampling for this type  of analysis, see Trindade et al., (2019).</p>
+<p>The ROF-based policies respond to each SOW by generating a unique infrastructure pathway - a sequence of infrastructure investment decisions over time. Infrastructure pathways over a set of 1,000 future SOWs are shown in Figure 2c. Infrastructure options are arrayed along the vertical axis and the sequence of infrastructure investments triggered by the policy is plotted as pathways over time.  Since the adaptive rule system generates a unique infrastructure sequence for each scenario, Figure 2c summarizes the ensemble of pathways by clustering SOWs according to infrastructure intensity. Dark green lines represent SOWs where the utilities heavily invest in new infrastructure, light green lines represent SOWs with low infrastructure investment and medium shaded lines represent moderate investment. The shading behind each pathway represents the frequency that each infrastructure option was triggered over time across sampled scenarios</p>
+<figure class="align-default" id="id3">
+<img alt="Figure 2" src="_images/Policy_MonteCarlo_Pathways_small.png" />
+<figcaption>
+<p><span class="caption-text">Figure 2</span><a class="headerlink" href="#id3" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="evaluating-robustness-over-time">
+<h3>Evaluating Robustness over time<a class="headerlink" href="#evaluating-robustness-over-time" title="Link to this heading">#</a></h3>
+<p>The two water utilities are interested in mainting both supply
+reliability and financial stability across the broadest set of plausible
+future SOWs. To measure the performance of the infrastructure pathway
+policy, they’ve defined five critical performance criteria:</p>
+<ul class="simple">
+<li><p>Reliability &gt; 99%</p></li>
+<li><p>Restriction Frequency &lt; 20%</p></li>
+<li><p>Peak Financial Cost &lt; 80% of annual revenue (a measure of debt
+service spending)</p></li>
+<li><p>Worst-case drought management cost &lt; 10% of annual revenue (a measure
+of unexpected drought costs)</p></li>
+<li><p>Unit Cost of Expansion &lt; 5 dollars/kgal</p></li>
+</ul>
+<p>To assess the robustness of the infrastructure pathway policy, the two
+utilities apply a satisficing metric, which measures the percentage of
+sampled SOWs where the pathway policy meets the peformance criteria:</p>
+<p><span class="math notranslate nohighlight">\(R =\frac{1}{N}\sum_{j=1}^{N}\Lambda_{\theta,j}\)</span></p>
+<p>Where, <span class="math notranslate nohighlight">\(\Lambda\_{\theta,j}=\)</span></p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases}
+1, \quad \textrm{if}\ F(\theta)_{j}\leq \Phi_j \\
+0, \quad \textrm{otherwise}
+\end{cases}\end{split}\]</div>
+<p>And <span class="math notranslate nohighlight">\(\Phi\)</span> is a vector of performance criteria for utility
+<span class="math notranslate nohighlight">\(j\)</span>, <span class="math notranslate nohighlight">\(\theta\)</span> is the portfolio and <span class="math notranslate nohighlight">\(N\)</span> is the total
+number of sampled SOWs.</p>
+<p>Below, we’ll visualize how robustness for the two utilities evolves over the 45-year planning horizon. We’ll assess robustness across three time periods, near-term (first 10 years), mid-term (22 years) and long term (45 years).</p>
+<p>We start by loading robustness values for the both utilities. These values are calculated by applying the robustness metric above across 2,000 simulated SOWs. To make this exercise computationally tractable, we’ve precomputed these values, which can be found in the files “short_term_robustness.csv”, “mid_term_robustness.csv” and “long_term_robustness.csv”. These values are calculated using the function “check_rdm_meet_criteria” within the helper functions.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
+<span class="kn">from</span> <span class="nn">functions.eBook_SD_helpers</span> <span class="kn">import</span> <span class="n">check_rdm_meet_criteria</span><span class="p">,</span> <span class="n">create_sd_input</span><span class="p">,</span> <span class="n">plot_selected_tree_maps</span><span class="p">,</span> <span class="n">get_factor_importances</span><span class="p">,</span> <span class="n">open_exploration</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+
+<span class="c1"># load Deeply uncertain factors</span>
+<span class="n">rdm_factors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/DU_Factors.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+
+<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
+<span class="n">short_term_robustness</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/short_term_robustness.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">mid_term_robustness</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/mid_term_robustness.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">long_term_robustness</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/long_term_robustness.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+
+<span class="c1"># plot robustness over time</span>
+<span class="n">fig</span> <span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span><span class="mi">22</span><span class="p">,</span><span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span><span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span> <span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span><span class="mi">22</span><span class="p">,</span><span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span><span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;Time Horizon (yrs)&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Robustness (% SOWs)&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">&#39;Bedford&#39;</span><span class="p">,</span> <span class="s1">&#39;Greene&#39;</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Robustness Over Time&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">107</span><span class="p">])</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">107.0</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_4_1.png" src="_images/discovery_4_1.png" />
+</section>
+<section id="exploring-performance-evolution">
+<h3>Exploring performance evolution<a class="headerlink" href="#exploring-performance-evolution" title="Link to this heading">#</a></h3>
+<p>The figure above reveals that the robustness of both water utilities degrades over time, with Bedford’s robustness declining further than Greene. This suggests that the proposed pathway policy is likely insufficient to meet the long-term needs of the two utilities. But how is the current policy insufficient? To answer that question we examine the performance measures that fail to meet performance criteria for each utility across the three planning horizons.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot the type of vulnerability over time</span>
+
+<span class="c1">### Bedford ###</span>
+<span class="n">plot_robustness_1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
+<span class="c1"># Determine the percentage of failure SOWs that violate each criterion (note some SOWS fail multiple criteria, so this may some to &gt;1)</span>
+<span class="n">criteria</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Reliability&#39;</span><span class="p">,</span> <span class="s1">&#39;Restriction Frequency&#39;</span><span class="p">,</span> <span class="s1">&#39;Peak Financial Cost&#39;</span><span class="p">,</span> <span class="s1">&#39;Worst-case drought</span><span class="se">\n</span><span class="s1">Management Cost&#39;</span><span class="p">,</span> <span class="s1">&#39;Stranded Assets&#39;</span><span class="p">]</span>
+<span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span>
+<span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">1</span><span class="p">,:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span>
+<span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">2</span><span class="p">,:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span>
+
+<span class="c1"># Plot over time</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">0</span><span class="p">,:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Fraction of failure SOWs&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">1</span><span class="p">,:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">2</span><span class="p">,:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Bedford&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="c1">### Greene ###</span>
+<span class="c1"># Determine the percentage of failure SOWs that violate each criterion (note some SOWS fail multiple criteria, so this may some to &gt;1)</span>
+<span class="n">plot_robustness_2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
+<span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">11</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">])</span>
+<span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">11</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">])</span>
+<span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">11</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">])</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Fraction of failure SOWs&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="p">:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Greene&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_6_0.png" src="_images/discovery_6_0.png" />
+<img alt="_images/discovery_6_1.png" src="_images/discovery_6_1.png" />
+<p>In the figures above, we observe that the vulnerability of both utilities changes in different ways. Early in the simulation period, Bedford is vulnerable to failures in reliability (though the robustness figure created in step B5.2 reveals that these failures are very rare). As the simulation period progresses, Bedford’s vulnerability expands to include failures in restriction frequency and worst-case cost. These failures indicate that the utility has an overall inability to manage drought conditions and future conditions progress.</p>
+<p>Greene shows a very different evolution in vulnerability. Early in the simulation period, failures manifest in the restriction frequency objective, suggesting that the utility must rely on water use restrictions to maintain supply reliability. As the simulation progresses however, the vulnerability evolves. When evaluated across the 45-year planning horizon, a new failure modes emerges - financial failure manifesting in peak financial cost and stranded assets. This suggests that the proposed pathway policy may be over-investing in new infrastructure, straining the utility’s budget with large debt payments that are unnecessary to maintain supply reliability.</p>
+</section>
+<section id="how-do-deep-uncertainties-generate-vulnerability">
+<h3>How do deep uncertainties generate vulnerability<a class="headerlink" href="#how-do-deep-uncertainties-generate-vulnerability" title="Link to this heading">#</a></h3>
+<p>While the evolution of robustness provides insight into how the system
+evolves over time, it does not reveal <em>why</em> each utility is vulnerable.
+To examine how deep uncertainties generate vulnerability over time for
+the two utilities, we perform scenario discovery (factor mapping,
+Chapter 4.3). Here we’ll utilize gradient boosted trees to identify
+regions of the uncertainty space that cause the utilities to fail to
+meet performance criteria.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># import the performance data across 2000 SOWs for three time periods</span>
+<span class="n">short_term_performance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/short_term_performance.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">mid_term_performance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/mid_term_performance.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">long_term_performance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/long_term_performance.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+
+<span class="n">satisficing_criteria</span> <span class="o">=</span> <span class="p">[</span><span class="mf">.98</span><span class="p">,</span> <span class="mf">.2</span><span class="p">,</span> <span class="mf">.8</span><span class="p">,</span> <span class="mf">.1</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>
+
+<span class="c1"># transform into scenario discovery input</span>
+<span class="n">short_term_SD_input</span> <span class="o">=</span> <span class="n">create_sd_input</span><span class="p">(</span><span class="n">short_term_performance</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">)</span>
+<span class="n">mid_term_SD_input</span> <span class="o">=</span> <span class="n">create_sd_input</span><span class="p">(</span><span class="n">mid_term_performance</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">)</span>
+<span class="n">long_term_SD_input</span> <span class="o">=</span> <span class="n">create_sd_input</span><span class="p">(</span><span class="n">long_term_performance</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">)</span>
+
+<span class="c1"># factor mapping Bedford</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;short_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;mid_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;long_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Bedford Factor Maps&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="c1"># factor mapping Greene</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="s1">&#39;short_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="s1">&#39;mid_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="s1">&#39;long_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Greene Factor Maps&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Greene</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Greene</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Greene</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_9_1.png" src="_images/discovery_9_1.png" />
+<img alt="_images/discovery_9_2.png" src="_images/discovery_9_2.png" />
+<p>In the figures above, we learn more about how the vulnerability of the two utilities evolves over time. Bedford begins with very few possible failures but appears vulnerable to high demand growth scenarios under future scenarios with high demands. When evaluated across a 22-year planning horizon, Bedford is vulnerable when the near-term demand growth is high and water use restrictions are less effective than predicted. Under the full 45-year planning horizon, Bedford is vulnerable to sustained high levels of demand growth, failing if either near-term or mid-term demand growth exceeds expected levels.</p>
+<p>Greene’s vulnerability evolves differently. It begins with vulnerability to high demand growth, but as the simulation progresses (and infrastructure is constructed), the utility becomes vulnerable to low-demand growth futures which cause the failures in financial criteria shown in section B.5.3. This indicates that the pathway policy over-builds in many SOWs, and becomes financially unstable if demand does not grow sufficiently to provide revenue to cover debt service payments.</p>
+</section>
+<section id="which-uncertainties-have-the-most-influence-on-time-evolving-performance">
+<h3>Which uncertainties have the most influence on time-evolving performance?<a class="headerlink" href="#which-uncertainties-have-the-most-influence-on-time-evolving-performance" title="Link to this heading">#</a></h3>
+<p>The factor maps generated in B.5.4 present the vulnerability generated by the two most important deep uncertainties as determined by Gradient Boosted Trees. Yet the factor prioritization shows that more than two uncertainties are influential to regional performance. Further, we can observe that individual uncertainties have different impacts on each performance obejctive, and these impacts may change over time. In the cells below, explore the impact of deep uncertainty by generating factor maps for different combinations of deep uncertain factors, objectives and time horizons.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+<span class="n">uncertainties</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;D1&#39;</span><span class="p">,</span> <span class="s1">&#39;D2&#39;</span><span class="p">,</span> <span class="s1">&#39;D3&#39;</span><span class="p">,</span> <span class="s1">&#39;BT&#39;</span><span class="p">,</span> <span class="s1">&#39;BM&#39;</span><span class="p">,</span> <span class="s1">&#39;DR&#39;</span><span class="p">,</span> <span class="s1">&#39;RE&#39;</span><span class="p">,</span> <span class="s1">&#39;EV&#39;</span><span class="p">,</span> <span class="s1">&#39;PM&#39;</span><span class="p">,</span> <span class="s1">&#39;CT&#39;</span><span class="p">,</span> <span class="s1">&#39;IA&#39;</span><span class="p">,</span> <span class="s1">&#39;IF&#39;</span><span class="p">,</span> <span class="s1">&#39;IP&#39;</span><span class="p">]</span>
+<span class="n">uncertainties</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Near-term demand&#39;</span><span class="p">,</span> <span class="s1">&#39;Mid-term demand&#39;</span><span class="p">,</span> <span class="s1">&#39;Long-term demand&#39;</span><span class="p">,</span> <span class="s1">&#39;Bond Term&#39;</span><span class="p">,</span> <span class="s1">&#39;Bond Rate&#39;</span><span class="p">,</span> <span class="s1">&#39;Discount Rate&#39;</span><span class="p">,</span> <span class="s1">&#39;Restriction Effectiveness&#39;</span><span class="p">,</span> <span class="s1">&#39;Evaporation Rate&#39;</span><span class="p">,</span> <span class="s1">&#39;Permitting time&#39;</span><span class="p">,</span> <span class="s1">&#39;Construction time&#39;</span><span class="p">,</span> <span class="s1">&#39;Inflow Amplitude&#39;</span><span class="p">,</span> <span class="s1">&#39;Inflow Frequency&#39;</span><span class="p">,</span> <span class="s1">&#39;Inflow Period&#39;</span><span class="p">]</span>
+
+<span class="n">u1_st_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">short_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="n">u1_mt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">mid_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="n">u1_lt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">long_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+
+<span class="n">u1_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">u1_st_FI</span><span class="p">,</span><span class="n">u1_mt_FI</span><span class="p">,</span> <span class="n">u1_lt_FI</span><span class="p">])</span>
+<span class="n">u1_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">u1_all</span><span class="p">)</span>
+
+<span class="c1"># factor ranking -- utility 2</span>
+<span class="n">u2_st_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">short_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
+<span class="n">u2_mt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">mid_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
+<span class="n">u2_lt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">long_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
+<span class="n">u2_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">u2_st_FI</span><span class="p">,</span><span class="n">u2_mt_FI</span><span class="p">,</span> <span class="n">u2_lt_FI</span><span class="p">])</span>
+<span class="n">u2_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">u2_all</span><span class="p">)</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span><span class="p">,</span> <span class="n">cax</span><span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span>
+                  <span class="n">gridspec_kw</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;width_ratios&quot;</span><span class="p">:[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">]})</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">wspace</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">im</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">u1_all</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;Reds&#39;</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mf">.3</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">13</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">(</span><span class="n">uncertainties</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Time Horizon&#39;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Bedford&#39;</span><span class="p">)</span>
+
+<span class="n">im1</span> <span class="o">=</span> <span class="n">ax2</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">u2_all</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;Reds&#39;</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mf">.3</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">13</span><span class="p">))</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">([])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Time Horizon&#39;</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Greene&#39;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="n">cax</span><span class="o">=</span><span class="n">cax</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Factor Importance&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_12_0.png" src="_images/discovery_12_0.png" />
+<p>The Figure above shows the factor importance as determined by gradient boosted trees for both utilities across the three planning horizons. While near-term demand growth is important for both utilities under all three planning horizons, the importance of other factors evolves over time. For example, restriction effectiveness plays an important role for Greene under the 22-year planning horizon but disappears under the 45-year planning horizon. In contrast, the bond interest rate is important for predicting success over the 45-year planning horizon, but does not appear important over the 10- or 22-year planning horizons. These findings highlight how assumptions about the planning period can have a large impact on modeling outcomes.</p>
+</section>
+<section id="open-exploration">
+<h3>Open exploration<a class="headerlink" href="#open-exploration" title="Link to this heading">#</a></h3>
+<p>In the cell below, use the function to explore how factor maps change
+for the two utilities based upon the uncertainties plotted, the
+objectives of interest and the time horizon.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># specify the utility (&quot;Bedford&quot; or &quot;Greene&quot;)</span>
+<span class="n">utility</span> <span class="o">=</span> <span class="s2">&quot;Bedford&quot;</span>
+
+<span class="c1"># specify which performance objectives to investigate (note that not all performance objectives have failures, which may result in a blank factor map)</span>
+<span class="c1"># set this to one of the following: &quot;Reliability&quot;, &quot;Restriction Frequency&quot;, &quot;Peak Financial Cost&quot;, &quot;Worst Case Cost&quot; or &quot;Unit Cost&quot;</span>
+<span class="n">objective</span> <span class="o">=</span> <span class="s2">&quot;Reliability&quot;</span>
+
+<span class="c1"># select uncertainties from the following list: &#39;D1&#39;, &#39;D2&#39;, &#39;D3&#39;, &#39;BT&#39;, &#39;BM&#39;, &#39;DR&#39;, &#39;RE&#39;, &#39;EV&#39;, &#39;PM&#39;, &#39;CT&#39;, &#39;IA&#39;, &#39;IF&#39;, &#39;IP&#39;</span>
+<span class="n">uncertainty_1</span> <span class="o">=</span> <span class="s1">&#39;D1&#39;</span>
+<span class="n">uncertainty_2</span> <span class="o">=</span> <span class="s1">&#39;D2&#39;</span>
+
+<span class="c1"># The code below will plot factor maps over the three planning horizons for the information above</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">open_exploration</span><span class="p">(</span><span class="n">utility</span><span class="p">,</span> <span class="n">objective</span><span class="p">,</span> <span class="s1">&#39;short_term&#39;</span><span class="p">,</span> <span class="n">uncertainty_1</span><span class="p">,</span> <span class="n">uncertainty_2</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">open_exploration</span><span class="p">(</span><span class="n">utility</span><span class="p">,</span> <span class="n">objective</span><span class="p">,</span> <span class="s1">&#39;mid_term&#39;</span><span class="p">,</span> <span class="n">uncertainty_1</span><span class="p">,</span> <span class="n">uncertainty_2</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">open_exploration</span><span class="p">(</span><span class="n">utility</span><span class="p">,</span> <span class="n">objective</span><span class="p">,</span> <span class="s1">&#39;long_term&#39;</span><span class="p">,</span> <span class="n">uncertainty_1</span><span class="p">,</span> <span class="n">uncertainty_2</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span><span class="p">,</span> <span class="n">reliability</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span><span class="p">,</span> <span class="n">reliability</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span><span class="p">,</span> <span class="n">reliability</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_16_1.png" src="_images/discovery_16_1.png" />
+</section>
+<section id="tips-to-apply-this-methodology-to-your-own-problem">
+<h3>Tips to apply this methodology to your own problem<a class="headerlink" href="#tips-to-apply-this-methodology-to-your-own-problem" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we demonstrated time-evolving scenario discovery for a
+cooperative water supply system. To apply this workflow to your own
+problem:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters of interest, which will
+represent uncertainties that characterize your system.</p></li>
+<li><p>Generate samples for these parameters (this can be done using the
+saltelli.sample function as in B.2 or done with another package).</p></li>
+<li><p>Define performance criteria for your problem</p></li>
+<li><p>Evaluate parameter sets through your model, and save performance
+measures across multiple time horizons</p></li>
+<li><p>Draw from the supporting code for this tutorial to perform scneario
+discovery and visualize results</p></li>
+</ol>
+</section>
+<section id="references">
+<h3>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h3>
+<p>Trindade, B. C., Reed, P. M., &amp; Characklis, G. W. (2019). Deeply uncertain pathways: Integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134, 103442.</p>
+</section>
+</section>
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+
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Time-evolving scenario discovery for infrastructure pathways</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#background">Background</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluating-robustness-over-time">Evaluating Robustness over time</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exploring-performance-evolution">Exploring performance evolution</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#how-do-deep-uncertainties-generate-vulnerability">How do deep uncertainties generate vulnerability</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#which-uncertainties-have-the-most-influence-on-time-evolving-performance">Which uncertainties have the most influence on time-evolving performance?</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#open-exploration">Open exploration</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-this-methodology-to-your-own-problem">Tips to apply this methodology to your own problem</a></li>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios">A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#background">Background</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s Get Started!</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#observed-record">Observed Record</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-stationary-generator-to-better-quantify-natural-variability">Synthetic Stationary Generator to Better Quantify Natural Variability</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#non-stationary-synthetic-generator-to-impose-climate-changes">Non-Stationary Synthetic Generator to Impose Climate Changes</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#placing-cmip5-projections-in-the-context-of-non-stationary-flows">Placing CMIP5 Projections in the Context of Non-Stationary Flows</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-create-an-hmm-based-generator-for-your-system">Tips to Create an HMM-Based Generator for your System</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#references">References</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios-tutorial">
+<h1>A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios Tutorial<a class="headerlink" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios-tutorial" title="Link to this heading">#</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/Hidden-Markov_Modeling_Approaches_to_Creating_Synthetic_Streamflow_Scenarios.ipynb">HMM Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios">
+<h2>A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios<a class="headerlink" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios" title="Link to this heading">#</a></h2>
+<p>In this notebook, we will be covering the basics of fitting a Hidden
+Markov Model-based synthetic streamflow generator for a single site in
+the Upper Colorado River Basin. First, we will characterize the observed
+historical flow in the basin from 1909-2013. Then, we will fit a
+synthetic streamflow generator to the observed flows in the basin in
+order to create stationary synthetic flows. Finally, we will create a
+non-stationary version of the generator to create flows that could be
+representative of plausible future climate in the region. We ultimately
+show how to place the synthetically generated flows in the context of
+physically-informed CMIP5 projections to compare the two methods.</p>
+<section id="background">
+<h3>Background<a class="headerlink" href="#background" title="Link to this heading">#</a></h3>
+<p>In the Western United States (US), and particularly the Colorado River
+Basin, a recent study used tree-ring reconstructions to suggest that the
+megadrought that has been occurring in the Southwest over the past 22
+years is the region’s worst drought since about 800 AD (Williams et al.,
+2022). The study’s lead author, UCLA climatologist Park Williams,
+suggested that had the sequence of wet-dry years occurred without
+anthropogenic forcing, the 2000s would have likely still been dry, but
+not on the same level as the worst of the last millennium’s
+megadroughts.</p>
+<p>The recent trend of warming and reduced soil moisture in the Southwest
+US is highly challenging from a water systems planning and management
+perspective for the Colorado River Basin. Given the wide recognition
+that the river is over-allocated, the most recent drought highlights the
+difficulty of sustaining the flow requirements as dictated by the
+Colorado Compact. Thus, there has been an increasing focus in
+exploratory modeling efforts to clarify how vulnerable water systems in
+this region are to plausible drought streamflow scenarios for the
+future. In this tutorial, we’ll discuss how to create these scenarios
+using a Hidden Markov Model (HMM)- based streamflow synthetic generator.
+As discussed in <a class="reference external" href="https://uc-ebook.org/docs/html/2_diagnostic_modeling_overview_and_perspectives.html#overview-of-model-diagnostics">Section
+2.1</a>
+and
+<a class="reference external" href="https://uc-ebook.org/docs/html/4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-dynamics-what-is-controlling-model-behaviors-of-interest">4.2</a>
+of the eBook, future climate conditions in the basin represent a deep
+uncertainty that can lead to highly consequential water scarcity
+outcomes. It is advantageous to create a model such as the HMM-based
+generator in order to facilitate the creation of many ensembles of
+streamflow that can ultimately be used to force regional water systems
+models to understand how variability and drought extremes affect
+regional water shortages, operations, and policies.</p>
+<figure>
+<center><img src="_static/1-lede_pix_-_usbr-powell-8517152024_aa66437c2e_o-cropped.jpg" style="width:50%"></center>
+<figcaption style="text-align:center"> Lake Powell shows persistent effects from drought (Source: U.S. Bureau of Reclamation) </figcaption>
+</figure></section>
+</section>
+<section id="lets-get-started">
+<h2>Let’s Get Started!<a class="headerlink" href="#lets-get-started" title="Link to this heading">#</a></h2>
+<section id="observed-record">
+<h3>Observed Record<a class="headerlink" href="#observed-record" title="Link to this heading">#</a></h3>
+<p>First, let’s take a look at the observed data from 1909-2013 for a
+specific site. In this example, we use the outlet gauge of the Upper
+Colorado River (USGS Gauge 09163500 at the Colorado-Utah state line).
+Below, we create a plot of the annual streamflow.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Import libraries</span>
+<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="nn">mpl</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">matplotlib.patches</span> <span class="k">as</span> <span class="nn">patches</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span>
+<span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">latin</span>
+<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">stats</span> <span class="k">as</span> <span class="n">ss</span>
+<span class="kn">import</span> <span class="nn">statistics</span>
+<span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>
+
+
+<span class="c1"># Import helper functions from local package</span>
+<span class="kn">from</span> <span class="nn">functions</span> <span class="kn">import</span> <span class="n">fitmodel</span>
+<span class="kn">from</span> <span class="nn">functions</span> <span class="kn">import</span> <span class="n">plotstates</span>
+<span class="kn">from</span> <span class="nn">functions</span> <span class="kn">import</span> <span class="n">plotdist</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Read in annual historical data</span>
+<span class="n">AnnualQ</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;data/uc_historical.csv&#39;</span><span class="p">)</span>
+<span class="n">AnnualQ</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+<span class="c1"># Plot a line graph</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper right&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_9_0.png" src="_images/hmm_9_0.png" />
+<p>Let’s calculate an 11-year rolling mean of the same data to get a sense
+of long-term trends.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+
+<span class="c1"># Plot the original line graph</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+         <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+         <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+         <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1"># Plot an 11-year rolling mean</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+         <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+         <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+         <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_11_0.png" src="_images/hmm_11_0.png" />
+<p>The Colorado Compact, which prescribes flows between the Upper and Lower
+Colorado Basins, was negotiated using data prior to 1922, a time period
+revealed by the above figure to be one of the consistently wetter
+periods on record. It’s clear today that since the 1980s, the Southwest
+US has been experiencing aridification (Overpeck et al., 2020) and that
+this observed record alone isn’t an accurate representation of what
+future climate might look like in this region.</p>
+<p>Let’s get a little more specific and formally quantify decadal droughts
+that have occurred in the observed period. We use a metric proposed in
+Ault et al. (2014). The authors define a decadal drought as when the
+11-year rolling mean falls below a threshold that is 1/2 a standard
+deviation below the overall mean of the record. We can then highlight
+the block of years that fall in a decadal drought using yellow
+rectangles below.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define drought threshold</span>
+<span class="n">std</span> <span class="o">=</span> <span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="n">threshold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+<span class="c1"># Find where the rolling mean dip below the threshold?</span>
+<span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+<span class="n">drought_years</span> <span class="o">=</span> <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">drought_instances</span><span class="p">]</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span><span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span><span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">threshold</span><span class="p">,</span>
+           <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
+           <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span>
+           <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Drought Threshold&#39;</span><span class="p">)</span>
+
+<span class="c1"># Visualize the drought periods as yellow rectangles</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">drought_years</span><span class="p">:</span>
+
+    <span class="c1"># Plot a box centered around those values and with 5 years on either side.</span>
+    <span class="n">rect</span> <span class="o">=</span> <span class="n">patches</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">((</span><span class="n">i</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="mi">11</span><span class="p">,</span><span class="mf">2e7</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">,</span> <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Add the patch to the Axes</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_14_0.png" src="_images/hmm_14_0.png" />
+<p>By this metric, the Upper Colorado Basin region has experienced two
+decadal droughts over the last century.</p>
+</section>
+<section id="synthetic-stationary-generator-to-better-quantify-natural-variability">
+<h3>Synthetic Stationary Generator to Better Quantify Natural Variability<a class="headerlink" href="#synthetic-stationary-generator-to-better-quantify-natural-variability" title="Link to this heading">#</a></h3>
+<p>It is important to remember that the streamflow that we have observed in
+the region over the last century is only one instance of the hydrology
+that could occur since the atmosphere is an inherently stochastic
+system. Thus, we require a tool that will allow us to see multiple
+plausible realizations of the streamflow record to understand the
+internal variability that characterizes the historical period. One
+observed realization of historical streamflow is limited in its ability
+to capture rare extremes; plausible (but not observed) alternative
+instances of streamflow records can help to fill this gap. The tool that
+we use to develop synthetic flows for the region is a Gaussian Hidden
+Markov Model (HMM). If a system follows a Markov process, it switches
+between a number of “hidden states” dictated by a transition matrix.
+Each state has its own Gaussian probability distribution (defined by a
+mean and standard deviation) and one can draw from this distribution to
+create synthetic flows that fit the properties of the historical
+distribution. HMMs are an attractive choice for this region because they
+can simulate persistence (i.e., long duration droughts), which is a
+characteristic of the region’s hydro-climatology. The figure below shows
+an example of a 2-state Gaussian HMM that we will be fitting for this
+example.</p>
+<figure>
+    <center><img src="_static/HMM_example.png" style="width:75%"></center>
+<figcaption style="text-align:center"> Two-state Gaussian HMM with mean and standard deviation parameters</figcaption>
+</figure><p>Below is the code that fits the HMM model to the last 2/3 of the
+historical record of log annual flows at the CO-UT stateline gauge and
+creates an alternative trace of 105 years. A subset of the dataset is
+chosen in order to minimize overfitting and to retain a set of data for
+validation of the model. When we fit our model, we utilize the
+Baum-Welch algorithm (a special version of the expectation-maximization
+algorithm) to find the optimal parameters that maximize the likelihood
+of seeing the observed flows. Ultimately, the algorithm will return a
+mean and standard deviation associated with each state (mus and sigmas
+defined below) and a 2x2 transition probability matrix that captures the
+likelihood of transitioning between states (P). We can also retrieve the
+annual hidden states across the observed series, also known as the
+Viterbi sequence of states, which classifies each year in a “wet” or
+“dry” state.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Number of years for alternative trace</span>
+<span class="n">n_years</span> <span class="o">=</span> <span class="mi">105</span>
+
+<span class="c1"># Import historical data that it used to fit HMM model</span>
+<span class="n">AnnualQ_h</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;data/uc_historical.csv&#39;</span><span class="p">)</span>
+
+<span class="c1"># Fit the model and pull out relevant parameters and samples</span>
+<span class="n">logQ</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">AnnualQ_h</span><span class="p">)</span>
+<span class="n">hidden_states</span><span class="p">,</span> <span class="n">mus</span><span class="p">,</span> <span class="n">sigmas</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">logProb</span><span class="p">,</span> <span class="n">samples</span><span class="p">,</span> <span class="n">model</span> <span class="o">=</span> <span class="n">fitmodel</span><span class="o">.</span><span class="n">fitHMM</span><span class="p">(</span><span class="n">logQ</span><span class="p">,</span> <span class="n">n_years</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>We’ve fit our HMM, but what does the model look like? Let’s plot the
+annual time series of hidden states, or the Viterbi sequence. In the
+code, above, we have defined that the drier state is always represented
+by state 0. Thus, we know that hidden_states = 0 corresponds to the dry
+state and hidden_states = 1 to the wet state.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot Vitebi sequence</span>
+<span class="n">plotstates</span><span class="o">.</span><span class="n">plotTimeSeries</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]),</span> <span class="n">hidden_states</span><span class="p">,</span> <span class="s1">&#39;Annual Flow (cube feet per year)&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_21_0.png" src="_images/hmm_21_0.png" />
+<p>In the figure above, we see that the years with the higher log flows
+tend to be classified in a “wet” state and the opposite is true of the
+“dry” state. We can also print the transition matrix, which shows the
+likelihood of transitioning between states. Note that the system has a
+high likelihood of persisting in the same state.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">transmat_</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[[</span><span class="mf">0.65095026</span> <span class="mf">0.34904974</span><span class="p">]</span>
+ <span class="p">[</span><span class="mf">0.3205531</span>  <span class="mf">0.6794469</span> <span class="p">]]</span>
+</pre></div>
+</div>
+<p>Let’s also plot the distribution of log annual flows associated with the
+wet and dry states.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot wet and dry state distributions</span>
+<span class="n">plotdist</span><span class="o">.</span><span class="n">plotDistribution</span><span class="p">(</span><span class="n">logQ</span><span class="p">,</span> <span class="n">mus</span><span class="p">,</span> <span class="n">sigmas</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_25_0.png" src="_images/hmm_25_0.png" />
+<p>The wet state distribution is characterized by a greater mean flow, but
+note that there is significant overlap in the tails of the distributions
+below which demonstrates why years with similiar flows can be classified
+in different states.</p>
+<p>Now let’s see what the drought dynamics look like in the synthetic
+scenario that we created using the same definition that we had used for
+the historical period.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Retrieve samples and back-transform out of log space</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">samples</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="p">)</span>
+<span class="n">AnnualQ_s</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+<span class="c1"># Define drought threshold</span>
+<span class="n">std</span><span class="o">=</span><span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="n">threshold</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+<span class="c1"># Where does the rolling mean dip below the threshold</span>
+<span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+<span class="n">drought_years</span> <span class="o">=</span> <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">drought_instances</span><span class="p">]</span>
+
+<span class="c1">#Visualize the streamflow scenario</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+
+<span class="c1">#Plot the original line graph</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1">#Plot a 11-year rolling mean</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">threshold</span><span class="p">,</span>
+           <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
+           <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span>
+           <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Drought Threshold&#39;</span><span class="p">)</span>
+
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">drought_years</span><span class="p">:</span>
+
+    <span class="c1">#Plot a box centered around those values and with 5 years on either side.</span>
+    <span class="n">rect</span> <span class="o">=</span> <span class="n">patches</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">((</span><span class="n">i</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span>
+                              <span class="mi">0</span><span class="p">),</span>
+                              <span class="mi">11</span><span class="p">,</span>
+                              <span class="mf">2e7</span><span class="p">,</span>
+                              <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+                              <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">,</span>
+                              <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Add the patch to the Axes</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow (Synthetic Stationary)&quot;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_28_0.png" src="_images/hmm_28_0.png" />
+<p>You can sample from the model and create more 105-year traces and note
+how the location and number of decadal droughts changes. This
+demonstrates how different the historical record can look just within
+the range of natural variability. It’s also important to remember that
+when droughts occur can also define the ultimate effect of the drought
+(i.e. is it a time when there is a large population growth or a time
+when humans can adapt by conserving or building more infrastructure?). A
+hydrologic drought need not manifest into an agricultural or operational
+drought of the same magnitude if stored surface water is available.</p>
+<p>We externally run the HMM many times to create a dataset of 100
+instances of the 105-year traces and 1000 instances of the 105-year
+traces that are available in the package
+(“synthetic_stationary_small_sample_100.csv”,“synthetic_stationary_large_sample_1000”).
+The shaded green lines correspond to the flow duration curves (FDCs) for
+the generated streamflow traces in comparison with the FDC of the
+historical record in beige.</p>
+<figure>
+    <center><img src="_static/stationary_synthetic_FDC.png" style="width:75%"></center>
+<figcaption style="text-align:center">Generated streamflow traces in comparison with the FDC of the historical record.</figcaption>
+</figure><p>As expected, the stationary synthetic FDCs envelope the historical FDC
+and particularly, the synthetic traces offer many more instances of low
+flow conditions that could lead to more extreme drought conditions than
+what has been observed historically. It is also useful to check for
+convergence of samples and to determine how many samples are needed to
+fully represent internal variability. Above we see that the extension to
+1000 instances of 105-year traces fills out regions of the FDC,
+including creating some more extreme drought conditions, but that
+additional samples will likely not fill out the FDC substantially more.</p>
+</section>
+<section id="non-stationary-synthetic-generator-to-impose-climate-changes">
+<h3>Non-Stationary Synthetic Generator to Impose Climate Changes<a class="headerlink" href="#non-stationary-synthetic-generator-to-impose-climate-changes" title="Link to this heading">#</a></h3>
+<p>Now, we create flows under non-stationary conditions to get a better
+understanding of what flows can look like under climate changes. In
+order to create flows under non-stationary conditions, we can toggle the
+parameters of the HMM model in order to create systematic changes to the
+model that can represent a changing climate. The HMM has 6 parameters
+that define it. When we fit the historical model, the parameters that
+are fit represent a baseline parameter value. In this non-stationary
+generator, we define a range to sample these parameters from.</p>
+<div class="pst-scrollable-table-container"><table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Parameter</p></th>
+<th class="head"><p>Current Value</p></th>
+<th class="head"><p>Lower Bound</p></th>
+<th class="head"><p>Upper Bound</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>Log-Space Wet State
+Mean Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.98</p></td>
+<td><p>1.02</p></td>
+</tr>
+<tr class="row-odd"><td><p>Log-Space Dry State
+Mean Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.98</p></td>
+<td><p>1.02</p></td>
+</tr>
+<tr class="row-even"><td><p>Log-Space Wet State
+Standard Deviation
+Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.75</p></td>
+<td><p>1.25</p></td>
+</tr>
+<tr class="row-odd"><td><p>Log-Space Dry State
+Standard Deviation
+Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.75</p></td>
+<td><p>1.25</p></td>
+</tr>
+<tr class="row-even"><td><p>Change in Dry-Dry
+Transition
+Probability</p></td>
+<td><p>0.00</p></td>
+<td><p>-0.30</p></td>
+<td><p>+0.30</p></td>
+</tr>
+<tr class="row-odd"><td><p>Change in Wet-Wet
+Transition
+Probability</p></td>
+<td><p>0.00</p></td>
+<td><p>-0.30</p></td>
+<td><p>+0.30</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>Now let’s sample 1000 times from these bounds to create 1000 new
+parameterizations of the model. Here we use SALib and the Latin
+Hypercube sample function.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create problem structure with parameters that we want to sample</span>
+<span class="n">problem</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">6</span><span class="p">,</span>
+    <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;wet_mu&#39;</span><span class="p">,</span> <span class="s1">&#39;dry_mu&#39;</span><span class="p">,</span> <span class="s1">&#39;wet_std&#39;</span><span class="p">,</span><span class="s1">&#39;dry_std&#39;</span><span class="p">,</span><span class="s1">&#39;dry_tp&#39;</span><span class="p">,</span><span class="s2">&quot;wet_tp&quot;</span><span class="p">],</span>
+    <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="mf">0.98</span><span class="p">,</span> <span class="mf">1.02</span><span class="p">],</span>
+               <span class="p">[</span><span class="mf">0.98</span><span class="p">,</span> <span class="mf">1.02</span><span class="p">],</span>
+               <span class="p">[</span><span class="mf">0.75</span><span class="p">,</span><span class="mf">1.25</span><span class="p">],</span>
+               <span class="p">[</span><span class="mf">0.75</span><span class="p">,</span><span class="mf">1.25</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.3</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.3</span><span class="p">]]</span>
+<span class="p">}</span>
+
+<span class="c1"># generate 1000 parameterizations</span>
+<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">1000</span>
+
+<span class="c1"># set random seed for reproducibility</span>
+<span class="n">seed_value</span> <span class="o">=</span> <span class="mi">123</span>
+
+<span class="c1"># Generate our samples</span>
+<span class="n">LHsamples</span> <span class="o">=</span> <span class="n">latin</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="n">n_samples</span><span class="p">,</span> <span class="n">seed_value</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now let’s look at what some of the traces look like in our
+non-stationary generator. Let’s choose a random instance from the
+1000-member space and adjust the parameters accordingly.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define static parameters</span>
+<span class="n">n_years</span> <span class="o">=</span> <span class="mi">105</span>
+
+<span class="c1"># Sample parameter; Adjust to any sample number from 0-999</span>
+<span class="n">sample</span> <span class="o">=</span> <span class="mi">215</span>
+
+<span class="c1"># Create empty arrays to store the new Gaussian HMM parameters for each SOW</span>
+<span class="n">Pnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">musNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">sigmasNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">logAnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+
+<span class="c1"># Calculate new transition matrix and stationary distribution of SOW at last node as well as new means and standard deviations</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">4</span><span class="p">]))</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">5</span><span class="p">]))</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+<span class="n">eigenvals</span><span class="p">,</span> <span class="n">eigenvecs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">))</span>
+<span class="n">one_eigval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">eigenvals</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
+<span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">divide</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span> <span class="n">one_eigval</span><span class="p">]),</span>
+                  <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span><span class="n">one_eigval</span><span class="p">])))</span>
+
+<span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
+<span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+<span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">3</span><span class="p">]</span>
+<span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
+
+<span class="c1"># Generate first state and log-space annual flow at last node</span>
+<span class="n">states</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+<span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">piNew</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
+    <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="k">else</span><span class="p">:</span>
+    <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+    <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="c1"># Generate remaining state trajectory and log space flows at last node</span>
+<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_years</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">Pnew</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">])]:</span>
+        <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+
+    <span class="k">if</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="c1"># Convert log-space flows to real-space flows</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">logAnnualQ_s</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
+</pre></div>
+</div>
+<p>Now let’s see what this synthetic trace looks like.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Retrieve samples and back-transform out of log space</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="p">)</span>
+<span class="n">AnnualQ_s</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+<span class="c1"># Define drought threshold</span>
+<span class="n">std</span> <span class="o">=</span> <span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="n">threshold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+<span class="c1"># Where does the rolling mean dip below the threshold</span>
+<span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+<span class="n">drought_years</span> <span class="o">=</span> <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">drought_instances</span><span class="p">]</span>
+
+<span class="c1"># Visualize the streamflow scenario</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+
+<span class="c1"># Plot the original line graph</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1"># Plot a 11-year rolling mean</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">threshold</span><span class="p">,</span>
+           <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
+           <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span>
+           <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Drought Threshold&#39;</span><span class="p">)</span>
+
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">drought_years</span><span class="p">:</span>
+
+    <span class="c1"># Plot a box centered around those values and with 5 years on either side.</span>
+    <span class="n">rect</span> <span class="o">=</span> <span class="n">patches</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">((</span><span class="n">i</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span>
+                             <span class="mi">11</span><span class="p">,</span>
+                             <span class="mf">2e7</span><span class="p">,</span>
+                             <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+                             <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">,</span>
+                             <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Add the patch to the Axes</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Annual Flow (Synthetic Non-Stationary)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper right&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_40_0.png" src="_images/hmm_40_0.png" />
+<p>Above is the example trace from the new non-stationary model. You may
+see fewer or more decadal drought instances. We can further summarize
+overall decadal drought characteristics across the samples. Let’s plot a
+histogram of the total number of times we go below the drought threshold
+across these realizations.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">decadal_drought_occurence</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">1000</span><span class="p">])</span>
+
+<span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">):</span>
+
+    <span class="c1"># Create empty arrays to store the new Gaussian HMM parameters for each SOW</span>
+    <span class="n">Pnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
+    <span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+    <span class="n">musNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+    <span class="n">sigmasNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+    <span class="n">logAnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+
+    <span class="c1"># Calculate new transition matrix and stationary distribution of SOW at last node</span>
+    <span class="c1"># as well as new means and standard deviations</span>
+
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">4</span><span class="p">]))</span>
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">5</span><span class="p">]))</span>
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+    <span class="n">eigenvals</span><span class="p">,</span> <span class="n">eigenvecs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">))</span>
+    <span class="n">one_eigval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">eigenvals</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
+    <span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">divide</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span> <span class="n">one_eigval</span><span class="p">]),</span>
+                      <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span> <span class="n">one_eigval</span><span class="p">])))</span>
+
+    <span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
+    <span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+    <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">3</span><span class="p">]</span>
+    <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
+
+    <span class="c1"># Generate first state and log-space annual flow at last node</span>
+    <span class="n">states</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+    <span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">piNew</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
+        <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+    <span class="c1"># generate remaining state trajectory and log space flows at last node</span>
+    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_years</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">Pnew</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">])]:</span>
+            <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+
+        <span class="k">if</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+    <span class="c1"># Convert log-space flows to real-space flows</span>
+    <span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">logAnnualQ_s</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="p">)</span>
+    <span class="n">AnnualQ_s</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+    <span class="c1"># Define drought threshold</span>
+    <span class="n">std</span> <span class="o">=</span> <span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+    <span class="n">threshold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+    <span class="c1"># Where does the rolling mean dip below the threshold</span>
+    <span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+    <span class="n">decadal_drought_occurence</span><span class="p">[</span><span class="n">y</span><span class="p">]</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">drought_instances</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">decadal_drought_occurence</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Non-Stationary generator&#39;</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s2">&quot;#005F73&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Number of Instances of Decadal Drought&#39;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Frequency&#39;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;-&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Observed&#39;</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span> <span class="o">=</span> <span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span> <span class="o">=</span> <span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_43_0.png" src="_images/hmm_43_0.png" />
+<p>Note how many more instances of the decadal droughts we are creating
+with the non-stationary generator than our observed 105-year trace which
+creates a rich space in which we can test our models. Just as we did
+with the stationary generator, we can externally run the non-stationary
+generator to create 10,000 instances of the 105-year traces that are
+available in the package
+(“synthetic_nonstationary_large_sample_10000.csv”). The shaded green and
+blue lines correspond to the FDCs for the stationary and non-stationary
+generated streamflow traces in comparison with the FDC of the historical
+record in beige. Note how the non-stationary generator produces even
+more drought extremes than the stationary non-synthetic traces.</p>
+<figure>
+    <center><img src="_static/nonstationary_synthetic_FDC.png" style="width:75%"></center>
+<figcaption style="text-align:center">Generated streamflow traces in comparison with the FDC of the historical record.</figcaption>
+</figure></section>
+<section id="placing-cmip5-projections-in-the-context-of-non-stationary-flows">
+<h3>Placing CMIP5 Projections in the Context of Non-Stationary Flows<a class="headerlink" href="#placing-cmip5-projections-in-the-context-of-non-stationary-flows" title="Link to this heading">#</a></h3>
+<p>We have broadened the drought conditions that we are creating which that
+can be very useful to understand how our water systems model performs
+under potentially extreme scenarios. However, it’s useful to compare our
+bottom-up synthetically generated flows in the context of global
+physically-driven CMIP5 projections to get a better understanding of how
+the two approaches compare. We first aquire 97 CMIP5 projections from
+the Colorado River Water Availability Study (CWCB, 2012). In each of
+these projections, monthly precipitation factor changes and temperature
+delta changes were computed between mean projected 2035–2065 climate
+statistics and mean historical climate statistics from 1950–2013. These
+97 different combinations of 12 monthly precipitation multipliers and 12
+monthly temperature delta shifts were applied to historical
+precipitation and temperature time series from 1950–2013. The resulting
+climate time series were run through a Variable Infiltration Capacity
+(VIC) model of the UCRB, resulting in 97 time series of projected future
+streamflows at the Colorado‐Utah state line.</p>
+<p>We fit an HMM to each trace of projected streamflow and get a set of
+corresponding HMM parameters. Then we take the ratio between these
+parameters and the baseline HMM parameters that we calculated earlier in
+the notebook in order to calculate the multipliers associated with each
+CMIP5 projection. This is all done externally, so we import the
+resulting multipliers in the next line.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Read in CMIP5 and paleo multipliers</span>
+<span class="n">CMIP5_multipliers</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;data/CMIP5_SOWs.txt&#39;</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s2">&quot; &quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Let’s plot a response surface that will allow us to see how combinations
+of HMM parameters tend to influence decadal drought. In order to get a
+continuous surface, we’ll fit a non-linear regression to the parameter
+values and then predict the decadal drought over a set of grid points.
+We fit the response surface for two parameters that should have an
+affect on decadal drought: the dry distribution mean and the dry-dry
+transition probabilites.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Choose two parameters to fit the response surface for</span>
+<span class="n">mu_dry</span><span class="o">=</span><span class="p">[</span><span class="n">i</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">LHsamples</span><span class="p">]</span>
+<span class="n">tp_dry</span><span class="o">=</span><span class="p">[</span><span class="n">i</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">LHsamples</span><span class="p">]</span>
+
+<span class="c1"># Create an interpolation grid</span>
+<span class="n">xgrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">),</span>
+                  <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">),</span>
+                  <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">))</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span>
+
+<span class="n">ygrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">),</span>
+                  <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">),</span>
+                  <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">))</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span>
+
+<span class="c1"># Fit regression</span>
+<span class="n">d</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;Dry_Tp&#39;</span><span class="p">:</span> <span class="n">tp_dry</span><span class="p">,</span>
+     <span class="s1">&#39;Dry_Mu&#39;</span><span class="p">:</span> <span class="n">mu_dry</span><span class="p">,</span>
+     <span class="s1">&#39;Drought_Occurrence&#39;</span><span class="p">:</span><span class="n">decadal_drought_occurence</span><span class="p">}</span>
+
+<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
+<span class="n">df</span><span class="p">[</span><span class="s1">&#39;Intercept&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">shape</span><span class="p">(</span><span class="n">df</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">df</span><span class="p">[</span><span class="s1">&#39;Interaction&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s1">&#39;Dry_Tp&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">df</span><span class="p">[</span><span class="s1">&#39;Dry_Mu&#39;</span><span class="p">]</span>
+<span class="n">cols</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Intercept&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s1">&#39;Dry_Mu&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s1">&#39;Dry_Tp&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s1">&#39;Interaction&#39;</span><span class="p">]</span>
+<span class="n">ols</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Drought_Occurrence&#39;</span><span class="p">],</span> <span class="n">df</span><span class="p">[</span><span class="n">cols</span><span class="p">])</span>
+<span class="n">result</span> <span class="o">=</span> <span class="n">ols</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
+
+<span class="c1"># Calculate drought occurence for each grid point</span>
+<span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">xgrid</span><span class="p">,</span> <span class="n">ygrid</span><span class="p">)</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
+<span class="n">y</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
+<span class="n">grid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">column_stack</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">*</span> <span class="n">y</span><span class="p">])</span>
+<span class="n">z</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
+<span class="n">z</span><span class="p">[</span><span class="n">z</span> <span class="o">&lt;</span> <span class="mf">0.0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="c1"># replace negative shortage predictions with 0</span>
+</pre></div>
+</div>
+<p>Let’s plot our results:</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set color gradient for response surface</span>
+<span class="n">drought_map</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">&#39;RdBu_r&#39;</span><span class="p">)</span>
+
+<span class="c1"># Reshape our predicted drought occurrence and define bounds of colors</span>
+<span class="n">Z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">shape</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
+<span class="n">vmin</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Drought_Occurrence&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">)])</span>
+<span class="n">vmax</span> <span class="o">=</span> <span class="mi">15</span>
+<span class="n">norm</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="p">,</span> <span class="n">vmax</span><span class="p">)</span>
+
+<span class="c1"># Plot response surface and CMIP5 projections</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">drought_map</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">norm</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">CMIP5_multipliers</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">7</span><span class="p">],</span>
+           <span class="n">CMIP5_multipliers</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">12</span><span class="p">],</span>
+           <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#ffffb3&#39;</span><span class="p">,</span>
+           <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;none&#39;</span><span class="p">,</span>
+           <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span>
+<span class="n">cbar</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">figure</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">ScalarMappable</span><span class="p">(</span><span class="n">norm</span><span class="o">=</span><span class="n">norm</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">drought_map</span><span class="p">),</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">nanmin</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">nanmax</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">nanmin</span><span class="p">(</span><span class="n">Y</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">nanmax</span><span class="p">(</span><span class="n">Y</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Dry State Mu&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Dry-Dry Transition Probability&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="n">labelsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Decadal Drought Occurrence&#39;</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=-</span><span class="mi">90</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">,</span> <span class="n">labelpad</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
+<span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;y&#39;</span><span class="p">,</span><span class="n">labelsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_53_0.png" src="_images/hmm_53_0.png" />
+<p>We see the influence of the dry state mean and dry-dry transition
+parameters. We’re likely to see more decadal droughts when we (1)
+increase the dry-dry transition probability, which inherently will
+increase persistence of the dry state, and (2) when we make the dry
+state log mean drier. Note that the CMIP5 scenarios tend to span the
+extent of the dry mean sample space, but are less representative of the
+dry transition probability sample space, which suggests that the types
+of hydrological droughts represented in the projections tend to only be
+wetter to slightly drier than our baseline. Both methods of producing
+these scenarios are valid, though studies have suggested that
+globally-resolved GCMs may be inappropriate to represent regional
+extremes. Ultimately, if your goal is to produce a variety of ensembles
+that are characterized by many different drought characteristics, you
+will likely find that a generator approach will serve this purpose
+better.</p>
+</section>
+<section id="tips-to-create-an-hmm-based-generator-for-your-system">
+<h3>Tips to Create an HMM-Based Generator for your System<a class="headerlink" href="#tips-to-create-an-hmm-based-generator-for-your-system" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we demonstrated how to fit an HMM-based generator for
+a single gauge located in the Upper Colorado River Basin. In order to
+apply this methodology to your problem, you will need to first ask:</p>
+<ol class="arabic simple">
+<li><p>Is this model appropriate for my location of interest? We have
+applied this style of generator to locations where persistent wet
+and dry states are characteristic, which tends to be in the Western
+US. Ultimately the best way to judge if an HMM is useful for your
+application is to fit the model and explore the resulting
+distributions. Are there two (or more) distinct states that emerge?
+If not, then your location may not exhibit the type of persistence
+that an HMM-based generator is useful for. You can consider
+exploring other styles of generators such as the Kirsch-Nowak
+generator (Kirsch et al., 2013).</p></li>
+<li><p>Do I have the right datasets? We use annual data for our location of
+interest. In this notebook, the HMM is fit to log annual flows.
+Ultimately, it can be disaggregated to daily flows (using a
+reference historical daily dataset) to be useful in water resources
+operational applications. You could also disaggregate to a finer
+resolution than daily if the historical dataset exists.</p></li>
+</ol>
+<p>If you meet these requirements, feel free to proceed through fitting the
+model using the code available in the notebook. Be sure to consider the
+appropirate number of samples to generate (both in a stationary and
+non-stationary case). Make sure that you test multiple sample sizes and
+continue to increase your sample size until you converge to a consistent
+representation of extremes. What is the appropriate number of LHS
+samples of the parameters to use? In this experiment we used 1,000
+samples of parameters due to extensive stability tests described in
+Quinn et al. (2020).</p>
+<p>Finally, to learn more about this test case refer to Hadmichael et
+al. (2020a) and Hadmichael et al. (2020b). For another study on
+synthetic drought generation to support vulnerability assessments in the
+Research Triangle region of North Carolina, please refer to Herman et
+al. (2016)</p>
+</section>
+<section id="references">
+<h3>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h3>
+<p>Ault, T. R., Cole, J. E., Overpeck, J. T., Pederson, G. T., &amp; Meko, D.
+M. (2014). Assessing the risk of persistent drought using climate model
+simulations and paleoclimate data. Journal of Climate, 27(20),
+7529-7549.</p>
+<p>CWCB (2012).Colorado River Water Availability Study Phase I Report.
+Colorado Water Conservation Board</p>
+<p>Hadjimichael, A., Quinn, J., Wilson, E., Reed, P., Basdekas, L., Yates,
+D., &amp; Garrison, M. (2020a). Defining robustness, vulnerabilities, and
+consequential scenarios for diverse stakeholder interests in
+institutionally complex river basins. Earth’s Future, 8(7),
+e2020EF001503.</p>
+<p>Hadjimichael, A., Quinn, J., &amp; Reed, P. (2020). Advancing diagnostic
+model evaluation to better understand water shortage mechanisms in
+institutionally complex river basins. Water Resources Research, 56(10),
+e2020WR028079.</p>
+<p>Herman, J. D., Zeff, H. B., Lamontagne, J. R., Reed, P. M., &amp;
+Characklis, G. W. (2016). Synthetic drought scenario generation to
+support bottom-up water supply vulnerability assessments. Journal of
+Water Resources Planning and Management, (11), 04016050.</p>
+<p>Kirsch, B. R., Characklis, G. W., &amp; Zeff, H. B. (2013). Evaluating the
+impact of alternative hydro-climate scenarios on transfer agreements:
+Practical improvement for generating synthetic streamflows. Journal of
+Water Resources Planning and Management, 139(4), 396-406.</p>
+<p>Overpeck, J.T. &amp; Udall, B. (2020) “Climate change and the aridification
+of North America.” Proceedings of the national academy of sciences
+117.22 11856-11858.</p>
+<p>Quinn, J. D., Hadjimichael, A., Reed,P. M., &amp; Steinschneider, S. (2020).
+Canexploratory modeling of water scarcity vulnerabilities and robustness
+bescenario neutral?Earth’s Future,8,e2020EF001650.
+<a class="reference external" href="https://doi.org/10.1029/2020EF001650Received">https://doi.org/10.1029/2020EF001650Received</a></p>
+<p>Williams, A. P., Cook, B. I., &amp; Smerdon, J. E. (2022). Rapid
+intensification of the emerging southwestern North American megadrought
+in 2020–2021. Nature Climate Change, 12(3), 232-234.</p>
+</section>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios">A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#background">Background</a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#observed-record">Observed Record</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-stationary-generator-to-better-quantify-natural-variability">Synthetic Stationary Generator to Better Quantify Natural Variability</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#non-stationary-synthetic-generator-to-impose-climate-changes">Non-Stationary Synthetic Generator to Impose Climate Changes</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#placing-cmip5-projections-in-the-context-of-non-stationary-flows">Placing CMIP5 Projections in the Context of Non-Stationary Flows</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-create-an-hmm-based-generator-for-your-system">Tips to Create an HMM-Based Generator for your System</a></li>
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+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fishery-dynamics-tutorial">B.1. Fishery Dynamics Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics">B.1.1. Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">B.1.1.1. Let’s get started!</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-identified-solutions-and-explore-performance">B.1.1.2. Step 1: Load identified solutions and explore performance</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis">B.1.1.3. Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-evaluate-the-system-over-all-generated-states-of-the-world">B.1.1.4. Step 3: Evaluate the system over all generated states of the world</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-calculate-sensitivity-indices">B.1.1.5. Step 4: Calculate sensitivity indices</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-explore-relationship-between-uncertain-factors-and-performance">B.1.1.6. Step 5: Explore relationship between uncertain factors and performance</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem">B.1.1.7. Tips to Apply Sobol SA and Scenario Discovery to your Problem</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sobol-sa-tutorial">B.2. Sobol SA Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa">B.2.1. Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id6">B.2.1.1. Let’s get started!</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-generate-the-problem-dictionary">B.2.1.2. Step 1: Generate the problem dictionary</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-generate-samples-using-the-saltelli-sampling-scheme">B.2.1.3. Step 2: Generate samples using the Saltelli sampling scheme</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-execute-the-ishigami-function-over-our-sample-set">B.2.1.4. Step 3: Execute the Ishigami function over our sample set</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method">B.2.1.5. Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-interpret-our-results">B.2.1.6. Step 5: Interpret our results</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-to-your-own-problem">B.2.1.7. Tips to Apply Sobol SA to Your Own Problem</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#logistic-regression-tutorial">B.3. Logistic Regression Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-logistic-regression-for-factor-mapping">B.3.1. Tutorial: Logistic Regression for Factor Mapping</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#background">B.3.1.1. Background</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id19">B.3.1.2. Let’s get started!</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-latin-hypercube-sample-and-set-up-problem">B.3.1.3. Step 1: Load Latin hypercube sample and set up problem</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-define-decision-relevant-metrics-for-illustration">B.3.1.4. Step 2: Define decision-relevant metrics for illustration</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-run-the-logistic-regression">B.3.1.5. Step 3: Run the logistic regression</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-factor-ranking">B.3.1.6. Step 4: Factor ranking</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-draw-factor-maps">B.3.1.7. Step 5: Draw factor maps</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-scenario-discovery-to-your-own-problem">B.3.1.8. Tips to Apply Scenario Discovery to Your Own Problem</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#hymod-dynamics-tutorial">B.4. HYMOD Dynamics Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-of-the-hymod-model">B.4.1. Tutorial: Sensitivity Analysis of the HYMOD Model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction-to-hymod">B.4.2. 1 - Introduction to HYMOD</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#global-sensitivity-analysis">B.4.3. 2- Global Sensitivity Analysis</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-this-methodology-to-your-own-problem">B.4.4. Tips to Apply this methodology to your own problem</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#time-evolving-scenario-discovery-for-infrastructure-pathways">B.5. Time-evolving scenario discovery for infrastructure pathways</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id24">B.5.1. Time-evolving scenario discovery for infrastructure pathways</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id25">B.5.1.1. Background</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluating-robustness-over-time">B.5.1.2. Evaluating Robustness over time</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#exploring-performance-evolution">B.5.1.3. Exploring performance evolution</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#how-do-deep-uncertainties-generate-vulnerability">B.5.1.4. How do deep uncertainties generate vulnerability</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#which-uncertainties-have-the-most-influence-on-time-evolving-performance">B.5.1.5. Which uncertainties have the most influence on time-evolving performance?</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#open-exploration">B.5.1.6. Open exploration</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id26">B.5.1.7. Tips to apply this methodology to your own problem</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#references">B.5.1.8. References</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios-tutorial">B.6. A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios">B.6.1. A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id29">B.6.1.1. Background</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id30">B.6.2. Let’s Get Started!</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#observed-record">B.6.2.1. Observed Record</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-stationary-generator-to-better-quantify-natural-variability">B.6.2.2. Synthetic Stationary Generator to Better Quantify Natural Variability</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#non-stationary-synthetic-generator-to-impose-climate-changes">B.6.2.3. Non-Stationary Synthetic Generator to Impose Climate Changes</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#placing-cmip5-projections-in-the-context-of-non-stationary-flows">B.6.2.4. Placing CMIP5 Projections in the Context of Non-Stationary Flows</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-create-an-hmm-based-generator-for-your-system">B.6.2.5. Tips to Create an HMM-Based Generator for your System</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id31">B.6.2.6. References</a></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="jupyter-notebook-tutorials">
+<span id="a2-jupyter-notebooks"></span><h1><span class="section-number">B. </span>Jupyter Notebook Tutorials<a class="headerlink" href="#jupyter-notebook-tutorials" title="Link to this heading">#</a></h1>
+<section id="fishery-dynamics-tutorial">
+<h2><span class="section-number">B.1. </span>Fishery Dynamics Tutorial<a class="headerlink" href="#fishery-dynamics-tutorial" title="Link to this heading">#</a></h2>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/fishery_dynamics.ipynb">Fishery Dynamics Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics">
+<h3><span class="section-number">B.1.1. </span>Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics<a class="headerlink" href="#tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics" title="Link to this heading">#</a></h3>
+<p>This notebook demonstrates the application of sensitivity analysis to
+discover factors that shape the behavior modes of a socio-ecological
+system with dynamic human action.</p>
+<p>The system of differential equations below represent a system of prey
+(defined in the equation below as x) and predator (defined as y) fish,
+with a human actor harvesting the prey fish. You can read more about
+this system at <a class="reference external" href="https://doi.org/10.1155/2020/4170453">Hadjimichael et
+al. (2020)</a>.</p>
+<img alt="_images/eqn2.png" src="_images/eqn2.png" />
+<p>The table below defines the parameters in the system and also denotes
+the baseline and ranges associated with each uncertain parameter.</p>
+<img alt="_images/table1.png" src="_images/table1.png" />
+<p>The system is simple but very rich in the dynamic behaviors it exhibits.
+This complexity is accompanied by the presence of several equilibria
+that come in and out of existence with different parameter values. The
+equilibria also change in their stability according to different
+parameter values, giving rise to different behavior modes as shown by
+the diverse predator and prey abundace trajectories in the figure below.</p>
+<img alt="_images/Figure_1.png" src="_images/Figure_1.png" />
+<p>In the unharvested system (without the human actor) the stability of
+several of these equilibria can be derived analytically. The task
+becomes significantly more difficult when the adaptive human actor is
+introduced, deciding to harvest the system at different rates according
+to their objectives and preferences.</p>
+<p>Sensitivity analysis methods can help us identify the factors that most
+control these dynamics by exploring the space of parameter values and
+seeing how system outputs change as a result.</p>
+<p>Through previously conducted optimization, there already exists a set of
+potential harvesting strategies that were identified in pursuit of five
+objectives:</p>
+<ul class="simple">
+<li><p>Maximize Harvesting Discounted Profits (Net Present Value)</p></li>
+<li><p>Minimize Prey Population Deficit</p></li>
+<li><p>Minimize Longest Duration of Consecutive Low Harvest</p></li>
+<li><p>Maximize Worst Harvest Instance</p></li>
+<li><p>Minimize Harvest Variance</p></li>
+</ul>
+<p>The identified harvesting strategies also meet the necessary constraint
+of not causing inadvertent predator collapse.</p>
+<p>We will be examining the effects of parametric uncertainty on these
+identified strategies, particularly focusing on two strategies: one
+selected to maximize harvesting profits and one identified through
+previous analysis to perform ‘well enough’ for all objectives across a
+wide range of states of the world (referred to as the ‘robust’
+harvesting policy).</p>
+<section id="lets-get-started">
+<h4><span class="section-number">B.1.1.1. </span>Let’s get started!<a class="headerlink" href="#lets-get-started" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we will be loading in data that has been produced in
+Hadjimichael et al. (2020). Before we start our analysis, we’ll load the
+relevant Python libraries. <strong>NOTE</strong>: To step through the notebook,
+execute each gray (code) box by typing “Shift+Enter”.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Import necessary libraries</span>
+
+<span class="kn">import</span> <span class="nn">msdbook</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">saltelli</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">sobol</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">patheffects</span> <span class="k">as</span> <span class="n">pe</span>
+
+<span class="c1"># load example data</span>
+<span class="n">msdbook</span><span class="o">.</span><span class="n">install_package_data</span><span class="p">()</span>
+
+<span class="o">%</span><span class="k">matplotlib</span> inline
+<span class="o">%</span><span class="k">config</span> InlineBackend.print_figure_kwargs = {&#39;bbox_inches&#39;:None}
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Downloading</span> <span class="n">example</span> <span class="n">data</span> <span class="k">for</span> <span class="n">msdbook</span> <span class="n">version</span> <span class="mf">0.1.5</span><span class="o">...</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">uncertain_params_bounds</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_metric_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">solutions</span><span class="o">.</span><span class="n">resultfile</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LHsamples_original_1000</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">param_values</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_params_256samples</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LeafCatch</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_simulations_256samples</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">Robustness</span><span class="o">.</span><span class="n">txt</span>
+</pre></div>
+</div>
+</section>
+<section id="step-1-load-identified-solutions-and-explore-performance">
+<h4><span class="section-number">B.1.1.2. </span>Step 1: Load identified solutions and explore performance<a class="headerlink" href="#step-1-load-identified-solutions-and-explore-performance" title="Link to this heading">#</a></h4>
+<p>Here we load in the solution set obtained in Hadjimichael et al. (2020).
+The solution set contains the decision variables and objectives
+associated with a variety of harvesting policies. For this tutorial, we
+focus on comparing two policies: harvesting profits and one that
+performs robustly across all objectives. Below, we are reading in the
+decision variables and objectives from an external file that can be
+found within the msdbook package data.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">robustness</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_robustness_data</span><span class="p">()</span>
+<span class="n">results</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_profit_maximization_data</span><span class="p">()</span>
+
+<span class="n">robust_solution</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">robustness</span><span class="p">[:,</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="c1">#pick robust solution</span>
+<span class="n">profit_solution</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">results</span><span class="p">[:,</span><span class="mi">6</span><span class="p">])</span> <span class="c1">#pick profitable solution</span>
+<span class="n">objective_performance</span> <span class="o">=</span> <span class="o">-</span><span class="n">results</span><span class="p">[:,</span><span class="mi">6</span><span class="p">:]</span> <span class="c1">#Retain objective values</span>
+
+<span class="c1"># Get decision variables for each of the policies</span>
+<span class="n">highprofitpolicy</span> <span class="o">=</span> <span class="n">results</span><span class="p">[</span><span class="n">profit_solution</span><span class="p">,</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">]</span>
+<span class="n">mostrobustpolicy</span> <span class="o">=</span> <span class="n">results</span><span class="p">[</span><span class="n">robust_solution</span><span class="p">,</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">]</span>
+</pre></div>
+</div>
+<p>Next we plot the identified solutions with regards to their objective
+performance in a parallel axis plot</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-objective-performance">plot_objective_performance</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ax</span><span class="p">,</span> <span class="n">ax1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_objective_performance</span><span class="p">(</span><span class="n">objective_performance</span><span class="p">,</span> <span class="n">profit_solution</span><span class="p">,</span> <span class="n">robust_solution</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/fishery_output_6_0.png" src="_images/fishery_output_6_0.png" />
+<p>The solution set from the optimization in Hadjimichael et al. (2020) are
+presented in a parallel axis plot where each of the five objectives (and
+one constraint) are represented as an axis. Each solution on the Pareto
+front is represented as a line where the color of the line indicates the
+value of the NPV objective. The preference for objective values is in
+the upward direction. Therefore, the ideal solution would be a line
+straight across the top of the plot that satisfies every objective.
+However, no such line exists because there are tradeoffs when sets of
+objectives are prioritized over the others. When lines cross in between
+axes, this indicates a tradeoff between objectives (as seen in the first
+two axes).The solution that is most robust in the NPV objective has the
+highest value on the first axis and is outlined in dark gold. The
+solution that is most robust across all objectives is outlined in a
+brighter yellow. A parallel axis is an effective visual to characterize
+high-dimensional tradeoffs in the system and visualize differences in
+performance across policies.</p>
+</section>
+<section id="step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis">
+<h4><span class="section-number">B.1.1.3. </span>Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis<a class="headerlink" href="#step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis" title="Link to this heading">#</a></h4>
+<p>In Step 1, we showed how the optimized harvesting policies performed in
+the objective space, which utilized the baseline parameters outlined in
+the table above. Now, we are interested in understanding how sensitive
+our two policies are to alternative states of the world that may be
+characterized by different parameter values. To do so, we first need to
+define the problem dictionary that allows us to generate these
+alternative states of the world.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set up SALib problem</span>
+<span class="n">problem</span> <span class="o">=</span> <span class="p">{</span>
+  <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">9</span><span class="p">,</span>
+  <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;a&#39;</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="s1">&#39;c&#39;</span><span class="p">,</span> <span class="s1">&#39;d&#39;</span><span class="p">,</span> <span class="s1">&#39;h&#39;</span><span class="p">,</span> <span class="s1">&#39;K&#39;</span><span class="p">,</span> <span class="s1">&#39;m&#39;</span><span class="p">,</span> <span class="s1">&#39;sigmaX&#39;</span><span class="p">,</span> <span class="s1">&#39;sigmaY&#39;</span><span class="p">],</span>
+  <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="mf">0.002</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.005</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+             <span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">5000</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">]]</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+<p>Then we use the following command to generate a Saltelli sample from
+these defined ranges:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">param_values</span> <span class="o">=</span> <span class="n">saltelli</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="mi">1024</span><span class="p">,</span> <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Generally, it is a good idea to save the result of the sample since it
+is often reused and regenerating it produces a different sample set. For
+this reason, we will load one from file that was previously generated.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously generated Saltelli sample from our msdbook package data</span>
+<span class="n">param_values</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_saltelli_param_values</span><span class="p">()</span>
+</pre></div>
+</div>
+</section>
+<section id="step-3-evaluate-the-system-over-all-generated-states-of-the-world">
+<h4><span class="section-number">B.1.1.4. </span>Step 3: Evaluate the system over all generated states of the world<a class="headerlink" href="#step-3-evaluate-the-system-over-all-generated-states-of-the-world" title="Link to this heading">#</a></h4>
+<p>Now we re-evaluate how well the policies do in the new states of the
+world. In order to characterize failure of a policy, we identify the
+states where the predator population collapses, as an inadvertent
+consequence of applying the harvesting strategy under a state of the
+world different from the one originally assumed. Due to how long this
+step takes to execute within the tutorial, we will read in the solutions
+from an external file. However, the block of code below shows how
+evaluation can be implemented.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># create array to store collapse values under both policies</span>
+<span class="n">collapse_days</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="nb">len</span><span class="p">(</span><span class="n">param_values</span><span class="p">),</span> <span class="mi">2</span><span class="p">])</span>
+
+<span class="c1"># evaluate performance under every state</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">param_values</span><span class="p">)):</span>
+
+    <span class="n">additional_inputs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="s1">&#39;Previous_Prey&#39;</span><span class="p">],</span>
+                                  <span class="p">[</span><span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">6</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">7</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span>
+
+    <span class="n">collapse_days</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span><span class="n">fish_game</span><span class="p">(</span><span class="n">highprofitpolicy</span><span class="p">,</span> <span class="n">additional_inputs</span><span class="p">)[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+    <span class="n">collapse_days</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">=</span><span class="n">fish_game</span><span class="p">(</span><span class="n">mostrobustpolicy</span><span class="p">,</span> <span class="n">additional_inputs</span><span class="p">)[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load the simulation data from our msdbook package data</span>
+<span class="n">collapse_days</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_collapse_data</span><span class="p">()</span>
+</pre></div>
+</div>
+</section>
+<section id="step-4-calculate-sensitivity-indices">
+<h4><span class="section-number">B.1.1.5. </span>Step 4: Calculate sensitivity indices<a class="headerlink" href="#step-4-calculate-sensitivity-indices" title="Link to this heading">#</a></h4>
+<p>Now we use a Sobol sensitivity analysis to calculate first-order,
+second-order, and total-order sensitivity indices for each parameter and
+for each of the two policies. These indicies help determine which
+factors explain the most variability in the number of days of predator
+population collapse.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Perform the Sobol SA for the profit-maximizing solution</span>
+<span class="n">Si_profit</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="n">collapse_days</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
+                          <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+                          <span class="n">conf_level</span><span class="o">=</span><span class="mf">0.95</span><span class="p">,</span>
+                          <span class="n">print_to_console</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Perform the Sobol SA for the robust solution</span>
+<span class="n">Si_robustness</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span>
+                              <span class="n">collapse_days</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
+                              <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+                              <span class="n">conf_level</span><span class="o">=</span><span class="mf">0.95</span><span class="p">,</span>
+                              <span class="n">print_to_console</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>              <span class="n">ST</span>   <span class="n">ST_conf</span>
+<span class="n">a</span>       <span class="mf">0.226402</span>  <span class="mf">0.036146</span>
+<span class="n">b</span>       <span class="mf">0.066819</span>  <span class="mf">0.013347</span>
+<span class="n">c</span>       <span class="mf">0.004395</span>  <span class="mf">0.004023</span>
+<span class="n">d</span>       <span class="mf">0.024509</span>  <span class="mf">0.006993</span>
+<span class="n">h</span>       <span class="mf">0.009765</span>  <span class="mf">0.005488</span>
+<span class="n">K</span>       <span class="mf">0.020625</span>  <span class="mf">0.009494</span>
+<span class="n">m</span>       <span class="mf">0.897971</span>  <span class="mf">0.066470</span>
+<span class="n">sigmaX</span>  <span class="mf">0.000136</span>  <span class="mf">0.000149</span>
+<span class="n">sigmaY</span>  <span class="mf">0.000739</span>  <span class="mf">0.001040</span>
+              <span class="n">S1</span>   <span class="n">S1_conf</span>
+<span class="n">a</span>       <span class="mf">0.087936</span>  <span class="mf">0.044236</span>
+<span class="n">b</span>       <span class="mf">0.000554</span>  <span class="mf">0.021474</span>
+<span class="n">c</span>      <span class="o">-</span><span class="mf">0.002970</span>  <span class="mf">0.004590</span>
+<span class="n">d</span>       <span class="mf">0.001206</span>  <span class="mf">0.015881</span>
+<span class="n">h</span>       <span class="mf">0.004554</span>  <span class="mf">0.007998</span>
+<span class="n">K</span>       <span class="mf">0.003843</span>  <span class="mf">0.012661</span>
+<span class="n">m</span>       <span class="mf">0.751301</span>  <span class="mf">0.071862</span>
+<span class="n">sigmaX</span> <span class="o">-</span><span class="mf">0.000325</span>  <span class="mf">0.001245</span>
+<span class="n">sigmaY</span> <span class="o">-</span><span class="mf">0.001887</span>  <span class="mf">0.002768</span>
+</pre></div>
+</div>
+<p>Looking at the total-order indices, (ST) factors <span class="math notranslate nohighlight">\(m\)</span>, <span class="math notranslate nohighlight">\(a\)</span>,
+<span class="math notranslate nohighlight">\(b\)</span>, <span class="math notranslate nohighlight">\(d\)</span> and <span class="math notranslate nohighlight">\(K\)</span> explain a non-negligible amount of
+variance therefore have an effect on the stability of this system.
+Looking at the first-order indices (S1), we also see that besides
+factors <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(a\)</span>, all other factors are important in this
+system through their interactions, which make up the difference between
+their S1 and ST indices. This shows the danger of limiting sensitivity
+analyses to first order effects, as factor importance might be
+significantly misjudged.</p>
+<p>These findings are supported by the analytical condition of equilibrium
+stability in this system:</p>
+<img alt="_images/eqn4.png" src="_images/eqn4.png" />
+<p>In an unharvested system, this condition is both necessary and
+sufficient for the equilibrium of the two species coexisting to be
+stable.</p>
+<p>When adaptive human action is introduced however, this condition is
+still necessary, but no longer sufficient, as harvesting reduces the
+numbers of prey fish and as a result reduces the resources for the
+predator fish. Since this harvesting value is not constant, but can
+dynamically adapt according to the harvester’s objectives, it cannot be
+introduced into this simple equation.</p>
+</section>
+<section id="step-5-explore-relationship-between-uncertain-factors-and-performance">
+<h4><span class="section-number">B.1.1.6. </span>Step 5: Explore relationship between uncertain factors and performance<a class="headerlink" href="#step-5-explore-relationship-between-uncertain-factors-and-performance" title="Link to this heading">#</a></h4>
+<p>In the following steps, we will use the results of our sensitivity
+analysis to investigate the relationships between parametric
+uncertainty, equilibrium stability and the performance of the two
+policies.</p>
+<p>We can use the top three factors identified (<span class="math notranslate nohighlight">\(m\)</span>, <span class="math notranslate nohighlight">\(a\)</span>, and
+<span class="math notranslate nohighlight">\(b\)</span>) to visualize the performance of our policies in this
+three-dimensional parametric space.</p>
+<p>We first define the stability condition, as a function of <span class="math notranslate nohighlight">\(b\)</span> and
+<span class="math notranslate nohighlight">\(m\)</span>, and calculate the corresponding values of <span class="math notranslate nohighlight">\(a\)</span>.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">inequality</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
+    <span class="k">return</span> <span class="p">((</span><span class="n">b</span><span class="o">**</span><span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">h</span><span class="o">*</span><span class="n">K</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">m</span><span class="p">))</span>
+
+<span class="c1"># boundary interval that separates successful and failed states of the world</span>
+<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mf">0.005</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">m</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">K</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">2000</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">b</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">inequality</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">clip</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-factor-performance">plot_factor_performance</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># generate plot</span>
+<span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_factor_performance</span><span class="p">(</span><span class="n">param_values</span><span class="p">,</span> <span class="n">collapse_days</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/fishery_output_22_0.png" src="_images/fishery_output_22_0.png" />
+<p>These figures show the combinations of factors that lead to success or
+failure in different states of the world for the profit-maximizing and
+robust policies. Each point is a state of the world, characterized by
+specific values of the parameters, and ideally, we would like the color
+of the point to be blue, to represent that there are a low number of
+days with a predator collapse in that world. The gray curve denotes the
+highly non-linear nature of the boundary, defined by the stability
+condition, that separates successful and failed states of the world. The
+figures demonstrate the following key points:</p>
+<p>First, as asserted above, the policies interact with the system in
+different and complex ways. In the presence of human action, the
+stability condition is not sufficient in determining whether the policy
+will succeed, even though it clearly shapes the system in a fundamental
+manner.</p>
+<p>Secondly, the robust policy manages to avoid collapse in many more of the sampled states of the world, indicated by the number of blue points. The robust policy avoids collapse in 31% of worlds versus 14% in the profit-maximizing policy.This presents a clear tradeoff between profit-maximizing performance androbustness against uncertainty.</p>
+</section>
+<section id="tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem">
+<h4><span class="section-number">B.1.1.7. </span>Tips to Apply Sobol SA and Scenario Discovery to your Problem<a class="headerlink" href="#tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we demonstrated a Sobol SA to identify the most
+important factors driving the behavior of a system (i.e. the number of
+the collapse days). In order to apply this methodology to your problem,
+you will need to have a set of optimized policies for your system that
+you are interested in analyzing. The general workflow is as follows:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters and set up the problem
+dictionary as in Step 2 above.</p></li>
+<li><p>Generate samples, or alternative states of the world using the
+<code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function.</p></li>
+<li><p>Evaluate your policies on the alternative states of the world. For
+your application, you will also need to develop a rule for
+determining success or failure of your policy in a new SOW. In this
+tutorial, success was denoted by a small number of collapse days.
+Ultimately, the rule will be specific to your application and can
+include various satisficing criteria.</p></li>
+<li><p>Calculate the Sobol indices and discover the most important
+parameters driving success and failure.</p></li>
+<li><p>Finally, use a similar plotting procedure as in step 5 to identify
+the combination of parameter values that lead to success and failure
+in the system.</p></li>
+</ol>
+</section>
+</section>
+</section>
+<section id="sobol-sa-tutorial">
+<h2><span class="section-number">B.2. </span>Sobol SA Tutorial<a class="headerlink" href="#sobol-sa-tutorial" title="Link to this heading">#</a></h2>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA Tutorial</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa">
+<h3><span class="section-number">B.2.1. </span>Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA<a class="headerlink" href="#tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we will set up a workflow to investigate how sensitive
+the output of a function is to its inputs. Why might you want to do
+this? Imagine that this function represents a complex system, such as
+the rainfall-runoff process of a watershed model, and that you, the
+researcher, want to investigate how your choice of input parameter
+values are affecting the model’s characterization of runoff in the
+watershed. Your parameter values are likely uncertain and can take on
+any value in a pre-defined range. Using a Sobol SA will allow you to
+sample different values of your parameters and calculate how sensitive
+your output of interest is to certain parameters. Below, we demonstrate
+Sobol SA for a simple function to illustrate the method, but the
+workflow can be applied to your own problem of interest!</p>
+<p>In order to conduct this analysis, we will use the popular Python
+Sensitivity Analysis Library
+(<a class="reference external" href="https://salib.readthedocs.io/en/latest/index.html">SALib</a>) to:</p>
+<ol class="arabic simple">
+<li><p>Generate a problem set as a dictionary for our Ishigami function that has three inputs</p></li>
+<li><p>Generate 2048 samples for our problem set using the Saltelli <a class="footnote-reference brackets" href="#id10" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id11" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a> sampling scheme</p></li>
+<li><p>Execute the Ishigami function for each of our samples and gather the outputs</p></li>
+<li><p>Compute the sensitivity analysis to generate first-order and total-order sensitivity indices using the Sobol <a class="footnote-reference brackets" href="#id12" id="id5" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a> method</p></li>
+<li><p>Interpret the meaning of our results</p></li>
+</ol>
+<section id="id6">
+<h4><span class="section-number">B.2.1.1. </span>Let’s get started!<a class="headerlink" href="#id6" title="Link to this heading">#</a></h4>
+<p><strong>NOTE</strong>: Content from this tutorial is taken directly from the SALib
+<a class="reference external" href="https://salib.readthedocs.io/en/latest/basics.html">“Basics”</a>
+walkthrough. To step through the notebook, execute each gray (code) box
+by typing “Shift+Enter”.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Import relevant libraries</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">from</span> <span class="nn">mpl_toolkits</span> <span class="kn">import</span> <span class="n">mplot3d</span>
+
+<span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">saltelli</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">sobol</span>
+<span class="kn">from</span> <span class="nn">SALib.test_functions</span> <span class="kn">import</span> <span class="n">Ishigami</span>
+</pre></div>
+</div>
+</section>
+<section id="step-1-generate-the-problem-dictionary">
+<h4><span class="section-number">B.2.1.2. </span>Step 1: Generate the problem dictionary<a class="headerlink" href="#step-1-generate-the-problem-dictionary" title="Link to this heading">#</a></h4>
+<p>The Ishigami function is of the form:</p>
+<div class="math notranslate nohighlight">
+\[f(x_1,x_2,x_3) = sin(x_1)+asin^2(x_2)+bx_3^4sin(x_1)\]</div>
+<p>The function has three inputs, 𝑥1, 𝑥2, 𝑥3 where 𝑥𝑖 ∈ [−𝜋, 𝜋]. The
+constants <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are defined as 7.0 and 0.1
+respectively.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Create a problem dictionary. Here we supply the number of variables, the names of each variable, and the bounds of the variables.</span>
+<span class="n">problem</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span>
+    <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;x1&#39;</span><span class="p">,</span> <span class="s1">&#39;x2&#39;</span><span class="p">,</span> <span class="s1">&#39;x3&#39;</span><span class="p">],</span>
+    <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="o">-</span><span class="mf">3.14159265359</span><span class="p">,</span> <span class="mf">3.14159265359</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">3.14159265359</span><span class="p">,</span> <span class="mf">3.14159265359</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">3.14159265359</span><span class="p">,</span> <span class="mf">3.14159265359</span><span class="p">]]</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+</section>
+<section id="step-2-generate-samples-using-the-saltelli-sampling-scheme">
+<h4><span class="section-number">B.2.1.3. </span>Step 2: Generate samples using the Saltelli sampling scheme<a class="headerlink" href="#step-2-generate-samples-using-the-saltelli-sampling-scheme" title="Link to this heading">#</a></h4>
+<p>Sobol SA requires the use of the Saltelli sampling scheme. The output of
+the <code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function is a NumPy array that is of shape 2048
+by 3. The sampler generates 𝑁∗(2𝐷+2) samples, where in this example, N
+is 256 (the argument we supplied) and D is 3 (the number of model
+inputs), yielding 2048 samples. The keyword argument
+<code class="docutils literal notranslate"><span class="pre">calc_second_order=False</span></code> will exclude second-order indices, resulting
+in a smaller sample matrix with 𝑁∗(𝐷+2) rows instead. Below, we plot the
+resulting Saltelli sample.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Generate parmeter values using the saltelli.sample function</span>
+<span class="n">param_values</span> <span class="o">=</span> <span class="n">saltelli</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="mi">256</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;`param_values` shape:  </span><span class="si">{</span><span class="n">param_values</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<pre class="literal-block"><cite>param_values</cite> shape:  (2048, 3)</pre>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Plot the 2048 samples of the parameters</span>
+
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
+<span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">axes</span><span class="p">(</span><span class="n">projection</span> <span class="o">=</span><span class="s2">&quot;3d&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter3D</span><span class="p">(</span><span class="n">param_values</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">param_values</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">param_values</span><span class="p">[:,</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;X1 Parameter&#39;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;X2 Parameter&#39;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s1">&#39;X3 Parameter&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Saltelli Sample of Parameter Values&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/output_7_0.png" src="_images/output_7_0.png" />
+</section>
+<section id="step-3-execute-the-ishigami-function-over-our-sample-set">
+<h4><span class="section-number">B.2.1.4. </span>Step 3: Execute the Ishigami function over our sample set<a class="headerlink" href="#step-3-execute-the-ishigami-function-over-our-sample-set" title="Link to this heading">#</a></h4>
+<p>SALib provides a nice wrapper to the Ishigami function that allows the
+user to directly pass the <code class="docutils literal notranslate"><span class="pre">param_values</span></code> array we just generated into
+the function directly.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Y</span> <span class="o">=</span> <span class="n">Ishigami</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">param_values</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method">
+<h4><span class="section-number">B.2.1.5. </span>Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method<a class="headerlink" href="#step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method" title="Link to this heading">#</a></h4>
+<p>The <code class="docutils literal notranslate"><span class="pre">sobol.analyze</span></code> function will use our problem dictionary and the
+result of the Ishigami runs (<code class="docutils literal notranslate"><span class="pre">Y</span></code>) to compute first-, second-, and
+total-order indicies.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Si</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><code class="docutils literal notranslate"><span class="pre">Si</span></code> is a Python dict with the keys “S1”, “S2”, “ST”, “S1_conf”,
+“S2_conf”, and “ST_conf”. The <code class="docutils literal notranslate"><span class="pre">_conf</span></code> keys store the corresponding
+confidence intervals, typically with a confidence level of 95%. Use the
+keyword argument <code class="docutils literal notranslate"><span class="pre">print_to_console=True</span></code> to print all indices. Or, we
+can print the individual values from <code class="docutils literal notranslate"><span class="pre">Si</span></code> as shown in the next step.</p>
+</section>
+<section id="step-5-interpret-our-results">
+<h4><span class="section-number">B.2.1.6. </span>Step 5: Interpret our results<a class="headerlink" href="#step-5-interpret-our-results" title="Link to this heading">#</a></h4>
+<p>We execute the following code and take a look at our first-order indices
+(<code class="docutils literal notranslate"><span class="pre">S1</span></code>) for each of our three inputs. These indicies can be interpreted
+as the fraction of variance in the output that is explained by each
+input individually.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">first_order</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">]</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;First-order:&#39;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1: </span><span class="si">{</span><span class="n">first_order</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">, x2: </span><span class="si">{</span><span class="n">first_order</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">, x3: </span><span class="si">{</span><span class="n">first_order</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">First</span><span class="o">-</span><span class="n">order</span><span class="p">:</span>
+<span class="n">x1</span><span class="p">:</span> <span class="mf">0.3184242969763115</span><span class="p">,</span> <span class="n">x2</span><span class="p">:</span> <span class="mf">0.4303808201623416</span><span class="p">,</span> <span class="n">x3</span><span class="p">:</span> <span class="mf">0.022687722804980225</span>
+</pre></div>
+</div>
+<p>If we were to rank the importance of the inputs in how much they
+individually explain the variance in the output, we would rank them from
+greatest to least importance as follows: 𝑥2, 𝑥1 and then 𝑥3. Since 𝑥3
+only explains 2% of the output variance, it does not explain output
+variability meaningfully. Thus, this indicates that there is
+contribution to the output variance by 𝑥2 and 𝑥1 independently, whereas
+𝑥3 does not contribute to the output variance. Determining what inputs
+are most important or what index value is meaningful is a common
+question, but one for which there is no general rule or threshold. This
+question is problem and context-dependent, but procedures have been
+identified to rank order influential inputs and which can be used to
+identify the least influential factors. These factors can be fixed to
+simplify the model <a class="footnote-reference brackets" href="#id13" id="id7" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id15" id="id8" role="doc-noteref"><span class="fn-bracket">[</span>5<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id16" id="id9" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a>.</p>
+<p>Next, we evaluate the total-order indices, which measure the
+contribution to the output variance caused by varying the model input,
+including both its first-order effects (the input varying alone) and all
+higher-order interactions across the input parameters.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">total_order</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">]</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Total-order:&#39;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1: </span><span class="si">{</span><span class="n">total_order</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">, x2: </span><span class="si">{</span><span class="n">total_order</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">, x3: </span><span class="si">{</span><span class="n">total_order</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Total</span><span class="o">-</span><span class="n">order</span><span class="p">:</span>
+<span class="n">x1</span><span class="p">:</span> <span class="mf">0.5184119098161343</span><span class="p">,</span> <span class="n">x2</span><span class="p">:</span> <span class="mf">0.41021260250026054</span><span class="p">,</span> <span class="n">x3</span><span class="p">:</span> <span class="mf">0.2299058431439953</span>
+</pre></div>
+</div>
+<p>The magnitude of the total order indices are substantially larger than
+the first-order indices, which reveals that higher-order interactions
+are occurring, i.e. that the interactions across inputs are also
+explaining some of the total variance in the output. Note that 𝑥3 has
+non-negligible total-order indices, which indicates that it is not a
+consequential parameter when considered in isolation, but becomes
+consequential and explains 25% of variance in the output through its
+interactions with 𝑥1 and 𝑥2.</p>
+<p>Finally, we can investigate these higher order interactions by viewing
+the second-order indices. The second-order indicies measure the
+contribution to the output variance caused by the interaction between
+any two model inputs. Some computing error can appear in these
+sensitivity indices, such as negative values. Typically, these computing
+errors shrink as the number of samples increases.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">second_order</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S2&#39;</span><span class="p">]</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Second-order:&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1-x2:  </span><span class="si">{</span><span class="n">second_order</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1-x3:  </span><span class="si">{</span><span class="n">second_order</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x2-x3:  </span><span class="si">{</span><span class="n">second_order</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Second</span><span class="o">-</span><span class="n">order</span><span class="p">:</span>
+<span class="n">x1</span><span class="o">-</span><span class="n">x2</span><span class="p">:</span>  <span class="o">-</span><span class="mf">0.043237389723234154</span>
+<span class="n">x1</span><span class="o">-</span><span class="n">x3</span><span class="p">:</span>  <span class="mf">0.17506452088709862</span>
+<span class="n">x2</span><span class="o">-</span><span class="n">x3</span><span class="p">:</span>  <span class="o">-</span><span class="mf">0.03430682392607577</span>
+</pre></div>
+</div>
+<p>We can see that there are strong interactions between 𝑥1 and 𝑥3. Note
+that in the Ishigami function, these two variables are multiplied in the
+last term of the function, which leads to interactive effects. If we
+were considering first order indices alone, we would erroneously assume
+that 𝑥3 explains no variance in the output, but the second-order and
+total order indices reveal that this is not the case. It’s easy to
+understand where we might see interactive effects in the case of the
+simple Ishigami function. However, it’s important to remember that in
+more complex systems, there may be many higher-order interactions that
+are not apparent, but could be extremely consequential in explaining the
+variance of the output.</p>
+</section>
+<section id="tips-to-apply-sobol-sa-to-your-own-problem">
+<h4><span class="section-number">B.2.1.7. </span>Tips to Apply Sobol SA to Your Own Problem<a class="headerlink" href="#tips-to-apply-sobol-sa-to-your-own-problem" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we demonstrated how to apply an SA analysis to a
+simple mathematical test function. In order to apply a Sobol SA to your
+own problem, you will follow the same general workflow that we defined
+above. You will need to:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters and set up the problem
+dictionary as in Step 1 above.</p></li>
+<li><p>Generate samples using the <code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function. This step is
+problem-dependent and note that the Sobol method can be
+computationally intensive depending on the model being analyzed. For
+example, for a simple rainfall-runoff model such as HYMOD, it has
+been recommended to run a sample size of at least N = 10,000 (which
+translates to 60,000 model runs). More complex models will be slower
+to run and will also require more samples to calculate accurate
+estimates of Sobol indices. Once you complete this process, pay
+attention to the confidence bounds on your sensitivity indices to see
+whether you need to run more samples.</p></li>
+<li><p>Run the parameter sets through your model. In the example above, the
+Ishigami function could be evaluated through SALib since it is a
+built in function. For your application, you will need to run these
+parameter sets through the problem externally and save the output.
+The output file should contain one row of output values for each
+model run.</p></li>
+<li><p>Calculate the Sobol indices. Now, the Y will be a numpy array with
+your external model output and you will need to include the parameter
+samples as an additional argument.</p></li>
+<li><p>Finally, we interpet the results. If the confidence intervals of your
+dominant indices are larger than roughly 10% of the value itself, you
+may want to consider increasing your sample size as computation
+permits. You should additionally read the references noted in Step 5
+above to understand more about identifying important factors.</p></li>
+</ol>
+<p><strong>References</strong></p>
+<aside class="footnote-list brackets">
+<aside class="footnote brackets" id="id10" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">1</a><span class="fn-bracket">]</span></span>
+<p>Saltelli, A. (2002). “Making best use of model evaluations to compute sensitivity indices.” Computer Physics Communications, 145(2):280-297, doi:<a class="reference external" href="https://doi.org/10.1016/S0010-4655(02)00280-1">10.1016/S0010-4655(02)00280-1</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id11" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">2</a><span class="fn-bracket">]</span></span>
+<p>Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and S. Tarantola (2010). “Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index.” Computer Physics Communications, 181(2):259-270, doi:<a class="reference external" href="https://doi.org/10.1016/j.cpc.2009.09.018">10.1016/j.cpc.2009.09.018</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id12" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id5">3</a><span class="fn-bracket">]</span></span>
+<p>Sobol, I. M. (2001). “Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates.” Mathematics and Computers in Simulation, 55(1-3):271-280, doi:<a class="reference external" href="https://doi.org/10.1016/S0378-4754(00)00270-6">10.1016/S0378-4754(00)00270-6</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id13" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id7">4</a><span class="fn-bracket">]</span></span>
+<p>T. H. Andres, “Sampling methods and sensitivity analysis for large parameter sets,” Journal of Statistical Computation and Simulation, vol. 57, no. 1–4, pp. 77–110, Apr. 1997, doi:<a class="reference external" href="https://doi.org/10.1080/00949659708811804">10.1080/00949659708811804</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id15" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id8">5</a><span class="fn-bracket">]</span></span>
+<p>Y. Tang, P. Reed, T. Wagener, and K. van Werkhoven, “Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation,” Hydrology and Earth System Sciences, vol. 11, no. 2, pp. 793–817, Feb. 2007, doi:<a class="reference external" href="https://doi.org/10.5194/hess-11-793-2007">10.5194/hess-11-793-2007</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id16" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id9">6</a><span class="fn-bracket">]</span></span>
+<p>J. Nossent, P. Elsen, and W. Bauwens, “Sobol’ sensitivity analysis of a complex environmental model,” Environmental Modelling &amp; Software, vol. 26, no. 12, pp. 1515–1525, Dec. 2011, doi:<a class="reference external" href="https://doi.org/10.1016/j.envsoft.2011.08.010">10.1016/j.envsoft.2011.08.010</a>.</p>
+</aside>
+</aside>
+</section>
+</section>
+</section>
+<section id="logistic-regression-tutorial">
+<h2><span class="section-number">B.3. </span>Logistic Regression Tutorial<a class="headerlink" href="#logistic-regression-tutorial" title="Link to this heading">#</a></h2>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/basin_users_logistic_regression.ipynb">Logistic Regression Tutorial</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-logistic-regression-for-factor-mapping">
+<h3><span class="section-number">B.3.1. </span>Tutorial: Logistic Regression for Factor Mapping<a class="headerlink" href="#tutorial-logistic-regression-for-factor-mapping" title="Link to this heading">#</a></h3>
+<p>This tutorial replicates a scenario discovery analysis performed in
+<a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020EF001503">Hadjimichael et
+al. (2020)</a>.</p>
+<section id="background">
+<h4><span class="section-number">B.3.1.1. </span>Background<a class="headerlink" href="#background" title="Link to this heading">#</a></h4>
+<p>Planners in the the Upper Colorado River Basin (UCRB, shown in the
+figure below) are seeking to understand the vulnerability of water users
+to uncertainties stemming from climate change, population growth and
+water policy changes. The UCRB spans 25,682 km2 in western Colorado and
+is home to approximately 300,000 residents and 1,012 km2 of irrigated
+land. Several thousand irrigation ditches divert water from the main
+river and its tributaties for irrigation (shown as small black dots in
+the figure). Transmountain diversions of approximately 567,400,000 m3
+per year are exported for irrigation, industrial and municipal uses in
+northern and eastern Colorado, serving the major population centers of
+Denver and Colorado Springs. These diversions are carried through
+tunnels, shown as large black dots in the figure.</p>
+<img alt="_images/basin_map.png" src="_images/basin_map.png" />
+<p>An important planning consideration is the water rights of each user,
+defined by seniority across all water uses (irrigation diversions,
+transboundary diversions, power plants etc.) in the basin. To assess the
+vulnerability of users with varying degrees of water rights seniority,
+planners simulate the system across an ensemble of scenarios using the
+state of Colorado’s StateMod platform. The model simulates streamflow,
+diversions, instream demands, and reservoir operations.</p>
+<p>Hadjimichael et al. (2020) employ an exploratory analysis by simulating
+a large ensemble of plausible scenarios using StateMod and then
+identifying consequential decision-relevant combinations of uncertain
+factors, termed scenario discovery. Focusing on decision-relevant
+metrics (metrics that are important to the user, the scenario discovery
+is applied to the water shortages experienced by each individual user
+(i.e., not on a single basin-wide or sector-wide metric). For this
+training example, we’ll be performing scenario discovery for three
+different water users: two irrigation users and one municipal user.</p>
+</section>
+<section id="id19">
+<h4><span class="section-number">B.3.1.2. </span>Let’s get started!<a class="headerlink" href="#id19" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we will be loading in data that has been produced in
+Hadjimichael et al. (2020). Before we start our analysis, we’ll load the
+relevant Python libraries, example data, and information for the three
+users.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#import libraries</span>
+<span class="kn">import</span> <span class="nn">msdbook</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="nn">mpl</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="c1"># load example data from Hadjimichael et al. (2020)</span>
+<span class="n">msdbook</span><span class="o">.</span><span class="n">install_package_data</span><span class="p">()</span>
+
+<span class="c1"># Select the IDs for the three users that we will perform the analysis for</span>
+<span class="n">all_IDs</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;7000550&#39;</span><span class="p">,</span><span class="s1">&#39;7200799&#39;</span><span class="p">,</span><span class="s1">&#39;3704614&#39;</span><span class="p">]</span>
+<span class="n">usernames</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Medium seniority irrigation&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Low seniority irrigation&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Transbasin municipal diversion&#39;</span><span class="p">]</span>
+<span class="n">nStructures</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Downloading</span> <span class="n">example</span> <span class="n">data</span> <span class="k">for</span> <span class="n">msdbook</span> <span class="n">version</span> <span class="mf">0.1.5</span><span class="o">...</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">uncertain_params_bounds</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_metric_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">solutions</span><span class="o">.</span><span class="n">resultfile</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LHsamples_original_1000</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">param_values</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_params_256samples</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LeafCatch</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_simulations_256samples</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">Robustness</span><span class="o">.</span><span class="n">txt</span>
+</pre></div>
+</div>
+</section>
+<section id="step-1-load-latin-hypercube-sample-and-set-up-problem">
+<h4><span class="section-number">B.3.1.3. </span>Step 1: Load Latin hypercube sample and set up problem<a class="headerlink" href="#step-1-load-latin-hypercube-sample-and-set-up-problem" title="Link to this heading">#</a></h4>
+<p>To examine regional vulnerability, we generate an ensemble of plausible
+future states of the world (SOWs) using Latin Hypercube Sampling. For
+this tutorial, we’ll load a file containing 1,000 samples across 14
+parameters. The sampled parameters encompass plausible changes to the
+future state of the basin, including changes to hydrology, water demands
+(irrigation, municipal &amp; industry, transbasin), and institutional and
+environmental factors (environmental flows, reservoir storage, operation
+of the Shoshone Power Plant). These samples are taken from ranges
+identified in <code class="docutils literal notranslate"><span class="pre">param_bounds</span></code>. Below we load in the 1000 samples, the
+range of values that the samples can take for each parameter, and the
+parameter names. More information on what each parameter constitutes can
+be found in Table 1 of Hadjimichael et al., 2020.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Identify the bounds for each of the 14 parameters</span>
+<span class="n">param_bounds</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_basin_param_bounds</span><span class="p">()</span>
+
+<span class="c1">#Load in the parameter samples</span>
+<span class="n">LHsamples</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_lhs_basin_sample</span><span class="p">()</span>
+
+<span class="c1">#Create an array of the parameter names</span>
+<span class="n">param_names</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Irrigation demand multiplier&#39;</span><span class="p">,</span><span class="s1">&#39;Reservoir loss&#39;</span><span class="p">,</span><span class="s1">&#39;Transbasin demand multiplier&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Municipal &amp; industrial multiplier&#39;</span><span class="p">,</span> <span class="s1">&#39;Shoshone&#39;</span><span class="p">,</span><span class="s1">&#39;Environmental flows&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Evaporation change&#39;</span><span class="p">,</span><span class="s1">&#39;Mean dry flow&#39;</span><span class="p">,</span><span class="s1">&#39;Dry flow variance&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Mean wet flow&#39;</span><span class="p">,</span><span class="s1">&#39;Wet flow variance&#39;</span><span class="p">,</span><span class="s1">&#39;Dry-dry probability&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Wet-wet probability&#39;</span><span class="p">,</span> <span class="s1">&#39;Snowmelt shift&#39;</span><span class="p">]</span>
+</pre></div>
+</div>
+</section>
+<section id="step-2-define-decision-relevant-metrics-for-illustration">
+<h4><span class="section-number">B.3.1.4. </span>Step 2: Define decision-relevant metrics for illustration<a class="headerlink" href="#step-2-define-decision-relevant-metrics-for-illustration" title="Link to this heading">#</a></h4>
+<p>Scenario discovery attempts to identify parametric regions that lead to
+‘success’ and ‘failure’. For this demonstration we’ll be defining
+‘success’ as states of the world where a shortage level doesn’t exceed
+its historical frequency.</p>
+</section>
+<section id="step-3-run-the-logistic-regression">
+<h4><span class="section-number">B.3.1.5. </span>Step 3: Run the logistic regression<a class="headerlink" href="#step-3-run-the-logistic-regression" title="Link to this heading">#</a></h4>
+<p>Logistic regression estimates the probability that a future SOW will be
+classified as a success or failure given a set of performance criteria.
+A logistic regression model is defined by:</p>
+<div class="math notranslate nohighlight">
+\[ln \bigg (\frac{p_i}{1-p_i} \bigg ) = X^T_i \beta\]</div>
+<p>where <span class="math notranslate nohighlight">\(p_i\)</span> is the probability the performance in the
+<span class="math notranslate nohighlight">\(i^{th}\)</span> SOW will be classified as a success, <span class="math notranslate nohighlight">\(X_i\)</span> is the
+vector of covariates describing the <span class="math notranslate nohighlight">\(i^{th}\)</span> SOW, and
+<span class="math notranslate nohighlight">\(\beta\)</span> is the vector of coefficients describing the relationship
+between the covariates and the response, which here will be estimated
+using maximum likelihood estimation.</p>
+<p>A logistic regression model was fit to the ensemble of SOWs using the
+performance criteria defined in step 2. Logistic regression modeling was
+conducted using the <a class="reference external" href="https://www.statsmodels.org/stable/index.html">Statsmodel
+Python</a> package. The
+data required for the full analysis is too large to include in this
+tutorial, but results can be found in the data file loaded below.</p>
+<p>The results files contain the occurence of different shortage frequency
+and magnitude combinations under the experiment, in increments of 10,
+between 0 and 100. These combinations (100 for each user) are
+alternative decision-relevant metrics that can be used for scenario
+discovery.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set arrays for shortage frequencies and magnitudes</span>
+<span class="n">frequencies</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">110</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+<span class="n">magnitudes</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">110</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+<span class="n">realizations</span> <span class="o">=</span> <span class="mi">10</span>
+
+<span class="c1"># Load performance and pseudo r scores for each of the users</span>
+<span class="n">results</span> <span class="o">=</span> <span class="p">[</span><span class="n">msdbook</span><span class="o">.</span><span class="n">load_user_heatmap_array</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">/</span> <span class="mi">100</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">))]</span>
+</pre></div>
+</div>
+</section>
+<section id="step-4-factor-ranking">
+<h4><span class="section-number">B.3.1.6. </span>Step 4: Factor ranking<a class="headerlink" href="#step-4-factor-ranking" title="Link to this heading">#</a></h4>
+<p>To rank the importance of each uncertain factor, we utilize McFadden’s
+psuedo-R2, a measure that quantifies the improvement of the model when
+utilizing each given predictor as compared to prediction using the mean
+of the data set:</p>
+<div class="math notranslate nohighlight">
+\[R^2_{McFadden}=1-\frac{ln \hat{L}(M_{full})}{ln \hat{L}(M_{intercept})}\]</div>
+<p>Here <span class="math notranslate nohighlight">\(ln \hat{L}(M_{full})\)</span> is the log likelihood of the full
+model (including the predictor) and <span class="math notranslate nohighlight">\(ln \hat{L}(M_{intercept})\)</span> is
+the log likelihood of the intercept model (which predicts the mean
+probability of success across all SOWs).</p>
+<p>Higher values of McFadden’s psuedo-R2 indicate higher factor importance
+(when the likelihood of the full model approaches one, the ratio of the
+likelihood of the full model compared to the intercept model will get
+very small).</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Load the pseudo-R^2 scores</span>
+<span class="n">scores</span> <span class="o">=</span> <span class="p">[</span><span class="n">msdbook</span><span class="o">.</span><span class="n">load_user_pseudo_scores</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">))]</span>
+
+<span class="c1"># Select indices of frequency and magnitudes that will be used for the visualization</span>
+<span class="n">freq</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
+<span class="n">mag</span> <span class="o">=</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">7</span><span class="p">]</span>
+</pre></div>
+</div>
+</section>
+<section id="step-5-draw-factor-maps">
+<h4><span class="section-number">B.3.1.7. </span>Step 5: Draw factor maps<a class="headerlink" href="#step-5-draw-factor-maps" title="Link to this heading">#</a></h4>
+<p>The McFadden’s psuedo-R2 scores files contain preliminary logistic
+regression results on parameter importance for each of these
+combinations. Using these psuedo-R2 scores we will identify the two most
+important factors for each metric which we’ll use to generate the final
+scenario discovery maps (note: there may be more than two important
+metrics for each user, but here we will demonstrate by mapping two).</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># setup figure</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">18</span><span class="p">),</span> <span class="n">tight_layout</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">patch</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">axes</span><span class="o">.</span><span class="n">flat</span><span class="p">)):</span>
+
+    <span class="n">ax</span> <span class="o">=</span> <span class="n">axes</span><span class="o">.</span><span class="n">flat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+    <span class="n">allSOWsperformance</span> <span class="o">=</span> <span class="n">results</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+    <span class="n">all_pseudo_r_scores</span> <span class="o">=</span> <span class="n">scores</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+    <span class="c1"># construct dataframe</span>
+    <span class="n">dta</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">LHsamples</span><span class="p">,</span> <span class="n">realizations</span><span class="p">,</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">),</span> <span class="n">columns</span><span class="o">=</span><span class="n">param_names</span><span class="p">)</span>
+    <span class="n">dta</span><span class="p">[</span><span class="s1">&#39;Success&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">allSOWsperformance</span><span class="p">[</span><span class="n">freq</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">mag</span><span class="p">[</span><span class="n">i</span><span class="p">],:]</span>
+
+    <span class="n">pseudo_r_scores</span> <span class="o">=</span> <span class="n">all_pseudo_r_scores</span><span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">frequencies</span><span class="p">[</span><span class="n">freq</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span><span class="o">+</span><span class="s1">&#39;yrs_&#39;</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">magnitudes</span><span class="p">[</span><span class="n">mag</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span><span class="o">+</span><span class="s1">&#39;prc&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>
+    <span class="n">top_predictors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">pseudo_r_scores</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">][:</span><span class="mi">2</span><span class="p">]</span> <span class="c1">#Sort scores and pick top 2 predictors</span>
+
+    <span class="c1"># define color map for dots representing SOWs in which the policy</span>
+    <span class="c1"># succeeds (light blue) and fails (dark red)</span>
+    <span class="n">dot_cmap</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">ListedColormap</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">227</span><span class="p">,</span><span class="mi">26</span><span class="p">,</span><span class="mi">28</span><span class="p">],[</span><span class="mi">166</span><span class="p">,</span><span class="mi">206</span><span class="p">,</span><span class="mi">227</span><span class="p">]])</span><span class="o">/</span><span class="mf">255.0</span><span class="p">)</span>
+
+    <span class="c1"># define color map for probability contours</span>
+    <span class="n">contour_cmap</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">&#39;RdBu&#39;</span><span class="p">)</span>
+
+    <span class="c1"># define probability contours</span>
+    <span class="n">contour_levels</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.05</span><span class="p">,</span><span class="mf">0.1</span><span class="p">)</span>
+
+    <span class="c1"># define base values of the predictors</span>
+    <span class="n">SOW_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span> <span class="c1"># default parameter values for base SOW</span>
+    <span class="n">base</span> <span class="o">=</span> <span class="n">SOW_values</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">]</span>
+    <span class="n">ranges</span> <span class="o">=</span> <span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">]</span>
+
+    <span class="c1"># define grid of x (1st predictor), and y (2nd predictor) dimensions</span>
+    <span class="c1"># to plot contour map over</span>
+    <span class="n">xgrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="mi">0</span><span class="p">],</span>
+                      <span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="mi">1</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">around</span><span class="p">((</span><span class="n">ranges</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">ranges</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span><span class="o">/</span><span class="mi">500</span><span class="p">,</span><span class="n">decimals</span><span class="o">=</span><span class="mi">4</span><span class="p">))</span>
+    <span class="n">ygrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">]][</span><span class="mi">0</span><span class="p">],</span>
+                      <span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">]][</span><span class="mi">1</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">around</span><span class="p">((</span><span class="n">ranges</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">ranges</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span><span class="o">/</span><span class="mi">500</span><span class="p">,</span><span class="n">decimals</span><span class="o">=</span><span class="mi">4</span><span class="p">))</span>
+    <span class="n">all_predictors</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dta</span><span class="o">.</span><span class="n">columns</span><span class="o">.</span><span class="n">tolist</span><span class="p">()[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">top_predictors</span><span class="p">]</span>
+    <span class="n">dta</span><span class="p">[</span><span class="s1">&#39;Interaction&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">dta</span><span class="p">[</span><span class="n">all_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span><span class="o">*</span><span class="n">dta</span><span class="p">[</span><span class="n">all_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
+
+    <span class="c1"># logistic regression here</span>
+    <span class="n">predictor_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">all_predictors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]]</span>
+    <span class="n">result</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">fit_logit</span><span class="p">(</span><span class="n">dta</span><span class="p">,</span> <span class="n">predictor_list</span><span class="p">)</span>
+
+    <span class="c1"># plot contour map</span>
+    <span class="n">contourset</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_contour_map</span><span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="n">dta</span><span class="p">,</span> <span class="n">contour_cmap</span><span class="p">,</span>
+                                          <span class="n">dot_cmap</span><span class="p">,</span> <span class="n">contour_levels</span><span class="p">,</span> <span class="n">xgrid</span><span class="p">,</span>
+                                          <span class="n">ygrid</span><span class="p">,</span> <span class="n">all_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">all_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">base</span><span class="p">)</span>
+
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">usernames</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+
+<span class="c1"># set up colorbar</span>
+<span class="n">cbar_ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_axes</span><span class="p">([</span><span class="mf">0.98</span><span class="p">,</span> <span class="mf">0.15</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">])</span>
+<span class="n">cbar</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">contourset</span><span class="p">,</span> <span class="n">cax</span><span class="o">=</span><span class="n">cbar_ax</span><span class="p">)</span>
+<span class="n">cbar_ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Probability of Success&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">cbar_ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;y&#39;</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">&#39;major&#39;</span><span class="p">,</span> <span class="n">labelsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">statsmodels</span><span class="o">/</span><span class="n">base</span><span class="o">/</span><span class="n">model</span><span class="o">.</span><span class="n">py</span><span class="p">:</span><span class="mi">127</span><span class="p">:</span> <span class="n">ValueWarning</span><span class="p">:</span> <span class="n">unknown</span> <span class="n">kwargs</span> <span class="p">[</span><span class="s1">&#39;disp&#39;</span><span class="p">]</span>
+  <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="n">msg</span><span class="p">,</span> <span class="n">ValueWarning</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Optimization</span> <span class="n">terminated</span> <span class="n">successfully</span><span class="o">.</span>
+         <span class="n">Current</span> <span class="n">function</span> <span class="n">value</span><span class="p">:</span> <span class="mf">0.378619</span>
+         <span class="n">Iterations</span> <span class="mi">8</span>
+<span class="n">Optimization</span> <span class="n">terminated</span> <span class="n">successfully</span><span class="o">.</span>
+         <span class="n">Current</span> <span class="n">function</span> <span class="n">value</span><span class="p">:</span> <span class="mf">0.397285</span>
+         <span class="n">Iterations</span> <span class="mi">8</span>
+<span class="n">Optimization</span> <span class="n">terminated</span> <span class="n">successfully</span><span class="o">.</span>
+         <span class="n">Current</span> <span class="n">function</span> <span class="n">value</span><span class="p">:</span> <span class="mf">0.377323</span>
+         <span class="n">Iterations</span> <span class="mi">8</span>
+</pre></div>
+</div>
+<img alt="_images/notebook_logistic_output_11_1.png" src="_images/notebook_logistic_output_11_1.png" />
+<p>The figure above demonstrates how different combinations of the
+uncertain factors lead to success or failure in different states of the
+world, which are denoted by the blue and red dots respectively. The
+probability of success and failure are further denoted by the contours
+in the figure. Several insights can be drawn from this figure.</p>
+<p>First, using metrics chosen to be decision-relevant (specific to each
+user) causes different factors to be identified as most important by
+this scenario-discovery exercise (the x- and y-axes for each of the
+subplots). In other words, depending on what the decision makers of this
+system want to prioritize they might choose to monitor different
+uncertain factors to track performance.</p>
+<p>Second, in the top panel, the two identified factors appear to also have
+an interactive effect on the metric used (shortages of a certain level
+and frequency in this example). In terms of scenario discovery, the
+Patient Rule Induction Method (PRIM) or Classification And Regression
+Trees (CART) would not be able to delineate this non-linear space and
+would therefore misclassify parameter combinations as ‘desirable’ when
+they were in fact undesirable, and vice versa.</p>
+<p>Lastly, logistic regression also produces contours of probability of
+success, i.e. different factor-value combinations are assigned different
+probabilities that a shortage level will be exceeded. This allows the
+decision makers to evaluate these insights while considering their risk
+aversion.</p>
+</section>
+<section id="tips-to-apply-scenario-discovery-to-your-own-problem">
+<h4><span class="section-number">B.3.1.8. </span>Tips to Apply Scenario Discovery to Your Own Problem<a class="headerlink" href="#tips-to-apply-scenario-discovery-to-your-own-problem" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we demonstrated how to perform a scenario discovery
+analysis for three different users in the UCRB. The analysis allowed us
+to determine which parameters the users would be most affected by and to
+visualize how different ranges of these parameters lead to success and
+failure for different users. This framework can be applicable to any
+other application where it is of interest to characterize success and
+failure based on uncertain parameter ranges. In order to apply the same
+framework to your own problem:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters of interest, which will
+represent uncertainties that characterize your system.</p></li>
+<li><p>Generate samples for these parameters (this can be done using the
+<code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function or externally).</p></li>
+<li><p>Define what constitutes success and failure in your problem. In this
+tutorial, success was defined based on not surpassing the historical
+drought frequency. Choose a metric that is relevant to your problem
+and decision-makers that might be involved. If your model involves an
+optimization, you can also define metrics based on meeting certain
+values of these objectives.</p></li>
+<li><p>Run the parameter sets through your model and calculate success and
+failure based on your metrics and across different users if
+applicable. This step will allow you to create the scatter plot part
+of the final figure.</p></li>
+<li><p>If it is of interest, the contours on the figure can be created by
+fitting the logistic regression model in a similiar manner as denoted
+in Steps 3 and 5 of the tutorial.</p></li>
+</ol>
+</section>
+</section>
+</section>
+<section id="hymod-dynamics-tutorial">
+<h2><span class="section-number">B.4. </span>HYMOD Dynamics Tutorial<a class="headerlink" href="#hymod-dynamics-tutorial" title="Link to this heading">#</a></h2>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">HYMOD Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-sensitivity-analysis-of-the-hymod-model">
+<h3><span class="section-number">B.4.1. </span>Tutorial: Sensitivity Analysis of the HYMOD Model<a class="headerlink" href="#tutorial-sensitivity-analysis-of-the-hymod-model" title="Link to this heading">#</a></h3>
+<p>The purpose of this tutorial is to demonstrate the global sensitivity
+analysis concepts and tools established in the Section 3.1 of the main
+text of this eBook. This demonstration will highlight the central role
+of design of experiments (Section 3.3), when implementing global
+sensitivity analysis tools described in Section 3.4.</p>
+<p>We’ll explore these tools and concepts using the HYdrological MODel
+(HYMOD), a rainfall-runoff model developed and used for river flow
+forecasting. HYMOD was chosen for demonstration because its purpose is
+to abstract highly complex and non-linear systems. The methods
+demonstrated in thistutorial can be applied to numerical models that
+simulate other complex non-linear systems.</p>
+<p>This tutorial will first introduce HYMOD and use it to simulate
+streamflows in a river basin. Next, we’ll employ sensitivity analysis
+concepets described in Section 3 of the main text to examine how values
+of HYMOD’s parameters impact streamflow predictions. We’ll then explore
+how the effects of these parameters may change over time using
+time-varying sensitivtiy analysis. Finally, we’ll demonstrate concepts
+presented in Chapter 7 through two ensemble-based methods of uncertainty
+quantification - Generalized Likelihood Uncertainty Estimation (GLUE)
+and Pre-Calibration.</p>
+<p>The tutorial includes the following steps:</p>
+<p>1.1 - Introduction to a simple hydrologic model (HYMOD)</p>
+<p>1.2 - Input Data</p>
+<p>1.3 - Running a basic simulation</p>
+<p>1.4 - Model outputs</p>
+<p>2.1 - Design of Experiments</p>
+<p>2.2 - Sensitivity analysis for one output</p>
+<p>2.3 - Sensitivity analysis across multiple outputs</p>
+<p>2.4 - Time-varying sensitivity analysis</p>
+</section>
+<section id="introduction-to-hymod">
+<h3><span class="section-number">B.4.2. </span>1 - Introduction to HYMOD<a class="headerlink" href="#introduction-to-hymod" title="Link to this heading">#</a></h3>
+<p id="hymod">1.1 Overview</p>
+<p>HYMOD is a hydrologic model (rainfall-runoff model) that simulates key
+hydrologic fluxes such as infiltration, streamflow and
+evapotranspiration. The model was originally developed and used for
+river flow forecasting, but it has also been been used to explore
+different sensitivity analysis (e.g., <a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/wrcr.20124">Herman et al.,
+2013</a>),
+uncertainty quantification (e.g., <a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2006WR005205">Smith et al.,
+2008</a>),
+and optimization (e.g., <a class="reference external" href="https://www.sciencedirect.com/science/article/pii/S0022169414006362?casa_token=IRqE19Hkfa8AAAAA:_fXOqfwpzxMpchvu8_0njCe0Ok9H29Gyw2F46l9PzG9UVODDTUg6wIOSiyp6uybGevNVnZ7N">Ye et al.,
+2014</a>)
+concepts.</p>
+<p>HYMOD accepts two inputs - daily precepitation and daily potential
+evapotranspiration (PET)- and generates predicitons of daily streamflow.
+HYMOD abstracts the highly non-linear process of runoff routing by
+dividing the flow into two components: quick flow, representing
+precipitation that quickly runs off the surface of the watershed into
+the stream, and slow flow, that moves through the soil and takes much
+longer to arrive at the stream.</p>
+<p>To generate streamflow predictions, HYMOD first models vertical
+processes within the watershed to determine how much water infiltrates
+and evaporates from the soil at a given time step. It then determines
+how much water should be partitioned into quick flow and slow flow
+processes. Within each process it abstracts residence time (the time it
+takes a unit volume of water to move through the watershed and into the
+stream) using a series of “reservoirs” each with a calibrated residence
+time.</p>
+<p>HYMOD’s representation of hydrologic processes are shown Figure 1 below
+and controlled by the following parameters:</p>
+<p><span class="math notranslate nohighlight">\(H_{uz}\)</span>: the maximum water storage capacity of the soil (mm)</p>
+<p><span class="math notranslate nohighlight">\(B_{exp}\)</span>: parameters describing the degree of spatial variability
+within the basin between 0 and Huz</p>
+<p><span class="math notranslate nohighlight">\(Alp\)</span>: Fraction of runoff contributing to quick flow</p>
+<p><span class="math notranslate nohighlight">\(K_q\)</span>: Quick flow residence time of linear infinite reservoir (the
+Kq values of all three linear reservoirs are the same)</p>
+<p><span class="math notranslate nohighlight">\(K_s\)</span>: Slow flow residence time of linear infinite reservoir</p>
+<img alt="_images/hymod_schematic-DAVE.png" src="_images/hymod_schematic-DAVE.png" />
+<p>HYMOD models the fraction of water that is stored in the soil
+<span class="math notranslate nohighlight">\((F(XH_{uz}))\)</span> using the following relationship:</p>
+<div class="math notranslate nohighlight">
+\[F(XH_{uz}) = 1 - (1 - \frac{XH_{uz}}{H_{uz}})^{B}\]</div>
+<p>where <span class="math notranslate nohighlight">\(XH_{uz}\)</span> is the water storage capacity of the soil;
+<span class="math notranslate nohighlight">\(H_{uz}\)</span> is the parameter describing basin maximum water
+storage capacity (mm); and <span class="math notranslate nohighlight">\(B\)</span> is the parameter describing the
+degree of spatial variability within the basin.</p>
+<p>The portion of precipitation that exceeds the water storage capacity is
+treated as runoff.</p>
+<p>To route runoff to streamflow, the excess runoff from the vertical
+processes is split into quick flow and slow flow. The proportion of
+runoff partitioned into quick flow and slow flow is determined by a
+parameter <span class="math notranslate nohighlight">\(Alp\)</span>, which ranges between 0 and 1. Quick flow is
+routed through <span class="math notranslate nohighlight">\(N\)</span> identical quick flow tanks <span class="math notranslate nohighlight">\(Q1, Q2... QN\)</span>
+(shown above as <span class="math notranslate nohighlight">\(N=3\)</span>). The rate at which runoff moves through the
+quick flow system is described by the residence time of the quick frlow
+tanks, <span class="math notranslate nohighlight">\(Kq\)</span> (day). Slow flow is routed through a parallel slow
+flow tank and the rate at which slow flow is routed is described by the
+slow flow residences time, <span class="math notranslate nohighlight">\(Ks\)</span> (day).</p>
+<p>Citation: Wagener, T., Boyle, D. P., Lees, M. J., Wheater, H. S., Gupta,
+H. V., &amp; Sorooshian, S. (2001). A framework for development and
+application of hydrological models. Hydrology and Earth System Sciences,
+5(1), 13-26.</p>
+<p id="inputs">1.2 Input data</p>
+<p>The HYMOD model only requires precipitation and potential
+evapotranspiration as inputs. For this example, we’ll run HYMOD using
+data from the Leaf River, a humid catchment located north of Collins
+Mississippi that has been widely used to explore HYMOD. The dataset also
+includes daily observed runoff that we later use to evaluate the
+performace of each sensitvity analysis sample set.</p>
+<p>In the following section of code, we’ll load the necessary python
+libraries and read in the input file. For this exercise we’ll only use
+the first eleven years of data. The first five rows of the input dataset
+are printed to show what they look like:</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">msdbook</span>
+
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+
+<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">metrics</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
+
+<span class="c1"># load example data</span>
+<span class="n">msdbook</span><span class="o">.</span><span class="n">install_package_data</span><span class="p">()</span>
+
+<span class="c1"># load the Leaf River HYMOD input file</span>
+<span class="n">leaf_data</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_input_file</span><span class="p">()</span>
+
+<span class="c1"># extract the first eleven years of data</span>
+<span class="n">leaf_data</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">4015</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Leaf River Data structure:&#39;</span><span class="p">)</span>
+
+<span class="c1"># There are only three columns in the file including precipitation, potential evapotranspiration and  streamflow</span>
+<span class="n">leaf_data</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Downloading</span> <span class="n">example</span> <span class="n">data</span> <span class="k">for</span> <span class="n">msdbook</span> <span class="n">version</span> <span class="mf">0.1.5</span><span class="o">...</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">uncertain_params_bounds</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_metric_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">solutions</span><span class="o">.</span><span class="n">resultfile</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LHsamples_original_1000</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">param_values</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_params_256samples</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LeafCatch</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_simulations_256samples</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">Robustness</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Leaf</span> <span class="n">River</span> <span class="n">Data</span> <span class="n">structure</span><span class="p">:</span>
+</pre></div>
+</div>
+<div>
+<style scoped>
+    .dataframe tbody tr th:only-of-type {
+        vertical-align: middle;
+    }
+
+    .dataframe tbody tr th {
+        vertical-align: top;
+    }
+
+    .dataframe thead th {
+        text-align: right;
+    }
+</style>
+<table border="1" class="dataframe">
+  <thead>
+    <tr style="text-align: right;">
+      <th></th>
+      <th>Precip</th>
+      <th>Pot_ET</th>
+      <th>Strmflw</th>
+    </tr>
+  </thead>
+  <tbody>
+    <tr>
+      <th>0</th>
+      <td>0.0</td>
+      <td>4.60</td>
+      <td>0.29</td>
+    </tr>
+    <tr>
+      <th>1</th>
+      <td>0.0</td>
+      <td>4.31</td>
+      <td>0.24</td>
+    </tr>
+    <tr>
+      <th>2</th>
+      <td>0.0</td>
+      <td>4.33</td>
+      <td>0.21</td>
+    </tr>
+    <tr>
+      <th>3</th>
+      <td>0.0</td>
+      <td>4.78</td>
+      <td>0.19</td>
+    </tr>
+    <tr>
+      <th>4</th>
+      <td>0.0</td>
+      <td>2.91</td>
+      <td>0.18</td>
+    </tr>
+  </tbody>
+</table>
+</div><p>To visualize catchment hydrology, streamflow and precipitation data are
+usually plotted together as a combined hydrograph (streamflow ) and
+hyetograph (rainfall, from Greek.hyetos, “rain”). Streamflow is plotted
+as a time series, while rainfall is shown as an inverted bar plot along
+the top of the graph. Streamflow labels are shown on the left y-axis,
+while rainfall labels are shown on the right y-axis.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># make an axis for the hydrograph</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">strmflw_ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">12</span><span class="p">,</span><span class="mi">6</span><span class="p">])</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">50</span><span class="p">])</span>
+
+<span class="c1">#make a second y-axis for the hyetograph</span>
+<span class="n">precip_ax</span> <span class="o">=</span> <span class="n">strmflw_ax</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">200</span><span class="p">])</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">invert_yaxis</span><span class="p">()</span>
+
+<span class="n">precip</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">]</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">])),</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;lightcoral&#39;</span><span class="p">)</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Streamflow (mm/day)&#39;</span><span class="p">)</span>
+
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">])),</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">],</span> <span class="n">width</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Rainfall (mm/day)&#39;</span><span class="p">)</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">&#39;Precip&#39;</span><span class="p">],</span> <span class="n">loc</span><span class="o">=</span><span class="s1">&#39;center right&#39;</span><span class="p">)</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">&#39;Streamflow&#39;</span><span class="p">],</span><span class="n">bbox_to_anchor</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.48</span><span class="p">))</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">&lt;</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">legend</span><span class="o">.</span><span class="n">Legend</span> <span class="n">at</span> <span class="mh">0x7f53b95c6850</span><span class="o">&gt;</span>
+</pre></div>
+</div>
+<img alt="_images/hymod1.png" src="_images/hymod1.png" />
+<p id="baseline">1.3 Running a Baseline Model Simulation</p>
+<p>We’ll start our experiment by running HYMOD using its default
+parameters.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># assign input parameters to generate a baseline simulated streamflow</span>
+<span class="n">Nq</span> <span class="o">=</span> <span class="mi">3</span>  <span class="c1"># Number of quickflow routing tanks</span>
+<span class="n">Kq</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="c1"># Quickflow routing tanks&#39; rate parameter</span>
+<span class="n">Ks</span> <span class="o">=</span>  <span class="mf">0.001</span> <span class="c1"># Slowflow routing tank&#39;s rate parameter</span>
+<span class="n">Alp</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="c1"># Quick/slow split parameter</span>
+<span class="n">Huz</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1"># Maximum height of soil moisture accounting tank</span>
+<span class="n">B</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="c1"># Scaled distribution function shape parameter</span>
+
+<span class="c1"># Note that the number of years is 11. One year of model warm-up and ten years are used for actual simulation</span>
+<span class="n">model</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">hymod</span><span class="p">(</span><span class="n">Nq</span><span class="p">,</span> <span class="n">Kq</span><span class="p">,</span> <span class="n">Ks</span><span class="p">,</span> <span class="n">Alp</span><span class="p">,</span> <span class="n">Huz</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">leaf_data</span><span class="p">,</span> <span class="n">ndays</span><span class="o">=</span><span class="mi">4015</span><span class="p">)</span>
+</pre></div>
+</div>
+<p id="outputs">1.4 Model Outputs</p>
+<p>Model outputs include actual evapotranspiration, quick and fast
+streamflow, and combined runoff. In this tutorial we focus on the total
+daily runoff, QQ (<span class="math notranslate nohighlight">\(m^3/s\)</span>). We can use the following script to
+plot simulated streamflow against observed streamflow.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-observed-vs-simulated-streamflow">plot_observed_vs_simulated_streamflow</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ax</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_observed_vs_simulated_streamflow</span><span class="p">(</span><span class="n">df</span><span class="o">=</span><span class="n">leaf_data</span><span class="p">,</span> <span class="n">hymod_dict</span><span class="o">=</span><span class="n">model</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod2.png" src="_images/hymod2.png" />
+<p>So how does our model perform? We can investigate model performance
+across several metrics:</p>
+<p>1: Mean Absolute Error (MAE); MAE conveys how the model performs on
+average across the 10 year simulation period, with smaller values
+indicating better performance. The absolute value is taken so that
+positive and negative errors do not cancel each other out.</p>
+<div class="math notranslate nohighlight">
+\[MAE = \frac{1}{N}\sum_{t=0}^N\left\lvert Q_{sim,t}-Q_{obs,t}\right\rvert\]</div>
+<p>2: Root Mean Square Error (RMSE); RMSE is sum of square errors across
+the 10 year simulation period. RMSE is sensitive to large errors between
+the historical record and the simulated flows, and thus is useful for
+highlighting the model’s ability to capture of extreme flood events.</p>
+<div class="math notranslate nohighlight">
+\[RMSE = \sqrt{\frac{1}{N}\sum_{t=1}^{N}(Q_{sim,t}-Q_{obs,t})^2}\]</div>
+<p>3: Log-Root Mean Square Error (Log(RMSE)) LOG(RMSE) focuses on model
+performance during low-flow events.</p>
+<div class="math notranslate nohighlight">
+\[LOG(RMSE) = log(RMSE)\]</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">mae</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">]</span> <span class="o">-</span> <span class="n">model</span><span class="p">[</span><span class="s1">&#39;Q&#39;</span><span class="p">]))</span>
+<span class="n">mse</span> <span class="o">=</span> <span class="n">metrics</span><span class="o">.</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">model</span><span class="p">[</span><span class="s1">&#39;Q&#39;</span><span class="p">],</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">])</span>
+<span class="n">rmse</span> <span class="o">=</span> <span class="n">mse</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;MAE: &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">mae</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">RMSE: &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">mse</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">LOG(RMSE): &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">rmse</span><span class="p">))</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">MAE</span><span class="p">:</span> <span class="mf">1.0787471470460999</span>
+<span class="n">RMSE</span><span class="p">:</span> <span class="mf">4.375695937555197</span>
+<span class="n">LOG</span><span class="p">(</span><span class="n">RMSE</span><span class="p">):</span> <span class="mf">2.0918164206151544</span>
+</pre></div>
+</div>
+<p>The error metrics show that HYMOD performs reasonably well, the MAE is
+around 1 <span class="math notranslate nohighlight">\(m^3/s\)</span>, the RMSE is on the order of 10% of the largest
+observed streamflow and the LOG(RMSE) is fairly low.</p>
+</section>
+<section id="global-sensitivity-analysis">
+<h3><span class="section-number">B.4.3. </span>2- Global Sensitivity Analysis<a class="headerlink" href="#global-sensitivity-analysis" title="Link to this heading">#</a></h3>
+<p id="sensitivity">2.1 Experimental Design and Setup</p>
+<p>Now we’ll examine how sensitive streamflow simulations generated by
+HYMOD are to the model’s input parameters. We’ll perform global
+sensitivity analysis (see Section 3.1 of the main text) using the SALib
+Python library.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">saltelli</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">sobol</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">delta</span>
+</pre></div>
+</div>
+<p>A first and critical step when conducting sensitivity analysis is
+determining the experimental design (see Design of Experiments, Section
+3.4 of the main text). Our experimental design involves defining the
+uncertainties that we’ll be examining, the output of interest, the
+ranges of each uncertainty that will be explored and the strategy for
+sampling the uncertainty space.</p>
+<p>For this experiment we’ll explore the five parameters highlighted in
+Figure 1. We’ll draw their ranges from existing literature on the model
+(note Jon H. paper). We’ll use a Sobol sampling an a quasi-random
+sampling with low sequences approach to sample the uncertainty space
+(Section 3.3.4).</p>
+<p>In this demonstration we’ll utilize Sobol Sensitivity Analysis, a
+variance based method (Section 3.4.5).</p>
+<p>To explore HYMOD’s behavoir, we’ll examine the sensitivity of four model
+ouputs to input parameters: 1) predicted flow, 2) Mean Absolute Error
+(compared with the calibaration data set), 3) Root Mean Square Error and
+4) Log Root Mean Square Error.</p>
+<p>This analysis will employ SALib, a Python implementation also utilized
+in the other SA tutorial (make this more formal).</p>
+<p>To start our analysis, we’ll create a dictionary that describes our
+model uncertainties and their ranges, this dictionary is named
+“problem_hymod” (SALib refers to these dictionaries as “problems”).</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">problem_hymod</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">5</span><span class="p">,</span>
+    <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">],</span>
+    <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>  <span class="c1"># Kq</span>
+               <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">],</span>  <span class="c1"># Ks</span>
+               <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>    <span class="c1"># Alp</span>
+               <span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">500</span><span class="p">],</span>  <span class="c1"># Huz</span>
+               <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">1.9</span><span class="p">]]</span>  <span class="c1"># B</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+<p>After defining our uncertainites and ranges, we’ll use SALib to sample
+the uncertainty space and run the model for each of the sample sets. We
+will load a sample that has already been created <code class="docutils literal notranslate"><span class="pre">param_values_hymod</span></code>
+for demonstration purposes. For HYMOD, literature recommends running at
+least N = 10,000 samples, to keep this demonstration easy to run
+however, we utilize only 256 sobol samples of uncertainties. To generate
+accurate approximations of second order sensitivity indicies SALib
+generates N*(2k+2) sets of samples, where N=256 and k=5 (number of
+uncertainties). For the math behind why this is needed, see (Saltelli,
+A., 2002. Making best use of model evaluations to compute sensitivity
+indices. Computer Physics Communications 145, 280–297.
+<a class="reference external" href="https://doi.org/10.1016/S0010-4655(02)00280-1">https://doi.org/10.1016/S0010-4655(02)00280-1</a>).</p>
+<p>The actual model simulation takes an extended period, so we also load
+the simulation data from a previous run. The following demonstrates how
+to conduct this analysis:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># generate 256 samples.</span>
+<span class="n">param_values_hymod</span> <span class="o">=</span> <span class="n">saltelli</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="mi">256</span><span class="p">)</span>
+
+<span class="c1"># dictionary to store outputs in</span>
+<span class="n">d_outputs</span> <span class="o">=</span> <span class="p">{}</span>
+
+<span class="c1"># run simulation for each parameter sample</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">param_values_hymod</span><span class="p">)):</span>
+
+    <span class="c1"># run model for each sensitivity analysis parameter sets</span>
+    <span class="n">hymod_output</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">hymod</span><span class="p">(</span><span class="n">Nq</span><span class="p">,</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>
+                                 <span class="n">leaf_data</span><span class="p">,</span>
+                                 <span class="n">ndays</span><span class="o">=</span><span class="mi">4015</span><span class="p">)</span>
+
+    <span class="c1"># store the simulated total flow discharge</span>
+    <span class="n">d_outputs</span><span class="p">[</span><span class="sa">f</span><span class="s2">&quot;Q</span><span class="si">{</span><span class="n">i</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">hymod_output</span><span class="p">[</span><span class="s2">&quot;Q&quot;</span><span class="p">]</span>
+
+
+<span class="n">Q_df_bw</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d_outputs</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously generated parameter values</span>
+<span class="n">param_values_hymod</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_params</span><span class="p">()</span>
+
+<span class="c1"># number of samples</span>
+<span class="n">n_samples</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">param_values_hymod</span><span class="p">)</span>
+
+<span class="c1"># load previously generated hymod simulated outputs</span>
+<span class="n">Q_df_bw</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_simulation</span><span class="p">()</span>
+
+<span class="c1"># column names of each sample simulation number</span>
+<span class="n">sample_column_names</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">Q_df_bw</span><span class="o">.</span><span class="n">columns</span> <span class="k">if</span> <span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s1">&#39;Q&#39;</span><span class="p">]</span>
+</pre></div>
+</div>
+<p>A hydrological model such as HYMOD usually includes ordinary
+differential equations that are sensitive to their initial condition.
+They also have components in their underlying formulation that have long
+memory such that prior time steps can affect their current simulations.
+For example, soil moisture or groundwater can hold water for a long time
+and therefore they are often considered to exhibit a long memory. This
+can affect the partitioning of water to runoff and infiltration, while
+also controlling the generation of base flow. Therefore, it is important
+to have a reasonable initial value for them. To achieve this,
+hydrologists usually extend their simulation period and after the
+simulations, they remove that extended time period that has unreasonable
+groundwater or surface water values. This time period is called the
+warm-up time period.</p>
+<p>Here we extended our simulation for one year (from 10 years to 11 years)
+and we removed the first year of simulation, therefore our warm-up
+period is one year.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># exclude the first year of simulation from the simulations and reset the index</span>
+<span class="n">Q_df</span> <span class="o">=</span> <span class="n">Q_df_bw</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">365</span><span class="p">:</span><span class="mi">4015</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">reset_index</span><span class="p">(</span><span class="n">drop</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="c1"># exclude the first year of the input data and reset the index</span>
+<span class="n">leaf_data</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">365</span><span class="p">:</span><span class="mi">4015</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">reset_index</span><span class="p">(</span><span class="n">drop</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now that HYMOD has been warmed up, we’ll examine how HYMOD’s streamflow
+outputs vary under different sample sets, and compare them with the
+observed streamflow.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># add date columns to our simulation data frame; for this data our start date is 1/1/2000</span>
+<span class="n">date_ts</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">date_range</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="s1">&#39;1/1/2000&#39;</span><span class="p">,</span> <span class="n">periods</span><span class="o">=</span><span class="mi">3650</span><span class="p">,</span> <span class="n">freq</span><span class="o">=</span><span class="s1">&#39;D&#39;</span><span class="p">)</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;date&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">year</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;month&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">month</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;day&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">day</span>
+
+<span class="c1"># aggregate the simulated observed streamflow to monthly mean</span>
+<span class="n">df_sim_mth_mean</span> <span class="o">=</span> <span class="n">Q_df</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">,</span> <span class="s1">&#39;month&#39;</span><span class="p">])[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+
+<span class="c1"># do the same for the observed data</span>
+<span class="n">date_ts</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">date_range</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="s1">&#39;1/1/2000&#39;</span><span class="p">,</span> <span class="n">periods</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">),</span> <span class="n">freq</span><span class="o">=</span><span class="s1">&#39;D&#39;</span><span class="p">)</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;date&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">year</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;month&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">month</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;day&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">day</span>
+
+<span class="c1"># aggregate the daily observed streamflow to monthly mean</span>
+<span class="n">df_obs_mth_mean</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">,</span> <span class="s1">&#39;month&#39;</span><span class="p">])</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-observed-vs-sensitivity-streamflow">plot_observed_vs_sensitivity_streamflow</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ax</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_observed_vs_sensitivity_streamflow</span><span class="p">(</span><span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_mth_mean</span><span class="p">,</span>
+                                                     <span class="n">df_sim</span><span class="o">=</span><span class="n">df_sim_mth_mean</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod3.png" src="_images/hymod3.png" />
+<p id="sametrics">2.2 Sensitivity of streamflows to model parameters</p>
+<p>Now we’ll examine how each of HYMOD’s parameters impact the variance of
+simulated streamflows. Using SALib we’ll calculate the first order and
+total order sensitivity indicies of each model parameter. The first
+order sensitivity index measure’s the individual impact that a given
+parameter has on the variance of the simulated streamflows. The total
+order index measures the impact of a given parameter along with all
+interactions that other parameters have with the given parameter on
+simulated streamflows.</p>
+<p>We’ll start with an matrix, Y, which contains our simulated streamflows
+for every uncertainty sample. We’ll then use the sobol.analyze function
+from SALib to calculate the sensitivity indicies (Si). The arguments for
+this function are the problem dictionary defined in part 2.2 of this
+tutorial, and the matrix of simulated streamflows, Y.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># overall aggregated indices</span>
+<span class="n">Y</span> <span class="o">=</span> <span class="n">Q_df</span><span class="p">[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+<span class="c1"># Perform analysis</span>
+<span class="n">Si</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now we can examine our results, we’ll print the first order and total
+order Si’s for each parameter, then visualize the results with bar plots</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;First order indices = &#39;</span><span class="p">,</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">])</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Total order indicies = &#39;</span><span class="p">,</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">])</span>
+
+<span class="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+<span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s1">&#39;&#39;</span><span class="p">,</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;First order Si&#39;</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s1">&#39;&#39;</span><span class="p">,</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Total order Si&#39;</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">First</span> <span class="n">order</span> <span class="n">indices</span> <span class="o">=</span>  <span class="p">[</span><span class="mf">9.55550001e-05</span> <span class="mf">7.49249463e-04</span> <span class="mf">5.62386413e-04</span> <span class="mf">7.03327551e-01</span>
+ <span class="mf">2.53701895e-01</span><span class="p">]</span>
+<span class="n">Total</span> <span class="n">order</span> <span class="n">indicies</span> <span class="o">=</span>  <span class="p">[</span><span class="mf">1.76174200e-06</span> <span class="mf">1.63288175e-03</span> <span class="mf">3.41378460e-04</span> <span class="mf">6.88983864e-01</span>
+ <span class="mf">2.53922146e-01</span><span class="p">]</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">ipykernel_launcher</span><span class="o">.</span><span class="n">py</span><span class="p">:</span><span class="mi">9</span><span class="p">:</span> <span class="ne">UserWarning</span><span class="p">:</span> <span class="n">FixedFormatter</span> <span class="n">should</span> <span class="n">only</span> <span class="n">be</span> <span class="n">used</span> <span class="n">together</span> <span class="k">with</span> <span class="n">FixedLocator</span>
+  <span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">&#39;__main__&#39;</span><span class="p">:</span>
+<span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">ipykernel_launcher</span><span class="o">.</span><span class="n">py</span><span class="p">:</span><span class="mi">15</span><span class="p">:</span> <span class="ne">UserWarning</span><span class="p">:</span> <span class="n">FixedFormatter</span> <span class="n">should</span> <span class="n">only</span> <span class="n">be</span> <span class="n">used</span> <span class="n">together</span> <span class="k">with</span> <span class="n">FixedLocator</span>
+  <span class="kn">from</span> <span class="nn">ipykernel</span> <span class="kn">import</span> <span class="n">kernelapp</span> <span class="k">as</span> <span class="n">app</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod4.png" src="_images/hymod4.png" />
+<p>Our findings indicate that in this instance, the streamflow estimate
+from HYMOD is highly sensitive to soil moisture parameters Huz and B and
+hardly affected by the routing parameters. Notably, there is very little
+interactions between parameters causing the total order indicies to be
+nearly identical to the first order indicies.</p>
+<p id="diffperformance">2.3 How do different performance metrics affect the results of our
+sensitivity analysis?</p>
+<p>Streamflow has many different properties. In this section, we discuss
+how the selection of metrics can lead to fundamentally different
+sensitivity analysis results. For example, one can only focus on
+aggregated streamflow metrics such as mean (what has been presented so
+far), or only on extreme events such as drought or floods.</p>
+<p>Here we compare three different metrics: 1- Mean error (ME) 2- Root Mean
+Square Error (RMSE) 3- Log-Root Mean Square Error (Log(RMSE))</p>
+<p>Each of these metrics focuses on a specific attribute of streamflow. For
+example, RMSE highlights the impacts of extreme flood events, while
+LOG(RMSE) focuses on model performance during low-flow events.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># calculate error metrics</span>
+<span class="n">mae</span> <span class="o">=</span> <span class="n">Q_df</span><span class="p">[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">leaf_data</span><span class="p">[</span><span class="s2">&quot;Strmflw&quot;</span><span class="p">]),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="n">mse</span> <span class="o">=</span> <span class="n">Q_df</span><span class="p">[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">metrics</span><span class="o">.</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s2">&quot;Strmflw&quot;</span><span class="p">]),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="n">rmse</span> <span class="o">=</span> <span class="n">mse</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="c1"># add error metrics to a dictionary</span>
+<span class="n">d_metrics</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;MAE&#39;</span><span class="p">:</span> <span class="n">mae</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">.</span><span class="n">values</span><span class="p">,</span>
+             <span class="s1">&#39;RMSE&#39;</span><span class="p">:</span> <span class="n">rmse</span><span class="o">.</span><span class="n">values</span><span class="p">,</span>
+             <span class="s1">&#39;LOG[RMSE]&#39;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">rmse</span><span class="o">.</span><span class="n">values</span><span class="p">)}</span>
+
+<span class="c1"># convert to a dataframe</span>
+<span class="n">df_metrics_SA</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d_metrics</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>We can use the following to calculate the SA indices for each metric and
+visualize it.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">df_metric_s1_result</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">df_metric_sT_result</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+
+<span class="c1"># conduct sensitivity analysis for each metric</span>
+<span class="k">for</span> <span class="n">index</span><span class="p">,</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">d_metrics</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+
+    <span class="c1"># get the data as a numpy array for the target metric</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">d_metrics</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+    <span class="c1"># use the metric to conduct SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add the sensitivity indices to the output data frame</span>
+    <span class="n">df_metric_s1_result</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">index</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">]</span>
+    <span class="n">df_metric_sT_result</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">index</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">]</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create seaborn heatmap with required labels</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+<span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="c1"># labels for y-axis</span>
+<span class="n">y_axis_labels</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Mean Absoulte Error&#39;</span><span class="p">,</span> <span class="s1">&#39;RSME&#39;</span><span class="p">,</span> <span class="s1">&#39;Log(RMSE)&#39;</span><span class="p">]</span>
+
+<span class="c1"># plot heatmap</span>
+<span class="n">ax1</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">df_metric_s1_result</span><span class="p">,</span> <span class="n">yticklabels</span><span class="o">=</span><span class="n">y_axis_labels</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>  <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;inferno_r&#39;</span><span class="p">,</span> <span class="n">cbar_kws</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;label&#39;</span><span class="p">:</span> <span class="s1">&#39;Si&#39;</span><span class="p">},</span> <span class="n">cbar</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">figure</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">label</span><span class="o">.</span><span class="n">set_size</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;First Order Sensitivity&#39;</span><span class="p">)</span>
+
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">df_metric_sT_result</span><span class="p">,</span> <span class="n">yticklabels</span><span class="o">=</span><span class="n">y_axis_labels</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>  <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;inferno_r&#39;</span><span class="p">,</span> <span class="n">cbar_kws</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;label&#39;</span><span class="p">:</span> <span class="s1">&#39;Si&#39;</span><span class="p">})</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">figure</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">label</span><span class="o">.</span><span class="n">set_size</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Total Order Sensitivity&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Text</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="s1">&#39;Total Order Sensitivity&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod5.png" src="_images/hymod5.png" />
+<p>The first order sensitivity indicies indicate that HYMOD’s sensitivity
+to its parameters is different depending on how its output is measured.
+Unsurprisingly, the mean absolute error is highly sensitive to the soil
+moisture accounting parameters Huz and B, just like the overall
+streamflow predictions above. However, when we examine RMSE and
+log(RMSE), the routing parameters Alp become sensitive, and the
+sensitivity to parameter B is reduced. As described above, RMSE and
+LOG(RMSE) respond to model performance in high-flow and low flow periods
+respectively. Our results indicate that for these flow regimes Alp, the
+parameter that governs the split between quick and slow flow is an
+important factor. While still the parameter with the highest most effect
+on all three measures, Huz is much less influential for RMSE and
+LOG(RMSE) than it is for MAE.</p>
+<p>The total order sensitivity indicies review a different, more complex
+story. While the MAE sensitivity is relatively governed by first order
+effects (like the streamflow predictions above), the RMSE and LOG(RMSE)
+error metrics show significant interactions. Alp has the highest total
+order sensitivity for RMSE and is eqal to Huz for Log(RMSE). Kq, which
+has a relatively low first order sensitivity index, shows strong
+contribution to variance when interactions are taken into account.</p>
+<p>Radial convergence plots are a helpful way to visualize the interactions
+between parameters. These plots array the model parameters in a circle
+and plot the first order, total order and second order Sobol sensitivity
+indices for each parameter. The first order sensitivity is shown as the
+size of a closed circle, the total order as the size of a larger open
+circle and the second order as the thickness of a line connecting two
+parameters. Below is an example of a radial convergence plot for the
+LOG(RMSE) measure. The plot indicates strong interactions between the
+Huz and Alp parameters, as well as Alp and Kq. There is also an
+interaction between Alp and Ks.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">itertools</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+<span class="kn">import</span> <span class="nn">math</span>
+<span class="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s1">&#39;whitegrid&#39;</span><span class="p">,</span> <span class="p">{</span><span class="s1">&#39;axes_linewidth&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="s1">&#39;axes.edgecolor&#39;</span><span class="p">:</span> <span class="s1">&#39;white&#39;</span><span class="p">})</span>
+
+<span class="k">def</span> <span class="nf">is_significant</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">confidence_interval</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="s2">&quot;conf&quot;</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">threshold</span> <span class="o">==</span> <span class="s2">&quot;conf&quot;</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">value</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">(</span><span class="n">confidence_interval</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">value</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">threshold</span><span class="p">))</span> <span class="o">&gt;</span> <span class="mi">0</span>
+
+<span class="k">def</span> <span class="nf">grouped_radial</span><span class="p">(</span><span class="n">SAresults</span><span class="p">,</span> <span class="n">parameters</span><span class="p">,</span> <span class="n">radSc</span><span class="o">=</span><span class="mf">2.0</span><span class="p">,</span> <span class="n">scaling</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">widthSc</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">STthick</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">varNameMult</span><span class="o">=</span><span class="mf">1.3</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">groups</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">gpNameMult</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="s2">&quot;conf&quot;</span><span class="p">):</span>
+    <span class="c1"># Derived from https://github.com/calvinwhealton/SensitivityAnalysisPlots</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+    <span class="n">color_map</span> <span class="o">=</span> <span class="p">{}</span>
+
+    <span class="c1"># initialize parameters and colors</span>
+    <span class="k">if</span> <span class="n">groups</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+
+        <span class="k">if</span> <span class="n">colors</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">colors</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;k&quot;</span><span class="p">]</span>
+
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">parameter</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+            <span class="n">color_map</span><span class="p">[</span><span class="n">parameter</span><span class="p">]</span> <span class="o">=</span> <span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)]</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">if</span> <span class="n">colors</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">colors</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">color_palette</span><span class="p">(</span><span class="s2">&quot;deep&quot;</span><span class="p">,</span> <span class="nb">max</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">groups</span><span class="p">)))</span>
+
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">groups</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+            <span class="c1">#parameters.extend(groups[key])</span>
+
+            <span class="k">for</span> <span class="n">parameter</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">[</span><span class="n">key</span><span class="p">]:</span>
+                <span class="n">color_map</span><span class="p">[</span><span class="n">parameter</span><span class="p">]</span> <span class="o">=</span> <span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)]</span>
+
+    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">parameters</span><span class="p">)</span>
+    <span class="n">angles</span> <span class="o">=</span> <span class="n">radSc</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">radSc</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angles</span><span class="p">)</span>
+    <span class="n">y</span> <span class="o">=</span> <span class="n">radSc</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angles</span><span class="p">)</span>
+
+    <span class="c1"># plot second-order indices</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">combinations</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="mi">2</span><span class="p">):</span>
+        <span class="c1">#key1 = parameters[i]</span>
+        <span class="c1">#key2 = parameters[j]</span>
+
+        <span class="k">if</span> <span class="n">is_significant</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S2&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S2_conf&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">threshold</span><span class="p">):</span>
+            <span class="n">angle</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">atan</span><span class="p">((</span><span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">-</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
+
+            <span class="k">if</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">-</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
+                <span class="n">angle</span> <span class="o">+=</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span>
+
+            <span class="n">line_hw</span> <span class="o">=</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S2&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">])</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
+
+            <span class="n">coords</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">((</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Polygon</span><span class="p">(</span><span class="n">coords</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;0.75&quot;</span><span class="p">))</span>
+
+    <span class="c1"># plot total order indices</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">is_significant</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST_conf&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">threshold</span><span class="p">):</span>
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Circle</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;w&#39;</span><span class="p">))</span>
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Circle</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="n">STthick</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;0.4&#39;</span><span class="p">,</span> <span class="n">fill</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
+
+    <span class="c1"># plot first-order indices</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">is_significant</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S1&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S1_conf&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">threshold</span><span class="p">):</span>
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Circle</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S1&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;0.4&#39;</span><span class="p">))</span>
+
+    <span class="c1"># add labels</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+        <span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">varNameMult</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">varNameMult</span><span class="o">*</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">key</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span>
+                <span class="n">rotation</span><span class="o">=</span><span class="n">angles</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="mi">360</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">-</span> <span class="mi">90</span><span class="p">,</span>
+                <span class="n">color</span><span class="o">=</span><span class="n">color_map</span><span class="p">[</span><span class="n">key</span><span class="p">])</span>
+
+    <span class="k">if</span> <span class="n">groups</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">group</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">groups</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+            <span class="nb">print</span><span class="p">(</span><span class="n">group</span><span class="p">)</span>
+            <span class="n">group_angle</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">angles</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">if</span> <span class="n">parameters</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">[</span><span class="n">group</span><span class="p">]])</span>
+
+            <span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">gpNameMult</span><span class="o">*</span><span class="n">radSc</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">group_angle</span><span class="p">),</span> <span class="n">gpNameMult</span><span class="o">*</span><span class="n">radSc</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">group_angle</span><span class="p">),</span> <span class="n">group</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span>
+                <span class="n">rotation</span><span class="o">=</span><span class="n">group_angle</span><span class="o">*</span><span class="mi">360</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">-</span> <span class="mi">90</span><span class="p">,</span>
+                <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)])</span>
+
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">([])</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;equal&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">])</span>
+    <span class="c1">#plt.show()</span>
+
+
+    <span class="k">return</span> <span class="n">fig</span>
+
+<span class="c1"># define groups for parameter uncertainties</span>
+<span class="n">groups</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;Soil Moisture&quot;</span> <span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Huz&quot;</span><span class="p">,</span> <span class="s2">&quot;B&quot;</span><span class="p">],</span>
+        <span class="s2">&quot;Routing&quot;</span> <span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Alp&quot;</span><span class="p">,</span> <span class="s2">&quot;Kq&quot;</span><span class="p">,</span> <span class="s2">&quot;Ks&quot;</span><span class="p">]}</span>
+
+
+<span class="n">fig</span> <span class="o">=</span> <span class="n">grouped_radial</span><span class="p">(</span><span class="n">Si</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">],</span> <span class="n">groups</span><span class="o">=</span><span class="n">groups</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="mf">0.025</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Soil</span> <span class="n">Moisture</span>
+<span class="n">Routing</span>
+</pre></div>
+</div>
+<img alt="_images/hymod6.png" src="_images/hymod6.png" />
+<p>2.4 Time-Varying Sensitivity Analysis</p>
+<p id="time">In section 2.5 we saw how performing sensitivity analysis on different
+measurements of model output can yeild in different results on the
+importance of each uncertain input. In this section we’ll examine how
+performing this analysis over time can yeild additional insight into the
+performance of HYMOD. We’ll first examine how model sensitivities vary
+by month, then examine how they change across each year of the
+simulation.</p>
+<p>For this demonstration, we’ll focus only on the monthly streamflow
+predictions generated by HYMOD.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># aggregate simulated streamflow data to monthly time series</span>
+<span class="n">df_sim_by_mth_mean</span> <span class="o">=</span> <span class="n">Q_df</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s1">&#39;month&#39;</span><span class="p">)[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+
+<span class="c1"># aggregate observed streamflow data to monthly time series</span>
+<span class="n">df_obs_by_mth_mean</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s1">&#39;month&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+<p>We can use the following to calculate the SA indices for each month and
+visualize it. Results are pre-loaded for efficiency.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># set up dataframes to store outputs</span>
+<span class="n">df_mth_s1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">12</span><span class="p">,</span><span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">df_mth_delta</span> <span class="o">=</span> <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="c1"># iterate through each month</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">12</span><span class="p">):</span>
+
+    <span class="c1"># generate the simulation data</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">df_sim_by_mth_mean</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+    <span class="c1"># run SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">delta</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">param_values_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add to output dataframes</span>
+    <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="n">df_mth_delta</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;delta&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+
+<span class="c1"># convert to arrays</span>
+<span class="n">arr_mth_s1</span> <span class="o">=</span> <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">values</span>
+<span class="n">arr_mth_delta</span> <span class="o">=</span> <span class="n">df_mth_delta</span><span class="o">.</span><span class="n">values</span>
+</pre></div>
+</div>
+<p>The following can be used to visualize the time-varying first-order
+indices. The first order represents the direct impacts of a specific
+parameter on model outputs.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-monthly-heatmap">plot_monthly_heatmap</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously ran data</span>
+<span class="n">arr_mth_delta</span><span class="p">,</span> <span class="n">arr_mth_s1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_monthly_simulations</span><span class="p">()</span>
+
+<span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_monthly_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_mth_s1</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                       <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_mth_mean</span><span class="p">,</span>
+                                       <span class="n">title</span><span class="o">=</span><span class="s1">&#39;First Order - Mean Monthly SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod7.png" src="_images/hymod7.png" />
+<p>This figure demonstrates the first order sensitivity indices when the
+streamflow data are aggregated by month. The purple line represents the
+observed monthly discharge. The figure indicates that the first order
+indices are highest for B and Huz across all months and lowest for Alp,
+Ks, and Kq. Note that in the months with the highest flow, Ks becomes an
+influential parameter.</p>
+<p>We can also focus on the total order sensitivity index that includes
+first-order SA indices and interactions between parameters</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_monthly_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_mth_delta</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                       <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_mth_mean</span><span class="p">,</span>
+                                       <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Total Order - Mean monthly SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod8.png" src="_images/hymod8.png" />
+<p>Notably, the total order sensitivity results are different than the
+first order sensitivity results, which indicates that interactions
+between the parameters (particularly in regards to routing parameters
+<span class="math notranslate nohighlight">\(Kq\)</span>, <span class="math notranslate nohighlight">\(Ks\)</span>, and <span class="math notranslate nohighlight">\(Alp\)</span>) contribute to changes in HYMOD
+output.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># group by year and get mean</span>
+<span class="n">df_sim_by_yr_mean</span> <span class="o">=</span> <span class="n">Q_df</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">])[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+
+<span class="c1"># group input data and get mean</span>
+<span class="n">df_obs_by_yr_mean</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">])</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+<p>We can also calculate the sensitivity analysis indices for each
+individual year. This will allow us to understand if model control
+changes during different years. The following code first aggregates the
+outputs to annual time steps, and then calculates the SA indices.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># set up dataframes to store outputs</span>
+<span class="n">df_yr_s1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">df_yr_delta</span> <span class="o">=</span> <span class="n">df_yr_s1</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="c1"># iterate through each year</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">):</span>
+
+    <span class="c1"># generate the simulation data</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">df_sim_by_yr_mean</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+    <span class="c1"># run SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">delta</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">param_values_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add to output dataframes</span>
+    <span class="n">df_yr_s1</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="n">df_yr_delta</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;delta&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+
+<span class="c1"># convert to arrays</span>
+<span class="n">arr_yr_s1</span> <span class="o">=</span> <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">values</span>
+<span class="n">arr_yr_delta</span> <span class="o">=</span> <span class="n">df_mth_delta</span><span class="o">.</span><span class="n">values</span>
+</pre></div>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-annual-heatmap">plot_annual_heatmap</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously ran data</span>
+<span class="n">arr_yr_delta</span><span class="p">,</span> <span class="n">arr_yr_s1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_annual_simulations</span><span class="p">()</span>
+
+<span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_annual_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_yr_s1</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_yr_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;First Order - Mean Annual SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod9.png" src="_images/hymod9.png" />
+<p>The first order sensitivities at the annual scale are not unlike the
+first order monthly sensitivities. Once again, sensitivities vary across
+year and Huz and B are the most consequential parameters.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_annual_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_yr_delta</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_yr_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Total Order - Mean Annual SA and Observed flow&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod10.png" src="_images/hymod10.png" />
+<p>Our results indicate that sensitivity analysis indices vary in different
+years and now that interactions are included, the Kq, Ks, and Alp
+variables impact the sensitivity of the streamflow output.</p>
+<p>Although time-varying sensitivity analysis (TVSA) at average monthly and
+average annual temporal resolutions is informative, TVSA is susceptible
+to the aggregation issue that we discussed earlier in section 3-2. To
+avoid that we can further discretize our time domain to zoom into
+individual months. This will provide us with even more information about
+model behavior and the sensitivity of different parameters in different
+states of the system. The block of code demonstrates how to implement
+the monthly TVSA.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># set up dataframes to store outputs</span>
+<span class="n">df_vary_s1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">df_obs_mth_mean</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">5</span><span class="p">)),</span>
+                          <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+
+<span class="n">df_vary_delta</span> <span class="o">=</span> <span class="n">df_vary_s1</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="c1"># iterate through each month</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">df_obs_mth_mean</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+
+    <span class="c1"># generate the simulation data</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">df_sim_mth_mean</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+    <span class="c1"># run SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">delta</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">param_values_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add to output dataframes</span>
+    <span class="n">df_vary_s1</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="n">df_vary_delta</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;delta&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+
+<span class="c1"># convert to arrays</span>
+<span class="n">arr_vary_s1</span> <span class="o">=</span> <span class="n">df_vary_s1</span><span class="o">.</span><span class="n">values</span>
+<span class="n">arr_vary_delta</span> <span class="o">=</span> <span class="n">df_vary_delta</span><span class="o">.</span><span class="n">values</span>
+</pre></div>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-varying-heatmap">plot_varying_heatmap</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load in previously ran data</span>
+<span class="n">arr_vary_delta</span><span class="p">,</span> <span class="n">arr_vary_s1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_varying_simulations</span><span class="p">()</span>
+
+<span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_varying_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_vary_s1</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_mth_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;First Order - Time-Varying SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod11.png" src="_images/hymod11.png" />
+<p>Compared to the TVSA when streamflow was aggregated, this figure
+suggests that Kq is indeed a relevant parameter for influencing
+streamflow output when individual months are considered.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_varying_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_vary_delta</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_mth_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Total Order - Time-Varying SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod12.png" src="_images/hymod12.png" />
+<p>As above, the total order sensitivities further indicate the importance
+of Kq that is not apparent if aggregation is utilized.</p>
+</section>
+<section id="tips-to-apply-this-methodology-to-your-own-problem">
+<h3><span class="section-number">B.4.4. </span>Tips to Apply this methodology to your own problem<a class="headerlink" href="#tips-to-apply-this-methodology-to-your-own-problem" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we demonstrated how to use global sensitivtiy analysis
+to explore a complex, non-linear model. We showed how measuring
+sensitivity across multiple measures of model performance and temporal
+aggregations yeilding differing results about model
+sensitivity/behavoir. While these results may seem contraditory, they
+provide useful insight into the behavoir of HYMOD. Would we expect the
+same parameters to control high flow and low flow regimes within the
+model? Maybe, depending on the system, but also, maybe not. This
+analysis can provide insight into how the model responds to its input
+parameters, allowing us to compare the results to our expectaions. This
+may allow us to find problems with our intial assumptions, or call
+attention to model features that can be improved or expanded upon.
+Depending on the model and context, it may also yield insight into the
+workings of the underlying system.</p>
+<p>To run this tutorial on your own model you will need to:</p>
+<ol class="arabic simple">
+<li><p>Design your experiment by choosing sampling bounds for your
+parameters and setting up the problem dictionary as in step 2-2</p></li>
+<li><p>Choose the parameters of interest</p></li>
+<li><p>Generate samples using the saltelli.sample function. This step is
+problem-dependent and note that the Sobol method can be
+computationally intensive depending on the model being analyzed. More
+complex models will be slower to run and will also require more
+samples to calculate accurate estimates of Sobol indices. Once you
+complete this process, pay attention to the confidence bounds on your
+sensitivity indices to see whether you need to run more samples.</p></li>
+<li><p>Run the parameter sets through your model and record each of the
+desired model outputs.</p></li>
+<li><p>Calculate the Sobol indices for each performance criteria. Now, the Y
+will be a numpy array with your external model output and you will
+need to include the parameter samples as an additional argument.</p></li>
+<li><p>Follow the procedure in step 2.6 to disaggregate performance across
+time</p></li>
+</ol>
+</section>
+</section>
+<section id="time-evolving-scenario-discovery-for-infrastructure-pathways">
+<h2><span class="section-number">B.5. </span>Time-evolving scenario discovery for infrastructure pathways<a class="headerlink" href="#time-evolving-scenario-discovery-for-infrastructure-pathways" title="Link to this heading">#</a></h2>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/Bedford_Greene_SD_tutorial.ipynb">Scenario Discovery Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="id24">
+<h3><span class="section-number">B.5.1. </span>Time-evolving scenario discovery for infrastructure pathways<a class="headerlink" href="#id24" title="Link to this heading">#</a></h3>
+<p>The purpose of this tutorial is to explore time-evolving vulnerability for systems that dynamically adapt to changing conditions. Using an example from water supply planning, we’ll first examine how performance of a dynamic infrastructure pathway policy changes over time, then use factor mapping (main text Chapter 4.3) to understand which combinations of uncertaities generate vulnerability for two water utilities. Next, we’ll perform factor prioritization (main text Chapter 4.3) to determine which uncertainties have the most influence on water supply performance. Finally, we’ll provide an open platform to explore vulnerability across multiple measures of performance and different combinations of uncertainties.</p>
+<section id="id25">
+<h4><span class="section-number">B.5.1.1. </span>Background<a class="headerlink" href="#id25" title="Link to this heading">#</a></h4>
+<p>The Bedford-Greene metropolitan area (Figure 1) is a stylized water resources test case where two urban water utilities seek to develop an infrastructure and investment and management strategy to confront growing demands and changing climate. The utilities have agreed to jointly construct a new water treatment plant on Lake Classon, a large regional resource. Both utilities have also identified a set of individual infrastructure options to construct if necessary.</p>
+<figure class="align-default" id="id32">
+<img alt="Figure 1" src="_images/Map_small.png" />
+<figcaption>
+<p><span class="caption-number">Fig. B.1 </span><span class="caption-text">Figure 1</span><a class="headerlink" href="#id32" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The utilities are formulating a cooperative and adaptive regional management strategy that uses a risk-of-failure (ROF) metric to trigger both short term drought mitigation actions (water use restrictions and treated transfers between utilities) and long-term infrastructure investment decisions (shown in Figure 2a). Both utilities have specified a set of risk triggers and developed a construction order for available infrastructure options.</p>
+<p>The utilities have run a Monte Carlo simulation to evaluate how these policies respond to a wide array of future States Of the World (SOWs). Each SOW represents a different combinations of thirteen uncertain system inputs including demand growth rates, changes to streamflows and financial variables. In this context, a fully specified SOW is composed of one sample of uncertain model inputs (e.g. one projection of demand growth rate coupled with one future streamflow scenario and one projection of future financial conditions). The water utilities used Latin Hypercube sampling (Chapter 3.3 of the main text) to develop an ensemble of 1,000 plausible future SOWs. The Monte Carlo simulation evaluates each candidate water supply infrastructure investment and management policy across all 1,000 SOWs, as shown in Figure 2b. For more details on the Monte Carlo sampling for this type  of analysis, see Trindade et al., (2019).</p>
+<p>The ROF-based policies respond to each SOW by generating a unique infrastructure pathway - a sequence of infrastructure investment decisions over time. Infrastructure pathways over a set of 1,000 future SOWs are shown in Figure 2c. Infrastructure options are arrayed along the vertical axis and the sequence of infrastructure investments triggered by the policy is plotted as pathways over time.  Since the adaptive rule system generates a unique infrastructure sequence for each scenario, Figure 2c summarizes the ensemble of pathways by clustering SOWs according to infrastructure intensity. Dark green lines represent SOWs where the utilities heavily invest in new infrastructure, light green lines represent SOWs with low infrastructure investment and medium shaded lines represent moderate investment. The shading behind each pathway represents the frequency that each infrastructure option was triggered over time across sampled scenarios</p>
+<figure class="align-default" id="id33">
+<img alt="Figure 2" src="_images/Policy_MonteCarlo_Pathways_small.png" />
+<figcaption>
+<p><span class="caption-number">Fig. B.2 </span><span class="caption-text">Figure 2</span><a class="headerlink" href="#id33" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="evaluating-robustness-over-time">
+<h4><span class="section-number">B.5.1.2. </span>Evaluating Robustness over time<a class="headerlink" href="#evaluating-robustness-over-time" title="Link to this heading">#</a></h4>
+<p>The two water utilities are interested in mainting both supply
+reliability and financial stability across the broadest set of plausible
+future SOWs. To measure the performance of the infrastructure pathway
+policy, they’ve defined five critical performance criteria:</p>
+<ul class="simple">
+<li><p>Reliability &gt; 99%</p></li>
+<li><p>Restriction Frequency &lt; 20%</p></li>
+<li><p>Peak Financial Cost &lt; 80% of annual revenue (a measure of debt
+service spending)</p></li>
+<li><p>Worst-case drought management cost &lt; 10% of annual revenue (a measure
+of unexpected drought costs)</p></li>
+<li><p>Unit Cost of Expansion &lt; 5 dollars/kgal</p></li>
+</ul>
+<p>To assess the robustness of the infrastructure pathway policy, the two
+utilities apply a satisficing metric, which measures the percentage of
+sampled SOWs where the pathway policy meets the peformance criteria:</p>
+<p><span class="math notranslate nohighlight">\(R =\frac{1}{N}\sum_{j=1}^{N}\Lambda_{\theta,j}\)</span></p>
+<p>Where, <span class="math notranslate nohighlight">\(\Lambda\_{\theta,j}=\)</span></p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases}
+1, \quad \textrm{if}\ F(\theta)_{j}\leq \Phi_j \\
+0, \quad \textrm{otherwise}
+\end{cases}\end{split}\]</div>
+<p>And <span class="math notranslate nohighlight">\(\Phi\)</span> is a vector of performance criteria for utility
+<span class="math notranslate nohighlight">\(j\)</span>, <span class="math notranslate nohighlight">\(\theta\)</span> is the portfolio and <span class="math notranslate nohighlight">\(N\)</span> is the total
+number of sampled SOWs.</p>
+<p>Below, we’ll visualize how robustness for the two utilities evolves over the 45-year planning horizon. We’ll assess robustness across three time periods, near-term (first 10 years), mid-term (22 years) and long term (45 years).</p>
+<p>We start by loading robustness values for the both utilities. These values are calculated by applying the robustness metric above across 2,000 simulated SOWs. To make this exercise computationally tractable, we’ve precomputed these values, which can be found in the files “short_term_robustness.csv”, “mid_term_robustness.csv” and “long_term_robustness.csv”. These values are calculated using the function “check_rdm_meet_criteria” within the helper functions.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
+<span class="kn">from</span> <span class="nn">functions.eBook_SD_helpers</span> <span class="kn">import</span> <span class="n">check_rdm_meet_criteria</span><span class="p">,</span> <span class="n">create_sd_input</span><span class="p">,</span> <span class="n">plot_selected_tree_maps</span><span class="p">,</span> <span class="n">get_factor_importances</span><span class="p">,</span> <span class="n">open_exploration</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+
+<span class="c1"># load Deeply uncertain factors</span>
+<span class="n">rdm_factors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/DU_Factors.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+
+<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
+<span class="n">short_term_robustness</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/short_term_robustness.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">mid_term_robustness</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/mid_term_robustness.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">long_term_robustness</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/long_term_robustness.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+
+<span class="c1"># plot robustness over time</span>
+<span class="n">fig</span> <span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span><span class="mi">22</span><span class="p">,</span><span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span><span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span> <span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span><span class="mi">22</span><span class="p">,</span><span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span><span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;Time Horizon (yrs)&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Robustness (% SOWs)&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">&#39;Bedford&#39;</span><span class="p">,</span> <span class="s1">&#39;Greene&#39;</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Robustness Over Time&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">107</span><span class="p">])</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">107.0</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_4_1.png" src="_images/discovery_4_1.png" />
+</section>
+<section id="exploring-performance-evolution">
+<h4><span class="section-number">B.5.1.3. </span>Exploring performance evolution<a class="headerlink" href="#exploring-performance-evolution" title="Link to this heading">#</a></h4>
+<p>The figure above reveals that the robustness of both water utilities degrades over time, with Bedford’s robustness declining further than Greene. This suggests that the proposed pathway policy is likely insufficient to meet the long-term needs of the two utilities. But how is the current policy insufficient? To answer that question we examine the performance measures that fail to meet performance criteria for each utility across the three planning horizons.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot the type of vulnerability over time</span>
+
+<span class="c1">### Bedford ###</span>
+<span class="n">plot_robustness_1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
+<span class="c1"># Determine the percentage of failure SOWs that violate each criterion (note some SOWS fail multiple criteria, so this may some to &gt;1)</span>
+<span class="n">criteria</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Reliability&#39;</span><span class="p">,</span> <span class="s1">&#39;Restriction Frequency&#39;</span><span class="p">,</span> <span class="s1">&#39;Peak Financial Cost&#39;</span><span class="p">,</span> <span class="s1">&#39;Worst-case drought</span><span class="se">\n</span><span class="s1">Management Cost&#39;</span><span class="p">,</span> <span class="s1">&#39;Stranded Assets&#39;</span><span class="p">]</span>
+<span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span>
+<span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">1</span><span class="p">,:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span>
+<span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">2</span><span class="p">,:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span>
+
+<span class="c1"># Plot over time</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">0</span><span class="p">,:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Fraction of failure SOWs&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">1</span><span class="p">,:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">2</span><span class="p">,:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Bedford&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="c1">### Greene ###</span>
+<span class="c1"># Determine the percentage of failure SOWs that violate each criterion (note some SOWS fail multiple criteria, so this may some to &gt;1)</span>
+<span class="n">plot_robustness_2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
+<span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">11</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">])</span>
+<span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">11</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">])</span>
+<span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">11</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">])</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Fraction of failure SOWs&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="p">:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Greene&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_6_0.png" src="_images/discovery_6_0.png" />
+<img alt="_images/discovery_6_1.png" src="_images/discovery_6_1.png" />
+<p>In the figures above, we observe that the vulnerability of both utilities changes in different ways. Early in the simulation period, Bedford is vulnerable to failures in reliability (though the robustness figure created in step B5.2 reveals that these failures are very rare). As the simulation period progresses, Bedford’s vulnerability expands to include failures in restriction frequency and worst-case cost. These failures indicate that the utility has an overall inability to manage drought conditions and future conditions progress.</p>
+<p>Greene shows a very different evolution in vulnerability. Early in the simulation period, failures manifest in the restriction frequency objective, suggesting that the utility must rely on water use restrictions to maintain supply reliability. As the simulation progresses however, the vulnerability evolves. When evaluated across the 45-year planning horizon, a new failure modes emerges - financial failure manifesting in peak financial cost and stranded assets. This suggests that the proposed pathway policy may be over-investing in new infrastructure, straining the utility’s budget with large debt payments that are unnecessary to maintain supply reliability.</p>
+</section>
+<section id="how-do-deep-uncertainties-generate-vulnerability">
+<h4><span class="section-number">B.5.1.4. </span>How do deep uncertainties generate vulnerability<a class="headerlink" href="#how-do-deep-uncertainties-generate-vulnerability" title="Link to this heading">#</a></h4>
+<p>While the evolution of robustness provides insight into how the system
+evolves over time, it does not reveal <em>why</em> each utility is vulnerable.
+To examine how deep uncertainties generate vulnerability over time for
+the two utilities, we perform scenario discovery (factor mapping,
+Chapter 4.3). Here we’ll utilize gradient boosted trees to identify
+regions of the uncertainty space that cause the utilities to fail to
+meet performance criteria.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># import the performance data across 2000 SOWs for three time periods</span>
+<span class="n">short_term_performance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/short_term_performance.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">mid_term_performance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/mid_term_performance.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">long_term_performance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/long_term_performance.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+
+<span class="n">satisficing_criteria</span> <span class="o">=</span> <span class="p">[</span><span class="mf">.98</span><span class="p">,</span> <span class="mf">.2</span><span class="p">,</span> <span class="mf">.8</span><span class="p">,</span> <span class="mf">.1</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>
+
+<span class="c1"># transform into scenario discovery input</span>
+<span class="n">short_term_SD_input</span> <span class="o">=</span> <span class="n">create_sd_input</span><span class="p">(</span><span class="n">short_term_performance</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">)</span>
+<span class="n">mid_term_SD_input</span> <span class="o">=</span> <span class="n">create_sd_input</span><span class="p">(</span><span class="n">mid_term_performance</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">)</span>
+<span class="n">long_term_SD_input</span> <span class="o">=</span> <span class="n">create_sd_input</span><span class="p">(</span><span class="n">long_term_performance</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">)</span>
+
+<span class="c1"># factor mapping Bedford</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;short_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;mid_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;long_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Bedford Factor Maps&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="c1"># factor mapping Greene</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="s1">&#39;short_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="s1">&#39;mid_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="s1">&#39;long_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Greene Factor Maps&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Greene</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Greene</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Greene</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_9_1.png" src="_images/discovery_9_1.png" />
+<img alt="_images/discovery_9_2.png" src="_images/discovery_9_2.png" />
+<p>In the figures above, we learn more about how the vulnerability of the two utilities evolves over time. Bedford begins with very few possible failures but appears vulnerable to high demand growth scenarios under future scenarios with high demands. When evaluated across a 22-year planning horizon, Bedford is vulnerable when the near-term demand growth is high and water use restrictions are less effective than predicted. Under the full 45-year planning horizon, Bedford is vulnerable to sustained high levels of demand growth, failing if either near-term or mid-term demand growth exceeds expected levels.</p>
+<p>Greene’s vulnerability evolves differently. It begins with vulnerability to high demand growth, but as the simulation progresses (and infrastructure is constructed), the utility becomes vulnerable to low-demand growth futures which cause the failures in financial criteria shown in section B.5.3. This indicates that the pathway policy over-builds in many SOWs, and becomes financially unstable if demand does not grow sufficiently to provide revenue to cover debt service payments.</p>
+</section>
+<section id="which-uncertainties-have-the-most-influence-on-time-evolving-performance">
+<h4><span class="section-number">B.5.1.5. </span>Which uncertainties have the most influence on time-evolving performance?<a class="headerlink" href="#which-uncertainties-have-the-most-influence-on-time-evolving-performance" title="Link to this heading">#</a></h4>
+<p>The factor maps generated in B.5.4 present the vulnerability generated by the two most important deep uncertainties as determined by Gradient Boosted Trees. Yet the factor prioritization shows that more than two uncertainties are influential to regional performance. Further, we can observe that individual uncertainties have different impacts on each performance obejctive, and these impacts may change over time. In the cells below, explore the impact of deep uncertainty by generating factor maps for different combinations of deep uncertain factors, objectives and time horizons.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+<span class="n">uncertainties</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;D1&#39;</span><span class="p">,</span> <span class="s1">&#39;D2&#39;</span><span class="p">,</span> <span class="s1">&#39;D3&#39;</span><span class="p">,</span> <span class="s1">&#39;BT&#39;</span><span class="p">,</span> <span class="s1">&#39;BM&#39;</span><span class="p">,</span> <span class="s1">&#39;DR&#39;</span><span class="p">,</span> <span class="s1">&#39;RE&#39;</span><span class="p">,</span> <span class="s1">&#39;EV&#39;</span><span class="p">,</span> <span class="s1">&#39;PM&#39;</span><span class="p">,</span> <span class="s1">&#39;CT&#39;</span><span class="p">,</span> <span class="s1">&#39;IA&#39;</span><span class="p">,</span> <span class="s1">&#39;IF&#39;</span><span class="p">,</span> <span class="s1">&#39;IP&#39;</span><span class="p">]</span>
+<span class="n">uncertainties</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Near-term demand&#39;</span><span class="p">,</span> <span class="s1">&#39;Mid-term demand&#39;</span><span class="p">,</span> <span class="s1">&#39;Long-term demand&#39;</span><span class="p">,</span> <span class="s1">&#39;Bond Term&#39;</span><span class="p">,</span> <span class="s1">&#39;Bond Rate&#39;</span><span class="p">,</span> <span class="s1">&#39;Discount Rate&#39;</span><span class="p">,</span> <span class="s1">&#39;Restriction Effectiveness&#39;</span><span class="p">,</span> <span class="s1">&#39;Evaporation Rate&#39;</span><span class="p">,</span> <span class="s1">&#39;Permitting time&#39;</span><span class="p">,</span> <span class="s1">&#39;Construction time&#39;</span><span class="p">,</span> <span class="s1">&#39;Inflow Amplitude&#39;</span><span class="p">,</span> <span class="s1">&#39;Inflow Frequency&#39;</span><span class="p">,</span> <span class="s1">&#39;Inflow Period&#39;</span><span class="p">]</span>
+
+<span class="n">u1_st_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">short_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="n">u1_mt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">mid_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="n">u1_lt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">long_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+
+<span class="n">u1_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">u1_st_FI</span><span class="p">,</span><span class="n">u1_mt_FI</span><span class="p">,</span> <span class="n">u1_lt_FI</span><span class="p">])</span>
+<span class="n">u1_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">u1_all</span><span class="p">)</span>
+
+<span class="c1"># factor ranking -- utility 2</span>
+<span class="n">u2_st_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">short_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
+<span class="n">u2_mt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">mid_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
+<span class="n">u2_lt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">long_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
+<span class="n">u2_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">u2_st_FI</span><span class="p">,</span><span class="n">u2_mt_FI</span><span class="p">,</span> <span class="n">u2_lt_FI</span><span class="p">])</span>
+<span class="n">u2_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">u2_all</span><span class="p">)</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span><span class="p">,</span> <span class="n">cax</span><span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span>
+                  <span class="n">gridspec_kw</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;width_ratios&quot;</span><span class="p">:[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">]})</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">wspace</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">im</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">u1_all</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;Reds&#39;</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mf">.3</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">13</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">(</span><span class="n">uncertainties</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Time Horizon&#39;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Bedford&#39;</span><span class="p">)</span>
+
+<span class="n">im1</span> <span class="o">=</span> <span class="n">ax2</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">u2_all</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;Reds&#39;</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mf">.3</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">13</span><span class="p">))</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">([])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Time Horizon&#39;</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Greene&#39;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="n">cax</span><span class="o">=</span><span class="n">cax</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Factor Importance&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_12_0.png" src="_images/discovery_12_0.png" />
+<p>The Figure above shows the factor importance as determined by gradient boosted trees for both utilities across the three planning horizons. While near-term demand growth is important for both utilities under all three planning horizons, the importance of other factors evolves over time. For example, restriction effectiveness plays an important role for Greene under the 22-year planning horizon but disappears under the 45-year planning horizon. In contrast, the bond interest rate is important for predicting success over the 45-year planning horizon, but does not appear important over the 10- or 22-year planning horizons. These findings highlight how assumptions about the planning period can have a large impact on modeling outcomes.</p>
+</section>
+<section id="open-exploration">
+<h4><span class="section-number">B.5.1.6. </span>Open exploration<a class="headerlink" href="#open-exploration" title="Link to this heading">#</a></h4>
+<p>In the cell below, use the function to explore how factor maps change
+for the two utilities based upon the uncertainties plotted, the
+objectives of interest and the time horizon.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># specify the utility (&quot;Bedford&quot; or &quot;Greene&quot;)</span>
+<span class="n">utility</span> <span class="o">=</span> <span class="s2">&quot;Bedford&quot;</span>
+
+<span class="c1"># specify which performance objectives to investigate (note that not all performance objectives have failures, which may result in a blank factor map)</span>
+<span class="c1"># set this to one of the following: &quot;Reliability&quot;, &quot;Restriction Frequency&quot;, &quot;Peak Financial Cost&quot;, &quot;Worst Case Cost&quot; or &quot;Unit Cost&quot;</span>
+<span class="n">objective</span> <span class="o">=</span> <span class="s2">&quot;Reliability&quot;</span>
+
+<span class="c1"># select uncertainties from the following list: &#39;D1&#39;, &#39;D2&#39;, &#39;D3&#39;, &#39;BT&#39;, &#39;BM&#39;, &#39;DR&#39;, &#39;RE&#39;, &#39;EV&#39;, &#39;PM&#39;, &#39;CT&#39;, &#39;IA&#39;, &#39;IF&#39;, &#39;IP&#39;</span>
+<span class="n">uncertainty_1</span> <span class="o">=</span> <span class="s1">&#39;D1&#39;</span>
+<span class="n">uncertainty_2</span> <span class="o">=</span> <span class="s1">&#39;D2&#39;</span>
+
+<span class="c1"># The code below will plot factor maps over the three planning horizons for the information above</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">open_exploration</span><span class="p">(</span><span class="n">utility</span><span class="p">,</span> <span class="n">objective</span><span class="p">,</span> <span class="s1">&#39;short_term&#39;</span><span class="p">,</span> <span class="n">uncertainty_1</span><span class="p">,</span> <span class="n">uncertainty_2</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">open_exploration</span><span class="p">(</span><span class="n">utility</span><span class="p">,</span> <span class="n">objective</span><span class="p">,</span> <span class="s1">&#39;mid_term&#39;</span><span class="p">,</span> <span class="n">uncertainty_1</span><span class="p">,</span> <span class="n">uncertainty_2</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">open_exploration</span><span class="p">(</span><span class="n">utility</span><span class="p">,</span> <span class="n">objective</span><span class="p">,</span> <span class="s1">&#39;long_term&#39;</span><span class="p">,</span> <span class="n">uncertainty_1</span><span class="p">,</span> <span class="n">uncertainty_2</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span><span class="p">,</span> <span class="n">reliability</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span><span class="p">,</span> <span class="n">reliability</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span><span class="p">,</span> <span class="n">reliability</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_16_1.png" src="_images/discovery_16_1.png" />
+</section>
+<section id="id26">
+<h4><span class="section-number">B.5.1.7. </span>Tips to apply this methodology to your own problem<a class="headerlink" href="#id26" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we demonstrated time-evolving scenario discovery for a
+cooperative water supply system. To apply this workflow to your own
+problem:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters of interest, which will
+represent uncertainties that characterize your system.</p></li>
+<li><p>Generate samples for these parameters (this can be done using the
+saltelli.sample function as in B.2 or done with another package).</p></li>
+<li><p>Define performance criteria for your problem</p></li>
+<li><p>Evaluate parameter sets through your model, and save performance
+measures across multiple time horizons</p></li>
+<li><p>Draw from the supporting code for this tutorial to perform scneario
+discovery and visualize results</p></li>
+</ol>
+</section>
+<section id="references">
+<h4><span class="section-number">B.5.1.8. </span>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h4>
+<p>Trindade, B. C., Reed, P. M., &amp; Characklis, G. W. (2019). Deeply uncertain pathways: Integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134, 103442.</p>
+</section>
+</section>
+</section>
+<section id="a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios-tutorial">
+<h2><span class="section-number">B.6. </span>A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios Tutorial<a class="headerlink" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios-tutorial" title="Link to this heading">#</a></h2>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/Hidden-Markov_Modeling_Approaches_to_Creating_Synthetic_Streamflow_Scenarios.ipynb">HMM Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios">
+<h3><span class="section-number">B.6.1. </span>A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios<a class="headerlink" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios" title="Link to this heading">#</a></h3>
+<p>In this notebook, we will be covering the basics of fitting a Hidden
+Markov Model-based synthetic streamflow generator for a single site in
+the Upper Colorado River Basin. First, we will characterize the observed
+historical flow in the basin from 1909-2013. Then, we will fit a
+synthetic streamflow generator to the observed flows in the basin in
+order to create stationary synthetic flows. Finally, we will create a
+non-stationary version of the generator to create flows that could be
+representative of plausible future climate in the region. We ultimately
+show how to place the synthetically generated flows in the context of
+physically-informed CMIP5 projections to compare the two methods.</p>
+<section id="id29">
+<h4><span class="section-number">B.6.1.1. </span>Background<a class="headerlink" href="#id29" title="Link to this heading">#</a></h4>
+<p>In the Western United States (US), and particularly the Colorado River
+Basin, a recent study used tree-ring reconstructions to suggest that the
+megadrought that has been occurring in the Southwest over the past 22
+years is the region’s worst drought since about 800 AD (Williams et al.,
+2022). The study’s lead author, UCLA climatologist Park Williams,
+suggested that had the sequence of wet-dry years occurred without
+anthropogenic forcing, the 2000s would have likely still been dry, but
+not on the same level as the worst of the last millennium’s
+megadroughts.</p>
+<p>The recent trend of warming and reduced soil moisture in the Southwest
+US is highly challenging from a water systems planning and management
+perspective for the Colorado River Basin. Given the wide recognition
+that the river is over-allocated, the most recent drought highlights the
+difficulty of sustaining the flow requirements as dictated by the
+Colorado Compact. Thus, there has been an increasing focus in
+exploratory modeling efforts to clarify how vulnerable water systems in
+this region are to plausible drought streamflow scenarios for the
+future. In this tutorial, we’ll discuss how to create these scenarios
+using a Hidden Markov Model (HMM)- based streamflow synthetic generator.
+As discussed in <a class="reference external" href="https://uc-ebook.org/docs/html/2_diagnostic_modeling_overview_and_perspectives.html#overview-of-model-diagnostics">Section
+2.1</a>
+and
+<a class="reference external" href="https://uc-ebook.org/docs/html/4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-dynamics-what-is-controlling-model-behaviors-of-interest">4.2</a>
+of the eBook, future climate conditions in the basin represent a deep
+uncertainty that can lead to highly consequential water scarcity
+outcomes. It is advantageous to create a model such as the HMM-based
+generator in order to facilitate the creation of many ensembles of
+streamflow that can ultimately be used to force regional water systems
+models to understand how variability and drought extremes affect
+regional water shortages, operations, and policies.</p>
+<figure>
+<center><img src="_static/1-lede_pix_-_usbr-powell-8517152024_aa66437c2e_o-cropped.jpg" style="width:50%"></center>
+<figcaption style="text-align:center"> Lake Powell shows persistent effects from drought (Source: U.S. Bureau of Reclamation) </figcaption>
+</figure></section>
+</section>
+<section id="id30">
+<h3><span class="section-number">B.6.2. </span>Let’s Get Started!<a class="headerlink" href="#id30" title="Link to this heading">#</a></h3>
+<section id="observed-record">
+<h4><span class="section-number">B.6.2.1. </span>Observed Record<a class="headerlink" href="#observed-record" title="Link to this heading">#</a></h4>
+<p>First, let’s take a look at the observed data from 1909-2013 for a
+specific site. In this example, we use the outlet gauge of the Upper
+Colorado River (USGS Gauge 09163500 at the Colorado-Utah state line).
+Below, we create a plot of the annual streamflow.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Import libraries</span>
+<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="nn">mpl</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">matplotlib.patches</span> <span class="k">as</span> <span class="nn">patches</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span>
+<span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">latin</span>
+<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">stats</span> <span class="k">as</span> <span class="n">ss</span>
+<span class="kn">import</span> <span class="nn">statistics</span>
+<span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>
+
+
+<span class="c1"># Import helper functions from local package</span>
+<span class="kn">from</span> <span class="nn">functions</span> <span class="kn">import</span> <span class="n">fitmodel</span>
+<span class="kn">from</span> <span class="nn">functions</span> <span class="kn">import</span> <span class="n">plotstates</span>
+<span class="kn">from</span> <span class="nn">functions</span> <span class="kn">import</span> <span class="n">plotdist</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Read in annual historical data</span>
+<span class="n">AnnualQ</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;data/uc_historical.csv&#39;</span><span class="p">)</span>
+<span class="n">AnnualQ</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+<span class="c1"># Plot a line graph</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper right&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_9_0.png" src="_images/hmm_9_0.png" />
+<p>Let’s calculate an 11-year rolling mean of the same data to get a sense
+of long-term trends.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+
+<span class="c1"># Plot the original line graph</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+         <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+         <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+         <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1"># Plot an 11-year rolling mean</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+         <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+         <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+         <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_11_0.png" src="_images/hmm_11_0.png" />
+<p>The Colorado Compact, which prescribes flows between the Upper and Lower
+Colorado Basins, was negotiated using data prior to 1922, a time period
+revealed by the above figure to be one of the consistently wetter
+periods on record. It’s clear today that since the 1980s, the Southwest
+US has been experiencing aridification (Overpeck et al., 2020) and that
+this observed record alone isn’t an accurate representation of what
+future climate might look like in this region.</p>
+<p>Let’s get a little more specific and formally quantify decadal droughts
+that have occurred in the observed period. We use a metric proposed in
+Ault et al. (2014). The authors define a decadal drought as when the
+11-year rolling mean falls below a threshold that is 1/2 a standard
+deviation below the overall mean of the record. We can then highlight
+the block of years that fall in a decadal drought using yellow
+rectangles below.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define drought threshold</span>
+<span class="n">std</span> <span class="o">=</span> <span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="n">threshold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+<span class="c1"># Find where the rolling mean dip below the threshold?</span>
+<span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+<span class="n">drought_years</span> <span class="o">=</span> <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">drought_instances</span><span class="p">]</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span><span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span><span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">threshold</span><span class="p">,</span>
+           <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
+           <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span>
+           <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Drought Threshold&#39;</span><span class="p">)</span>
+
+<span class="c1"># Visualize the drought periods as yellow rectangles</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">drought_years</span><span class="p">:</span>
+
+    <span class="c1"># Plot a box centered around those values and with 5 years on either side.</span>
+    <span class="n">rect</span> <span class="o">=</span> <span class="n">patches</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">((</span><span class="n">i</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="mi">11</span><span class="p">,</span><span class="mf">2e7</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">,</span> <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Add the patch to the Axes</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_14_0.png" src="_images/hmm_14_0.png" />
+<p>By this metric, the Upper Colorado Basin region has experienced two
+decadal droughts over the last century.</p>
+</section>
+<section id="synthetic-stationary-generator-to-better-quantify-natural-variability">
+<h4><span class="section-number">B.6.2.2. </span>Synthetic Stationary Generator to Better Quantify Natural Variability<a class="headerlink" href="#synthetic-stationary-generator-to-better-quantify-natural-variability" title="Link to this heading">#</a></h4>
+<p>It is important to remember that the streamflow that we have observed in
+the region over the last century is only one instance of the hydrology
+that could occur since the atmosphere is an inherently stochastic
+system. Thus, we require a tool that will allow us to see multiple
+plausible realizations of the streamflow record to understand the
+internal variability that characterizes the historical period. One
+observed realization of historical streamflow is limited in its ability
+to capture rare extremes; plausible (but not observed) alternative
+instances of streamflow records can help to fill this gap. The tool that
+we use to develop synthetic flows for the region is a Gaussian Hidden
+Markov Model (HMM). If a system follows a Markov process, it switches
+between a number of “hidden states” dictated by a transition matrix.
+Each state has its own Gaussian probability distribution (defined by a
+mean and standard deviation) and one can draw from this distribution to
+create synthetic flows that fit the properties of the historical
+distribution. HMMs are an attractive choice for this region because they
+can simulate persistence (i.e., long duration droughts), which is a
+characteristic of the region’s hydro-climatology. The figure below shows
+an example of a 2-state Gaussian HMM that we will be fitting for this
+example.</p>
+<figure>
+    <center><img src="_static/HMM_example.png" style="width:75%"></center>
+<figcaption style="text-align:center"> Two-state Gaussian HMM with mean and standard deviation parameters</figcaption>
+</figure><p>Below is the code that fits the HMM model to the last 2/3 of the
+historical record of log annual flows at the CO-UT stateline gauge and
+creates an alternative trace of 105 years. A subset of the dataset is
+chosen in order to minimize overfitting and to retain a set of data for
+validation of the model. When we fit our model, we utilize the
+Baum-Welch algorithm (a special version of the expectation-maximization
+algorithm) to find the optimal parameters that maximize the likelihood
+of seeing the observed flows. Ultimately, the algorithm will return a
+mean and standard deviation associated with each state (mus and sigmas
+defined below) and a 2x2 transition probability matrix that captures the
+likelihood of transitioning between states (P). We can also retrieve the
+annual hidden states across the observed series, also known as the
+Viterbi sequence of states, which classifies each year in a “wet” or
+“dry” state.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Number of years for alternative trace</span>
+<span class="n">n_years</span> <span class="o">=</span> <span class="mi">105</span>
+
+<span class="c1"># Import historical data that it used to fit HMM model</span>
+<span class="n">AnnualQ_h</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;data/uc_historical.csv&#39;</span><span class="p">)</span>
+
+<span class="c1"># Fit the model and pull out relevant parameters and samples</span>
+<span class="n">logQ</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">AnnualQ_h</span><span class="p">)</span>
+<span class="n">hidden_states</span><span class="p">,</span> <span class="n">mus</span><span class="p">,</span> <span class="n">sigmas</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">logProb</span><span class="p">,</span> <span class="n">samples</span><span class="p">,</span> <span class="n">model</span> <span class="o">=</span> <span class="n">fitmodel</span><span class="o">.</span><span class="n">fitHMM</span><span class="p">(</span><span class="n">logQ</span><span class="p">,</span> <span class="n">n_years</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>We’ve fit our HMM, but what does the model look like? Let’s plot the
+annual time series of hidden states, or the Viterbi sequence. In the
+code, above, we have defined that the drier state is always represented
+by state 0. Thus, we know that hidden_states = 0 corresponds to the dry
+state and hidden_states = 1 to the wet state.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot Vitebi sequence</span>
+<span class="n">plotstates</span><span class="o">.</span><span class="n">plotTimeSeries</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]),</span> <span class="n">hidden_states</span><span class="p">,</span> <span class="s1">&#39;Annual Flow (cube feet per year)&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_21_0.png" src="_images/hmm_21_0.png" />
+<p>In the figure above, we see that the years with the higher log flows
+tend to be classified in a “wet” state and the opposite is true of the
+“dry” state. We can also print the transition matrix, which shows the
+likelihood of transitioning between states. Note that the system has a
+high likelihood of persisting in the same state.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">transmat_</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[[</span><span class="mf">0.65095026</span> <span class="mf">0.34904974</span><span class="p">]</span>
+ <span class="p">[</span><span class="mf">0.3205531</span>  <span class="mf">0.6794469</span> <span class="p">]]</span>
+</pre></div>
+</div>
+<p>Let’s also plot the distribution of log annual flows associated with the
+wet and dry states.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot wet and dry state distributions</span>
+<span class="n">plotdist</span><span class="o">.</span><span class="n">plotDistribution</span><span class="p">(</span><span class="n">logQ</span><span class="p">,</span> <span class="n">mus</span><span class="p">,</span> <span class="n">sigmas</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_25_0.png" src="_images/hmm_25_0.png" />
+<p>The wet state distribution is characterized by a greater mean flow, but
+note that there is significant overlap in the tails of the distributions
+below which demonstrates why years with similiar flows can be classified
+in different states.</p>
+<p>Now let’s see what the drought dynamics look like in the synthetic
+scenario that we created using the same definition that we had used for
+the historical period.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Retrieve samples and back-transform out of log space</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">samples</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="p">)</span>
+<span class="n">AnnualQ_s</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+<span class="c1"># Define drought threshold</span>
+<span class="n">std</span><span class="o">=</span><span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="n">threshold</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+<span class="c1"># Where does the rolling mean dip below the threshold</span>
+<span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+<span class="n">drought_years</span> <span class="o">=</span> <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">drought_instances</span><span class="p">]</span>
+
+<span class="c1">#Visualize the streamflow scenario</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+
+<span class="c1">#Plot the original line graph</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1">#Plot a 11-year rolling mean</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">threshold</span><span class="p">,</span>
+           <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
+           <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span>
+           <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Drought Threshold&#39;</span><span class="p">)</span>
+
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">drought_years</span><span class="p">:</span>
+
+    <span class="c1">#Plot a box centered around those values and with 5 years on either side.</span>
+    <span class="n">rect</span> <span class="o">=</span> <span class="n">patches</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">((</span><span class="n">i</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span>
+                              <span class="mi">0</span><span class="p">),</span>
+                              <span class="mi">11</span><span class="p">,</span>
+                              <span class="mf">2e7</span><span class="p">,</span>
+                              <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+                              <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">,</span>
+                              <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Add the patch to the Axes</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow (Synthetic Stationary)&quot;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_28_0.png" src="_images/hmm_28_0.png" />
+<p>You can sample from the model and create more 105-year traces and note
+how the location and number of decadal droughts changes. This
+demonstrates how different the historical record can look just within
+the range of natural variability. It’s also important to remember that
+when droughts occur can also define the ultimate effect of the drought
+(i.e. is it a time when there is a large population growth or a time
+when humans can adapt by conserving or building more infrastructure?). A
+hydrologic drought need not manifest into an agricultural or operational
+drought of the same magnitude if stored surface water is available.</p>
+<p>We externally run the HMM many times to create a dataset of 100
+instances of the 105-year traces and 1000 instances of the 105-year
+traces that are available in the package
+(“synthetic_stationary_small_sample_100.csv”,“synthetic_stationary_large_sample_1000”).
+The shaded green lines correspond to the flow duration curves (FDCs) for
+the generated streamflow traces in comparison with the FDC of the
+historical record in beige.</p>
+<figure>
+    <center><img src="_static/stationary_synthetic_FDC.png" style="width:75%"></center>
+<figcaption style="text-align:center">Generated streamflow traces in comparison with the FDC of the historical record.</figcaption>
+</figure><p>As expected, the stationary synthetic FDCs envelope the historical FDC
+and particularly, the synthetic traces offer many more instances of low
+flow conditions that could lead to more extreme drought conditions than
+what has been observed historically. It is also useful to check for
+convergence of samples and to determine how many samples are needed to
+fully represent internal variability. Above we see that the extension to
+1000 instances of 105-year traces fills out regions of the FDC,
+including creating some more extreme drought conditions, but that
+additional samples will likely not fill out the FDC substantially more.</p>
+</section>
+<section id="non-stationary-synthetic-generator-to-impose-climate-changes">
+<h4><span class="section-number">B.6.2.3. </span>Non-Stationary Synthetic Generator to Impose Climate Changes<a class="headerlink" href="#non-stationary-synthetic-generator-to-impose-climate-changes" title="Link to this heading">#</a></h4>
+<p>Now, we create flows under non-stationary conditions to get a better
+understanding of what flows can look like under climate changes. In
+order to create flows under non-stationary conditions, we can toggle the
+parameters of the HMM model in order to create systematic changes to the
+model that can represent a changing climate. The HMM has 6 parameters
+that define it. When we fit the historical model, the parameters that
+are fit represent a baseline parameter value. In this non-stationary
+generator, we define a range to sample these parameters from.</p>
+<div class="pst-scrollable-table-container"><table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Parameter</p></th>
+<th class="head"><p>Current Value</p></th>
+<th class="head"><p>Lower Bound</p></th>
+<th class="head"><p>Upper Bound</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>Log-Space Wet State
+Mean Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.98</p></td>
+<td><p>1.02</p></td>
+</tr>
+<tr class="row-odd"><td><p>Log-Space Dry State
+Mean Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.98</p></td>
+<td><p>1.02</p></td>
+</tr>
+<tr class="row-even"><td><p>Log-Space Wet State
+Standard Deviation
+Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.75</p></td>
+<td><p>1.25</p></td>
+</tr>
+<tr class="row-odd"><td><p>Log-Space Dry State
+Standard Deviation
+Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.75</p></td>
+<td><p>1.25</p></td>
+</tr>
+<tr class="row-even"><td><p>Change in Dry-Dry
+Transition
+Probability</p></td>
+<td><p>0.00</p></td>
+<td><p>-0.30</p></td>
+<td><p>+0.30</p></td>
+</tr>
+<tr class="row-odd"><td><p>Change in Wet-Wet
+Transition
+Probability</p></td>
+<td><p>0.00</p></td>
+<td><p>-0.30</p></td>
+<td><p>+0.30</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>Now let’s sample 1000 times from these bounds to create 1000 new
+parameterizations of the model. Here we use SALib and the Latin
+Hypercube sample function.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create problem structure with parameters that we want to sample</span>
+<span class="n">problem</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">6</span><span class="p">,</span>
+    <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;wet_mu&#39;</span><span class="p">,</span> <span class="s1">&#39;dry_mu&#39;</span><span class="p">,</span> <span class="s1">&#39;wet_std&#39;</span><span class="p">,</span><span class="s1">&#39;dry_std&#39;</span><span class="p">,</span><span class="s1">&#39;dry_tp&#39;</span><span class="p">,</span><span class="s2">&quot;wet_tp&quot;</span><span class="p">],</span>
+    <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="mf">0.98</span><span class="p">,</span> <span class="mf">1.02</span><span class="p">],</span>
+               <span class="p">[</span><span class="mf">0.98</span><span class="p">,</span> <span class="mf">1.02</span><span class="p">],</span>
+               <span class="p">[</span><span class="mf">0.75</span><span class="p">,</span><span class="mf">1.25</span><span class="p">],</span>
+               <span class="p">[</span><span class="mf">0.75</span><span class="p">,</span><span class="mf">1.25</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.3</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.3</span><span class="p">]]</span>
+<span class="p">}</span>
+
+<span class="c1"># generate 1000 parameterizations</span>
+<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">1000</span>
+
+<span class="c1"># set random seed for reproducibility</span>
+<span class="n">seed_value</span> <span class="o">=</span> <span class="mi">123</span>
+
+<span class="c1"># Generate our samples</span>
+<span class="n">LHsamples</span> <span class="o">=</span> <span class="n">latin</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="n">n_samples</span><span class="p">,</span> <span class="n">seed_value</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now let’s look at what some of the traces look like in our
+non-stationary generator. Let’s choose a random instance from the
+1000-member space and adjust the parameters accordingly.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define static parameters</span>
+<span class="n">n_years</span> <span class="o">=</span> <span class="mi">105</span>
+
+<span class="c1"># Sample parameter; Adjust to any sample number from 0-999</span>
+<span class="n">sample</span> <span class="o">=</span> <span class="mi">215</span>
+
+<span class="c1"># Create empty arrays to store the new Gaussian HMM parameters for each SOW</span>
+<span class="n">Pnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">musNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">sigmasNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">logAnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+
+<span class="c1"># Calculate new transition matrix and stationary distribution of SOW at last node as well as new means and standard deviations</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">4</span><span class="p">]))</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">5</span><span class="p">]))</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+<span class="n">eigenvals</span><span class="p">,</span> <span class="n">eigenvecs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">))</span>
+<span class="n">one_eigval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">eigenvals</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
+<span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">divide</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span> <span class="n">one_eigval</span><span class="p">]),</span>
+                  <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span><span class="n">one_eigval</span><span class="p">])))</span>
+
+<span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
+<span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+<span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">3</span><span class="p">]</span>
+<span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
+
+<span class="c1"># Generate first state and log-space annual flow at last node</span>
+<span class="n">states</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+<span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">piNew</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
+    <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="k">else</span><span class="p">:</span>
+    <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+    <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="c1"># Generate remaining state trajectory and log space flows at last node</span>
+<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_years</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">Pnew</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">])]:</span>
+        <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+
+    <span class="k">if</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="c1"># Convert log-space flows to real-space flows</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">logAnnualQ_s</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
+</pre></div>
+</div>
+<p>Now let’s see what this synthetic trace looks like.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Retrieve samples and back-transform out of log space</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="p">)</span>
+<span class="n">AnnualQ_s</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+<span class="c1"># Define drought threshold</span>
+<span class="n">std</span> <span class="o">=</span> <span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="n">threshold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+<span class="c1"># Where does the rolling mean dip below the threshold</span>
+<span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+<span class="n">drought_years</span> <span class="o">=</span> <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">drought_instances</span><span class="p">]</span>
+
+<span class="c1"># Visualize the streamflow scenario</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+
+<span class="c1"># Plot the original line graph</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1"># Plot a 11-year rolling mean</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">threshold</span><span class="p">,</span>
+           <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
+           <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span>
+           <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Drought Threshold&#39;</span><span class="p">)</span>
+
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">drought_years</span><span class="p">:</span>
+
+    <span class="c1"># Plot a box centered around those values and with 5 years on either side.</span>
+    <span class="n">rect</span> <span class="o">=</span> <span class="n">patches</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">((</span><span class="n">i</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span>
+                             <span class="mi">11</span><span class="p">,</span>
+                             <span class="mf">2e7</span><span class="p">,</span>
+                             <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+                             <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">,</span>
+                             <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Add the patch to the Axes</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Annual Flow (Synthetic Non-Stationary)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper right&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_40_0.png" src="_images/hmm_40_0.png" />
+<p>Above is the example trace from the new non-stationary model. You may
+see fewer or more decadal drought instances. We can further summarize
+overall decadal drought characteristics across the samples. Let’s plot a
+histogram of the total number of times we go below the drought threshold
+across these realizations.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">decadal_drought_occurence</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">1000</span><span class="p">])</span>
+
+<span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">):</span>
+
+    <span class="c1"># Create empty arrays to store the new Gaussian HMM parameters for each SOW</span>
+    <span class="n">Pnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
+    <span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+    <span class="n">musNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+    <span class="n">sigmasNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+    <span class="n">logAnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+
+    <span class="c1"># Calculate new transition matrix and stationary distribution of SOW at last node</span>
+    <span class="c1"># as well as new means and standard deviations</span>
+
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">4</span><span class="p">]))</span>
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">5</span><span class="p">]))</span>
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+    <span class="n">eigenvals</span><span class="p">,</span> <span class="n">eigenvecs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">))</span>
+    <span class="n">one_eigval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">eigenvals</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
+    <span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">divide</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span> <span class="n">one_eigval</span><span class="p">]),</span>
+                      <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span> <span class="n">one_eigval</span><span class="p">])))</span>
+
+    <span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
+    <span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+    <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">3</span><span class="p">]</span>
+    <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
+
+    <span class="c1"># Generate first state and log-space annual flow at last node</span>
+    <span class="n">states</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+    <span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">piNew</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
+        <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+    <span class="c1"># generate remaining state trajectory and log space flows at last node</span>
+    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_years</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">Pnew</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">])]:</span>
+            <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+
+        <span class="k">if</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+    <span class="c1"># Convert log-space flows to real-space flows</span>
+    <span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">logAnnualQ_s</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="p">)</span>
+    <span class="n">AnnualQ_s</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+    <span class="c1"># Define drought threshold</span>
+    <span class="n">std</span> <span class="o">=</span> <span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+    <span class="n">threshold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+    <span class="c1"># Where does the rolling mean dip below the threshold</span>
+    <span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+    <span class="n">decadal_drought_occurence</span><span class="p">[</span><span class="n">y</span><span class="p">]</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">drought_instances</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">decadal_drought_occurence</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Non-Stationary generator&#39;</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s2">&quot;#005F73&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Number of Instances of Decadal Drought&#39;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Frequency&#39;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;-&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Observed&#39;</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span> <span class="o">=</span> <span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span> <span class="o">=</span> <span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_43_0.png" src="_images/hmm_43_0.png" />
+<p>Note how many more instances of the decadal droughts we are creating
+with the non-stationary generator than our observed 105-year trace which
+creates a rich space in which we can test our models. Just as we did
+with the stationary generator, we can externally run the non-stationary
+generator to create 10,000 instances of the 105-year traces that are
+available in the package
+(“synthetic_nonstationary_large_sample_10000.csv”). The shaded green and
+blue lines correspond to the FDCs for the stationary and non-stationary
+generated streamflow traces in comparison with the FDC of the historical
+record in beige. Note how the non-stationary generator produces even
+more drought extremes than the stationary non-synthetic traces.</p>
+<figure>
+    <center><img src="_static/nonstationary_synthetic_FDC.png" style="width:75%"></center>
+<figcaption style="text-align:center">Generated streamflow traces in comparison with the FDC of the historical record.</figcaption>
+</figure></section>
+<section id="placing-cmip5-projections-in-the-context-of-non-stationary-flows">
+<h4><span class="section-number">B.6.2.4. </span>Placing CMIP5 Projections in the Context of Non-Stationary Flows<a class="headerlink" href="#placing-cmip5-projections-in-the-context-of-non-stationary-flows" title="Link to this heading">#</a></h4>
+<p>We have broadened the drought conditions that we are creating which that
+can be very useful to understand how our water systems model performs
+under potentially extreme scenarios. However, it’s useful to compare our
+bottom-up synthetically generated flows in the context of global
+physically-driven CMIP5 projections to get a better understanding of how
+the two approaches compare. We first aquire 97 CMIP5 projections from
+the Colorado River Water Availability Study (CWCB, 2012). In each of
+these projections, monthly precipitation factor changes and temperature
+delta changes were computed between mean projected 2035–2065 climate
+statistics and mean historical climate statistics from 1950–2013. These
+97 different combinations of 12 monthly precipitation multipliers and 12
+monthly temperature delta shifts were applied to historical
+precipitation and temperature time series from 1950–2013. The resulting
+climate time series were run through a Variable Infiltration Capacity
+(VIC) model of the UCRB, resulting in 97 time series of projected future
+streamflows at the Colorado‐Utah state line.</p>
+<p>We fit an HMM to each trace of projected streamflow and get a set of
+corresponding HMM parameters. Then we take the ratio between these
+parameters and the baseline HMM parameters that we calculated earlier in
+the notebook in order to calculate the multipliers associated with each
+CMIP5 projection. This is all done externally, so we import the
+resulting multipliers in the next line.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Read in CMIP5 and paleo multipliers</span>
+<span class="n">CMIP5_multipliers</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;data/CMIP5_SOWs.txt&#39;</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s2">&quot; &quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Let’s plot a response surface that will allow us to see how combinations
+of HMM parameters tend to influence decadal drought. In order to get a
+continuous surface, we’ll fit a non-linear regression to the parameter
+values and then predict the decadal drought over a set of grid points.
+We fit the response surface for two parameters that should have an
+affect on decadal drought: the dry distribution mean and the dry-dry
+transition probabilites.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Choose two parameters to fit the response surface for</span>
+<span class="n">mu_dry</span><span class="o">=</span><span class="p">[</span><span class="n">i</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">LHsamples</span><span class="p">]</span>
+<span class="n">tp_dry</span><span class="o">=</span><span class="p">[</span><span class="n">i</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">LHsamples</span><span class="p">]</span>
+
+<span class="c1"># Create an interpolation grid</span>
+<span class="n">xgrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">),</span>
+                  <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">),</span>
+                  <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">))</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span>
+
+<span class="n">ygrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">),</span>
+                  <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">),</span>
+                  <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">))</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span>
+
+<span class="c1"># Fit regression</span>
+<span class="n">d</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;Dry_Tp&#39;</span><span class="p">:</span> <span class="n">tp_dry</span><span class="p">,</span>
+     <span class="s1">&#39;Dry_Mu&#39;</span><span class="p">:</span> <span class="n">mu_dry</span><span class="p">,</span>
+     <span class="s1">&#39;Drought_Occurrence&#39;</span><span class="p">:</span><span class="n">decadal_drought_occurence</span><span class="p">}</span>
+
+<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
+<span class="n">df</span><span class="p">[</span><span class="s1">&#39;Intercept&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">shape</span><span class="p">(</span><span class="n">df</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">df</span><span class="p">[</span><span class="s1">&#39;Interaction&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s1">&#39;Dry_Tp&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">df</span><span class="p">[</span><span class="s1">&#39;Dry_Mu&#39;</span><span class="p">]</span>
+<span class="n">cols</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Intercept&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s1">&#39;Dry_Mu&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s1">&#39;Dry_Tp&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s1">&#39;Interaction&#39;</span><span class="p">]</span>
+<span class="n">ols</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Drought_Occurrence&#39;</span><span class="p">],</span> <span class="n">df</span><span class="p">[</span><span class="n">cols</span><span class="p">])</span>
+<span class="n">result</span> <span class="o">=</span> <span class="n">ols</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
+
+<span class="c1"># Calculate drought occurence for each grid point</span>
+<span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">xgrid</span><span class="p">,</span> <span class="n">ygrid</span><span class="p">)</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
+<span class="n">y</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
+<span class="n">grid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">column_stack</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">*</span> <span class="n">y</span><span class="p">])</span>
+<span class="n">z</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
+<span class="n">z</span><span class="p">[</span><span class="n">z</span> <span class="o">&lt;</span> <span class="mf">0.0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="c1"># replace negative shortage predictions with 0</span>
+</pre></div>
+</div>
+<p>Let’s plot our results:</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set color gradient for response surface</span>
+<span class="n">drought_map</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">&#39;RdBu_r&#39;</span><span class="p">)</span>
+
+<span class="c1"># Reshape our predicted drought occurrence and define bounds of colors</span>
+<span class="n">Z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">shape</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
+<span class="n">vmin</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Drought_Occurrence&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">)])</span>
+<span class="n">vmax</span> <span class="o">=</span> <span class="mi">15</span>
+<span class="n">norm</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="p">,</span> <span class="n">vmax</span><span class="p">)</span>
+
+<span class="c1"># Plot response surface and CMIP5 projections</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">drought_map</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">norm</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">CMIP5_multipliers</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">7</span><span class="p">],</span>
+           <span class="n">CMIP5_multipliers</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">12</span><span class="p">],</span>
+           <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#ffffb3&#39;</span><span class="p">,</span>
+           <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;none&#39;</span><span class="p">,</span>
+           <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span>
+<span class="n">cbar</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">figure</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">ScalarMappable</span><span class="p">(</span><span class="n">norm</span><span class="o">=</span><span class="n">norm</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">drought_map</span><span class="p">),</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">nanmin</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">nanmax</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">nanmin</span><span class="p">(</span><span class="n">Y</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">nanmax</span><span class="p">(</span><span class="n">Y</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Dry State Mu&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Dry-Dry Transition Probability&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="n">labelsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Decadal Drought Occurrence&#39;</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=-</span><span class="mi">90</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">,</span> <span class="n">labelpad</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
+<span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;y&#39;</span><span class="p">,</span><span class="n">labelsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_53_0.png" src="_images/hmm_53_0.png" />
+<p>We see the influence of the dry state mean and dry-dry transition
+parameters. We’re likely to see more decadal droughts when we (1)
+increase the dry-dry transition probability, which inherently will
+increase persistence of the dry state, and (2) when we make the dry
+state log mean drier. Note that the CMIP5 scenarios tend to span the
+extent of the dry mean sample space, but are less representative of the
+dry transition probability sample space, which suggests that the types
+of hydrological droughts represented in the projections tend to only be
+wetter to slightly drier than our baseline. Both methods of producing
+these scenarios are valid, though studies have suggested that
+globally-resolved GCMs may be inappropriate to represent regional
+extremes. Ultimately, if your goal is to produce a variety of ensembles
+that are characterized by many different drought characteristics, you
+will likely find that a generator approach will serve this purpose
+better.</p>
+</section>
+<section id="tips-to-create-an-hmm-based-generator-for-your-system">
+<h4><span class="section-number">B.6.2.5. </span>Tips to Create an HMM-Based Generator for your System<a class="headerlink" href="#tips-to-create-an-hmm-based-generator-for-your-system" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we demonstrated how to fit an HMM-based generator for
+a single gauge located in the Upper Colorado River Basin. In order to
+apply this methodology to your problem, you will need to first ask:</p>
+<ol class="arabic simple">
+<li><p>Is this model appropriate for my location of interest? We have
+applied this style of generator to locations where persistent wet
+and dry states are characteristic, which tends to be in the Western
+US. Ultimately the best way to judge if an HMM is useful for your
+application is to fit the model and explore the resulting
+distributions. Are there two (or more) distinct states that emerge?
+If not, then your location may not exhibit the type of persistence
+that an HMM-based generator is useful for. You can consider
+exploring other styles of generators such as the Kirsch-Nowak
+generator (Kirsch et al., 2013).</p></li>
+<li><p>Do I have the right datasets? We use annual data for our location of
+interest. In this notebook, the HMM is fit to log annual flows.
+Ultimately, it can be disaggregated to daily flows (using a
+reference historical daily dataset) to be useful in water resources
+operational applications. You could also disaggregate to a finer
+resolution than daily if the historical dataset exists.</p></li>
+</ol>
+<p>If you meet these requirements, feel free to proceed through fitting the
+model using the code available in the notebook. Be sure to consider the
+appropirate number of samples to generate (both in a stationary and
+non-stationary case). Make sure that you test multiple sample sizes and
+continue to increase your sample size until you converge to a consistent
+representation of extremes. What is the appropriate number of LHS
+samples of the parameters to use? In this experiment we used 1,000
+samples of parameters due to extensive stability tests described in
+Quinn et al. (2020).</p>
+<p>Finally, to learn more about this test case refer to Hadmichael et
+al. (2020a) and Hadmichael et al. (2020b). For another study on
+synthetic drought generation to support vulnerability assessments in the
+Research Triangle region of North Carolina, please refer to Herman et
+al. (2016)</p>
+</section>
+<section id="id31">
+<h4><span class="section-number">B.6.2.6. </span>References<a class="headerlink" href="#id31" title="Link to this heading">#</a></h4>
+<p>Ault, T. R., Cole, J. E., Overpeck, J. T., Pederson, G. T., &amp; Meko, D.
+M. (2014). Assessing the risk of persistent drought using climate model
+simulations and paleoclimate data. Journal of Climate, 27(20),
+7529-7549.</p>
+<p>CWCB (2012).Colorado River Water Availability Study Phase I Report.
+Colorado Water Conservation Board</p>
+<p>Hadjimichael, A., Quinn, J., Wilson, E., Reed, P., Basdekas, L., Yates,
+D., &amp; Garrison, M. (2020a). Defining robustness, vulnerabilities, and
+consequential scenarios for diverse stakeholder interests in
+institutionally complex river basins. Earth’s Future, 8(7),
+e2020EF001503.</p>
+<p>Hadjimichael, A., Quinn, J., &amp; Reed, P. (2020). Advancing diagnostic
+model evaluation to better understand water shortage mechanisms in
+institutionally complex river basins. Water Resources Research, 56(10),
+e2020WR028079.</p>
+<p>Herman, J. D., Zeff, H. B., Lamontagne, J. R., Reed, P. M., &amp;
+Characklis, G. W. (2016). Synthetic drought scenario generation to
+support bottom-up water supply vulnerability assessments. Journal of
+Water Resources Planning and Management, (11), 04016050.</p>
+<p>Kirsch, B. R., Characklis, G. W., &amp; Zeff, H. B. (2013). Evaluating the
+impact of alternative hydro-climate scenarios on transfer agreements:
+Practical improvement for generating synthetic streamflows. Journal of
+Water Resources Planning and Management, 139(4), 396-406.</p>
+<p>Overpeck, J.T. &amp; Udall, B. (2020) “Climate change and the aridification
+of North America.” Proceedings of the national academy of sciences
+117.22 11856-11858.</p>
+<p>Quinn, J. D., Hadjimichael, A., Reed,P. M., &amp; Steinschneider, S. (2020).
+Canexploratory modeling of water scarcity vulnerabilities and robustness
+bescenario neutral?Earth’s Future,8,e2020EF001650.
+<a class="reference external" href="https://doi.org/10.1029/2020EF001650Received">https://doi.org/10.1029/2020EF001650Received</a></p>
+<p>Williams, A. P., Cook, B. I., &amp; Smerdon, J. E. (2022). Rapid
+intensification of the emerging southwestern North American megadrought
+in 2020–2021. Nature Climate Change, 12(3), 232-234.</p>
+</section>
+</section>
+</section>
+</section>
+
+
+                </article>
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+    <i class="fa-solid fa-list"></i> Contents
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+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fishery-dynamics-tutorial">B.1. Fishery Dynamics Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics">B.1.1. Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">B.1.1.1. Let’s get started!</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-identified-solutions-and-explore-performance">B.1.1.2. Step 1: Load identified solutions and explore performance</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis">B.1.1.3. Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-evaluate-the-system-over-all-generated-states-of-the-world">B.1.1.4. Step 3: Evaluate the system over all generated states of the world</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-calculate-sensitivity-indices">B.1.1.5. Step 4: Calculate sensitivity indices</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-explore-relationship-between-uncertain-factors-and-performance">B.1.1.6. Step 5: Explore relationship between uncertain factors and performance</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem">B.1.1.7. Tips to Apply Sobol SA and Scenario Discovery to your Problem</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sobol-sa-tutorial">B.2. Sobol SA Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa">B.2.1. Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id6">B.2.1.1. Let’s get started!</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-generate-the-problem-dictionary">B.2.1.2. Step 1: Generate the problem dictionary</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-generate-samples-using-the-saltelli-sampling-scheme">B.2.1.3. Step 2: Generate samples using the Saltelli sampling scheme</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-execute-the-ishigami-function-over-our-sample-set">B.2.1.4. Step 3: Execute the Ishigami function over our sample set</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method">B.2.1.5. Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-interpret-our-results">B.2.1.6. Step 5: Interpret our results</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-to-your-own-problem">B.2.1.7. Tips to Apply Sobol SA to Your Own Problem</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#logistic-regression-tutorial">B.3. Logistic Regression Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-logistic-regression-for-factor-mapping">B.3.1. Tutorial: Logistic Regression for Factor Mapping</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#background">B.3.1.1. Background</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id19">B.3.1.2. Let’s get started!</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-latin-hypercube-sample-and-set-up-problem">B.3.1.3. Step 1: Load Latin hypercube sample and set up problem</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-define-decision-relevant-metrics-for-illustration">B.3.1.4. Step 2: Define decision-relevant metrics for illustration</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-run-the-logistic-regression">B.3.1.5. Step 3: Run the logistic regression</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-factor-ranking">B.3.1.6. Step 4: Factor ranking</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-draw-factor-maps">B.3.1.7. Step 5: Draw factor maps</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-scenario-discovery-to-your-own-problem">B.3.1.8. Tips to Apply Scenario Discovery to Your Own Problem</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#hymod-dynamics-tutorial">B.4. HYMOD Dynamics Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-of-the-hymod-model">B.4.1. Tutorial: Sensitivity Analysis of the HYMOD Model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction-to-hymod">B.4.2. 1 - Introduction to HYMOD</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#global-sensitivity-analysis">B.4.3. 2- Global Sensitivity Analysis</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-this-methodology-to-your-own-problem">B.4.4. Tips to Apply this methodology to your own problem</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#time-evolving-scenario-discovery-for-infrastructure-pathways">B.5. Time-evolving scenario discovery for infrastructure pathways</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id24">B.5.1. Time-evolving scenario discovery for infrastructure pathways</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id25">B.5.1.1. Background</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluating-robustness-over-time">B.5.1.2. Evaluating Robustness over time</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#exploring-performance-evolution">B.5.1.3. Exploring performance evolution</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#how-do-deep-uncertainties-generate-vulnerability">B.5.1.4. How do deep uncertainties generate vulnerability</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#which-uncertainties-have-the-most-influence-on-time-evolving-performance">B.5.1.5. Which uncertainties have the most influence on time-evolving performance?</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#open-exploration">B.5.1.6. Open exploration</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id26">B.5.1.7. Tips to apply this methodology to your own problem</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#references">B.5.1.8. References</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios-tutorial">B.6. A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios">B.6.1. A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id29">B.6.1.1. Background</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id30">B.6.2. Let’s Get Started!</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#observed-record">B.6.2.1. Observed Record</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-stationary-generator-to-better-quantify-natural-variability">B.6.2.2. Synthetic Stationary Generator to Better Quantify Natural Variability</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#non-stationary-synthetic-generator-to-impose-climate-changes">B.6.2.3. Non-Stationary Synthetic Generator to Impose Climate Changes</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#placing-cmip5-projections-in-the-context-of-non-stationary-flows">B.6.2.4. Placing CMIP5 Projections in the Context of Non-Stationary Flows</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-create-an-hmm-based-generator-for-your-system">B.6.2.5. Tips to Create an HMM-Based Generator for your System</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id31">B.6.2.6. References</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="A1_Uncertainty_Quantification.html">A. Uncertainty Quantification</a></li>
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+<li class="toctree-l1 current active"><a class="current reference internal" href="#">C. Plotting Code Samples</a></li>
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+    <h1>Plotting Code Samples</h1>
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+            
+            <div>
+                <h2> Contents </h2>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#hymod-ipynb">C.1. hymod.ipynb</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-observed-vs-simulated-streamflow">C.1.1. <strong>plot_observed_vs_simulated_streamflow()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-observed-vs-sensitivity-streamflow">C.1.2. <strong>plot_observed_vs_sensitivity_streamflow()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-monthly-heatmap">C.1.3. <strong>plot_monthly_heatmap()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-annual-heatmap">C.1.4. <strong>plot_annual_heatmap()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-varying-heatmap">C.1.5. <strong>plot_varying_heatmap()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-precalibration-flow">C.1.6. <strong>plot_precalibration_flow()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-precalibration-glue">C.1.7. <strong>plot_precalibration_glue()</strong></a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fishery-dynamics-ipynb">C.2. fishery_dynamics.ipynb</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-objective-performance">C.2.1. <strong>plot_objective_performance()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-factor-performance">C.2.2. <strong>plot_factor_performance()</strong></a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="plotting-code-samples">
+<h1><span class="section-number">C. </span>Plotting Code Samples<a class="headerlink" href="#plotting-code-samples" title="Link to this heading">#</a></h1>
+<section id="hymod-ipynb">
+<h2><span class="section-number">C.1. </span>hymod.ipynb<a class="headerlink" href="#hymod-ipynb" title="Link to this heading">#</a></h2>
+<p>The following are the plotting functions as described in the <code class="docutils literal notranslate"><span class="pre">hymod.ipynb</span></code> Jupyter notebook tutorial.</p>
+<p>The following are the necessary package imports to run these functions:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="kn">from</span> <span class="nn">matplotlib.lines</span> <span class="kn">import</span> <span class="n">Line2D</span>
+</pre></div>
+</div>
+<section id="plot-observed-vs-simulated-streamflow">
+<h3><span class="section-number">C.1.1. </span><strong>plot_observed_vs_simulated_streamflow()</strong><a class="headerlink" href="#plot-observed-vs-simulated-streamflow" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_observed_vs_simulated_streamflow</span><span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="n">hymod_dict</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">12</span><span class="p">,</span> <span class="mi">6</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot observed versus simulated streamflow.</span>
+
+<span class="sd">    :param df:              Dataframe of hymod input data including columns for precip, potential evapotranspiration,</span>
+<span class="sd">                            and streamflow</span>
+
+<span class="sd">    :param hymod_dict:      A dictionary of hymod outputs</span>
+<span class="sd">    :type hymod_dict:       dict</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set plot style</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">style</span><span class="o">.</span><span class="n">use</span><span class="p">(</span><span class="s1">&#39;seaborn-white&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># plot observed streamflow</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">])),</span> <span class="n">df</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;pink&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plot simulated streamflow</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">])),</span> <span class="n">hymod_dict</span><span class="p">[</span><span class="s1">&#39;Q&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set axis labels</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Streamflow($m^3/s$)&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Days&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set plot title</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Observed vs. Simulated Streamflow&#39;</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">ax</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-observed-vs-sensitivity-streamflow">
+<h3><span class="section-number">C.1.2. </span><strong>plot_observed_vs_sensitivity_streamflow()</strong><a class="headerlink" href="#plot-observed-vs-sensitivity-streamflow" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_observed_vs_sensitivity_streamflow</span><span class="p">(</span><span class="n">df_obs</span><span class="p">,</span> <span class="n">df_sim</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot observed streamflow versus simulations generated from sensitivity analysis.</span>
+
+<span class="sd">    :param df_obs:          Dataframe of mean monthly hymod input data including columns for precip,</span>
+<span class="sd">                            potential evapotranspiration, and streamflow</span>
+
+<span class="sd">    :param df_sim:          Dataframe of mean monthly simulation data from sensitivity analysis</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">month_list</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df_sim</span><span class="p">))</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># set labels</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Days&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Flow Discharge (m^3/s)&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plots all simulated streamflow cases under different sample sets</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">df_sim</span><span class="o">.</span><span class="n">columns</span><span class="p">:</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">month_list</span><span class="p">,</span> <span class="n">df_sim</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;pink&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
+
+    <span class="c1"># plot observed streamflow</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">month_list</span><span class="p">,</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Observed vs. Sensitivity Analysis Outputs&#39;</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">ax</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-monthly-heatmap">
+<h3><span class="section-number">C.1.3. </span><strong>plot_monthly_heatmap()</strong><a class="headerlink" href="#plot-monthly-heatmap" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_monthly_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="p">,</span> <span class="n">df_obs</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s1">&#39;&#39;</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">14</span><span class="p">,</span> <span class="mi">6</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot a sensitivity metric overlain by observed flow.</span>
+
+<span class="sd">    :param arr_sim:         Numpy array of simulated metrics</span>
+
+<span class="sd">    :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis</span>
+
+<span class="sd">    :param title:           Title of plot</span>
+<span class="sd">    :type title:            str</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># plot heatmap</span>
+    <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="p">,</span>
+                <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span>
+                <span class="n">yticklabels</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">],</span>
+                <span class="n">cmap</span><span class="o">=</span><span class="n">sns</span><span class="o">.</span><span class="n">color_palette</span><span class="p">(</span><span class="s2">&quot;ch:s=-.2,r=.6&quot;</span><span class="p">))</span>
+
+    <span class="c1"># setup overlay axis</span>
+    <span class="n">ax2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+
+    <span class="c1"># plot line</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">12.5</span><span class="p">),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;slateblue&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plot points on line</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">12.5</span><span class="p">),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;slateblue&#39;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set axis limits and labels</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">12</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s1">&#39;jan&#39;</span><span class="p">,</span> <span class="s1">&#39;feb&#39;</span><span class="p">,</span> <span class="s1">&#39;mar&#39;</span><span class="p">,</span> <span class="s1">&#39;apr&#39;</span><span class="p">,</span> <span class="s1">&#39;may&#39;</span><span class="p">,</span> <span class="s1">&#39;jun&#39;</span><span class="p">,</span> <span class="s1">&#39;jul&#39;</span><span class="p">,</span> <span class="s1">&#39;aug&#39;</span><span class="p">,</span> <span class="s1">&#39;sep&#39;</span><span class="p">,</span> <span class="s1">&#39;oct&#39;</span><span class="p">,</span> <span class="s1">&#39;nov&#39;</span><span class="p">,</span> <span class="s1">&#39;dec&#39;</span><span class="p">])</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Flow Discharge($m^3/s$)&#39;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+
+    <span class="k">return</span> <span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-annual-heatmap">
+<h3><span class="section-number">C.1.4. </span><strong>plot_annual_heatmap()</strong><a class="headerlink" href="#plot-annual-heatmap" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_annual_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="p">,</span> <span class="n">df_obs</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s1">&#39;&#39;</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">14</span><span class="p">,</span><span class="mi">5</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot a sensitivity metric overlain by observed flow..</span>
+
+<span class="sd">    :param arr_sim:         Numpy array of simulated metrics</span>
+
+<span class="sd">    :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis</span>
+
+<span class="sd">    :param title:           Title of plot</span>
+<span class="sd">    :type title:            str</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># plot heatmap</span>
+    <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">sns</span><span class="o">.</span><span class="n">color_palette</span><span class="p">(</span><span class="s2">&quot;YlOrBr&quot;</span><span class="p">))</span>
+
+    <span class="c1"># setup overlay axis</span>
+    <span class="n">ax2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+
+    <span class="c1"># plot line</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">10.5</span><span class="p">),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;slateblue&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plot points on line</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">10.5</span><span class="p">),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;slateblue&#39;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set up axis lables and limits</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">([</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">2000</span><span class="p">,</span> <span class="mi">2010</span><span class="p">))</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Flow Discharge($m^3/s$)&#39;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-varying-heatmap">
+<h3><span class="section-number">C.1.5. </span><strong>plot_varying_heatmap()</strong><a class="headerlink" href="#plot-varying-heatmap" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_varying_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="p">,</span> <span class="n">df_obs</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s1">&#39;&#39;</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">14</span><span class="p">,</span><span class="mi">5</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot a sensitivity metric overlain by observed flow..</span>
+
+<span class="sd">    :param arr_sim:         Numpy array of simulated metrics</span>
+
+<span class="sd">    :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis</span>
+
+<span class="sd">    :param title:           Title of plot</span>
+<span class="sd">    :type title:            str</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># plot heatmap</span>
+    <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="p">,</span>
+                <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span>
+                <span class="n">yticklabels</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">],</span>
+                <span class="n">cmap</span><span class="o">=</span><span class="n">sns</span><span class="o">.</span><span class="n">light_palette</span><span class="p">(</span><span class="s2">&quot;seagreen&quot;</span><span class="p">,</span> <span class="n">as_cmap</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span>
+
+    <span class="n">n_years</span> <span class="o">=</span> <span class="n">df_obs</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+
+    <span class="c1"># setup overlay axis</span>
+    <span class="n">ax2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+
+    <span class="c1"># plot line</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n_years</span><span class="p">),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;slateblue&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plot points on line</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n_years</span><span class="p">),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;slateblue&#39;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set up axis lables and limits</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">119.5</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Flow Discharge&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Number of Months&#39;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-precalibration-flow">
+<h3><span class="section-number">C.1.6. </span><strong>plot_precalibration_flow()</strong><a class="headerlink" href="#plot-precalibration-flow" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_precalibration_flow</span><span class="p">(</span><span class="n">df_sim</span><span class="p">,</span> <span class="n">df_obs</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot flow discharge provided by the ensemble of parameters sets from Pre-Calibration versus the observed</span>
+<span class="sd">    flow data.</span>
+
+<span class="sd">    :param df_sim:          Dataframe of simulated metrics</span>
+
+<span class="sd">    :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># set axis labels</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Days&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Flow Discharge&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plot pre-calibration results</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">df_sim</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df_sim</span><span class="p">)),</span> <span class="n">df_sim</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span>  <span class="n">color</span><span class="o">=</span><span class="s2">&quot;lightgreen&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
+
+    <span class="c1"># plot observed</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df_sim</span><span class="p">)),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span>  <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Observed vs. Pre-Calibration Outputs&#39;</span><span class="p">)</span>
+
+    <span class="c1"># customize legend</span>
+    <span class="n">custom_lines</span> <span class="o">=</span> <span class="p">[</span><span class="n">Line2D</span><span class="p">([</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span>  <span class="n">color</span><span class="o">=</span><span class="s2">&quot;lightgreen&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">),</span>
+                    <span class="n">Line2D</span><span class="p">([</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">)]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">custom_lines</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;Pre-Calibration&#39;</span><span class="p">,</span> <span class="s1">&#39;Observed&#39;</span><span class="p">])</span>
+
+    <span class="k">return</span> <span class="n">ax</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-precalibration-glue">
+<h3><span class="section-number">C.1.7. </span><strong>plot_precalibration_glue()</strong><a class="headerlink" href="#plot-precalibration-glue" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_precalibration_glue</span><span class="p">(</span><span class="n">df_precal</span><span class="p">,</span> <span class="n">df_glue</span><span class="p">,</span> <span class="n">df_obs</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot flow discharge provided by the ensemble of parameters sets from Pre-Calibration versus the observed</span>
+<span class="sd">    flow data.</span>
+
+<span class="sd">    :param df_sim:          Dataframe of simulated metrics</span>
+
+<span class="sd">    :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># set axis labels</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Days&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Flow Discharge&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plot pre-calibration results</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">df_precal</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df_precal</span><span class="p">)),</span> <span class="n">df_precal</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span>  <span class="n">color</span><span class="o">=</span><span class="s2">&quot;lightgreen&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
+
+    <span class="c1"># plot glue</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">df_glue</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df_glue</span><span class="p">)),</span> <span class="n">df_glue</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;lightblue&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
+
+    <span class="c1"># plot observed</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df_precal</span><span class="p">)),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span>  <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Observed vs. Sensitivity Analysis Outputs across GLUE/Pre-Calibration&#39;</span><span class="p">)</span>
+
+    <span class="c1"># customize legend</span>
+    <span class="n">custom_lines</span> <span class="o">=</span> <span class="p">[</span><span class="n">Line2D</span><span class="p">([</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span>  <span class="n">color</span><span class="o">=</span><span class="s2">&quot;lightgreen&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">),</span>
+                    <span class="n">Line2D</span><span class="p">([</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;lightblue&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">),</span>
+                    <span class="n">Line2D</span><span class="p">([</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">)]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">custom_lines</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;Pre-Calibration&#39;</span><span class="p">,</span> <span class="s1">&#39;GLUE&#39;</span><span class="p">,</span> <span class="s1">&#39;Observed&#39;</span><span class="p">])</span>
+
+    <span class="k">return</span> <span class="n">ax</span>
+</pre></div>
+</div>
+</section>
+</section>
+<section id="fishery-dynamics-ipynb">
+<h2><span class="section-number">C.2. </span>fishery_dynamics.ipynb<a class="headerlink" href="#fishery-dynamics-ipynb" title="Link to this heading">#</a></h2>
+<p>The following are the plotting functions as described in the <code class="docutils literal notranslate"><span class="pre">fishery_dynamics.ipynb</span></code> Jupyter notebook tutorial.</p>
+<p>The following are the necessary package imports to run these functions:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">patheffects</span> <span class="k">as</span> <span class="n">pe</span>
+</pre></div>
+</div>
+<section id="plot-objective-performance">
+<h3><span class="section-number">C.2.1. </span><strong>plot_objective_performance()</strong><a class="headerlink" href="#plot-objective-performance" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_objective_performance</span><span class="p">(</span><span class="n">objective_performance</span><span class="p">,</span> <span class="n">profit_solution</span><span class="p">,</span> <span class="n">robust_solution</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">18</span><span class="p">,</span> <span class="mi">9</span><span class="p">)):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot the identified solutions with regards to their objective performance</span>
+<span class="sd">    in a parallel axis plot</span>
+
+<span class="sd">    :param objective_performance:               Objective performance array</span>
+<span class="sd">    :param profit_solution:                     Profitable solutions array</span>
+<span class="sd">    :param robust_solution:                     Robust solutions array</span>
+<span class="sd">    :param figsize:                             Figure size</span>
+<span class="sd">    :type figsize:                              tuple</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># create the figure object</span>
+    <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># set up subplot axis object</span>
+    <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+
+    <span class="c1"># labels where constraint is always 0</span>
+    <span class="n">objs_labels</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Net present</span><span class="se">\n</span><span class="s1">value (NPV)&#39;</span><span class="p">,</span>
+                   <span class="s1">&#39;Prey population deficit&#39;</span><span class="p">,</span>
+                   <span class="s1">&#39;Longest duration</span><span class="se">\n</span><span class="s1">of low harvest&#39;</span><span class="p">,</span>
+                   <span class="s1">&#39;Worst harvest instance&#39;</span><span class="p">,</span>
+                   <span class="s1">&#39;Variance of harvest&#39;</span><span class="p">,</span>
+                   <span class="s1">&#39;Duration of predator</span><span class="se">\n</span><span class="s1">population collapse&#39;</span><span class="p">]</span>
+
+    <span class="c1"># normalization across objectives</span>
+    <span class="n">mins</span> <span class="o">=</span> <span class="n">objective_performance</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+    <span class="n">maxs</span> <span class="o">=</span> <span class="n">objective_performance</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+    <span class="n">norm_reference</span> <span class="o">=</span> <span class="n">objective_performance</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
+        <span class="n">mm</span> <span class="o">=</span> <span class="n">objective_performance</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">min</span><span class="p">()</span>
+        <span class="n">mx</span> <span class="o">=</span> <span class="n">objective_performance</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
+        <span class="k">if</span> <span class="n">mm</span> <span class="o">!=</span> <span class="n">mx</span><span class="p">:</span>
+            <span class="n">norm_reference</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">objective_performance</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">mm</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">mx</span> <span class="o">-</span> <span class="n">mm</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">norm_reference</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+
+    <span class="c1"># colormap from matplotlib</span>
+    <span class="n">cmap</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s2">&quot;Blues&quot;</span><span class="p">)</span>
+
+    <span class="c1"># plot all solutions</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">norm_reference</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])):</span>
+        <span class="n">ys</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">norm_reference</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:],</span> <span class="mf">1.0</span><span class="p">)</span>
+        <span class="n">xs</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ys</span><span class="p">))</span>
+        <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">cmap</span><span class="p">(</span><span class="n">ys</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+
+    <span class="c1"># to highlight robust solutions</span>
+    <span class="n">ys</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">norm_reference</span><span class="p">[</span><span class="n">profit_solution</span><span class="p">,</span> <span class="p">:],</span> <span class="mf">1.0</span><span class="p">)</span>  <span class="c1"># Most profitable</span>
+    <span class="n">xs</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ys</span><span class="p">))</span>
+    <span class="n">l1</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">],</span>
+                 <span class="n">ys</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">],</span>
+                 <span class="n">c</span><span class="o">=</span><span class="n">cmap</span><span class="p">(</span><span class="n">ys</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
+                 <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span>
+                 <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Most robust in NPV&#39;</span><span class="p">,</span>
+                 <span class="n">path_effects</span><span class="o">=</span><span class="p">[</span><span class="n">pe</span><span class="o">.</span><span class="n">Stroke</span><span class="p">(</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="n">foreground</span><span class="o">=</span><span class="s1">&#39;darkgoldenrod&#39;</span><span class="p">),</span> <span class="n">pe</span><span class="o">.</span><span class="n">Normal</span><span class="p">()])</span>
+
+    <span class="n">ys</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">norm_reference</span><span class="p">[</span><span class="n">robust_solution</span><span class="p">,</span> <span class="p">:],</span> <span class="mf">1.0</span><span class="p">)</span>  <span class="c1"># Most robust in all criteria</span>
+    <span class="n">xs</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ys</span><span class="p">))</span>
+    <span class="n">l2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">],</span>
+                 <span class="n">ys</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">],</span>
+                 <span class="n">c</span><span class="o">=</span><span class="n">cmap</span><span class="p">(</span><span class="n">ys</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
+                 <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span>
+                 <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Most robust across criteria&#39;</span><span class="p">,</span>
+                 <span class="n">path_effects</span><span class="o">=</span><span class="p">[</span><span class="n">pe</span><span class="o">.</span><span class="n">Stroke</span><span class="p">(</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="n">foreground</span><span class="o">=</span><span class="s1">&#39;gold&#39;</span><span class="p">),</span> <span class="n">pe</span><span class="o">.</span><span class="n">Normal</span><span class="p">()])</span>
+
+    <span class="c1"># build colorbar</span>
+    <span class="n">sm</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">ScalarMappable</span><span class="p">(</span><span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">)</span>
+    <span class="n">sm</span><span class="o">.</span><span class="n">set_array</span><span class="p">([</span><span class="n">objective_performance</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">objective_performance</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">max</span><span class="p">()])</span>
+    <span class="n">cbar</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">sm</span><span class="p">)</span>
+    <span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">Net present value (NPV)&quot;</span><span class="p">)</span>
+
+    <span class="c1"># tick values</span>
+    <span class="n">minvalues</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;</span><span class="si">{0:.3f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mins</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
+                 <span class="s2">&quot;</span><span class="si">{0:.3f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="o">-</span><span class="n">mins</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
+                 <span class="nb">str</span><span class="p">(</span><span class="o">-</span><span class="n">mins</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span>
+                 <span class="s2">&quot;</span><span class="si">{0:.3f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="o">-</span><span class="n">mins</span><span class="p">[</span><span class="mi">3</span><span class="p">]),</span>
+                 <span class="s2">&quot;</span><span class="si">{0:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="o">-</span><span class="n">mins</span><span class="p">[</span><span class="mi">4</span><span class="p">]),</span>
+                 <span class="nb">str</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span>
+
+    <span class="n">maxvalues</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;</span><span class="si">{0:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">maxs</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
+                 <span class="s2">&quot;</span><span class="si">{0:.3f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="o">-</span><span class="n">maxs</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
+                 <span class="nb">str</span><span class="p">(</span><span class="o">-</span><span class="n">maxs</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span>
+                 <span class="s2">&quot;</span><span class="si">{0:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">maxs</span><span class="p">[</span><span class="mi">3</span><span class="p">]),</span>
+                 <span class="s2">&quot;</span><span class="si">{0:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="o">-</span><span class="n">maxs</span><span class="p">[</span><span class="mi">4</span><span class="p">]),</span>
+                 <span class="nb">str</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span>
+
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Preference -&gt;&quot;</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">([])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="n">minvalues</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span> <span class="o">+</span> <span class="n">objs_labels</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">objs_labels</span><span class="p">))])</span>
+
+    <span class="c1"># make a twin axis for toplabels</span>
+    <span class="n">ax1</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twiny</span><span class="p">()</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">([])</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="n">maxvalues</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">maxs</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
+
+    <span class="k">return</span> <span class="n">ax</span><span class="p">,</span> <span class="n">ax1</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-factor-performance">
+<h3><span class="section-number">C.2.2. </span><strong>plot_factor_performance()</strong><a class="headerlink" href="#plot-factor-performance" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_factor_performance</span><span class="p">(</span><span class="n">param_values</span><span class="p">,</span> <span class="n">collapse_days</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Visualize the performance of our policies in three-dimensional</span>
+<span class="sd">    parametric space.</span>
+
+<span class="sd">    :param param_values:                Saltelli sample array</span>
+<span class="sd">    :param collapse_days:               Simulation array</span>
+<span class="sd">    :param b:                           b parameter boundary interval</span>
+<span class="sd">    :param m:                           m parameter boundary interval</span>
+<span class="sd">    :param a:                           a parameter boundary interval</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set colormap</span>
+    <span class="n">cmap</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s2">&quot;RdBu_r&quot;</span><span class="p">)</span>
+
+    <span class="c1"># build figure object</span>
+    <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">figaspect</span><span class="p">(</span><span class="mf">0.5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">600</span><span class="p">,</span> <span class="n">constrained_layout</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+    <span class="c1"># set up scalable colormap</span>
+    <span class="n">sm</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">ScalarMappable</span><span class="p">(</span><span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">)</span>
+
+    <span class="c1"># set up subplot for profit maximizing policy</span>
+    <span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">projection</span><span class="o">=</span><span class="s1">&#39;3d&#39;</span><span class="p">)</span>
+
+    <span class="c1"># add point data for profit plot</span>
+    <span class="n">sows</span> <span class="o">=</span> <span class="n">ax1</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">param_values</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+                       <span class="n">param_values</span><span class="p">[:,</span><span class="mi">6</span><span class="p">],</span>
+                       <span class="n">param_values</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+                       <span class="n">c</span><span class="o">=</span><span class="n">collapse_days</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+                       <span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">,</span>
+                       <span class="n">s</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
+
+    <span class="c1"># add surface data for boundary separating successful and failed states of the world</span>
+    <span class="n">pts_ineq</span> <span class="o">=</span> <span class="n">ax1</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+
+    <span class="c1"># add reference point to plot</span>
+    <span class="n">pt_ref</span> <span class="o">=</span> <span class="n">ax1</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.005</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+
+    <span class="c1"># set up plot aesthetics and labels</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;b&quot;</span><span class="p">)</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;m&quot;</span><span class="p">)</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s2">&quot;a&quot;</span><span class="p">)</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_zlim</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">])</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">])</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_view_interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span>  <span class="mf">0.5</span><span class="p">)</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">view_init</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="o">-</span><span class="mi">17</span><span class="p">)</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Profit maximizing policy&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set up subplot for robust policy</span>
+    <span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">projection</span><span class="o">=</span><span class="s1">&#39;3d&#39;</span><span class="p">)</span>
+
+    <span class="c1"># add point data for robust plot</span>
+    <span class="n">sows</span> <span class="o">=</span> <span class="n">ax2</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">param_values</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+                       <span class="n">param_values</span><span class="p">[:,</span><span class="mi">6</span><span class="p">],</span>
+                       <span class="n">param_values</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+                       <span class="n">c</span><span class="o">=</span><span class="n">collapse_days</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+                       <span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">,</span>
+                       <span class="n">s</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
+
+    <span class="c1"># add surface data for boundary separating successful and failed states of the world</span>
+    <span class="n">pts_ineq</span> <span class="o">=</span> <span class="n">ax2</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+
+    <span class="c1"># add reference point to plot</span>
+    <span class="n">pt_ref</span> <span class="o">=</span> <span class="n">ax2</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.005</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+
+    <span class="c1"># set up plot aesthetics and labels</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;b&quot;</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;m&quot;</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s2">&quot;a&quot;</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_zlim</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">])</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">])</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_view_interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">view_init</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="o">-</span><span class="mi">17</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Robust policy&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set up colorbar</span>
+    <span class="n">sm</span><span class="o">.</span><span class="n">set_array</span><span class="p">([</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">collapse_days</span><span class="o">.</span><span class="n">max</span><span class="p">()])</span>
+    <span class="n">cbar</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">sm</span><span class="p">)</span>
+    <span class="n">cbar</span><span class="o">.</span><span class="n">set_label</span><span class="p">(</span><span class="s1">&#39;Days with predator collapse&#39;</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span>
+</pre></div>
+</div>
+</section>
+</section>
+</section>
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+    <i class="fa-solid fa-list"></i> Contents
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#hymod-ipynb">C.1. hymod.ipynb</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-observed-vs-simulated-streamflow">C.1.1. <strong>plot_observed_vs_simulated_streamflow()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-observed-vs-sensitivity-streamflow">C.1.2. <strong>plot_observed_vs_sensitivity_streamflow()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-monthly-heatmap">C.1.3. <strong>plot_monthly_heatmap()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-annual-heatmap">C.1.4. <strong>plot_annual_heatmap()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-varying-heatmap">C.1.5. <strong>plot_varying_heatmap()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-precalibration-flow">C.1.6. <strong>plot_precalibration_flow()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-precalibration-glue">C.1.7. <strong>plot_precalibration_glue()</strong></a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fishery-dynamics-ipynb">C.2. fishery_dynamics.ipynb</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-objective-performance">C.2.1. <strong>plot_objective_performance()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-factor-performance">C.2.2. <strong>plot_factor_performance()</strong></a></li>
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+.. _introduction:
+
+************
+Introduction
+************
+
+This guidance text has been developed in support of the Integrated Multisector Multiscale Modeling (IM3) Science Focus Area’s objective to formally integrate uncertainty into its research tasks. IM3 is focused on innovative modeling to explore how human and natural system landscapes in the United States co-evolve in response to short-term shocks and long-term influences. The project’s challenging scope is to advance our ability to study the interactions between energy, water, land, and urban systems, at scales ranging from local (~1km) to the contiguous United States, while consistently addressing influences such as population change, technology change, heat waves, and drought. Uncertainty and careful model-driven scientific insights are central to the project's science objectives shown below.
+
+**IM3 Key MSD Science Objectives:**
+
+*Develop flexible, open-source, and integrated modeling capabilities that capture the structure, dynamic behavior, and emergent properties of the multiscale interactions within and between human and natural systems.*
+
+*Use these capabilities to study the evolution, vulnerability, and resilience of interacting human and natural systems and landscapes from local to continental scales, including their responses to the compounding effects of long-term influences and short-term shocks.*
+
+*Understand the implications of uncertainty in data, observations, models, and model coupling approaches for projections of human-natural system dynamics.*
+
+Addressing the objectives above poses a strong transdisciplinary challenge that depends on a diversity of models and, more specifically, a consistent framing for making model-based science inferences. The term transdisciplinary science as used here formally implies a deep integration of disciplines to aid our hypothesis-driven understanding of coupled human-natural systems--bridging differences in theory, hypothesis generation, modeling, and modes of inference :cite:p:`national2014convergence`. The IM3 MSD research foci and questions require a deep integration across disciplines, where new modes of analysis can emerge that rapidly synthesize and exploit advances for making decision-relevant insights that at minimum acknowledge uncertainty and more ideally promote a rigorous quantitative mapping of its effects on the generality of claimed scientific insights. More broadly, diverse scientific disciplines engaged in the science of coupled human-natural systems, ranging from natural sciences to engineering and economics, employ a diversity of numerical computer models to study and understand their underlying systems of focus. The utility of these computer models hinges on their ability to represent the underlying real systems with sufficient fidelity and enable the inference of novel insights. This is particularly challenging in the case of coupled human-natural systems where there exists a multitude of interdependent human and natural processes taking place that could potentially be represented. These processes usually translate into modeled representations that are highly complex, non-linear, and exhibit strong interactions and threshold behaviors :cite:p:`elsawah2020eight, haimes2018risk,helbing2013globally`. Model complexity and detail have also been increasing as a result of our improving understanding of these processes, the availability of data, and the rapid growth in computing power :cite:p:`saltelli2019so`. As model complexity grows, modelers need to specify a lot more information than before: additional model inputs and relationships as more processes are represented, higher resolution data as more observations are collected, new coupling relationships and interactions as diverse models are being used in combination to answer multisector questions (e.g., the land-water-energy nexus). Typically, not all of this information is well known, nor is the impact of these many uncertainties on model outputs well understood. It is further especially difficult to distinguish the effects of individual as well as interacting sources of uncertainty when modeling coupled systems with multisector and multiscale dynamics :cite:p:`wirtz2017rocky`.
+
+Given the challenge and opportunity posed by the disciplinary diversity of IM3, we utilized an informal team-wide survey to understand how the various disciplines typically address uncertainty, emphasizing key literature examples and domain-specific reviews. The feedback received provided perspectives across diverse areas within the Earth sciences, different engineering fields, as well as economics. Although our synthesis of this survey information highlighted some commonality across areas (e.g., the frequent use of scenario-based modeling), we identified key differences in vocabulary, the frequency with which formal uncertainty analysis appears in the disciplinary literature, and technical approaches. The IM3 team’s responses captured a very broad conceptual continuum of methodological traditions, ranging from deterministic (no uncertainty) modeling to the theoretical case of fully engaging in modeling sources of uncertainty. Overall, error-driven analyses that focus on replicating prior observed conditions were reported to be the most prevalent types of studies for all disciplines. It was generally less common for studies to strongly engage with analyzing uncertainty via more formal ensemble analyses and design of experiments, though some areas did show significantly higher levels of activity. Another notable finding from our survey was the apparent lack of focus on understanding how model coupling relationships shape uncertainty. Although these observations are limited to the scope of feedback attained in the team-wide IM3 survey responses and the bodies of literature reported by respondents, we believe they reflect challenges that are common across the MSD community. 
+
+In the IM3 uncertainty-related research that has occurred since this survey, we have observed that differences in terminology and interpretation of terminology across modeling teams can be confounding. One of the goals of this eBook is to provide a common language for uncertainty analysis within IM3 and, hopefully, for the broader MSD community. While individual scientific disciplines would be expected to retain their own terminology, by providing explicit definitions of terms we can facilitate the translation of concepts across transdisciplinary science teams. To begin, we use the term Uncertainty Analysis (UA) as an umbrella phrase covering all methods in this eBook. Next, we distinguish the key terms of uncertainty quantification (UQ) and uncertainty characterization (UC). UQ refers to the formal focus on the full specification of likelihoods as well as the distributional forms necessary to infer the joint probabilistic response across all modeled factors of interest :cite:p:`cooke1991experts`. UC refers to exploratory modeling of alternative hypotheses to understand the co-evolutionary dynamics of influences and stressors, as well as path dependent changes in the form and function of modelled systems :cite:p:`moallemi2020exploratory,walker2003defining`. As discussed in later sections, the choice of UC or UQ depends on the specific goals of studies, the availability of data, the types of uncertainties (e.g., well-characterized or deep), and the complexity of underlying models as well as computational limits. Definitions of key uncertainty analysis terms used in this eBook appear below, and our Glossary (:numref:`glossary`) contains a complete list of terms.
+
+* **Exploratory modeling**: Use of large ensembles of uncertain conditions to discover decision-relevant combinations of uncertain factors
+* **Factor**: Any model component that can affect model outputs: inputs, resolution levels, coupling relationships, model relationships and parameters. In models with acceptable model fidelity these factors may represent elements of the real-world system under study.
+* **Sensitivity analysis**: Model evaluation to understand the factors and processes that most (or least) control a model’s outputs
+    * **Local sensitivity analysis**: Varying uncertain factors around specific reference values
+    * **Global sensitivity analysis**: Varying uncertain factors throughout their entire feasible value space
+* **Uncertainty characterization**: Model evaluation under alternative factor hypotheses to explore their implications for model output uncertainty
+* **Uncertainty quantification**: Representation of model output uncertainty using probability distributions
+
+At present, there is no singular guide for confronting the computational and conceptual challenges of the multi-model, transdisciplinary workflows that characterize ambitious projects such as IM3 :cite:p:`saltelli2015climate`. The primary aim of this text is to begin to address this gap and provide guidance for facing these challenges. :numref:`2_diagnostic_modeling` provides an overview of diagnostic modeling and the different perspectives for how we should evaluate our models, :numref:`3_sensitivity_analysis_the_basics` summarizes basic methods and concepts for sensitivity analysis, and :numref:`4_sensitivity_analysis` delves into more technical applications of sensitivity analysis to support diagnostic model evaluation and exploratory modeling. Finally, :numref:`5_conclusion` provides some concluding remarks across the UC and UQ topics covered in this text. The appendices of this text include a glossary of the key concepts, an overview of UQ methods, and coding-based illustrative examples of key UC concepts discussed in earlier chapters.
diff --git a/dev/docs/html/_sources/2.1_overview_of_model_diagnostics.rst b/dev/docs/html/_sources/2.1_overview_of_model_diagnostics.rst
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+Overview of model diagnostics
+#############################
+
+Model diagnostics provide a rich basis for hypothesis testing, model innovation, and improved inferences when classifying what is controlling highly consequential results (e.g., vulnerability or resilience in coupled human-natural systems). :numref:`Figure_2_1`, adapted from :cite:p:`saltelli2019so`, presents idealized illustrations of the relationship between UC and global sensitivity analysis for two coupled simulation models. The figure illustrates how UC can be used to address how uncertainties in various modeling decisions (e.g., data inputs, parameters, model structures, coupling relationships) can be sampled and simulated to yield the empirical model output distribution(s) of interest. Monte Carlo frameworks allow us to sample and propagate (or integrate) the ensemble response of the model(s) of focus. The first step of any UC analysis is the specification of the initial input distributions as illustrated in :numref:`Figure_2_1`. The second step is to perform the Monte Carlo simulations. The question can then be raised, which of the modeling assumptions in our Monte Carlo experiment are the most responsible for the resulting output uncertainty. We can answer this question using global sensitivity analysis as illustrated in :numref:`Figure_2_1`. Global sensitivity analysis can be defined as a formal Monte Carlo sampling and analysis of modeling choices (structures, parameters, inputs) to quantify their influence on direct model outputs (or output-informed metrics). UC experiments by themselves do not explain why a particular uncertain outcome is produced, but produce distributions of model outcomes, as portrayed by the yellow curve. The pie chart shown in :numref:`Figure_2_1` is a conceptual representation of the results of using a global sensitivity analysis to identify those factors that are most dominantly influencing results, either individually or interactively :cite:`saltelli_global_2008`.
+
+.. _Figure_2_1:
+.. figure:: _static/figure2_1_idealized_uc.png
+    :alt: Figure 2.1
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Idealized uncertainty characterization and global sensitivity analysis for two coupled simulation models. Uncertainty coming from various sources (e.g., inputs, model structures, coupling relationships) is propagated through the coupled model(s) to generate empirical distributions of outputs of interest (uncertainty characterization). This model output uncertainty can be decomposed to its origins, by means of sensitivity analysis. Figure adapted from :cite:t:`saltelli2019so`.
+
+UC and global sensitivity analysis are not independent modeling analyses. As illustrated here, any global sensitivity analysis requires an initial UC hypothesis in the form of statistical assumptions and representations for the modeling choices of focus (structural, parametric, and data inputs). Information from these two model diagnostic tools can then be used to inform data needs for future model runs, experiments to reduce the uncertainty present, or the simplification or enhancement of the model where necessary. Together UC and global sensitivity analysis provide a foundation for diagnostic exploratory modeling that has a consistent focus on the assumptions, structural model forms, alternative parameterizations, and input data sets that are used to characterize the behavioral space of one or more models.
diff --git a/dev/docs/html/_sources/2.2_perspectives_on_diagnostic_model_evaluation.rst b/dev/docs/html/_sources/2.2_perspectives_on_diagnostic_model_evaluation.rst
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+.. _perspectives:
+
+Perspectives on diagnostic model evaluation
+###########################################
+
+When we judge or diagnose models, the terms "verification" and "validation" are commonly used. However, their appropriateness in the context of numerical models representing complex coupled human-natural systems is questionable :cite:`beven2002towards, oreskes_verification_1994`. The core issue relates to the fact that these systems are often not fully known or perfectly implemented when modeled. Rather, they are defined within specific system framings and boundary conditions in an evolving learning process with the goal of making continual progress towards attaining higher levels of fidelity in capturing behaviors or properties of interest. Evaluating the fidelity of a model's performance can be highly challenging. For example, the observations used to evaluate the fidelity of parameterized processes are often measured at a finer resolution than what is represented in the model, creating the challenge of how to manage their relative scales when performing evaluation. In other cases, numerical models may neglect or simplify system processes because sufficient data is not available or the physical mechanisms are not fully known. If sufficient agreement between prediction and observation is not achieved, it is challenging to know whether these types of modeling choices are the cause, or if other issues, such as deficiencies in the input parameters and/or other modeling assumptions are the true cause of errors. Even if there is high agreement between prediction and observation, the model cannot necessarily be considered validated, as it is always possible that the right values were produced for the wrong reasons. For example, low error can stem from a situation where different errors in underlying assumptions or parameters cancel each other out (“compensatory errors”). Furthermore, coupled human-natural system models are often subject to “equifinality”, a situation where multiple parameterized formulations can produce similar outputs or equally acceptable representations of the observed data. There is therefore no uniquely “true” or validated model, and the common practice of selecting “the best” deterministic calibration set is more of an assumption than a finding :cite:`beven1993prophecy, beven_future_1992`. The situation becomes even more tenuous when observational data is limited in its scope and/or quality to be insufficient to distinguish model representations or their performance differences.
+
+These limitations on model verification undermine any purely positivist treatment of model validity: that a model should correctly and precisely represent reality to be valid. Under this perspective, closely related to empiricism, statistical tests should be used to compare the model’s output with observations and only through empirical verification can a model or theory be deemed credible. A criticism to this viewpoint (besides the aforementioned challenges for model verification) is that it reduces the justification of a model to the single criterion of predictive ability and accuracy :cite:`barlas_philosophical_1990`. Authors have argued that this ignores the explanatory power held in models and other procedures, which can also advance scientific knowledge :cite:`toulmin_form_1977`. These views gave rise to relativist perspectives of science, which instead place more value on model utility in terms of fitness for a specific purpose or inquiry, rather than representational accuracy and predictive ability :cite:`kleindorfer_validation_1998`. This viewpoint appears to be most prevalent among practitioners seeking decision-relevant insights (i.e., inspire new views vs. predict future conditions). The relativist perspective argues for the use of models as heuristics that can enhance our understanding and conceptions of system behaviors or possibilities :cite:`eker2018practice`. In contrast, natural sciences favor a positivist perspective, emphasizing similarity between simulation and observation even in application contexts where it is clear that projections are being made for conditions that have never been observed and the system of focus will have evolved structurally beyond the model representation being employed (e.g., decadal to centennial evolution of human-natural systems).
+
+These differences in prevalent perspectives are mirrored in how model validation is defined by the two camps: From the relativist perspective, validation is seen as a process of incremental “confidence building” in a model as a mechanism for insight :cite:`barlas_formal_1996`, whereas in natural sciences validation is framed as a way to classify a model as having an acceptable representation of physical reality :cite:`oreskes_verification_1994`. Even though the relativist viewpoint does not dismiss the importance of representational accuracy, it does place it within a larger process of establishing confidence through a variety of tools. These tools, not necessarily quantitative, include communicating information between practitioners and modelers, interpreting a multitude of model outputs, and contrasting preferences and viewpoints.
+
+On the technical side of the argument, differing views on the methodology of model validation appear as early as in the 1960’s. :cite:t:`naylor_verification_1967` argue that model validation should not be limited to a single metric or test of performance (e.g., a single error metric), but should rather be extended to multiple tests that reflect different aspects of a model’s structure and behavior. This and similar arguments are made in literature to this day :cite:`beven2018hypothesis, gupta2012towards, gupta_reconciling_2008, kumar_typology_2011, nearing_does_2020` and are primarily founded on two premises. First, that even though modelers widely recognize that their models are abstractions of the truth, they still make truth claims based on traditional performance metrics that measure the divergence of their model from observation :cite:`nearing_does_2020`. Second, that the natural systems mimicked by the models contain many processes that exhibit significant heterogeneity at various temporal and spatial scales. This heterogeneity is lost when a single performance measure is used, as a result of the inherent loss of process information occurring when transitioning from a highly dimensional and interactive system to the dimension of a single metric :cite:`beven2002towards`. These arguments are further elaborated in :numref:`4_sensitivity_analysis`.
+
+Multiple authors have proposed that the traditional reliance on single measures of model performance should be replaced by the evaluation of several model signatures (characteristics) to identify model structural errors and achieve a sufficient assessment of model performance :cite:`gupta_toward_1998, gupta_reconciling_2008, pianosi_understanding_2017, rouge_coordination_2019`. There is however a point of departure here, especially when models are used to produce inferences that can inform decisions. When agencies and practitioners use models of their systems for public decisions, those models have already met sufficient conditions for credibility (e.g., acceptable representational fidelity), but may face broader tests on their salience and legitimacy in informing negotiated decisions :cite:`cash_knowledge_2003, eker2018practice, white_credibility_2010`. This presents a new challenge to model validation, that of selecting decision-relevant performance metrics, reflective of the system’s stakeholders' viewpoints, so that the most consequential uncertainties are identified and addressed :cite:`saltelli_when_2014`. For complex multisector models at the intersection of climatic, hydrologic, agricultural, energy, or other processes, the output space is made up of a multitude of states and variables, with very different levels of salience to the system's stakeholders and to their goals being achieved :cite:`wagener_what_2019`. This is further complicated when such systems are also institutionally and dynamically complex. As a result, a broader set of qualitative and quantitative performance metrics is necessary to evaluate models of such complex systems, one that embraces the plurality of value systems, agencies and perspectives present. For IM3, even though the goal is to develop better projections of future vulnerability and resilience in co-evolving human-natural systems and not to provide decision support per se, it is critical for our multisector, multiscale model evaluation processes to represent stakeholders’ adaptive decision processes credibly.
+
+As a final point, when a model is used in a projection mode, its results are also subject to additional uncertainty, as there is no guarantee that the model’s functionality and predictive ability will stay the same as the baseline, where the verification and validation tests were conducted. This challenge requires an additional expansion of the scope of model evaluation: a broader set of uncertain conditions needs to be explored, spanning beyond historical observation and exploring a wide range of unprecedented conditions. This perspective on modeling, termed exploratory :cite:`bankes_exploratory_1993`, views models as computational experiments that can be used to explore vast ensembles of potential scenarios to identify those with consequential effects. Exploratory modeling literature explicitly orients experiments toward stakeholder consequences and decision-relevant inferences and shifts the focus from predicting future conditions to *discovering* which conditions lead to undesirable or desirable consequences.
+
+This evolution in modeling perspectives can be mirrored by the IM3 family of models in a progression from evaluating models relative to observed history to advanced formalized analyses to make inferences on multisector, multiscale vulnerabilities and resilience. Exploratory modeling approaches can help fashion experiments with large numbers of alternative hypotheses on the co-evolutionary dynamics of influences, stressors, as well as path-dependent changes in the form and function of human-natural systems :cite:`weaver_improving_2013`. The aim of this text is to therefore guide the reader through the use of sensitivity analysis (SA) methods across these perspectives on diagnostic and exploratory modeling.
diff --git a/dev/docs/html/_sources/2_diagnostic_modeling_overview_and_perspectives.rst b/dev/docs/html/_sources/2_diagnostic_modeling_overview_and_perspectives.rst
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+.. _2_diagnostic_modeling:
+
+*********************************************
+Diagnostic Modeling Overview and Perspectives
+*********************************************
+
+This text prescribes a formal model diagnostic approach that is a deliberative and iterative combination of state-of-the-art UC and global sensitivity analysis techniques that progresses from observed history-based fidelity evaluations to forward looking resilience and vulnerability inferences :cite:`gupta_reconciling_2008, hadjimichael_advancing_2020`.
+
+.. include:: 2.1_overview_of_model_diagnostics.rst
+
+.. include:: 2.2_perspectives_on_diagnostic_model_evaluation.rst
+
+.. note::
+    The following articles are suggested as fundamental reading for the information presented in this section:
+
+    * Naomi Oreskes, Kristin Shrader–Frechette, and Kenneth Belitz. Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences. *Science*, 263 (5147): 641-646, February 1994. URL: https://science.sciencemag.org/content/263/5147/641. DOI: https://doi.org/10.1126/science.263.5147.641.
+    * Keith Beven. Towards a coherent philosophy for modelling the environment. *Proceedings of the Royal Society of London*. Series A: mathematical, physical and engineering sciences, 458 (2026): 2465-2484, 2002.
+    * Eker, S., Rovenskaya, E., Obersteiner, M., Langan, S., 2018. Practice and perspectives in the validation of resource management models. *Nature Communications* 9, 1–10. https://doi.org/10.1038/s41467-018-07811-9
+
+    The following articles can be used as supplemental reading:
+
+    * Canham, C.D., Cole, J.J., Lauenroth, W.K. (Eds.), 2003. Models in Ecosystem Science. Princeton University Press. https://doi.org/10.2307/j.ctv1dwq0tq
+
+
+
diff --git a/dev/docs/html/_sources/3.1_global_versus_local_sensitivity.rst b/dev/docs/html/_sources/3.1_global_versus_local_sensitivity.rst
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+.. _global_vs_local:
+
+Global Versus Local Sensitivity
+###############################
+
+Out of the several definitions for sensitivity analysis presented in the literature, the most widely used has been proposed by :cite:t:`saltelli2004sensitivity` as “the study of how uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input”. In other words, sensitivity analysis explores the relationship between the model’s :math:`N` input variables, :math:`x=[x_1,x_2,...,x_N]`, and :math:`M` output variables, :math:`y=[y_1,y_2,...,y_M]` with :math:`y=g(x)`, where :math:`g` is the model that maps the model inputs to the outputs :cite:p:`borgonovo2016sensitivity`.
+
+Historically, there have been two broad categories of sensitivity analysis techniques: local and global. Local sensitivity analysis is performed by varying model parameters around specific reference values, with the goal of exploring how small input perturbations influence model performance. Due to its ease-of-use and limited computational demands, this approach has been widely used in literature, but has important limitations :cite:p:`rakovec2014distributed,saltelli2010avoid`. If the model is not linear, the results of local sensitivity analysis can be heavily biased, as they are strongly influenced by independence assumptions and a limited exploration of model inputs (e.g., :cite:t:`tang2007comparing`). If the model’s factors interact, local sensitivity analysis will underestimate their importance, as it does not account for those effects (e.g., :cite:p:`hamm2006variance`). In general, as local sensitivity analysis only partially and locally explores a model's parametric space, it is not considered a valid approach for nonlinear models :cite:p:`saltelli2019`. This is illustrated in :numref:`Figure_3_1` (a-b), presenting contour plots of a model response (:math:`y`) with an additive linear model (a) and with a nonlinear model (b). In a linear model without interactions between the input terms :math:`x_1` and :math:`x_2`, local sensitivity analysis (assuming deviations from some reference values) can produce appropriate sensitivity indices (:numref:`Figure_3_1` (a)). If however, factors :math:`x_1` and :math:`x_2` interact, the local and partial consideration of the space can not properly account for each factor’s effects on the model response (:numref:`Figure_3_1` (b)), as it is only informative at the reference value where it is applied. In contrast, a global sensitivity analysis varies uncertain factors within the entire feasible space of variable model responses (:numref:`Figure_3_1` (c)). This approach reveals the global effects of each parameter on the model output, including any interactive effects. For models that cannot be proven linear, global sensitivity analysis is preferred and this text is primarily discussing global sensitivity analysis methods. In the text that follows, whenever we use the term sensitivity analysis we are referring to its global application.
+
+.. _Figure_3_1:
+.. figure:: _static/figure3_1_global_versus_local.png
+    :alt: Figure 3.1
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Treatment of a two-dimensional space of variability by local (panels a-b) and global (panel c) sensitivity analyses. Panels depict contour plots with the value of a model response (:math:`y`) changing with changes in the values of input terms :math:`x_1` and :math:`x_2`. Local sensitivity analysis is only an appropriate approach to sensitivity in the case of linear models without interactions between terms, for example in panel (a), where :math:`y=3x_1+5x_2`. In the case of more complex models, for example in panels (b-c), where :math:`y={1 \above 1pt e^{x^2_1+x^2_2}} + {50 \above 1pt e^{(0.1x_1)^2+(0.1x_2)^3}}`, local sensitivity will miscalculate sensitivity indices as the assessed changes in the value :math:`y` depend on the assumed base values chose for :math:`x_1` and :math:`x_2` (panel (b)). In these cases, global sensitivity methods should be used instead (panel (c)). The points in panel (c) are generated using a uniform random sample of :math:`n=50`, but many other methods are available.
diff --git a/dev/docs/html/_sources/3.2_why_perform_sensitivity_analysis.rst b/dev/docs/html/_sources/3.2_why_perform_sensitivity_analysis.rst
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+++ b/dev/docs/html/_sources/3.2_why_perform_sensitivity_analysis.rst
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+.. _why_SA:
+
+Why Perform Sensitivity Analysis
+################################
+
+It is important to understand the many ways in which a SA might be of use to your modeling effort. Most commonly, one might be motivated to perform sensitivity analysis for the following reasons:
+
+*Model evaluation*: Sensitivity analysis can be used to gauge model inferences when assumptions about the structure of the model or its parameterization are dubious or have changed. For instance, consider a numerical model that uses a set of calibrated parameter values to produce outputs, which we then use to inform decisions about the real-world system represented. One might like to know if small changes in these parameter values significantly change this model’s output and the decisions it informs or if, instead, our parameter inferences yield stable model behavior regardless of the uncertainty present in the specific parameterized processes or properties. This can either discredit or lend credence to the model at hand, as well as any inferences drawn that are founded on its accurate representation of the system. Sensitivity analysis can identify which uncertain model factors cause this undesirable model behavior.
+
+*Model simplification*: Sensitivity analysis can also be used to identify factors or components of the model that appear to have limited effects on direct outputs or metrics of interest. Consider a model that has been developed in an organization for the purposes of a specific research question and is later used in the context of a different application. Some processes represented in significant detail might no longer be of the same importance while consuming significant data or computational resources, as different outputs might be pertinent to the new application. Sensitivity analysis can be used to identify unimportant model components and simplify them to nominal values and reduced model forms. Model complexity and computational costs can therefore be reduced.
+
+*Model refinement*: Alternatively, sensitivity analysis can reveal the factors or processes that are highly influential to the outputs or metrics of interest, by assessing their relative importance. In the context of model evaluation, this can inform which model components warrant additional investigation or measurement so the uncertainty surrounding them and the resulting model outputs or metrics of interest can be reduced.
+
+*Exploratory modeling*: When sufficient credence has been established in the model, sensitivity analysis can be applied to a host of other inquiries. Inferences about the factors and processes that most (or least) control a model’s outputs of interest can be extrapolated to the real system they represent and be used in a heuristic manner to inform model-based inferences. On this foundation, a model paired with the advanced techniques presented in this text can be used to “discover” decision relevant and highly consequential outcomes (i.e., scenario discovery, discussed in more detail in :numref:`consequential_scenarios` :cite:p:`bankes_exploratory_1993, bryant2010thinking`).
+
+The nature and context of the model shapes the specific objectives of applying a sensitivity analysis, as well as methods and tools most appropriate and defensible for each application setting :cite:`saltelli2004sensitivity, saltelli2002relative, wagener_what_2019`. The three most common sensitivity analysis modes (*Factor Prioritization*, *Factor Fixing*, and *Factor Mapping*) are presented below, but the reader should be aware that other uses have been proposed in the literature (e.g., :cite:`anderson2014uncertainty,borgonovo2010sensitivity`).
+
+*Factor prioritization*: This sensitivity analysis application mode (also referred to as *factor ranking*) refers to when one would like to identify the uncertain factors that have the greatest impact on the variability of the output, and which, when fixed to their true value (i.e., if there were no uncertainty regarding their value), would lead to the greatest reduction in output variability :cite:`saltelli2008global`. Information from this type of analysis can be crucial to model improvement as these factors can become the focus of future measurement campaigns or numerical experiments so that uncertainty in the model output can be reduced. The impact of each uncertain input on the variance of the model output is often used as the criterion for factor prioritization. :numref:`Figure_3_2` (a) shows the effects of three uncertain variables (:math:`X_1`, :math:`X_2`, and :math:`X_3`) on the variance of output :math:`Y`. :math:`V(E(Y|X_i))` indicates the variance in :math:`Y` if factor :math:`X_i` is left to vary freely while all other factors remain fixed to nominal values. In this case, factor :math:`X_2` makes the largest contribution to the variability of output :math:`Y` and it should therefore be prioritized. In the context of risk analysis, factor prioritization can be used to reduce output variance to below a given tolerable threshold (also known as variance cutting).
+
+*Factor fixing*: This mode of sensitivity analysis (also referred to as *factor screening*) aims to identify the model components that have a negligible effect or make no significant contributions to the variability of the outputs or metrics of interest (usually referred to as non-influential :cite:`saltelli2008global`). In the stylized example of :numref:`Figure_3_2` (a), :math:`X_1` makes the smallest contribution to the variability of output :math:`Y` suggesting that the uncertainty in its value could be negligible and the factor itself fixed in subsequent model executions. Eliminating these factors or processes in the model or fixing them to a nominal value can help reduce model complexity as well as the unnecessary computational burden of subsequent model runs, results processing, or other sensitivity analyses (the fewer uncertain factors considered, the fewer runs are necessary to illuminate their effects on the output). Significance of the outcome can be gauged in a variety of manners, depending on the application. For instance, if applying a variance-based method, a minimum threshold value of contribution to the variance could be considered as a significance ‘cutoff’, and factors with indices below that value can be considered non-influential. Conclusions about factor fixing should be made carefully, considering all of the effects a factor has, individually and in interaction with other factors (explained in more detail in the :numref:`variance-based_methods`).
+
+*Factor mapping*: Finally, factor mapping can be used to pinpoint which values of uncertain factors lead to model outputs within a given range of the output space :cite:`saltelli2008global`. In the context of model diagnostics, it is possible that the model’s output changes in ways considered impossible based on the represented processes, or other observed evidence. In this situation, factor mapping can be used to identify which uncertain model factors cause this undesirable model behavior by ‘filtering’ model runs that are considered ‘non-behavioral’ :cite:`edwards2011precalibrating,pianosi2016sensitivity,spear1980eutrophication`. In :numref:`Figure_3_2` (b), region :math:`B` of the output space :math:`Y` denotes the set of behavioral model outcomes and region :math:`\bar{B}` denotes the set of non-behavioral outcomes, resulting from the entirety of input space :math:`X`. Factor mapping refers to the process of tracing which factor values of input space :math:`X` produce the behavioral model outcomes in the output space.
+
+.. _Figure_3_2:
+.. figure:: _static/figure3_2_factor_mapping.png
+    :alt: Figure 3.2
+    :width: 500px
+    :figclass: margin-caption
+    :align: center
+
+    Factor prioritization, factor fixing and factor mapping settings of sensitivity analysis.
+
+The language used above reflects a use of sensitivity analysis for model fidelity evaluation and refinement. However, as previously mentioned, when a model has been established as a sufficiently accurate representation of the system, sensitivity analysis can produce additional inferences (i.e., exploratory modeling and scenario discovery). For instance, under the factor mapping use, the analyst can now focus on undesirable system states and discover which factors are most responsible for them: for instance, “population growth of above 25% would be responsible for unacceptably high energy demands”. Factor prioritization and factor fixing can be used to make equivalent inferences, such as “growing populations and increasing temperatures are the leading factors for changing energy demands” (prioritizing of factors) or “changing dietary needs are inconsequential to increasing energy demands for this region” (a factor that can be fixed in subsequent model runs). All these inferences hinge on the assumption that the real system’s stakeholders consider the model states faithful enough representations of system states. As elaborated in :numref:`perspectives`, this view on sensitivity analysis is founded on a relativist perspective on modeling, which tends to place more value on model usefulness rather than strict accuracy of representation in terms of error. As such, sensitivity analysis performed with decision-making relevance in mind will focus on model outputs or metrics that are consequential and decision relevant (e.g., energy demand in the examples above).
\ No newline at end of file
diff --git a/dev/docs/html/_sources/3.3.1_one_at_a_time_oat.rst b/dev/docs/html/_sources/3.3.1_one_at_a_time_oat.rst
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+One-At-a-Time (OAT)
+*******************
+
+In this approach, only one model factor is changed at a time while all others are kept fixed across each iteration in a sampling sequence. The OAT method assumes that model factors of focus are linearly independent (i.e., there are no interactions) and can analyze how factors individually influence model outputs or metrics of interest. While popular given its ease of implementation, OAT  is ultimately limited in its exploration of a model’s sensitivities :cite:p:`saltelli2008global`. It is primarily used with local sensitivity techniques with similar criticisms: applying this sampling scheme on a system with nonlinear and interactive processes will miss important information on the effect uncertain factors have on the model. OAT samplings can be repeated multiple times in a more sophisticated manner and across different locations of the parameter space to overcome some of these challenges, which would increase computational costs and negate the main reasons for its selection. Given these limitations OAT methods could be used as preliminary, low-cost analyses of the factors' individual effects, but should ultimately be complemented with more sophisticated methods.
diff --git a/dev/docs/html/_sources/3.3.2_full_fractional_factorial_sampling.rst b/dev/docs/html/_sources/3.3.2_full_fractional_factorial_sampling.rst
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+++ b/dev/docs/html/_sources/3.3.2_full_fractional_factorial_sampling.rst
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+Full and Fractional Factorial Sampling
+**************************************
+
+In full factorial sampling each factor is treated as being discrete by considering two or more levels (or intervals) of its values. The sampling process then generates samples within each possible combination of levels, corresponding to each parameter. This scheme produces a more comprehensive sampling of the factors’ variability space, as it accounts for all candidate combinations of factor levels (:numref:`Figure_3_3` (a)). If the number of levels is the same across all factors, the number of generated samples is estimated using :math:`n^k`, where :math:`n` is the number of levels and :math:`k` is the number of factors. For example, :numref:`Figure_3_3` (a) presents a full factorial sampling of three uncertain factors :math:`(x_1,` :math:`x_2,` and :math:`x_3)`, each considered as having four discrete levels. The total number of samples necessary for such an experiment is :math:`4^3=64`. As the number of factors increases, the number of simulations necessary will also grow exponentially, making full factorial sampling computationally burdensome (:numref:`Figure_3_3` (b)). As a result, it is common in the literature to apply full factorial sampling at only two levels per factor, typically the two extremes :cite:`montgomery2017design`. This significantly reduces computational burden but is only considered appropriate in cases where factors can indeed only assume two discrete values (e.g., when testing the effects of epistemic uncertainty and comparing between model structure A and model structure B). In the case of physical parameters on continuous distributions (e.g., when considering the effects of measurement uncertainty in a temperature sensor), discretizing the range of a factor to only extreme levels can bias its estimated importance.
+
+Fractional factorial sampling is a widely used alternative to full factorial sampling that allows the analyst to significantly reduce the number of simulations by focusing on the main effects of a factor and seeking to avoid model runs that yield redundant response information :cite:p:`saltelli2008global`. In other words, if one can reasonably assume that higher-order interactions are negligible, information about the most significant effects and lower-order interactions (e.g., effects from pairs of factors) can be obtained using a fraction of the full factorial design. Traditionally, fractional factorial design has also been limited to two levels :cite:p:`montgomery2017design`, referred to as Fractional Factorial designs 2k-p :cite:p:`box19612`. Recently, Generalized Fractional Factorial designs have also been proposed that allow for the structured generation of samples at more than two levels per factor :cite:p:`surowiec2017generalized`. Consider a case where the modeling team dealing with the problem in :numref:`Figure_3_3` (a) cannot afford to perform 64 simulations of their model. They can afford 32 runs for their experiment and instead decide to fractionally sample the variability space of their factors. A potential design of such a sampling strategy is presented in :numref:`Figure_3_3` (c).
+
+.. _Figure_3_3:
+.. figure:: _static/figure3_3_alternative_designs.png
+    :alt: Figure 3.3
+    :width: 500px
+    :figclass: margin-caption
+    :align: center
+
+    Alternative designs of experiments and their computational costs for three uncertain factors :math:`(x_1,` :math:`x_2,` and :math:`x_3)`. (a) Full factorial design sampling of three factors at four levels, at a total of 64 samples; (b) exponential growth of necessary number of samples when applying full factorial design at four levels; (c) fractional factorial design of three factors at four levels, at a total of 32 samples; and (d) Latin Hypercube sample of three factors with uniform distributions, at a total of 32 samples.
diff --git a/dev/docs/html/_sources/3.3.3_latin_hypercube_sampling.rst b/dev/docs/html/_sources/3.3.3_latin_hypercube_sampling.rst
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+Latin Hypercube Sampling (LHS)
+*****************************
+
+Latin hypercube sampling (LHS) :cite:p:`mckay1979` is one of the most common methods in space-filling experimental designs. With this sampling technique, for :math:`N` uncertain factors, an :math:`N`-dimensional hypercube is generated, with each factor divided into an equal number of levels depending on the total number of samples to be generated. Equal numbers of samples are then randomly generated at each level, across all factors. In this manner, latin hypercube design guarantees sampling from every level of the variability space and without any overlaps. When the number of samples generated is much larger than the number of uncertain factors, LHS can be very effective in examining the effects of each factor :cite:p:`saltelli2008global`. LHS is an attractive technique, because it guarantees a diverse coverage of the space, through the use of subintervals, without being constrained to discrete levels for each factor - compare :numref:`Figure_3_3` (c) with :numref:`Figure_3_3` (d) for the same number of samples.
+
+LHS is less effective when the number of samples is not much larger than the number of uncertain factors, and the effects of each factor cannot be appropriately distinguished. The samples between factors can also be highly correlated, biasing any subsequent sensitivity analysis results. To address this, the sampling scheme can be modified to control for the correlation in parameters while maximizing the information derived. An example of such modification is through the use of orthogonal arrays :cite:p:`tang1993orthogonal`.
diff --git a/dev/docs/html/_sources/3.3.4_low_discrepancy_sequences.rst b/dev/docs/html/_sources/3.3.4_low_discrepancy_sequences.rst
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+Low-Discrepancy Sequences
+*************************
+
+Low-discrepancy sequences is another sampling technique that employs a pseudo-random generator for Monte Carlo sampling :cite:p:`dalal2008low, zaremba1968mathematical`. These quasi-Monte Carlo methods eliminate 'lumpiness' across samples (i.e, the presence of gaps and clusters) by minimizing discrepancy across the hypercube samples. Discrepancy can be quantitatively measured using the deviations of sampled points from a uniform distribution :cite:`kucherenko2015exploring, dalal2008low`. Low-discrepancy sequences ensure that the number of samples in any subspace of the variability hypercube is approximately the same. This is not something guaranteed by Latin Hypercube sampling, and even though its design can be improved through optimization with various criteria, such adjustments are limited to small sample sizes and low dimensions :cite:`iooss2010numerical,jin2003efficient,kucherenko2015exploring,morris1995exploratory,park1994optimal`. In contrast, the Sobol sequence :cite:p:`sobol1976uniformly,sobol1967distribution`, one of the most widely used sampling techniques, utilizes the low-discrepancy approach to uniformly fill the sampled factor space. A core advantage of this style of sampling is that it takes far fewer samples (i.e., simulations) to attain a much lower level of error in estimating model output statistics (e.g., the mean and variance of outputs).
+
+.. note::
+
+    Put this into practice! Click the following link to try out an interactive tutorial which uses Sobol sequence sampling for the purposes of a Sobol sensitivity analysis:  `Sobol SA using SALib Jupyter Notebook <nb_saltelli_sobol_>`_
diff --git a/dev/docs/html/_sources/3.3.5_other_types_of_sampling.rst b/dev/docs/html/_sources/3.3.5_other_types_of_sampling.rst
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+++ b/dev/docs/html/_sources/3.3.5_other_types_of_sampling.rst
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+Other types of sampling
+***********************
+
+The sampling techniques mentioned so far are general sampling methods useful for a variety of applications beyond sensitivity analysis. There are however techniques that have been developed for specific sensitivity analysis methods. Examples of these methods include the Morris One-At-a-Time :cite:p:`morris1991factorial`, Fourier Amplitude Sensitivity Test (FAST; :cite:p:`cukier1973study`), Extended FAST :cite:p:`saltelli1999quantitative`, and Extended Sobol methods :cite:p:`sobol2001global`. For example, the Morris sampling strategy builds a number of trajectories (usually referred to as repetitions and denoted by :math:`r`) in the input space each composed of :math:`N+1` factor points, where :math:`N` is the number of uncertain factors. The first point of the trajectory is selected randomly and the subsequent :math:`N` points are generated by moving one factor at a time by a fixed amount. Each factor is perturbed once along the trajectory, while the starting points of all of the trajectories are randomly and uniformly distributed. Several variations of this strategy also exist in the literature; for more details on each approach and their differences the reader is directed to :cite:t:`pianosi2016sensitivity`.
diff --git a/dev/docs/html/_sources/3.3.6_synthetic_generation_of_input_time_series.rst b/dev/docs/html/_sources/3.3.6_synthetic_generation_of_input_time_series.rst
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--- /dev/null
+++ b/dev/docs/html/_sources/3.3.6_synthetic_generation_of_input_time_series.rst
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+Synthetic generation of input time series
+*****************************************
+
+Models often have input time series or processes with strong temporal and/or spatial correlations (e.g., streamflow, energy demand, pricing of commodities, etc.) that, while they might not immediately come to mind as factors to be examined in sensitivity analysis, can be treated as such. Synthetic input time series are used for a variety of reasons, for example, when observations are not available or are limited, or when past observations are not considered sufficiently representative to capture rare or extreme events of interest :cite:p:`herman2016synthetic,milly2008stationarity`. Synthetic generation of input time series provides a valuable tool to consider non-stationarity and incorporate potential stressors, such as climate change impacts into input time series :cite:p:`borgomeo2015numerical`. For example, a century of record will be insufficient to capture very high impact rare extreme events (e.g., persistent multi-year droughts). A large body of statistical literature exists focusing on the topics of synthetic weather :cite:p:`herrera2017review,wilks1999weather` and streamflow :cite:p:`lamontagne2018generating,medda2019comparison` generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound (hot, severe drought) extremes. It is beyond the scope of this text to review these methods, but readers are encouraged to explore the studies cited above as well as the following publications for discussions and comparisons of these methods: :cite:p:`borgomeo2015numerical,herman2016synthetic,kirsch2013evaluating,loucks2017water,steinschneider2015integrated,vogel2017stochastic,vogel1988value`. The use of these methods for the purposes of exploratory modeling, especially in the context of well-characterized versus deep uncertainty, is further discussed in :numref:`consequential_scenarios`.
diff --git a/dev/docs/html/_sources/3.3_design_of_experiments.rst b/dev/docs/html/_sources/3.3_design_of_experiments.rst
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+.. _design_of_experiments:
+
+Design of Experiments
+#####################
+
+Before conducting a sensitivity analysis, the first element that needs to be clarified is the uncertainty space of the model :cite:p:`helton2006survey,pianosi2016sensitivity`. In other words, how many and which factors making up the mathematical model are considered uncertain and can potentially affect the model output and the inferences drawn from it. Uncertain factors can be model parameters, model structures, inputs, or alternative model resolution levels (scales), all of which can be assessed through the tools presented in this text. Depending on the kind of factor, its variability can be elicited through various means: expert opinion, values reported in the literature, historical observations, its physical meaning (e.g., population values in a city can never be negative), or through the use of more formal UQ methods (:numref:`A1_uncertainty_quantification`). The model uncertainty space represents the entire space of variability present in each of the uncertain factors of a model. The complexity of most real-world models means that the response function, :math:`y=g(x)`, mapping inputs to outputs, is hardly ever available in an analytical form and therefore analytically computing the sensitivity of the output to each uncertain factor becomes impossible. In these cases, sensitivity analysis is only feasible through numerical procedures that employ different strategies to sample the uncertainty space and calculate sensitivity indices.
+
+A sampling strategy is often referred to as a *design of experiments* and represents a methodological choice made before conducting any sensitivity analysis. Experimental design was first introduced by :cite:t:`fisher1936design` in the context of laboratory or field-based experiments. Its application in sensitivity analysis is similar to setting up a physical experiment in that it is used to discover the behavior of a system under specific conditions. An ideal design of experiments should provide a framework for the extraction of all plausible information about the impact of each factor on the output of the model. The design of experiments is used to set up a simulation platform with the minimum computational cost to answer specific questions that cannot be readily drawn from the data through analytical or common data mining techniques. Models representing coupled human-natural systems usually have a large number of inputs, state variables and parameters, but not all of them exert fundamental control over the numerical process, despite their uncertainty, nor have substantial impacts on the model output, either independently or through their interactions. Each factor influences the model output in different ways that need to be discovered. For example, the influence of a parameter on model output can be linear or non-linear and can be continuous or only be active during specific times or at particular states of the system :cite:p:`herman2013time, massmann2014new`. An effective and efficient design of experiments allows the analyst to explore these complex relationships and evaluate different behaviors of the model for various scientific questions :cite:p:`van2016sensitivity`. The rest of this section overviews some of the most commonly used designs of experiments. Table 1 summarizes the designs discussed.
+
+.. list-table:: Summary of designs of experiments overviewed in this section. * Depends on the sample size.
+  :header-rows: 1
+
+  * - *Design of experiments*
+    - *Factor interactions considered*
+    - *Treatment of factor domains*
+  * - One-At-a-Time (OAT)
+    - No - main effects only
+    - Continuous (distributions)
+  * - Full Factorial Sampling
+    - Yes - including total effects
+    - Discrete (levels)
+  * - Fractional Factorial Sampling
+    - Yes - only lower-order effects*
+    - Discrete (levels)
+  * - Latin Hypercube (LH) Sampling
+    - Yes - including total effects*
+    - Continuous (distributions)
+  * - Quasi-Random Sampling with Low-Discrepancy Sequences
+    - Yes - including total effects*
+    - Continuous (distributions)
+
+There are a few different approaches to the design of experiments, closely related to the chosen sensitivity analysis approach, which is in turn shaped by the research motivations, scientific questions, and computational constraints at hand (additional discussion of this can be found at the end of :numref:`3_sensitivity_analysis_the_basics`). For example, in a sensitivity analysis using perturbation and derivatives methods, the model input parameters vary from their nominal values one at a time, something that the design of experiments needs to reflect. If, instead, one were to perform sensitivity analysis using a multiple-starts perturbation method, the design of experiments needs to consider that multiple points across the factor space are used. The design of experiments specifically defines two key characteristics of samples that are fed to the numerical model: the number of samples and the range of each factor.
+
+Generally, sampling can be performed randomly or by applying a stratifying approach. In random sampling, such as Monte Carlo :cite:p:`metropolis1949monte`, samples are randomly generated by a pseudo-random number generator with an a-priori assumption about the distribution of parameters and their possible ranges. Random seeds can also be used to ensure consistency and higher control over the random process. However, this method could leave some gaps in the parameter space and cause clustering in some spaces, especially for a large number of parameters :cite:p:`norton2015introduction`. Most sampling strategies use stratified sampling to mitigate these disadvantages. Stratified sampling techniques divide the domain of each factor into subintervals, often of equal lengths. From each subinterval, an equal number of samples is drawn randomly, or based on the specific locations within the subintervals :cite:p:`saltelli2008global`.
+
+.. include:: 3.3.1_one_at_a_time_oat.rst
+
+.. include:: 3.3.2_full_fractional_factorial_sampling.rst
+
+.. include:: 3.3.3_latin_hypercube_sampling.rst
+
+.. include:: 3.3.4_low_discrepancy_sequences.rst
+
+.. include:: 3.3.5_other_types_of_sampling.rst
+
+.. include:: 3.3.6_synthetic_generation_of_input_time_series.rst
diff --git a/dev/docs/html/_sources/3.4.1_derivative_based_methods.rst b/dev/docs/html/_sources/3.4.1_derivative_based_methods.rst
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+Derivative-based Methods
+************************
+
+Derivative-based methods explore how model outputs are affected by perturbations in a single model input around a particular input value. These methods are local and are performed using OAT sampling. For simplicity of mathematical notations, let us assume that the model :math:`g(X)` only returns one output. Following :cite:p:`borgonovo2008sensitivity` and :cite:p:`pianosi2016sensitivity`, the sensitivity index, :math:`S_i` , of the model’s *i*-th input factor, :math:`x_i` , can be measured using the partial derivative evaluated at a nominal value, :math:`\bar{x}`, of the vector of inputs:
+
+.. math::
+
+  S_i (\bar{x}) = \frac{\partial g}{\partial x} |_{\bar{x}{^c{_i}}}
+
+where *c*\ :sub:`i`\  is the scaling factor. In most applications however, the relationship :math:`g(X)` is not fully known in its analytical form, and therefore the above partial derivative is usually approximated: 
+
+.. math::
+
+  S_i (\bar{x}) = \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i
+
+Using this approximation, the *i*-th input factor is perturbed by a magnitude of :math:`\Delta_i`, and its relative importance is calculated. Derivative-based methods are some of the oldest sensitivity analysis methods as they only require :math:`N+1` model evaluations to estimate indices for :math:`N` uncertain factors. As described above, being computationally very cheap comes at the cost of not being able to explore the entire input space, but only (local) perturbations to the nominal value. Additionally, as these methods examine the effects of each input factor one at a time, they cannot assess parametric interactions or capture the interacting nature of many real systems and the models that abstract them.
diff --git a/dev/docs/html/_sources/3.4.2_elementary_effect_methods.rst b/dev/docs/html/_sources/3.4.2_elementary_effect_methods.rst
new file mode 100644
index 0000000..f3bc67c
--- /dev/null
+++ b/dev/docs/html/_sources/3.4.2_elementary_effect_methods.rst
@@ -0,0 +1,29 @@
+Elementary Effect Methods
+*************************
+
+Elementary effect (EE) SA methods provide a solution to the local nature of the derivative-based methods by exploring the entire parametric range of each input parameter :cite:p:`iooss2015review`. However, EE methods still use OAT sampling and do not vary all input parameters simultaneously while exploring the parametric space. The OAT nature of EEs methods therefore prevents them from properly capturing the interactions between uncertain factors. EEs methods are computationally efficient compared to their All-At-a-Time (AAT) counterparts, making them more suitable when computational capacity is a limiting factor, while still allowing for some inferences regarding factor interactions.
+The most popular EE method is the Method of Morris :cite:p:`morris1991factorial`. Following the notation by :cite:p:`pianosi2016sensitivity`, this method calculates global sensitivity using the mean of the EEs (finite differences) of each parameter at different locations:
+
+.. math::
+
+  S_i = \mu_i^* = \frac{1}{r}\sum_{j=1}^r EE^j_i = \frac{1}{r}\sum_{j=1}^r \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i
+
+
+with :math:`r` representing the number of sample repetitions (also refered to as trajectories) in the input space, usually set between 4 and 10 :cite:`saltelli2004sensitivity`. Each :math:`x_j` represents the points of each trajectory, with :math:`j=1,…, r`, selected as described in the sampling strategy for this method, found above. This method also produces the standard deviation of the EEs:
+
+.. math::
+
+  \sigma_i = \sqrt{\frac{1}{r}\sum_{j=1}^r(EE_i^j-\frac{1}{r}\sum_{j=1}^r EE^j_i)^2}
+
+
+which is a measure of parametric interactions. Higher values of :math:`\sigma_i` suggest model responses at different levels of factor :math:`x_i` are significantly different, which indicates considerable interactions between that and other uncertain factors. The values of :math:`\mu_i^*` and :math:`\sigma_i` for each factor allow us to draw several different conclusions, illustrated in :numref:`Figure_3_4`, following the example by :cite:`iooss2015review`. In this example, factors :math:`x_1`, :math:`x_2`, :math:`x_4`, and :math:`x_5` can be said to have an influence on the model outputs, with :math:`x_1`, :math:`x_4`, and :math:`x_5` having some interactive or non-linear effects. Depending on the orders of magnitude of :math:`\mu_i^*` and :math:`\sigma_i` one can indirectly deduce whether the factors have strong interactive effects, for example if a factor :math:`\sigma_i << \mu_i^*` then the relationship between that factor and the output can be assumed to be largely linear (note that this is still an OAT method and assumptions on factor interactions should be strongly caveated). Extensions of the Method of Morris have also been developed specifically for the purposes of factor fixing and explorations of parametric interactions (e.g., :cite:p:`borgonovo2010sensitivity,campolongo1999use,cropp2002new`).
+
+
+.. _Figure_3_4:
+.. figure:: _static/figure3_4_morris_method.png
+    :alt: Figure 3.4
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Illustrative results of the Morris Method. Factors :math:`x_1`, :math:`x_2`, :math:`x_4`, and :math:`x_5` have an influence on the model outputs, with :math:`x_1`, :math:`x_4`, and :math:`x_5` having interactive or non-linear effects. Whether or not a factor should be considered influential to the output depends on the output selected and is specific to the research context and purpose of the analysis, as discussed in :numref:`why_SA`.
diff --git a/dev/docs/html/_sources/3.4.3_regression_based_methods.rst b/dev/docs/html/_sources/3.4.3_regression_based_methods.rst
new file mode 100644
index 0000000..722cb9c
--- /dev/null
+++ b/dev/docs/html/_sources/3.4.3_regression_based_methods.rst
@@ -0,0 +1,27 @@
+Regression-based Methods
+************************
+Regression analysis is one of the oldest ways of investigating parametric importance and sensitivity :cite:p:`saltelli2004sensitivity`. Here, we describe some of the most popular regression-based sensitivity indices. One of the main sensitivity indices of this category is the standardized regression coefficient (SRC). To calculate SRC, a linear regression relationship needs to be fitted between the input vector, :math:`x`, and the model output of interest by using a least-square minimizing method:
+
+.. math::
+  y = b_0 + \sum_{i=1}^N b_ix_i
+
+where :math:`b_0` and :math:`b_i` (corresponding to the *i*-th model input) are regression coefficients. The following relationship can then be used to calculate the SRCs for different input values:
+
+.. math::
+  S_i=SRC_i=b_i\frac{\sigma_i}{\sigma_y}
+
+where :math:`\sigma_i` and :math:`\sigma_y` are standard deviations of *i*-th model input and output, respectively.
+
+Several other regression-based indices explore the correlation between input and output parameters as a proxy to model parametric sensitivity :cite:`helton1993uncertainty,iooss2015review,manache2008identification`. The Pearson correlation coefficient (PCC) can be used when a linear relationship exists between an uncertain factor, :math:`x_i`, and the output :math:`y`:
+
+.. math::
+  S_i=PCC=\frac{cov(x_i,y)}{\sigma_i\sigma_y}
+
+In cases when there are outliers in the data or the relationship between the uncertain factors and the output is not linear, rank-based correlation coefficients are preferred, for example, Spearman’s rank correlation coefficient (SRCC):
+
+.. math::
+  S_i=SRCC=\frac{cov(rx_i,ryi)}{\sigma_{ri}\sigma_{ry}}
+
+where the raw values of :math:`x_i` and :math:`y` and converted to ranks :math:`rx_i` and :math:`ry` respectively, which instead represent a measurement of the strength of the monotonic relationship, rather than linear relationship, between the input and output. Other regression-based metrics include the partial correlations coefficient, the partial rank correlations coefficient, and the Nash-Sutcliffe coefficient, more discussion on which can be found in :cite:p:`iooss2015review,borgonovo2016sensitivity`.
+
+Tree-based regression techniques have also been used for sensitivity analysis in an effort to address the challenges faced with nonlinear models :cite:p:`pappenberger2006ignorance`. Examples of these methods include the Patient Rule Induction Method (PRIM; :cite:p:`friedman1999bump`) and Classification And Regression Trees (CART; :cite:p:`breiman1984classification`). CART-based approaches also include boosting and bagging extensions :cite:p:`freund1999short,breiman1996bagging`. These methods are particularly useful when sensitivity analysis is used for factor mapping (i.e., when trying to identify which uncertain model factors produce a certain model behavior). :numref:`consequential_scenarios` elaborates on the use of these methods. Regression-based sensitivity analysis methods are global by nature and can explore the entire space of variables. However, the true level of comprehensiveness depends on the design of experiments and the number of simulations providing data to establish the regression relationships. Although they are usually computationally efficient, they do not produce significant information about parametric interactions :cite:p:`borgonovo2016sensitivity,saltelli2004sensitivity`.
diff --git a/dev/docs/html/_sources/3.4.4_regional_sensitivity_analysis.rst b/dev/docs/html/_sources/3.4.4_regional_sensitivity_analysis.rst
new file mode 100644
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+++ b/dev/docs/html/_sources/3.4.4_regional_sensitivity_analysis.rst
@@ -0,0 +1,10 @@
+Regional Sensitivity Analysis
+*****************************
+Another method primarily applied for basic factor mapping applications is Regional Sensitivity Analysis (RSA; :cite:p:`hornberger1981approach`). RSA is a global sensitivity analysis method that is typically implemented using standard sampling methods such as latin hypercube sampling. It is performed by specifying a condition on the output space (e.g., an upper threshold) and classifying outputs that meet the condition as behavioral and the ones that fail it as non-behavioral (illustrated in :numref:`Figure_3_2` (b)). Note that the specified threshold depends on the nature of the problem, model, and the research question. It can reflect model-performance metrics (such as errors) or consequential decision-relevant metrics (such as unacceptable system outcomes). The behavioral and non-behavioral outputs are then traced back to their originating sampled factors, where differences between the distributions of samples can be used to determine their significance in producing each part of the output. The Kolmogorov-Smirnov divergence is commonly used to quantify the difference between the distribution of behavioral and non-behavioral parameters :cite:p:`pianosi2016sensitivity`:
+
+.. math::
+  S_i=|F_{x_i|y_b} (y \in Y_b)-F_{x_i|y_{nb}} (y \in Y_{nb})|
+
+where :math:`Y_b` represents the set of behavioral outputs, and :math:`F_{x_i|y_b}` is the empirical cumulative distribution function of the values of :math:`x_i` associated with values of :math:`y` that belong in the behavioral set. The :math:`nb` notation indicates the equivalent elements related to the non-behavioral set. Large differences between the two distributions indicate stronger effects by the parameters on the respective part of the output space.
+
+Used in a factor mapping setting, RSA can be applied for scenario discovery :cite:p:`lempert2006general,groves2007new`, the Generalized Likelihood Uncertainty Estimation method (GLUE; :cite:p:`beven_future_1992,beven2014glue,blasone2008generalized`) and other hybrid sensitivity analysis methods (e.g., :cite:p:`cryer1999regional,wei2013monte`). The fundamental shortcomings of RSA are that, in some cases, it could be hard to interpret the difference between behavioral and non-behavioral sample sets, and that insights about parametric correlations and interactions cannot always be uncovered :cite:p:`saltelli2004sensitivity`. For more elaborate discussions and illustrations of the RSA method, readers are directed to :cite:t:`saltelli2008global,tang2007comparing,young1999data` and references therein.
diff --git a/dev/docs/html/_sources/3.4.5_variance_based_methods.rst b/dev/docs/html/_sources/3.4.5_variance_based_methods.rst
new file mode 100644
index 0000000..73ba3ba
--- /dev/null
+++ b/dev/docs/html/_sources/3.4.5_variance_based_methods.rst
@@ -0,0 +1,32 @@
+.. _variance-based_methods:
+
+Variance-based Methods
+**********************
+Variance-based sensitivity analysis methods hypothesize that various specified model factors contribute differently to the variation of model outputs; therefore, decomposition and analysis of output variance can determine a model’s sensitivity to input parameters :cite:p:`sobol2001global,saltelli2004sensitivity`. The most popular variance-based method is the Sobol method, which is a global sensitivity analysis method that takes into account complex and nonlinear factor interaction when calculating sensitivity indices, and employs more sophisticated sampling methods (e.g., the Sobol sampling method). The Sobol method is able to calculate three types of sensitivity indices that provide different types of information about model sensitivities. These indices include first-order, higher-order (e.g., second-, third-, etc. orders), and total-order sensitivities.
+
+The first-order sensitivity index indicates the percent of model output variance contributed by a factor individually (i.e., the effect of varying :math:`x_i` alone) and is obtained using the following :cite:p:`sobol2001global,saltelli2002making`:
+
+.. math::
+  S_i^1=\frac{V_{x_i}[E_{x_{\sim i}}(x_i)]}{V(y)}
+
+with :math:`E` and :math:`V` denoting the expected value and the variance, respectively. :math:`x_{\sim i}` denotes all factors expect for :math:`x_i`. The first-order sensitivity index (:math:`S_i^1`) can therefore also be thought of as the portion of total output variance (:math:`V_y`) that can be reduced if the uncertainty in factor :math:`x_i` is eliminated :cite:p:`saltelli2002sensitivity`. First-order sensitivity indices are usually used to understand the independent effect of a factor and to distinguish its individual versus interactive influence. It would be expected for linearly independent factors that they would only have first order indices (no interactions) that should correspond well with sensitivities obtained from simpler methods using OAT sampling.
+
+Higher-order sensitivity indices explore the interaction between two or more parameters that contribute to model output variations. For example, a second-order index indicates how interactions between a pair of factors can lead to change in model output variance and is calculated using the following relationship:
+
+.. math::
+  S_{ij}^2=\frac{V_{x_{i,j}}[E_{x_{\sim i,j}}(x_i,x_j)]}{V(y)}
+
+with :math:`i \ne j`. Higher order indices can be calculated by similar extensions (i.e., fixing additional operators together), but it is usually computationally expensive in practice.
+
+The total sensitivity analysis index represents the entire influence of an input factor on model outputs including all of its interactions with other factors :cite:p:`homma1996importance`. In other words, total-order indices include first-order and all higher-order interactions associated with each factor and can be estimated calculated using the following:
+
+.. math::
+  S_i^T= \frac{E_{x_{\sim i}}[V_{x_i}(x_{\sim i})]}{V(y)} = 1 - \frac{V_{x_{\sim i}}[E_{x_{i}}(x_{\sim i})]}{V(y)}
+
+This index reveals the expected portion of variance that remains if uncertainty is eliminated in all factors but :math:`x_i` :cite:p:`saltelli2002sensitivity`. The total sensitivity index is the overall best measure of sensitivity as it captures the full individual and interactive effects of model factors.
+
+Besides the Sobol method, there are some other variance-based sensitivity analysis methods, such as the Fourier amplitude sensitivity test (FAST; :cite:`cukier1973study, mcrae1982development`) and extended-FAST :cite:`saltelli1998alternative,vazquez2014global`, that have been used by the scientific community. However, Sobol remains by far the most common method of this class. Variance-based techniques have been widely used and have proved to be powerful in a variety of applications. Despite their popularity, some authors have expressed concerns about the methods’ appropriateness in some settings. Specifically, the presence of heavy-tailed distributions or outliers, or when model outputs are multimodal can bias the sensitivity indices produced by these methods :cite:p:`auder2008global,zadeh2017comparison,pianosi2015simple`. Moment-independent measures, discussed below, attempt to overcome these challenges.
+
+.. note::
+
+    Put this into practice! Click the following link to try out an interactive tutorial which demonstrates the application of a Sobol sensitivity analysis:  `Sobol SA using SALib Jupyter Notebook <nb_saltelli_sobol_>`_
diff --git a/dev/docs/html/_sources/3.4.6_analysis_of_variance_anova.rst b/dev/docs/html/_sources/3.4.6_analysis_of_variance_anova.rst
new file mode 100644
index 0000000..b359cd8
--- /dev/null
+++ b/dev/docs/html/_sources/3.4.6_analysis_of_variance_anova.rst
@@ -0,0 +1,3 @@
+Analysis of Variance (ANOVA)
+****************************
+Analysis of Variance (ANOVA) was first introduced by :cite:t:`fisher1934statistical` and has since become a popular factor analysis method in physical experiments. ANOVA can be used as a sensitivity analysis method in computational experiments with a factorial design of experiment (referred to as factorial ANOVA). Note that Sobol can also be categorized as an ANOVA sensitivity analysis method, and that is why Sobol is sometimes referred to as a functional ANOVA :cite:p:`brevault2013comparison`. Factorial ANOVA methods are particularly suited for models and problems that have discrete input spaces, significantly reducing the computational time. More information about these methods can be found in :cite:p:`archer1997sensitivity,brevault2013comparison,owen2013variance`.
\ No newline at end of file
diff --git a/dev/docs/html/_sources/3.4.7_moment_independent_density_based_methods.rst b/dev/docs/html/_sources/3.4.7_moment_independent_density_based_methods.rst
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+++ b/dev/docs/html/_sources/3.4.7_moment_independent_density_based_methods.rst
@@ -0,0 +1,8 @@
+Moment-Independent (Density-Based) Methods
+******************************************
+These methods typically compare the entire distribution (i.e., not just the variance) of input and output parameters in order to determine the sensitivity of the output to a particular input variable. Several moment-independent sensitivity analysis methods have been proposed in recent years. The delta (:math:`\delta`) moment-independent method calculates the difference between unconditional and conditional cumulative distribution functions of the output. The method was first introduced by :cite:`borgonovo2006measuring,borgonovo2007new` and has become widely used in various disciplines. The :math:`\delta` sensitivity index is defined as follows:
+
+.. math::
+  S_i=\delta_i=\frac{1}{2}E_{x_i}|f_y(y)-f_{y|x_i}(y)|dy
+
+where :math:`f_y(y)` is the probability density function of the entire model output :math:`y`, and :math:`f_{y|x_i}(y)` is the conditional density of :math:`y`, given that factor :math:`x_i` assumes a fixed value. The :math:`\delta_i` sensitivity indicator therefore represents the normalized expected shift in the distribution of :math:`y` provoked by :math:`x_i`. Moment-independent methods are advantageous in cases where we are concerned about the entire distribution of events, such as when uncertain factors lead to more extreme events in a system :cite:p:`hadjimichael_advancing_2020`. Further, they can be used with a pre-existing sample of data, without requiring a specific sampling scheme, unlike the previously reviewed methods :cite:p:`plischke2013global`. The :math:`\delta` sensitivity index does not include interactions between factors and it is therefore akin to the first order index produced by the Sobol method. Interactions between factors can still be estimated using this method, by conditioning the calculation on more than one uncertain factor being fixed :cite:p:`borgonovo2007new`.
diff --git a/dev/docs/html/_sources/3.4_sensitivity_analysis_methods.rst b/dev/docs/html/_sources/3.4_sensitivity_analysis_methods.rst
new file mode 100644
index 0000000..87aa851
--- /dev/null
+++ b/dev/docs/html/_sources/3.4_sensitivity_analysis_methods.rst
@@ -0,0 +1,20 @@
+.. _sensitivity_analysis_methods:
+
+Sensitivity Analysis Methods
+############################
+
+In this section, we describe some of the most widely applied sensitivity analysis methods along with their mathematical definitions. We also provide a detailed discussion on applying each method, as well as a comparison of and their features and limitations.
+
+.. include:: 3.4.1_derivative_based_methods.rst
+
+.. include:: 3.4.2_elementary_effect_methods.rst
+
+.. include:: 3.4.3_regression_based_methods.rst
+
+.. include:: 3.4.4_regional_sensitivity_analysis.rst
+
+.. include:: 3.4.5_variance_based_methods.rst
+
+.. include:: 3.4.6_analysis_of_variance_anova.rst
+
+.. include:: 3.4.7_moment_independent_density_based_methods.rst
diff --git a/dev/docs/html/_sources/3.5_how_to_choose_a_sensitivity_analysis_method_model_traits_and_dimensionality.rst b/dev/docs/html/_sources/3.5_how_to_choose_a_sensitivity_analysis_method_model_traits_and_dimensionality.rst
new file mode 100644
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--- /dev/null
+++ b/dev/docs/html/_sources/3.5_how_to_choose_a_sensitivity_analysis_method_model_traits_and_dimensionality.rst
@@ -0,0 +1,17 @@
+How To Choose A Sensitivity Analysis Method: Model Traits And Dimensionality
+############################################################################
+
+:numref:`Figure_3_5`, synthesized from variants found in :cite:p:`iooss2015review, pianosi2016sensitivity`, presents a graphical synthesis of the methods overviewed in this section, with regards to their appropriateness of application based on the complexity of the model at hand and the computational limits on the number of model evaluations afforded. The bars below each method also indicate the sensitivity analysis purposes they are most appropriate to address, which are in turn a reflection of the motivations and research questions the sensitivity analysis is called to address. Computational intensity is measured as a multiple of the number of model factors that are considered uncertain (:math:`d`). Increasing model complexity mandates that more advanced sensitivity analysis methods are applied to address potential nonlinearities, factor interactions, and discontinuities. Such methods can only be performed at increasing computational expense. For example, computationally cheap linear regression should not be used to assess factors’ importance if the model cannot be proven linear and the factors independent, because important relationships will invariably be missed (recall the example in :numref:`Figure_3_5`). When computational limits do constrain applications to make simplified assumptions and sensitivity techniques, any conclusions in such cases should be delivered with clear statements of the appropriate caveats.
+
+.. _Figure_3_5:
+.. figure:: _static/figure3_5classificationofmethods.png
+    :alt: Figure 3_5
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Classification of the sensitivity analysis methods overviewed in this section, with regards to their computational cost (horizontal axis), their appropriateness to model complexity (vertical axis), and the purpose they can be used for (colored bars). d: number of uncertain factors considered; ANOVA: Analysis of Variance; FAST: Fourier Amplitude Sensitivity Test; PRIM: Patient Rule Induction Method; CART: Classification and Regression Trees; SRCC: Spearman’s rank correlation coefficient: NSE: Nash–Sutcliffe efficiency; SRC: standardized regression coefficient; PCC: Pearson correlation coefficient. This figure is synthesized from variants found in :cite:p:`iooss2015review, pianosi2016sensitivity`.
+
+The reader should also be aware that the estimates of computational intensity that are given here are indicative of magnitude and would vary depending on the sampling technique, model complexity and the level of information being asked. For example, a Sobol sensitivity analysis typically requires a sample of size :math:`n * d+2` to produce first- and total-order indices, where :math:`d` is the number of uncertain factors and :math:`n` is a scaling factor, selected ad hoc, depending on model complexity :cite:p:`saltelli2002relative`. The scaling factor :math:`n` is typically set to at least 1000, but it should most appropriately be set on the basis of index convergence. In other words, a prudent analyst would perform the analysis several times with increasing :math:`n` and observe at what level the indices converge to stable values :cite:p:`NOSSENT20111515`. The level should be the minimum sample size used in subsequent sensitivity analyses of the same system. Furthermore, if the analyst would like to better understand the degrees of interaction between factors, requiring second-order indices, the sample size would have to increase to :math:`n * 2d+2` :cite:p:`saltelli2002relative`.
+
+Another important consideration is that methods that do not require specific sampling schemes can be performed in conjunction with others without requiring additional model evaluations. None of the regression-based methods, for example, require samples of specific structures or sizes, and can be combined with other methods for complementary purposes. For instance, one could complement a Sobol analysis with an application of CART, using the same data, but to address questions relating to factor mapping (e.g., we know factor :math:`x_i`  is important for a model output, but we would like to also know which of its values specifically push the output to undesirable states). Lastly, comparing results from different methods performed together can be especially useful in model diagnostic settings. For example, :cite:p:`hadjimichael_advancing_2020` used :math:`\delta` indices, first-order Sobol indices, and :math:`R^2`  values from linear regression, all performed on the same factors, to derive insights about the effects on factors on different moments of the output distribution and about the linearity of their relationship.
diff --git a/dev/docs/html/_sources/3.6_software_toolkits.rst b/dev/docs/html/_sources/3.6_software_toolkits.rst
new file mode 100644
index 0000000..0b42a98
--- /dev/null
+++ b/dev/docs/html/_sources/3.6_software_toolkits.rst
@@ -0,0 +1,19 @@
+.. _software_toolkits:
+
+Software Toolkits
+#################
+
+This section presents available open source sensitivity analysis software tools, based on the programming language they use and the methods they support :numref:`Figure_3_6`. Our review covers five widely used programming languages: R, MATLAB, Julia, Python, and C++, as well as one tool that provides a graphical user interface (GUI). Each available SA tool was assessed on the number of SA methods and design of experiments methods it supports. For example, the *sensobol* package in R only supports the variance-based Sobol method. However, it is the only package we came across that calculates third-order interactions among parameters. On the other side of the spectrum, there are SA software packages that contain several popular SA methods. For example, *SALib* in Python :cite:`herman2017salib` supports seven different SA methods. The *DifferentialEquations* package is a comprehensive package developed for Julia, and *GlobalSensitivityAnalysis* is another Julia package that has mostly adapted SALib methods. :numref:`Figure_3_6` also identifies the SA packages that have been updated since 2018, indicating active support and development.
+
+
+.. _Figure_3_6:
+.. figure:: _static/figure3_6_softwaretoolkits.png
+    :alt: Figure 3_6
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Sensitivity analysis packages available in different programming language platforms (R, Python, Julia, MATLAB, and C++), with the number of methods they support. Packages supporting more than five methods are indicated in pink. Packages updated since 2018 are indicated with asterisks. 
+
+
+
diff --git a/dev/docs/html/_sources/3_sensitivity_analysis_the_basics.rst b/dev/docs/html/_sources/3_sensitivity_analysis_the_basics.rst
new file mode 100644
index 0000000..8fee574
--- /dev/null
+++ b/dev/docs/html/_sources/3_sensitivity_analysis_the_basics.rst
@@ -0,0 +1,30 @@
+.. _3_sensitivity_analysis_the_basics:
+
+********************************
+Sensitivity Analysis: The Basics
+********************************
+
+.. include:: 3.1_global_versus_local_sensitivity.rst
+
+.. include:: 3.2_why_perform_sensitivity_analysis.rst
+
+.. include:: 3.3_design_of_experiments.rst
+
+.. include:: 3.4_sensitivity_analysis_methods.rst
+
+.. include:: 3.5_how_to_choose_a_sensitivity_analysis_method_model_traits_and_dimensionality.rst
+
+.. include:: 3.6_software_toolkits.rst
+
+.. note::
+    The following articles are suggested as fundamental reading for the information presented in this section:
+
+    * Wagener, T., Pianosi, F., 2019. What has Global Sensitivity Analysis ever done for us? A systematic review to support scientific advancement and to inform policy-making in earth system modelling. *Earth-Science Reviews* 194, 1–18. https://doi.org/10.1016/j.earscirev.2019.04.006
+    * Pianosi, F., Beven, K., Freer, J., Hall, J.W., Rougier, J., Stephenson, D.B., Wagener, T., 2016. Sensitivity analysis of environmental models: A systematic review with practical workflow. Environmental Modelling & Software 79, 214–232. https://doi.org/10.1016/j.envsoft.2016.02.008
+
+    The following articles can be used as supplemental reading:
+
+    * Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S., 2008. Global Sensitivity Analysis: The Primer, 1st edition. ed. Wiley-Interscience, Chichester, England ; Hoboken, NJ.
+    * Montgomery, D.C., 2017. Design and analysis of experiments. John Wiley & Sons.
+    * Iooss, B., Lemaître, P., 2015. A Review on Global Sensitivity Analysis Methods, in: Dellino, G., Meloni, C. (Eds.), Uncertainty Management in Simulation-Optimization of Complex Systems: Algorithms and Applications, Operations Research/Computer Science Interfaces Series. Springer US, Boston, MA, pp. 101–122. https://doi.org/10.1007/978-1-4899-7547-8_5
+
diff --git a/dev/docs/html/_sources/4.1_understanding_errors_what_is_controlling_model_performance.rst b/dev/docs/html/_sources/4.1_understanding_errors_what_is_controlling_model_performance.rst
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+.. _4_1_understanding_errors:
+
+Understanding Errors: What Is Controlling Model Performance?
+############################################################
+
+Sensitivity analysis is a diagnostic tool when reconciling model outputs with observed data. It is helpful for clarifying how and under what conditions modeling choices (structure, parameterization, data inputs, etc.) propagate through model components and manifest in their effects on model outputs. This exploration is performed through carefully designed sampling of multiple combinations of input parameters and subsequent evaluation of the model structures that are emerging as controlling factors. Model structure and parameterization are two of the most commonly explored aspects of models that have been a central focus when evaluating their performance relative to available observations :cite:`beven1993prophecy`. Addressing these issues plays an important role in establishing credibility in model predictions, particularly in the positivist natural sciences literature. Traditional model evaluations compare the model with observed data, and then rely on expert judgements of its acceptability based on the closeness between simulation and observation with one or a small number of selected metrics. This approach can be myopic, as it is often impossible to use one metric to attribute a certain error and to link that with different parts of the model and its parameters :cite:`gupta2008reconciling`. This means that, even when the error or fitting measure between the model estimates and observations is very small, it is not guaranteed that all the components in the model accurately represent the conceptual reality that the model is abstracting: propagated errors in different parts of the model might cancel each other out, or multiple parameterized implementations of the model can yield similar performance (i.e., equifinality :cite:`beven1993prophecy`).
+
+The inherent complexity of a system hinders accepting or rejecting a model based on one performance measure and different types of measures can aid in evaluating the various components of a model as essentially a multiobjective problem :cite:`gupta2012towards,beven2018hypothesis`. In addition, natural systems mimicked by the models contain various interacting components that might act differently across spatial and temporal domains :cite:`herman2013time,pianosi2016sensitivity`. This heterogeneity is lost when a single performance measure is used, as a highly dimensional and interactive system becomes aggregated through the averaging of spatial or temporal output errors :cite:`beven2002towards`. Therefore, diagnostic error analyses should consider multiple error signatures across different scales and states of concern when seeking to understand how model performance relates to observed data (:numref:`Figure_4_1`). Diverse error signatures can be used to measure the consistency of underlying processes and behaviors of the model and to evaluate the dynamics of model controls under changing temporal and spatial conditions :cite:`beven2001equifinality`. Within this framework, even minimal information extracted from the data can be beneficial as it helps us unearth structural inadequacies in the model. In this context, proper selection of measures of model performance and the number of measures could play consequential roles in our understanding of the model and its predictions :cite:`mcphail2018robustness`.
+
+As discussed earlier, instead of the traditional focus on using deterministic prediction that results in a single error measure, many plausible states and spaces could be searched for making different inferences and quantifying uncertainties. This process also requires estimates of prior probability distributions of all the important parameters and quantification of model behavior across input space. One strategy to reduce the search space is filtering of some model alternatives that are not consistent with observations and known system behaviors. Those implausible parts of the search space can be referred to as non-physical or non-behavioral alternatives :cite:`edwards2011precalibrating, beven2014glue`. This step is conducted before the Bayesian calibration exercise (see :numref:`A1_uncertainty_quantification`).
+
+A comprehensive model diagnostic workflow typically entails the components demonstrated in :numref:`Figure_4_1`. The workflow begins with the selection of model input parameters and their plausible ranges. After the parameter selection, we need to specify the design of experiment (:numref:`design_of_experiments`) and the sensitivity analysis method (:numref:`sensitivity_analysis_methods`) to be used. As previously discussed, these methods require different numbers of model simulations, and each method provides a different insights into the direct effects and interactions of the uncertain factors. In addition, the simulation time of the model and the available computational resources are two of the primary considerations that influence these decisions. After identifying the appropriate methods, we generate a matrix of input parameters, where each set of input parameters will be used to conduct a model simulation. The model can include one or more output variables that fluctuate in time and space. The next step is to analyze model performance by comparing model outputs with observations. As discussed earlier, the positivist model evaluation paradigm focuses on a single model performance metric (error), leading to a loss of information about model parameters and the suitability of the model's structure. However, a thorough investigation of the temporal and spatial signatures of model outputs using various performance metrics or time- and space-varying sensitivity analyses can shed more light on the fitness of each parameter set and the model’s internal structure. This analysis provides diagnostic feedback on the importance and range of model parameters and can guide further improvement of the model algorithm.
+
+.. _Figure_4_1:
+.. figure:: _static/figure4_1_diagnostic_workflow.png
+    :alt: Figure 4.1
+    :width: 500px
+    :figclass: margin-caption
+    :align: center
+
+    Diagnostic evaluation of model fidelity using sensitivity analysis methods.
+
+.. note::
+
+    Put this into practice! Click the following badge to try out an interactive tutorial on implementing a time-varying sensitivity analysis of HYMOD model parameters:  `HYMOD Jupyter Notebook <nb_hymod_>`_
diff --git a/dev/docs/html/_sources/4.2_consequential_dynamics_what_is_controlling_model_behaviors_of_interest.rst b/dev/docs/html/_sources/4.2_consequential_dynamics_what_is_controlling_model_behaviors_of_interest.rst
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+Consequential Dynamics: What is Controlling Model Behaviors of Interest?
+########################################################################
+
+Consequential changes in dynamic systems can take many forms, but most dynamic behavior can be categorized in a few basic patterns. Feedback structures inherent in a system, be they positive or negative, generate these patterns which can be grouped into three groups for the simplest of systems: exponential growth, goal seeking, and oscillation (:numref:`Figure_4_2`). A positive (or self-reinforcing) feedback gives rise to exponential growth, a negative (or self-correcting) feedback gives rise to a goal seeking mode, and negative feedbacks with time delays give rise to oscillatory behavior. Nonlinear interactions between the system’s feedback structures can give rise to more complex dynamic behavior modes, examples of which are also shown in :numref:`Figure_4_2`, adapted from :cite:t:`sterman2001system`.
+
+.. _Figure_4_2:
+.. figure:: _static/figure4_2_behavior_modes.png
+    :alt: Figure 4.2
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Common modes of behavior in dynamic systems, occurring based on the presence of positive and negative feedback relationships, and linear and non-linear interactions. Adapted from :cite:t:`sterman2001system`.
+
+The nature of feedback processes in a dynamic system shapes its fundamental behavior: positive feedbacks generate their own growth, negative feedbacks self-limit, seeking balance and equilibrium. In this manner, feedback processes give rise to different regimes, multiples of which could be present in each mode of behavior. Consider a population of mammals growing exponentially until it reaches the carrying capacity of its environment (referred to as S-shaped growth). When the population is exponentially growing, the regime is dominated by positive feedback relationships that reinforce its growth. As the population approaches its carrying capacity limit, negative feedback structures begin to dominate, counteracting the growth and establishing a stable equilibrium. Shifts between regimes can be thought of as tipping points, mathematically defined as unstable equilibria, where the presence of positive feedbacks amplifies disturbances and moves the system to a new equilibrium point. In the case of stable equilibria, the presence of negative feedbacks dampens any small disturbance and maintains the system at a stable state. As different feedback relationships govern each regime, different factors (those making up each feedback mechanism) are activated and shape the states the system is found in, as well as define the points of equilibria.
+
+For simple stylized models with a small number of states, system dynamics analysis can analytically derive these equilibria, the conditions for their stability and the factors determining them. The ability for this to be performed is, however, significantly challenged when it comes to systems that attempt to more closely resemble real complex systems. We argue this is the case for several reasons. First, besides generally exhibiting complex nonlinear dynamics, real world systems are also made up from larger numbers of interacting elements, which often makes the analytic derivation of system characteristics intractable :cite:p:`anderies2019knowledge,muneepeerakul2020emergence`. Second, human-natural systems temporally evolve and transform when human state-aware action is present. Consider, for instance, humans recreationally hunting the aforementioned population of mammals. Humans act based on the mammal population levels by enforcing hunting quotas or establishing protected territories or eliminating other predators. The mammal population reacts in a response, giving birth to ever changing state-action-consequence feedbacks, the path dependencies of which become difficult to diagnose and understand (e.g., :cite:p:`hadjimichael2020navigating`). Trying to simulate the combination of these two challenges (large numbers of state-aware agents interacting with a natural resource and with each other) produces intractable models that require advanced heuristics to analyze their properties and establish useful inferences.
+
+Sensitivity analysis paired with exploratory modeling methods offers a promising set of tools to address these challenges. We present a simple demonstrative application based on :cite:t:`quinn2017direct`. This stylistic example was first developed by :cite:t:`carpenter_management_1999` and represents a town that must balance its agricultural and industrial productivity with the pollution it creates in a downstream lake. Increased productivity allows for increased profits, which the town aims to maximize, but it also produces more pollution for the lake. Too much phosphorus pollution can cause irreversible eutrophication, a process known as “tipping” the lake. The model of phosphorus in the lake :math:`X_t` at time :math:`t` is governed by:
+
+.. math::
+
+  X_{t+1}= X_{t}+a_{t}+\frac{X_{t}^q} {1+X_{t}^q}-bX_t+\varepsilon
+
+where :math:`a_t \in [0,0.1]` is the town’s pollution release at each timestep, :math:`b` is the natural decay rate of phosphorus in the lake, :math:`q` defines the lake’s recycling rate (primarily through sediments), and :math:`\varepsilon` represents uncontrollable natural inflows of pollution modeled as a log-normal distribution with a given mean, :math:`\mu`, and standard deviation :math:`\sigma`.
+
+Panels (a-c) in :numref:`Figure_4_3` plot the fluxes of phosphorus into the lake versus the mass accumulation of phosphorus in the lake. The red line corresponds to the phosphorus sinks in the lake (natural decay), given by :math:`bX_t`. The grey shaded area represents the lake’s phosphorus recycling flux, given by :math:`\frac{X_{t}^q} {1+X_{t}^q}`. The points of intersection indicate the system’s equilibria, two of which are stable, and one is unstable (also known as the tipping point). The stable equilibrium in the bottom left of the figure reflects an oligotrophic lake, whereas the stable equilibrium in the top right represents a eutrophic lake. With increasing phosphorus values, the tipping point can be crossed, and the lake will experience irreversible eutrophication, as the recycling rate would exceed the removal rate even if the town’s pollution became zero. In the absence of anthropogenic and natural inflows of pollution in the lake (:math:`a_t` and :math:`\varepsilon` respectively), the area between the bottom-left black point and the white point in the middle can be considered as the safe operating space, before emission levels cross the tipping point.
+
+.. _Figure_4_3:
+.. figure:: _static/figure4_3_lake_problem_fluxes.png
+    :alt: Figure 4.3
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Fluxes of phosphorus with regards to mass of phosphorus in the lake and sensitivity analysis results, assuming :math:`b=0.42` and :math:`q=2`. (a) Fluxes of phosphorus assuming no emmisions policy and no natural inflows. (b-c) Fluxes phosphorus when applying two different emissions policies. The "Best economic policy" and the "Most reliable policy" have been identified by :cite:t:`quinn2017direct` and can be found at :cite:t:`quinn_julianneqlake_problem_dps_2017`. (d) Results of a sensitivity analysis on the parameters of the model most consequential to the reliability of the "Most reliable policy". The code to replicate the sensitivity analysis can be found at :cite:t:`hadka_project-platypusrhodium_2017`. Panels (a-c) are used courtesy of Julianne Quinn, University of Virginia.
+
+The town has identified two potential policies that can be used to manage this lake, one that maximizes its economic profits (“best economic policy”) and one that maximizes the time below the tipping point (“most reliable policy”). Panels (b-c) in :numref:`Figure_4_3` add the emissions from these policies to the recycling flux and show how the equilibria points shift as a result. In both cases the stable oligotrophic equilibrium increases and the tipping point decreases, narrowing the safe operating space :cite:`anderies2019knowledge, carpenter_allowing_2015`. The best economic policy results in a much narrower space of action, with the tipping point very close to the oligotrophic equilibrium. The performance of both policies depends significantly on the system parameters. For example, a higher value of :math:`b`, the natural decay rate, would shift the red line upward, moving the equilibria points and widening the safe operating space. Inversely, a higher value of :math:`q`, the lake’s recycling rate, would shift the recycling line upward, moving the tipping point lower and decreasing the safe operating space. The assumptions under which these policies were identified are therefore critical to their performance and any potential uncertainty in the parameter values could be detrimental to the system’s objectives being met.
+
+Sensitivity analysis can be used to clarify the role these parameters play on policy performance. :numref:`Figure_4_3` (d) shows the results of a Sobol sensitivity analysis on the reliability of the “most reliable” policy in a radial convergence diagram. The significance of each parameter is indicated by the size of circles corresponding to it. The size of the interior dark circle indicates the parameter’s first-order effects and the size of the exterior circle indicates the parameter’s total-order effects. The thickness of the lines between two parameters indicated the extent of their interaction (second-order effects). In this case, parameters :math:`b` and :math:`q` appear to have the most significant importance on the system, followed by the mean, :math:`\mu`, of the natural inflows. All these parameters function in a manner that shifts the location of the three equilibria and therefore policies that are identified ignoring this parametric uncertainty might fail to meet their intended goals.
+
+It is worth mentioning that current sensitivity analysis methods are somewhat challenged in addressing several system dynamics analysis questions. The fundamental reason is that sensitivity analysis methods and tools have been developed to gauge numerical sensitivity of model output to changes in factor values. This is natural, as most simulation studies (e.g., all aforementioned examples) have been traditionally concerned with this type of sensitivity. In system dynamics modeling, however, a more important and pertinent concern is changes between regimes or between behavior modes (also known as bifurcations) as a result of changes in model factors :cite:p:`hekimouglu2016sensitivity, sterman2001system`. This poses two new challenges. First, identifying a change in regime depends on several characteristics besides a change in output value, like the rate and direction of change. Second, behavior mode changes are qualitative and discontinuous, as equilibria change in stability but also move in and out of existence.
+
+Despite these challenges, recent advanced sensitivity analysis methods can help illuminate which factors in a system are most important in shaping boundary conditions (tipping points) between different regimes and determining changes in behavior modes. Reviewing such methods is outside the scope of this text, but the reader is directed to the examples of :cite:t:`eker2018practice` and :cite:t:`hadjimichael2020navigating`, who apply parameterised perturbation on the functional relationships of a system to study the effects of model structural uncertainty on model outputs and bifurcations, and :cite:t:`hekimouglu2016sensitivity` and :cite:t:`steinmann2020behavior` who, following wide sampling of uncertain inputs, cluster the resulting time series in modes of behavior and identify most important factors for each.
+
+.. note::
+
+    Put this into practice! Click the following badge to try out an interactive tutorial on performing a sensitivity analysis to discover consequential dynamics:  `Factor Discovery Jupyter Notebook <nb_fishery_dynamics_>`_
diff --git a/dev/docs/html/_sources/4.3_consequential_scenarios_what_is_controlling_consequential_outcomes.rst b/dev/docs/html/_sources/4.3_consequential_scenarios_what_is_controlling_consequential_outcomes.rst
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+.. _consequential_scenarios:
+
+Consequential Scenarios: What is Controlling Consequential Outcomes?
+####################################################################
+
+As overviewed in :numref:`perspectives`, most models are abstractions of systems in the real world. When sufficient confidence has been established in a model, it can then act as a surrogate for the actual system, in that the consequences of potential stressors, proposed actions or other changes can be evaluated by computer model simulations :cite:`bankes_exploratory_1993`. A model simulation then represents a computational experiment, which can be used to assess how the modeled system would behave should the various changes come to be. Steven Bankes coined the term exploratory modeling to describe the use of large sets of such computational experiments to investigate their implications on the system. :numref:`Figure_4_4` presents a typical workflow of an exploratory modeling application. Exploratory modeling approaches typically use sampling designs to generate large ensembles of states that represent combinations of changes happening together, spanning the entire range of potential values a factor might take (indicated in :numref:`Figure_4_4` by numbers 2-5). This perspective on modeling is particularly relevant to studies making long term projections into the future.
+
+.. _Figure_4_4:
+.. figure:: _static/figure4_4_exploratory_workflow.png
+    :alt: Figure 4_4
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    A typical exploratory modeling workflow
+
+
+In the long-term policy analysis literature, exploratory modeling has prominently placed itself as an alternative to traditional narrative scenario or assumptions-based planning approaches, in what can be summarized in the following two-pronged critique :cite:`bankes_exploratory_1993, bankes2001computer, lempert2003shaping`. The most prevalent criticism sees that the future and how it might evolve is both highly complex and deeply uncertain. Despite its benefits for interpretation and intuitive appeal, a small number of scenarios invariably misses many other potential futures that did not get selected as sufficiently representative of the future. This is especially the case for aggregate, narrative scenarios that describe simultaneous changes in multiple sectors together (e.g., "increased energy demand, combined with high agricultural land use and large economic growth"), such as the emission scenarios produced by the Intergovernmental Panel on Climate Change :cite:`lamontagne2018large`. The bias introduced by this reduced set of potential changes can skew inferences drawn from the model, particularly when the original narrative scenarios are focused on a single or narrow set of measures of system behavior.
+
+The second main criticism of traditional narrative scenario-based planning methods is that they provide no systematic way to distinguish which of the constituent factors lead to the undesirable consequences produced by a scenario. Narrative scenarios (e.g., the scenario matrix framework of RCPs-SSPs-SPAs; :cite:`o2014new`) encompass multiple changes happening together selected to span the range of potential changes but are not typically generated in a systematic factorial manner that considers the multiple ways the factors can be combined. This has two critical limitations. It obfuscates the role each component factor plays in the system, both in isolation and in combination with others (e.g., "is it the increased energy demand or the high agricultural land use that cause unbearable water stress?"). It also renders the delineation of how much change in a factor is critical near impossible. Consider, for example, narrative scenario A with a 5% increase in energy demand, and scenario B with a 30% increase in energy demand, which would have dire consequences. At which point between 5% and 30% do the dire consequences actually begin to occur? Such questions cannot be answered without a wide exploration of the space of potential changes. It should be noted that for some levels of model complexity and computational demands (e.g., global-scale models) there is little feasible recourse beyond the use of narrative scenarios.
+
+Exploratory modeling is typically paired with scenario discovery methods (indicated by number 9 in :numref:`Figure_4_4`) that identify which of the scenarios (also known as states of the world) generated indeed have consequences of interest for stakeholders and policy makers, in an approach referred to as ensemble-based scenario-discovery :cite:`bryant2010thinking, groves2007new, lempert2006general`. This approach therefore flips the planning analysis from one that attempts to predict future system conditions to one that attempts to discover the (un)desirable future conditions. Ensemble-based scenario discovery can thus inform what modeling choices yield the most consequential behavioral changes or outcomes, especially when considering deeply uncertain, scenario-informed projections :cite:`walker2013adapt,moallemi2020exploratory`. The relative likelihoods and relevance of the discovered scenarios can be subsequently evaluated by the practitioners a posteriori, within a richer context of knowing the wider set of potential consequences :cite:`dessai2009climate`. This can include changing how an analysis is framed (number 10 in :numref:`Figure_4_4`). For instance, one could initially focus on ensemble modeling of vulnerability using a single uncertain factor that is assumed to be well characterized by historical observations (e.g., streamflow; this step is represented by numbers 2-3 in :numref:`Figure_4_4`). The analysis can then shift to include projections of more factors treated as deeply uncertain (e.g., urbanization, population demands, temperature, and snow-melt) to yield a far wider space of challenging projected futures. UC experiments contrasting these two framings can be highly valuable for tracing how vulnerability inferences change as the modeled space of futures expands from the historical baseline :cite:`quinn2017direct`.
+
+An important nuance to be clarified here is that the focus or purpose of a modeling exercise plays a major role in whether a given factor of interest is considered well-characterized or deeply uncertain. Take the example context of characterizing temperature or streamflow extremes, where for each state variable of interest for a given location of focus there is a century of historical observations. Clearly, the observation technologies will have evolved over time uniquely for temperature and streamflow measurements and they likely lack replicate experiments (data uncertainty). A century of record will be insufficient to capture very high impact and rare extreme events (i.e., increasingly poor structural/parametric inference for the distributions of specific extreme single or compound events). The mechanistic processes as well as their evolving variability will be interdependent but uniquely different for each of these state variables. A large body of statistical literature exists focusing on the topics of synthetic weather :cite:p:`wilks1999weather,herrera2017review` or streamflow :cite:p:`lamontagne2018generating,medda2019comparison` generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound extremes. These history-focused approaches can be viewed as providing well-characterized quantifications of streamflow or temperature distributions; however, they do not capture how coupled natural-human processes can fundamentally change their dynamics when transitioning to projections of longer-term futures (e.g., streamflow and temperature in 2055). Consequently, changing the focus of the modeling to making long term projections of future streamflow or temperature now makes these processes deeply uncertain.
+
+Scenario discovery methods (number 9 in :numref:`Figure_4_4`) can be qualitative or quantitative and they generally attempt to distinguish futures in which a system or proposed policies to manage the system meet or miss their goals :cite:`groves2007new`. The emphasis placed by exploratory modeling on model outputs that have decision relevant consequences represents a shift toward a broader class of metrics that are reflective of the stakeholders’ concerns, agency and preferences (also discussed in :numref:`perspectives`). As a result, sensitivity analysis and scenario discovery methods in this context are therefore applied to performance metrics that go beyond model error but are rather focused on broader metrics such as the resilience of a sector, the reliability of a process, or the vulnerability of a population in the face of uncertainty. In exploratory modeling literature, this metric is most typically—but not always—a measure of robustness (number 8 in Fig. 13). Robustness is a property of a system or a design choice capturing its insensitivity to uncertainty and can be measured via a variety of means, most recently reviewed by :cite:t:`herman2015should` and :cite:t:`mcphail2018robustness`.
+
+Scenario discovery is typically performed through the use of algorithms applied on large databases of model runs, generated through exploratory modeling, with each model run representing the performance of the system in one potential state of the world. The algorithms seek to identify the combinations of factor values (e.g., future conditions) that best distinguish the cases in which the system does or does not meet its objectives. The most widely known classification algorithms are the Patient Rule Induction Method (PRIM; :cite:p:`friedman1999bump`) and Classification and Regression Trees (CART; :cite:p:`breiman1984classification`). These factor mapping algorithms create orthogonal boundaries (multi-dimensional hypercubes) between states of the world that are successful or unsuccessful in meeting the system’s goals :cite:p:`dalal2008low`. The algorithms attempt to strike a balance between simplicity of classification (and as a result, interpretability) and accuracy :cite:p:`groves2007new, bryant2010thinking, lempert2019robust`.
+
+Even though these approaches have been shown to yield interpretable and relevant scenarios :cite:p:`kwakkel2019supporting`, several authors have pointed out the limitations of these methods with regards to their division of space in orthogonal behavioral and non-behavioral regions :cite:p:`trindade2019deeply`. Due to their reliance on boundaries orthogonal to the uncertainty axes, PRIM and CART cannot capture interactions between the various uncertain factors considered, which can often be significant :cite:p:`quinn2018exploring`. More advanced methods have been proposed to address this drawback, with logistic regression being perhaps the most prominent :cite:p:`gold2019identifying, lamontagne2019robust, quinn2018exploring`. Logistic regression can produce boundaries that are not necessarily orthogonal to each uncertainty axis, nor necessarily linear, if interactive terms between two parameters are used to build the regression model. It also describes the probability that a state of the world belongs to the scenarios that lead to failure. This feature allows users to define regions of success based on a gradient of estimated probability of success in those worlds, unlike PRIM which only classifies states of the world in two regions :cite:p:`hadjimichael_defining_2020, quinn2018exploring`.
+
+Another more advanced factor mapping method is boosted trees :cite:p:`drucker1996boosting, freund1999short`. Boosted trees can avoid two limitations inherent to the application of logistic regression: i) to build a nonlinear classification model the interactive term between two uncertainties needs to be pre-specified and cannot be discovered (e.g., we need to know a-priori whether factor :math:`x_1` interacts with :math:`x_2` in a relationship that looks like :math:`x_1`·:math:`x_2` or :math:`x_1^{x_2}`); and ii) the subspaces defined are always convex. The application of such a factor mapping algorithm is limited in the presence of threshold-based rules with discrete actions in a modeled system (e.g., “if network capacity is low, build new infrastructure”), which results in failure regions that are nonlinear and non-convex :cite:p:`trindade2019deeply`. Boosting works by creating an ensemble of classifiers and forcing some of them to focus on the hard-to-learn parts of the problem, and others to focus on the easy-to-learn parts. Boosting applied to CART trees can avoid the aforementioned challenges faced by other scenario discovery methods, while resisting overfitting :cite:p:`murphy2012machine`, assuring the identified success and failure regions are still easy to interpret.
+
+Below we provide an example application of two scenario discovery methods, PRIM and logistic regression, using the lake problem introduced in the previous section. From the sensitivity analysis results presented in :numref:`Figure_4_3` (d), we can already infer that parameters :math:`b` and :math:`q` have important effects on model outputs (i.e., we have performed factor prioritization). Scenario discovery (i.e., factor mapping) complements this analysis by further identifying the specific values of b and q that can lead to consequential and undesirable outcomes. For the purposes of demonstration, we can assume the undesirable outcome in this case is defined as the management policy failing to achieve 90% reliability in a state of the world.
+
+.. _Figure_4_5:
+.. figure:: _static/figure14lake_problem_SD.png
+    :alt: Figure 14.
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Scenario discovery for the lake problem, using (a) PRIM and (b) logistic regression.
+
+:numref:`Figure_4_5` shows the results of scenario discovery, performed through (a) PRIM and (b) logistic regression. Each point in the two panels indicates a potential state of the world, generated through Latin Hypercube Sampling. Each point is colored by whether the policy meets the above performance criterion, with blue indicating success and red indicating failure. PRIM identifies several orthogonal areas of interest, one of which is shown in panel (a). As discussed above, this necessary orthogonality limits how PRIM identifies areas of success (the area within the box). As factors  :math:`b` and :math:`q` interact in this system, the transition boundary between the regions of success and failure is not orthogonal to any of the axes. As a result, a large number of points in the bottom right and the top left of the figure are left outside of the identified region. Logistic regression can overcome this limitation by identifying a diagonal boundary between the two regions, seen in panel (b). This method also produces a gradient of estimated probability of success across these regions.
+
+.. note::
+
+    Put this into practice! Click the following link to try out an interactive tutorial on performing factor mapping using logistic regression:  `Logistic Regression Jupyter Notebook <nb_logistic_regression_>`_
diff --git a/dev/docs/html/_sources/4_sensitivity_analysis_diagnostic_and_exploratory_modeling.rst b/dev/docs/html/_sources/4_sensitivity_analysis_diagnostic_and_exploratory_modeling.rst
new file mode 100644
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--- /dev/null
+++ b/dev/docs/html/_sources/4_sensitivity_analysis_diagnostic_and_exploratory_modeling.rst
@@ -0,0 +1,24 @@
+.. _4_sensitivity_analysis:
+
+*******************************************************
+Sensitivity Analysis: Diagnostic & Exploratory Modeling
+*******************************************************
+
+.. include:: 4.1_understanding_errors_what_is_controlling_model_performance.rst
+
+.. include:: 4.2_consequential_dynamics_what_is_controlling_model_behaviors_of_interest.rst
+
+.. include:: 4.3_consequential_scenarios_what_is_controlling_consequential_outcomes.rst
+
+.. note::
+    The following articles are suggested as fundamental reading for the information presented in this section:
+
+    * Gupta, H.V., Wagener, T., Liu, Y., 2008. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. *Hydrological Processes: An International Journal* 22, 3802–3813.
+    * Bankes, S., 1993. Exploratory Modeling for Policy Analysis. *Operations Research* 41, 435–449. https://doi.org/10.1287/opre.41.3.435
+    * Groves, D.G., Lempert, R.J., 2007. A new analytic method for finding policy-relevant scenarios. *Global Environmental Change* 17, 73–85. https://doi.org/10.1016/j.gloenvcha.2006.11.006
+
+
+    The following articles can be used as supplemental reading:
+
+    * Marchau, V.A.W.J., Walker, W.E., Bloemen, P.J.T.M., Popper, S.W. (Eds.), 2019. Decision Making under Deep Uncertainty: From Theory to Practice. Springer International Publishing. https://doi.org/10.1007/978-3-030-05252-2
+    * Sterman, J.D., 1994. Learning in and about complex systems. System Dynamics Review 10, 291–330. https://doi.org/10.1002/sdr.4260100214
diff --git a/dev/docs/html/_sources/5_conclusion.rst b/dev/docs/html/_sources/5_conclusion.rst
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+.. _5_conclusion:
+
+**********
+Conclusion
+**********
+
+As noted in the Introduction (:numref:`introduction`), the computational and conceptual challenges of the multi-model, transdisciplinary workflows that characterize ambitious projects such as IM3 have limited UC and UQ analyses. Moreover, the very nature and purpose of modeling and diagnostic model evaluation can have very diverse philosophical framings depending on the disciplines involved (see :numref:`Figure_1_1` and :numref:`perspectives`). The guidance provided in this text can be used to frame consistent and rigorous experimental designs for better understanding the consequences and insights from our modeling choices when seeking to capture complex human-natural systems. The progression of sections of this text provide a thorough introduction of the concepts and definitions of diagnostic model evaluation, sensitivity analysis and UC. In addition, we comprehensively discuss how specific modeling objectives and applications should guide the selection of appropriate techniques; broadly, these can include model diagnostics, in-depth analysis of the behavior of the abstracted system, and projections under conditions of deep uncertainty. This text also contains a detailed presentation of the main sensitivity analysis methods and a discussion of their features and main limitations. Readers are also provided with an overview of computer tools and platforms that have been developed and could be considered in addressing IM3 scientific questions. The appendices of this text include an overview of UQ methods, a terminology glossary of the key concepts as well as example test cases and scripts to showcase various UC related capabilities.
+
+Although we distinguish the UC and UQ model diagnostics, the reader should note that we suggest an overall consistent approach to both in this text by emphasizing “exploratory modeling” (see review by :cite:t:`moallemi2020exploratory`). Although data support, model complexity, and computational limits strongly distinguish the feasibility and appropriateness of various UC diagnostic tools (e.g., see :numref:`Figure_3_5`), we overall recommend that modelers view their work through the lens of cycles of learning. Iterative and deliberative exploration of model-based hypotheses and inferences for transdisciplinary teams is non-trivial and ultimately critical for mapping where innovations or insights are most consequential. Overall, we recommend approaching modeling with an openness to the diverse disciplinary perspectives such as those mirrored by the IM3 family of models in a progression from evaluating models relative to observed history to advanced formalized analyses to make inferences on multi-sector, multi-scale vulnerabilities and resilience. Exploratory modeling approaches can help fashion experiments with large numbers of alternative hypotheses on the co-evolutionary dynamics of influences, stressors, as well as path-dependent changes in the form and function of coupled human-natural systems :cite:`weaver_improving_2013`. This text guides the reader through the use of sensitivity analysis and uncertainty methods across the diverse perspectives that have shaped modern diagnostic and exploratory modeling.
diff --git a/dev/docs/html/_sources/6_glossary.rst b/dev/docs/html/_sources/6_glossary.rst
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+.. _glossary:
+
+========
+Glossary
+========
+
+**Design of experiment**: Provides a framework for the extraction of all plausible information about the impact of each factor on the output of the numerical model
+
+**Exploratory modeling**: Use of large ensembles of uncertain conditions to discover decision-relevant combinations of uncertain factors 
+
+**Factor**: Any model component that can affect model outputs: inputs, resolution levels, coupling relationships, model relationships and parameters. In models with acceptable model fidelity these factors may represent elements of the real-world system under study. 
+
+**Factor mapping**: A technique to identify which uncertain model factors lead to certain model behavior
+
+**Factor prioritization**: A technique to identify the uncertain factors which, when fixed to their true value, would lead to the greatest reduction in output variability
+
+**Factor screening**: A technique to identify model components that have a negligible effect or make no significant contributions to the variability of the outputs or metrics of interest 
+
+**First-, second-, total-order effects**: First-order effects indicate the percent of model output variance contributed by a factor individually. Second-order effects capture how interactions between a pair of parameter input variables can lead to change in model output. Total-order effects consider all the effects a factor has, individually and in interaction with other factors. 
+
+**Hindcasting**: A type of predictive check that uses the model to estimate output for past events to see how well the output matches the known results. 
+
+**Pre-calibration**: A hybrid uncertainty assessment method that involves identifying a plausible set of parameters using some prespecified screening criterion, such as the distance from the model results to the observations. 
+
+**Prior**: The best assessment of the probability of an event based on existing knowledge before a new experiment is conducted
+
+**Posterior**: The revised or updated probability of an event after taking into account new information 
+
+**Probabilistic inversion**: Uses additional information, for instance, a probabilistic expert assessment or survey result, to update an existing prior distribution
+
+**Return level**: A value that is expected to be equaled or exceeded on average once every interval of time (T) (with a probability of 1/T)
+
+**Return period**: The estimated time interval between events of a similar size or intensity/ 
+
+**Sampling**: The process of selecting model parameters or inputs that characterize the model uncertainty space. 
+
+**Scenario discovery**: Use of large ensembles of uncertain conditions to discover decision-relevant combinations of uncertain factors 
+
+**Sensitivity analysis**: Conducted to understand the factors and processes that most (or least) control a model’s outputs
+
+	*Local sensitivity analysis*: Model evaluation performed by varying uncertain factors around specific reference values
+
+	*Global sensitivity analysis*: Model evaluation performed by varying uncertain factors throughout their entire feasible value space
+
+**Uncertainty**
+
+*Deep uncertainty*: Refers to situations where expert opinions consulted on a decision do not know or cannot agree on system boundaries, or the outcomes of interest and their relative importance, or the prior probability Distribution for the various uncertain factors present
+
+*Epistemic uncertainty*: Systematic uncertainty that comes about due to the lack of knowledge or data to choose the best model
+
+*Ontological uncertainty*: Uncertainties due to processes, interactions, or futures, that are not contained within current conceptual models
+
+*Aleatory uncertainty*: Uncertainty due to natural randomness in processes 
+
+*Uncertainty characterization*: Model evaluation under alternative factor hypotheses to explore their implications for model output uncertainty
+
+*Uncertainty quantification*: Representation of model output uncertainty using probability distributions
+
+**Variance decomposition**: A technique to partition how much of the variability in a model’s output is due to different explanatory variables.  
diff --git a/dev/docs/html/_sources/A1.1_UQ_Introduction.rst b/dev/docs/html/_sources/A1.1_UQ_Introduction.rst
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+Introduction
+############
+
+As defined in :numref:`introduction`, uncertainty quantification (UQ) refers to the formal focus on the full specification of likelihoods as well as distributional forms necessary to infer the joint  probabilistic response across all modeled factors of interest :cite:`cooke1991experts`. This is in contrast to UC (the primary focus of the main document of this book), which is instead aimed at identifying which modeling choices yield the most consequential changes or outcomes and exploring alternative hypotheses related to the form and function of modeled systems :cite:`moallemi2020exploratory, walker2003defining`.
+
+UQ is important for quantifying the relative merits of hypotheses for at least three main reasons. First, identifying model parameters that are consistent with observations is an important part of model development. Due to several effects, including correlations between parameters, simplified or incomplete model structures (relative to the full real-world dynamics), and uncertainty in the observations, many different combinations of parameter values can be consistent with the model structure and the observations to varying extents. Accounting for this uncertainty is conceptually preferable to selecting a single “best fit” parameter vector, particularly as consistency with historical or present observations does not necessarily  guarantee skillful future projections.
+
+The act of quantification requires specific assumptions about distributional forms and likelihoods, which may be more or less justified depending on prior information about the system or model behavior. As a result, UQ is well-suited for studies accounting for or addressing hypotheses related to systems with a relatively large amount of available data and models which are computationally inexpensive, particularly when the emphasis is on prediction. As shown in :numref:`Figure_A1_1`, there is a fundamental tradeoff between the available number of model evaluations (for a fixed computational budget) and the number of parameters treated as uncertain. Sensitivity analyses are therefore part of a typical UQ workflow to identify which factors can be fixed and which ought to be prioritized in the UQ.
+
+.. _Figure_A1_1:
+.. figure:: _static/figureA1_1_UQ_approaches.png
+    :alt: Figure A1.1
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Overview of selected existing approaches for uncertainty quantification and their appropriateness given the number of uncertain model parameters and the number of available model simulations. Green shading denotes regions suitable for uncertainty quantification and red shading indicates regions more appropriate for uncertainty characterization.
+
+The choice of a particular UQ method depends on both the desired level of quantification and the ability to navigate the tradeoff between computational expense and the number of uncertain parameters (:numref:`Figure_A1_1`). For example, Markov chain Monte Carlo with a full system model can provide an improved representation of uncertainty compared to the coarser pre-calibration approach :cite:`ruckert_assessing_2017`, but requires many more model evaluations. The use of a surrogate model to approximate the full system model can reduce the number of needed model evaluations by several orders of magnitude, but the uncertainty quantification can only accommodate a limited number of parameters.
+
+The remainder of this appendix will focus on introducing workflows for particular UQ methods, including a brief discussion of advantages and limitations.
\ No newline at end of file
diff --git a/dev/docs/html/_sources/A1.2_Parametric_Bootstrap.rst b/dev/docs/html/_sources/A1.2_Parametric_Bootstrap.rst
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+Parametric Bootstrap
+####################
+
+The parametric bootstrap :cite:p:`efron_bootstrap_1986` refers to a process of model recalibration to alternate realizations of the data. The bootstrap was originally developed to estimate standard errors and confidence intervals without ascertaining key assumptions that might not hold given the available data. In a setting where observations can be viewed as independent realizations of an underlying stochastic process, a sufficiently rich dataset can be treated as a population representing the data distribution. New datasets are then generated by resampling from the data with replacement, and the model can be refit to each new dataset using maximum-likelihood estimation. The resulting distribution of estimates can then be viewed as a representation of parametric uncertainty.
+
+A typical workflow for the parametric bootstrap is shown in :numref:`Figure_A1_2`. After identifying outputs of interest and preparing the data, the parametric model is fit by some procedure such as minimizing root-mean-square-error or maximizing the likelihood. Alternate datasets are constructed by resampling from the population or by generating new samples from the fitted data-generating process. It is important at this step that the resampled quantities are independent of one another. For example, in the context of temporally- or spatially-correlated data, such as time series, the raw observations cannot be treated as independent realizations. However, the residuals resulting from fitting the model to the data could be (depending on their structure). For example, if the residuals are treated as independent, they can then be resampled with replacement, and these residuals added to the original model fit to create new realizations. If the residuals are assumed to be the result of an autoregressive process, this process could be fit to the original residual series and new residuals be created using this model :cite:`sriver_characterizing_2018`. The model is then refit to each new realization.
+
+.. _Figure_A1_2:
+.. figure:: _static/figureA1_2_bootstrap_workflow.png
+    :alt: Figure A1.2
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Workflow for the parametric bootstrap.
+
+The bootstrap is computationally convenient, particularly as the process of fitting the model to each realization can be easily parallelized. This approach also requires minimal prior assumptions. However, due to the assumption that the available data are representative of the underlying data distribution, the bootstrap can neglect key uncertainties which might influence the results. For example, when using an autoregressive process to generate new residuals, uncertainty in the autocorrelation parameter and innovation variance is neglected, which may bias estimates of, for example, low-probability but high-impact events :cite:`ruckert_effects_2017`.
diff --git a/dev/docs/html/_sources/A1.3_Precalibration.rst b/dev/docs/html/_sources/A1.3_Precalibration.rst
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+Pre-Calibration
+###############
+
+Pre-calibration :cite:`beven_future_1992, edwards_precalibrating_2011, boukouvalas_bayesian_2014` involves the identification of a plausible set of parameters using some prespecified screening criterion, such as the distance from the model results to the observations (based on an appropriate metric for the desired matching features, such as root-mean-squared error). A typical workflow is shown in :numref:`Figure_A1_3`. Parameter values are obtained by systematically sampling the input space (see :numref:`design_of_experiments`). After the model is evaluated at the samples, only those passing the distance criterion are retained. This selects a subset of the parameter space as “plausible” based on the screening criterion, though there is no assignment of probabilities within this plausible region.
+
+.. _Figure_A1_3:
+.. figure:: _static/figureA1_3_precal_workflow.png
+    :alt: Figure A1.3
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Workflow for pre-calibration.
+
+Pre-calibration can be useful for models which are inexpensive enough that a reasonable
+number of samples can be used to represent the parameter space, but which are too expensive to facilitate full uncertainty quantification. High-dimensional parameter spaces, which can be problematic for the uncertainty quantification methods below, may also be explored using pre-calibration. One key prerequisite to using this method is the ability to place a meaningful distance metric on the output space.
+
+However, pre-calibration results in a very coarse characterization of uncertainty, especially when considering a large number of parameters, as more samples are needed to fully characterize the parameter space. Due to the inability to evaluate the relative probability of regions of the parameter space beyond the binary plausible-and-implausible characterization, pre-calibration can also result in degraded hindcast and projection skills and parameter estimates :cite:`makowski_using_2002, shafii_uncertainty-based_2014, ruckert_assessing_2017`.
+
+A related method, widely used in hydrological studies, is generalized likelihood uncertainty estimation, or GLUE :cite:`beven_future_1992`. Unlike pre-calibration, the underlying argument for GLUE relies on the concept of equifinality :cite:`beven_equifinality_2001`, which posits that it is impossible to find a uniquely well-performing parameter vector for models of abstract environmental systems :cite:`beven_equifinality_2001, vrugt_embracing_2018`. In other words, there exist multiple parameter vectors which perform equally or similarly well. As with pre-calibration, GLUE uses a goodness-of-fit measure (though this is called a “likelihood” in the GLUE literature, as opposed to a statistical likelihood function :cite:`stedinger_appraisal_2008`) to evaluate samples. After setting a threshold of acceptable performance with respect to that measure, samples are evaluated and classified into “behavioral” or “non-behavioral” according to the threshold.
+
+.. note::
+
+    Put this into practice! Click the following badge to try out an interactive tutorial on utilizing Pre-Calibration and GLUE for HYMOD model calibration:  `Pre-Calibration Jupyter Notebook <nb_hymod_>`_
diff --git a/dev/docs/html/_sources/A1.4_Markov_Chain_Monte_Carlo.rst b/dev/docs/html/_sources/A1.4_Markov_Chain_Monte_Carlo.rst
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+Markov Chain Monte Carlo
+########################
+
+Markov chain Monte Carlo (MCMC) is a “gold standard” approach to full uncertainty quantification. MCMC refers to a category of algorithms which systematically sample from a target distribution (in this case, the posterior distribution) by constructing a Markov chain. A Markov chain is a probabilistic structure consisting of a state space, an initial probability distribution over the states, and a transition distribution between states. If a Markov chain satisfies certain properties :cite:p:`robert_monte_2013, robert_metropolishastings_2015`, the probability of being in each state will eventually converge to a stable, or stationary, distribution, regardless of the initial probabilities.
+
+MCMC algorithms construct a Markov chain of samples from a parameter space (the combination of model and statistical parameters). This Markov chain is constructed so that the stationary distribution is a target distribution, in this case the (Bayesian) posterior distribution. As a result, after the transient period, the resulting samples can be viewed as a set of dependent samples from the posterior (the dependence is due to the autocorrelation between samples resulting from the Markov chain transitions). Expected values can be computed from these samples (for example, using batch-means estimators :cite:p:`flegal_markov_2008`), or the chain can be sub-sampled or thinned and the resulting samples used as independent Monte Carlo samples due to the reduced or eliminated autocorrelation.
+
+A general workflow for MCMC is shown in :numref:`Figure_A1_4`. The first decision is whether to use the full model or a surrogate model (or emulator). Typical surrogates include Gaussian process emulation :cite:p:`currin_bayesian_1991, sacks_design_1989`, polynomial chaos expansions :cite:p:`ghanem_spectral_1991, xiu_wiener--askey_2002`, support vector machines :cite:p:`ciccazzo_svm_2016, pruett_creation_2016`, and neural networks :cite:p:`eason_adaptive_2014, gorissen_sequential_2009`. Surrogate modeling can be faster, but requires a sufficient number of model evaluations for the surrogate to accurately represent the model’s response surface, and this typically limits the number of parameters which can be included in the analysis.
+
+.. _Figure_A1_4:
+.. figure:: _static/figureA1_4_mcmc_workflow.png
+    :alt: Figure A1.4
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Workflow for Markov chain Monte Carlo.
+
+After selecting the variables which will be treated as uncertain, the next step is to specify the likelihood based on the selected surrogate model or the structure of the data-model residuals. For example, it may not always be appropriate to treat the residuals as independent and identically distributed (as is commonly done in linear regression). A mis-specification of the residual structure can result in biases and over- or under-confident inferences and projections :cite:p:`brynjarsdottir_learning_2014`.
+
+After specifying the prior distributions (see :numref:`critical_first_step`), the selected MCMC algorithm should be used to draw samples from the posterior distribution. There are many MCMC algorithms, all of which have advantages and disadvantages for a particular problem. These include the Metropolis-Hastings algorithm :cite:p:`robert_metropolishastings_2015` and Hamiltonian Monte Carlo :cite:p:`betancourt_conceptual_2017, neal_mcmc_2011`. Software packages typically implement one MCMC method, sometimes designed for a particular problem setting or likelihood specification. For example, R’s *adaptMCMC* implements an adaptive Metropolis-Hastings algorithm :cite:p:`vihola_robust_2012`, while *NIMBLE* :cite:p:`valpine_programming_2017, nimble_development_team_nimble_2021` uses a user-customizable Metropolis-Hastings implementation, as well as functionality for Gibbs sampling (which is a special case of Metropolis-Hastings where the prior distribution has a convenient mathematical form). Some recent implementations, such as *Stan* :cite:p:`stan_development_team_stan_2021`, *pyMC3* :cite:p:`salvatier_probabilistic_2016`, and *Turing* :cite:p:`ge_turing_2018` allow different algorithms to be used.
+
+A main consideration when using MCMC algorithms is testing for convergence to the target distribution. As convergence is guaranteed only for a sufficiently large number of transitions, it is impossible to conclude for certain that a chain has converged for a fixed number of iterations. However, several heuristics have been developed :cite:p:`flegal_markov_2008, gelman_inference_1992` to increase evidence that convergence has occurred.
diff --git a/dev/docs/html/_sources/A1.5_Other_methods.rst b/dev/docs/html/_sources/A1.5_Other_methods.rst
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+Other Methods
+#############
+
+Other common methods for UQ exist. These include sequential Monte Carlo, otherwise known as particle filtering :cite:p:`del_moral_sequential_2006, doucet_sequential_2000, liu_combined_2001`, where a number of particles are used to evaluate samples. An advantage of sequential Monte Carlo is that the vast majority of the computation can be parallelized, unlike with standard MCMC. A major weakness is the potential for degeneracy :cite:p:`doucet_sequential_2000`, where many particles have extremely small weights, resulting in the effective use of only a few samples.
+
+Another method is approximate Bayesian computation (ABC) :cite:p:`cabras_approximate_2015, lintusaari_fundamentals_2017, sunnaker_approximate_2013`. ABC is a likelihood-free approach that compares model output to a set of summary statistics. ABC is therefore well-suited for models and residual structures which do not lend themselves to a computationally-tractable likelihood, but the resulting inferences are known to be biased if the set of summary statistics is not sufficient, which can be difficult to know a-priori.
diff --git a/dev/docs/html/_sources/A1.6_Critical_first_step.rst b/dev/docs/html/_sources/A1.6_Critical_first_step.rst
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+.. _critical_first_step:
+
+The Critical First Step: How to Choose a Prior Distribution
+###########################################################
+
+Prior distributions play an important role in Bayesian uncertainty quantification, particularly when data is limited relative to the dimension of the model. Bayesian updating can be thought of as an information filter, where each additional datum is added to the information contained in the prior; eventually, the prior makes relatively little impact. In real world problems, it can be extremely difficult to assess how much data is required for the choice of prior to become less relevant. The choice of prior can also be influential when conducting SA prior to or without UQ. This is because a prior distribution for a sensitivity analysis for a given parameter which is much wider than the region where the model response surface is sensitive to the parameter value might cause the sensitivity calculation to underestimate the response in that potentially critical region. Similarly, a prior which is too narrow may miss regions where the model responds to the parameter altogether.
+
+Ideally, prior distributions are constructed independently of any analysis of the new data considered. This is because using data to inform the prior as well as to compute the likelihood reuses information in a potentially inappropriate way, which can lead to overconfident inferences. Following :cite:t:`jaynes_probability_1996, gelman_prior_2017` refers to the ideal prior as one which encodes all available information about the model. For practical reasons (difficulty of construction or computational inconvenience), most priors fail to achieve this ideal. These compromises mean that priors should be transparently articulated and justified, so that the impact of the choice of prior can be fully understood. When there is ambiguity about an appropriate prior, such as how fat the tails should be, an analyst should examine how sensitive the UQ results are to the choice of prior.
+
+Priors can also be classified in terms of the information encoded by them, demonstrated in :numref:`Figure_A1_5`. Non-informative priors (illustrated in :numref:`Figure_A1_5` (a)) allegedly correspond to (and are frequently justified by) a position of ignorance. A classic example is the use of a uniform distribution. A uniform prior can, however, be problematic, as it can lead to improper inferences by giving extremely large values the same prior probability as values which may seem more likely :cite:p:`gelman_prior_2017, robert_bayesian_2007`, and therefore does not really reflect a state of complete ignorance. In the extreme case of a uniform prior over the entire real line, every particular region has effectively a prior weight of zero, even though not all regions are a priori unlikely :cite:p:`robert_bayesian_2007`. Moreover, a uniform prior which excludes possible parameter values is not actually noninformative, as it assigns zero probability to those values while jumping to a nonzero probability as soon as the boundary is crossed. While a uniform prior can be problematic for the task of uncertainty quantification, it may be useful for an initial sensitivity analysis to identify the boundary of any regions where the model is sensitive to the parameter.
+
+.. _Figure_A1_5:
+.. figure:: _static/figureA1_5_priors_posteriors.png
+    :alt: Figure A1.5
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Impact of priors on posterior inferences. These plots show the results of inference for a linear regression model with 15 data points. The true value of the parameter is equal to -3. All priors have mean 0. In panel (a), a non-informative prior allows the tails of the posterior to extend freely, which may result in unreasonably large parameter values. In panel (b), a weakly informative prior constrains the tails more, but allows them to extend without too much restriction. In panel (c), an informative prior strongly constrains the tails of the posterior and biases the inference closer towards the prior mean (the posterior mean is -0.89 in this case, and closer to -3 in the other two cases).
+
+Informative priors strongly bound the range of probable values (illustrated in :numref:`Figure_A1_5` (c)). One example is a Gaussian distribution with a relatively small standard deviation, so that large values are assigned a close to null prior probability. Another example is the jump from zero to non-zero probability occurring at the truncation point of a truncated Gaussian, which could be justified based on information that the parameter cannot take on values beyond this point. Without this type of justification, however, priors may be too informative, failing to allow the information contained in the available data to update them.
+
+Finally, weakly informative priors (illustrated in :numref:`Figure_A1_5` (b)) fall in between :cite:p:`gelman_prior_2017`. They regularize better than non-informative priors, but allow for more inference flexibility than fully informative priors. An example might be a Gaussian distribution with a moderate standard deviation, which still assigns negligible probability for values far away from the mean, but is less constrained than a narrow Gaussian for a reasonably large area. A key note is that it is not necessarily better to be more informative if this cannot be justified by the available information.
\ No newline at end of file
diff --git a/dev/docs/html/_sources/A1.7_Critical_final_step.rst b/dev/docs/html/_sources/A1.7_Critical_final_step.rst
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+The Critical Final Step: Predictive Checks
+##########################################
+
+Every UQ workflow requires a number of choices, potentially including selecting prior distributions, the likelihood specification, and any used numerical models. Checking the appropriateness of these choices is an essential step for sound inferences, as misspecification can produce biased results :cite:p:`brynjarsdottir_learning_2014`. Model checking in this fashion is part of an iterative UQ process, as the results can reveal adjustments to the statistical model or the need to select a different numerical model :cite:p:`gelman_posterior_1996, gelman_bayesian_2020, gelman_philosophy_2013`.
+
+A classic example is the need to check the structure of residuals for correlations. Many standard statistical models, such as linear regression, assume that the residuals are independent and identically distributed from the error distribution. The presence of correlations, including temporal autocorrelations and spatial correlations, indicates a structural mismatch between the likelihood and the data. In these cases, the likelihood should be adjusted to account for these correlations.
+
+Checking residuals in this fashion is one example of a predictive check (or a posterior predictive check in the Bayesian setting). One way to view UQ is as a means to recover data-generating processes (associated with each parameter vector) consistent with the observations. Predictive checks compare the inferred data-generating process to the observations to determine whether the model is capable of appropriately capturing uncertainty. After conducting the UQ analysis, alternatively realized datasets are simulated from sampled parameters. These alternative datasets, or their summary statistics, can be tested against the observations to determine adequacy of the fit. Predictive checks are therefore a way of probing various model components to identify shortcomings that might result in biased inferences or poor projections, depending on the goal of the analysis.
+
+One example of a graphical predictive check for time series models is hindcasting, where predictive intervals are constructed from the alternative datasets and plotted along with the data. Hindcasts demonstrate how well the model is capable of capturing the broader dynamics of the data, as well as whether the parameter distributions produce appropriate levels of output uncertainty. A related quantitative check is the surprise index, which calculates the percentage of data points located within a fixed predictive interval. For example, the 90% predictive interval should contain approximately 90% of the data. More uncertainty than this reflects underconfidence, while less uncertainty reflects overconfidence. This could be the result of priors that are not appropriately informative, or a likelihood that does not account for correlations between data points appropriately. It could also be the result of a numerical model that isn’t sufficiently sensitive to the parameters that are treated as uncertain.
diff --git a/dev/docs/html/_sources/A1.8_Key_take_home.rst b/dev/docs/html/_sources/A1.8_Key_take_home.rst
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--- /dev/null
+++ b/dev/docs/html/_sources/A1.8_Key_take_home.rst
@@ -0,0 +1,16 @@
+Key Take-Home Points
+####################
+
+When appropriate, UQ is an important component of the exploratory modeling workflow. While a number of parameter sets could be consistent with observations, they may result in divergent model outputs when exposed to different future conditions. This can result in identifying risks which are not visible when selecting a “best fit” parameterization. Quantifying uncertainties also allows us to quantify the support for hypotheses, which is an essential part of the scientific process.
+
+Due to the scale and complexity of the experiments taking place in IM3, UQ has not been extensively used. The tradeoff between the available number of function evaluations and the number of uncertain parameters illustrated in :numref:`Figure_A1_1` is particularly challenging due to the increasing complexity of state-of-the-art models and the movement towards coupled, multisector models. This tradeoff can be addressed somewhat through the use of emulators and parallelizable methods. In particular, when attempting to navigate this tradeoff by limiting the number of uncertain parameters, it is important to carefully iterate with sensitivity analyses to ensure that critical parameters are identified.
+
+Specifying prior distributions and likelihoods is an ongoing challenge. Prior distributions, in particular, should be treated as deeply uncertain when appropriate. One key advantage of the methods described in this chapter is that they have the potential for increased transparency. When it is not possible to conduct a sensitivity analysis on a number of critical priors due to limited computational budgets, fully specifying and providing a justification for the utilized distributions allows other researchers to identify key assumptions and build on existing work. The same is true for the specification of likelihoods---while likelihood-free methods avoid the need to specify a likelihood function, they require other assumptions or choices, which should be described and justified as transparently as possible.
+
+We conclude this appendix with some key recommendations:
+1. UQ analysis does not require full confidence in priors and likelihoods. Rather, UQ should be treated as part of an exploratory modeling workflow, where hypotheses related to model structures, prior distributions, and likelihoods can be tested.
+2. For complex multisectoral models, UQ will typically require the use of a reduced set of parameters, either through emulation or by fixing the others to their best-fit values. These parameters should be selected through a careful sensitivity analysis.
+3. Avoid the use of supposedly “non-informative” priors, such as uniform priors, whenever possible. In the absence of strong information about parameter values, the use of weakly informative priors, such as diffuse normals, is preferable.
+4. Be cognizant of the limitations of conclusions that can be drawn by using each method. The bootstrap, for example, may result in overconfidence if the dataset is limited and is not truly representative of the underlying stochastic process.
+5. When using MCMC, Markov chains can not be shown to have converged to the target distribution, but rather evidence can be collected to demonstrate that it is likely that they have.
+6. Conduct predictive checks based on the assumptions underlying the choices made in the analysis, and iteratively update those choices if the assumptions prove to be ill-suited for the problem at hand.
diff --git a/dev/docs/html/_sources/A1_Uncertainty_Quantification.rst b/dev/docs/html/_sources/A1_Uncertainty_Quantification.rst
new file mode 100644
index 0000000..68718d4
--- /dev/null
+++ b/dev/docs/html/_sources/A1_Uncertainty_Quantification.rst
@@ -0,0 +1,18 @@
+.. raw:: latex
+
+    \appendix
+
+.. _A1_uncertainty_quantification:
+
+**************************
+Uncertainty Quantification
+**************************
+
+.. include:: A1.1_UQ_Introduction.rst
+.. include:: A1.2_Parametric_Bootstrap.rst
+.. include:: A1.3_Precalibration.rst
+.. include:: A1.4_Markov_Chain_Monte_Carlo.rst
+.. include:: A1.5_Other_methods.rst
+.. include:: A1.6_Critical_first_step.rst
+.. include:: A1.7_Critical_final_step.rst
+.. include:: A1.8_Key_take_home.rst
diff --git a/dev/docs/html/_sources/A2.1_fishgame.rst b/dev/docs/html/_sources/A2.1_fishgame.rst
new file mode 100644
index 0000000..783399b
--- /dev/null
+++ b/dev/docs/html/_sources/A2.1_fishgame.rst
@@ -0,0 +1,416 @@
+Fishery Dynamics Tutorial
+*************************
+
+.. note::
+    
+    | Run the tutorial interactively:  `Fishery Dynamics Notebook <nb_fishery_dynamics_>`_.
+    | Please be aware that notebooks can take a couple minutes to launch.
+    | To run the notebooks yourself, download the files `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks>`_ and use these `requirements <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt>`_.
+
+
+Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics
+======================================================================================
+
+This notebook demonstrates the application of sensitivity analysis to
+discover factors that shape the behavior modes of a socio-ecological
+system with dynamic human action.
+
+The system of differential equations below represent a system of prey
+(defined in the equation below as x) and predator (defined as y) fish,
+with a human actor harvesting the prey fish. You can read more about
+this system at `Hadjimichael et
+al. (2020) <https://doi.org/10.1155/2020/4170453>`__.
+
+
+.. image:: _static/eqn2.png
+
+The table below defines the parameters in the system and also denotes
+the baseline and ranges associated with each uncertain parameter.
+
+.. image:: _static/table1.png
+
+The system is simple but very rich in the dynamic behaviors it exhibits.
+This complexity is accompanied by the presence of several equilibria
+that come in and out of existence with different parameter values. The
+equilibria also change in their stability according to different
+parameter values, giving rise to different behavior modes as shown by
+the diverse predator and prey abundace trajectories in the figure below.
+
+.. image:: _static/Figure_1.png
+
+In the unharvested system (without the human actor) the stability of
+several of these equilibria can be derived analytically. The task
+becomes significantly more difficult when the adaptive human actor is
+introduced, deciding to harvest the system at different rates according
+to their objectives and preferences.
+
+Sensitivity analysis methods can help us identify the factors that most
+control these dynamics by exploring the space of parameter values and
+seeing how system outputs change as a result.
+
+Through previously conducted optimization, there already exists a set of
+potential harvesting strategies that were identified in pursuit of five
+objectives:
+
+-  Maximize Harvesting Discounted Profits (Net Present Value)
+-  Minimize Prey Population Deficit
+-  Minimize Longest Duration of Consecutive Low Harvest
+-  Maximize Worst Harvest Instance
+-  Minimize Harvest Variance
+
+The identified harvesting strategies also meet the necessary constraint
+of not causing inadvertent predator collapse.
+
+We will be examining the effects of parametric uncertainty on these
+identified strategies, particularly focusing on two strategies: one
+selected to maximize harvesting profits and one identified through
+previous analysis to perform ‘well enough’ for all objectives across a
+wide range of states of the world (referred to as the ‘robust’
+harvesting policy).
+
+Let’s get started!
+------------------
+
+In this tutorial, we will be loading in data that has been produced in
+Hadjimichael et al. (2020). Before we start our analysis, we’ll load the
+relevant Python libraries. **NOTE**: To step through the notebook,
+execute each gray (code) box by typing “Shift+Enter”.
+
+.. code:: ipython3
+
+    #Import necessary libraries
+    
+    import msdbook
+    import numpy as np
+    import matplotlib.pyplot as plt
+    from SALib.sample import saltelli
+    from SALib.analyze import sobol
+    from matplotlib import patheffects as pe
+    
+    # load example data
+    msdbook.install_package_data()
+    
+    %matplotlib inline
+    %config InlineBackend.print_figure_kwargs = {'bbox_inches':None}
+
+.. parsed-literal::
+
+    Downloading example data for msdbook version 0.1.5...
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/uncertain_params_bounds.txt
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_metric_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_vary_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_mth_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/solutions.resultfile
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/3704614_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/LHsamples_original_1000.txt
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/3704614_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/param_values.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_yr_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_mth_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7000550_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/collapse_days.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/hymod_params_256samples.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_vary_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7000550_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7200799_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_yr_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7200799_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/LeafCatch.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/hymod_simulations_256samples.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/Robustness.txt
+
+Step 1: Load identified solutions and explore performance
+---------------------------------------------------------
+
+Here we load in the solution set obtained in Hadjimichael et al. (2020).
+The solution set contains the decision variables and objectives
+associated with a variety of harvesting policies. For this tutorial, we
+focus on comparing two policies: harvesting profits and one that
+performs robustly across all objectives. Below, we are reading in the
+decision variables and objectives from an external file that can be
+found within the msdbook package data.
+
+.. code:: ipython3
+
+    robustness = msdbook.load_robustness_data()
+    results = msdbook.load_profit_maximization_data()
+    
+    robust_solution = np.argmax(robustness[:,-1]) #pick robust solution
+    profit_solution = np.argmin(results[:,6]) #pick profitable solution
+    objective_performance = -results[:,6:] #Retain objective values 
+    
+    # Get decision variables for each of the policies
+    highprofitpolicy = results[profit_solution,0:6]
+    mostrobustpolicy = results[robust_solution,0:6]
+
+
+Next we plot the identified solutions with regards to their objective
+performance in a parallel axis plot
+
+.. tip:: View the source code used to create this plot here: `plot_objective_performance <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-objective-performance>`_
+
+.. code:: ipython3
+
+    ax, ax1 = msdbook.plot_objective_performance(objective_performance, profit_solution, robust_solution)
+
+
+.. image:: _static/fishery_output_6_0.png
+
+
+The solution set from the optimization in Hadjimichael et al. (2020) are
+presented in a parallel axis plot where each of the five objectives (and
+one constraint) are represented as an axis. Each solution on the Pareto
+front is represented as a line where the color of the line indicates the
+value of the NPV objective. The preference for objective values is in
+the upward direction. Therefore, the ideal solution would be a line
+straight across the top of the plot that satisfies every objective.
+However, no such line exists because there are tradeoffs when sets of
+objectives are prioritized over the others. When lines cross in between
+axes, this indicates a tradeoff between objectives (as seen in the first
+two axes).The solution that is most robust in the NPV objective has the
+highest value on the first axis and is outlined in dark gold. The
+solution that is most robust across all objectives is outlined in a
+brighter yellow. A parallel axis is an effective visual to characterize
+high-dimensional tradeoffs in the system and visualize differences in
+performance across policies.
+
+Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis
+-----------------------------------------------------------------------
+
+In Step 1, we showed how the optimized harvesting policies performed in
+the objective space, which utilized the baseline parameters outlined in
+the table above. Now, we are interested in understanding how sensitive
+our two policies are to alternative states of the world that may be
+characterized by different parameter values. To do so, we first need to
+define the problem dictionary that allows us to generate these
+alternative states of the world.
+
+.. code:: ipython3
+
+    # Set up SALib problem
+    problem = {
+      'num_vars': 9,
+      'names': ['a', 'b', 'c', 'd', 'h', 'K', 'm', 'sigmaX', 'sigmaY'],
+      'bounds': [[0.002, 2], [0.005, 1], [0.2, 1], [0.05, 0.2], [0.001, 1],
+                 [100, 5000], [0.1, 1.5], [0.001, 0.01], [0.001, 0.01]]
+    }
+
+
+Then we use the following command to generate a Saltelli sample from
+these defined ranges:
+
+.. code:: python
+
+   param_values = saltelli.sample(problem, 1024, calc_second_order=False)
+
+Generally, it is a good idea to save the result of the sample since it
+is often reused and regenerating it produces a different sample set. For
+this reason, we will load one from file that was previously generated.
+
+.. code:: ipython3
+
+    # load previously generated Saltelli sample from our msdbook package data
+    param_values = msdbook.load_saltelli_param_values()
+
+
+Step 3: Evaluate the system over all generated states of the world
+------------------------------------------------------------------
+
+Now we re-evaluate how well the policies do in the new states of the
+world. In order to characterize failure of a policy, we identify the
+states where the predator population collapses, as an inadvertent
+consequence of applying the harvesting strategy under a state of the
+world different from the one originally assumed. Due to how long this
+step takes to execute within the tutorial, we will read in the solutions
+from an external file. However, the block of code below shows how
+evaluation can be implemented.
+
+.. code:: python
+
+   # create array to store collapse values under both policies
+   collapse_days = np.zeros([len(param_values), 2])
+
+   # evaluate performance under every state
+   for i in range(len(param_values)):
+       
+       additional_inputs = np.append(['Previous_Prey'],
+                                     [param_values[i,0],
+                                      param_values[i,1],
+                                      param_values[i,2],
+                                      param_values[i,3],
+                                      param_values[i,4],
+                                      param_values[i,5],
+                                      param_values[i,6],
+                                      param_values[i,7],
+                                      param_values[i,8]])
+       
+       collapse_days[i,0]=fish_game(highprofitpolicy, additional_inputs)[1][0]
+       collapse_days[i,1]=fish_game(mostrobustpolicy, additional_inputs)[1][0]
+
+.. code:: ipython3
+
+    # load the simulation data from our msdbook package data
+    collapse_days = msdbook.load_collapse_data()
+
+
+Step 4: Calculate sensitivity indices
+-------------------------------------
+
+Now we use a Sobol sensitivity analysis to calculate first-order,
+second-order, and total-order sensitivity indices for each parameter and
+for each of the two policies. These indicies help determine which
+factors explain the most variability in the number of days of predator
+population collapse.
+
+.. code:: ipython3
+
+    #Perform the Sobol SA for the profit-maximizing solution
+    Si_profit = sobol.analyze(problem, collapse_days[:, 0], 
+                              calc_second_order=False, 
+                              conf_level=0.95, 
+                              print_to_console=True)
+
+
+.. code:: ipython3
+
+    #Perform the Sobol SA for the robust solution
+    Si_robustness = sobol.analyze(problem, 
+                                  collapse_days[:, 1], 
+                                  calc_second_order=False, 
+                                  conf_level=0.95, 
+                                  print_to_console=True)
+
+
+.. parsed-literal::
+
+                  ST   ST_conf
+    a       0.226402  0.036146
+    b       0.066819  0.013347
+    c       0.004395  0.004023
+    d       0.024509  0.006993
+    h       0.009765  0.005488
+    K       0.020625  0.009494
+    m       0.897971  0.066470
+    sigmaX  0.000136  0.000149
+    sigmaY  0.000739  0.001040
+                  S1   S1_conf
+    a       0.087936  0.044236
+    b       0.000554  0.021474
+    c      -0.002970  0.004590
+    d       0.001206  0.015881
+    h       0.004554  0.007998
+    K       0.003843  0.012661
+    m       0.751301  0.071862
+    sigmaX -0.000325  0.001245
+    sigmaY -0.001887  0.002768
+
+
+Looking at the total-order indices, (ST) factors :math:`m`, :math:`a`,
+:math:`b`, :math:`d` and :math:`K` explain a non-negligible amount of
+variance therefore have an effect on the stability of this system.
+Looking at the first-order indices (S1), we also see that besides
+factors :math:`m` and :math:`a`, all other factors are important in this
+system through their interactions, which make up the difference between
+their S1 and ST indices. This shows the danger of limiting sensitivity
+analyses to first order effects, as factor importance might be
+significantly misjudged.
+
+These findings are supported by the analytical condition of equilibrium
+stability in this system:
+
+.. image:: _static/eqn4.png
+
+In an unharvested system, this condition is both necessary and
+sufficient for the equilibrium of the two species coexisting to be
+stable.
+
+When adaptive human action is introduced however, this condition is
+still necessary, but no longer sufficient, as harvesting reduces the
+numbers of prey fish and as a result reduces the resources for the
+predator fish. Since this harvesting value is not constant, but can
+dynamically adapt according to the harvester’s objectives, it cannot be
+introduced into this simple equation.
+
+Step 5: Explore relationship between uncertain factors and performance
+----------------------------------------------------------------------
+
+In the following steps, we will use the results of our sensitivity
+analysis to investigate the relationships between parametric
+uncertainty, equilibrium stability and the performance of the two
+policies.
+
+We can use the top three factors identified (:math:`m`, :math:`a`, and
+:math:`b`) to visualize the performance of our policies in this
+three-dimensional parametric space.
+
+We first define the stability condition, as a function of :math:`b` and
+:math:`m`, and calculate the corresponding values of :math:`a`.
+
+.. code:: ipython3
+
+    def inequality(b, m, h, K):
+        return ((b**m)/(h*K)**(1-m))
+    
+    # boundary interval that separates successful and failed states of the world
+    b = np.linspace(start=0.005, stop=1, num=1000)
+    m = np.linspace(start=0.1, stop=1.5, num=1000)
+    h = np.linspace(start=0.001, stop=1, num=1000)
+    K = np.linspace(start=100, stop=2000, num=1000)
+    b, m = np.meshgrid(b, m)
+    a = inequality(b, m, h, K)
+    a = a.clip(0,2)
+
+
+.. tip:: View the source code used to create this plot here: `plot_factor_performance <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-factor-performance>`_
+
+.. code:: ipython3
+
+    # generate plot
+    ax1, ax2 = msdbook.plot_factor_performance(param_values, collapse_days, b, m, a)
+
+    
+.. image:: _static/fishery_output_22_0.png
+
+
+These figures show the combinations of factors that lead to success or
+failure in different states of the world for the profit-maximizing and
+robust policies. Each point is a state of the world, characterized by
+specific values of the parameters, and ideally, we would like the color
+of the point to be blue, to represent that there are a low number of
+days with a predator collapse in that world. The gray curve denotes the
+highly non-linear nature of the boundary, defined by the stability
+condition, that separates successful and failed states of the world. The
+figures demonstrate the following key points:
+
+First, as asserted above, the policies interact with the system in
+different and complex ways. In the presence of human action, the
+stability condition is not sufficient in determining whether the policy
+will succeed, even though it clearly shapes the system in a fundamental
+manner.
+
+Secondly, the robust policy manages to avoid collapse in many more of the sampled states of the world, indicated by the number of blue points. The robust policy avoids collapse in 31% of worlds versus 14% in the profit-maximizing policy.This presents a clear tradeoff between profit-maximizing performance androbustness against uncertainty.
+
+Tips to Apply Sobol SA and Scenario Discovery to your Problem
+-------------------------------------------------------------
+
+In this tutorial, we demonstrated a Sobol SA to identify the most
+important factors driving the behavior of a system (i.e. the number of
+the collapse days). In order to apply this methodology to your problem,
+you will need to have a set of optimized policies for your system that
+you are interested in analyzing. The general workflow is as follows:
+
+1. Choose sampling bounds for your parameters and set up the problem
+   dictionary as in Step 2 above.
+2. Generate samples, or alternative states of the world using the
+   ``saltelli.sample`` function.
+3. Evaluate your policies on the alternative states of the world. For
+   your application, you will also need to develop a rule for
+   determining success or failure of your policy in a new SOW. In this
+   tutorial, success was denoted by a small number of collapse days.
+   Ultimately, the rule will be specific to your application and can
+   include various satisficing criteria.
+4. Calculate the Sobol indices and discover the most important
+   parameters driving success and failure.
+5. Finally, use a similar plotting procedure as in step 5 to identify
+   the combination of parameter values that lead to success and failure
+   in the system.
diff --git a/dev/docs/html/_sources/A2.2_saltelli.rst b/dev/docs/html/_sources/A2.2_saltelli.rst
new file mode 100644
index 0000000..409681a
--- /dev/null
+++ b/dev/docs/html/_sources/A2.2_saltelli.rst
@@ -0,0 +1,302 @@
+Sobol SA Tutorial
+*************************
+
+.. note::
+    
+    | Run the tutorial interactively:  `Sobol SA Tutorial <nb_saltelli_sobol_>`_.
+    | Please be aware that notebooks can take a couple minutes to launch.
+    | To run the notebooks yourself, download the files `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks>`_ and use these `requirements <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt>`_.
+
+Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA
+====================================================================================
+
+In this tutorial, we will set up a workflow to investigate how sensitive
+the output of a function is to its inputs. Why might you want to do
+this? Imagine that this function represents a complex system, such as
+the rainfall-runoff process of a watershed model, and that you, the
+researcher, want to investigate how your choice of input parameter
+values are affecting the model’s characterization of runoff in the
+watershed. Your parameter values are likely uncertain and can take on
+any value in a pre-defined range. Using a Sobol SA will allow you to
+sample different values of your parameters and calculate how sensitive
+your output of interest is to certain parameters. Below, we demonstrate
+Sobol SA for a simple function to illustrate the method, but the
+workflow can be applied to your own problem of interest!
+
+In order to conduct this analysis, we will use the popular Python
+Sensitivity Analysis Library
+(`SALib <https://salib.readthedocs.io/en/latest/index.html>`__) to:
+
+#. Generate a problem set as a dictionary for our Ishigami function that has three inputs
+#. Generate 2048 samples for our problem set using the Saltelli [1]_ [2]_ sampling scheme
+#. Execute the Ishigami function for each of our samples and gather the outputs
+#. Compute the sensitivity analysis to generate first-order and total-order sensitivity indices using the Sobol [3]_ method
+#. Interpret the meaning of our results
+
+Let’s get started!
+------------------
+
+**NOTE**: Content from this tutorial is taken directly from the SALib
+`“Basics” <https://salib.readthedocs.io/en/latest/basics.html>`__
+walkthrough. To step through the notebook, execute each gray (code) box
+by typing “Shift+Enter”.
+
+.. code:: ipython3
+
+    #Import relevant libraries
+    import numpy as np
+    import matplotlib.pyplot as plt 
+    from mpl_toolkits import mplot3d
+    
+    from SALib.sample import saltelli
+    from SALib.analyze import sobol
+    from SALib.test_functions import Ishigami
+
+Step 1: Generate the problem dictionary
+---------------------------------------
+
+The Ishigami function is of the form:
+
+.. math:: f(x_1,x_2,x_3) = sin(x_1)+asin^2(x_2)+bx_3^4sin(x_1)
+
+The function has three inputs, 𝑥1, 𝑥2, 𝑥3 where 𝑥𝑖 ∈ [−𝜋, 𝜋]. The
+constants :math:`a` and :math:`b` are defined as 7.0 and 0.1
+respectively.
+
+.. code:: ipython3
+
+    #Create a problem dictionary. Here we supply the number of variables, the names of each variable, and the bounds of the variables.
+    problem = {
+        'num_vars': 3,
+        'names': ['x1', 'x2', 'x3'],
+        'bounds': [[-3.14159265359, 3.14159265359],
+                   [-3.14159265359, 3.14159265359],
+                   [-3.14159265359, 3.14159265359]]
+    }
+
+Step 2: Generate samples using the Saltelli sampling scheme
+-----------------------------------------------------------
+
+Sobol SA requires the use of the Saltelli sampling scheme. The output of
+the ``saltelli.sample`` function is a NumPy array that is of shape 2048
+by 3. The sampler generates 𝑁∗(2𝐷+2) samples, where in this example, N
+is 256 (the argument we supplied) and D is 3 (the number of model
+inputs), yielding 2048 samples. The keyword argument
+``calc_second_order=False`` will exclude second-order indices, resulting
+in a smaller sample matrix with 𝑁∗(𝐷+2) rows instead. Below, we plot the
+resulting Saltelli sample.
+
+.. code:: ipython3
+
+    #Generate parmeter values using the saltelli.sample function
+    param_values = saltelli.sample(problem, 256)
+    
+    print(f"`param_values` shape:  {param_values.shape}")
+
+
+
+.. parsed-literal::
+
+    `param_values` shape:  (2048, 3)
+
+
+.. code:: ipython3
+
+    #Plot the 2048 samples of the parameters 
+    
+    fig = plt.figure(figsize = (7, 5))
+    ax = plt.axes(projection ="3d")
+    ax.scatter3D(param_values[:,0], param_values[:,1], param_values[:,2])
+    ax.set_xlabel('X1 Parameter')
+    ax.set_ylabel('X2 Parameter')
+    ax.set_zlabel('X3 Parameter')
+    plt.title("Saltelli Sample of Parameter Values")
+    
+    plt.show()
+
+
+
+.. image:: _static/output_7_0.png
+
+
+Step 3: Execute the Ishigami function over our sample set
+---------------------------------------------------------
+
+SALib provides a nice wrapper to the Ishigami function that allows the
+user to directly pass the ``param_values`` array we just generated into
+the function directly.
+
+.. code:: ipython3
+
+    Y = Ishigami.evaluate(param_values)
+
+Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method
+-------------------------------------------------------------------------------------------
+
+The ``sobol.analyze`` function will use our problem dictionary and the
+result of the Ishigami runs (``Y``) to compute first-, second-, and
+total-order indicies.
+
+.. code:: ipython3
+
+    Si = sobol.analyze(problem, Y)
+
+``Si`` is a Python dict with the keys “S1”, “S2”, “ST”, “S1_conf”,
+“S2_conf”, and “ST_conf”. The ``_conf`` keys store the corresponding
+confidence intervals, typically with a confidence level of 95%. Use the
+keyword argument ``print_to_console=True`` to print all indices. Or, we
+can print the individual values from ``Si`` as shown in the next step.
+
+Step 5: Interpret our results
+-----------------------------
+
+We execute the following code and take a look at our first-order indices
+(``S1``) for each of our three inputs. These indicies can be interpreted
+as the fraction of variance in the output that is explained by each
+input individually.
+
+.. code:: ipython3
+
+    first_order = Si['S1']
+    
+    print('First-order:')
+    print(f"x1: {first_order[0]}, x2: {first_order[1]}, x3: {first_order[2]}")
+
+
+.. parsed-literal::
+
+    First-order:
+    x1: 0.3184242969763115, x2: 0.4303808201623416, x3: 0.022687722804980225
+
+
+If we were to rank the importance of the inputs in how much they
+individually explain the variance in the output, we would rank them from
+greatest to least importance as follows: 𝑥2, 𝑥1 and then 𝑥3. Since 𝑥3
+only explains 2% of the output variance, it does not explain output
+variability meaningfully. Thus, this indicates that there is
+contribution to the output variance by 𝑥2 and 𝑥1 independently, whereas
+𝑥3 does not contribute to the output variance. Determining what inputs
+are most important or what index value is meaningful is a common
+question, but one for which there is no general rule or threshold. This
+question is problem and context-dependent, but procedures have been
+identified to rank order influential inputs and which can be used to
+identify the least influential factors. These factors can be fixed to
+simplify the model [4]_ [5]_ [6]_.
+
+Next, we evaluate the total-order indices, which measure the
+contribution to the output variance caused by varying the model input,
+including both its first-order effects (the input varying alone) and all
+higher-order interactions across the input parameters.
+
+.. code:: ipython3
+
+    total_order = Si['ST']
+    
+    print('Total-order:')
+    print(f"x1: {total_order[0]}, x2: {total_order[1]}, x3: {total_order[2]}")
+
+
+.. parsed-literal::
+
+    Total-order:
+    x1: 0.5184119098161343, x2: 0.41021260250026054, x3: 0.2299058431439953
+
+
+The magnitude of the total order indices are substantially larger than
+the first-order indices, which reveals that higher-order interactions
+are occurring, i.e. that the interactions across inputs are also
+explaining some of the total variance in the output. Note that 𝑥3 has
+non-negligible total-order indices, which indicates that it is not a
+consequential parameter when considered in isolation, but becomes
+consequential and explains 25% of variance in the output through its
+interactions with 𝑥1 and 𝑥2.
+
+Finally, we can investigate these higher order interactions by viewing
+the second-order indices. The second-order indicies measure the
+contribution to the output variance caused by the interaction between
+any two model inputs. Some computing error can appear in these
+sensitivity indices, such as negative values. Typically, these computing
+errors shrink as the number of samples increases.
+
+.. code:: ipython3
+
+    second_order = Si['S2']
+    
+    print("Second-order:")
+    print(f"x1-x2:  {second_order[0,1]}")
+    print(f"x1-x3:  {second_order[0,2]}")
+    print(f"x2-x3:  {second_order[1,2]}")
+
+
+
+.. parsed-literal::
+
+    Second-order:
+    x1-x2:  -0.043237389723234154
+    x1-x3:  0.17506452088709862
+    x2-x3:  -0.03430682392607577
+
+
+We can see that there are strong interactions between 𝑥1 and 𝑥3. Note
+that in the Ishigami function, these two variables are multiplied in the
+last term of the function, which leads to interactive effects. If we
+were considering first order indices alone, we would erroneously assume
+that 𝑥3 explains no variance in the output, but the second-order and
+total order indices reveal that this is not the case. It’s easy to
+understand where we might see interactive effects in the case of the
+simple Ishigami function. However, it’s important to remember that in
+more complex systems, there may be many higher-order interactions that
+are not apparent, but could be extremely consequential in explaining the
+variance of the output.
+
+Tips to Apply Sobol SA to Your Own Problem
+------------------------------------------
+
+In this tutorial, we demonstrated how to apply an SA analysis to a
+simple mathematical test function. In order to apply a Sobol SA to your
+own problem, you will follow the same general workflow that we defined
+above. You will need to:
+
+1. Choose sampling bounds for your parameters and set up the problem
+   dictionary as in Step 1 above.
+2. Generate samples using the ``saltelli.sample`` function. This step is
+   problem-dependent and note that the Sobol method can be
+   computationally intensive depending on the model being analyzed. For
+   example, for a simple rainfall-runoff model such as HYMOD, it has
+   been recommended to run a sample size of at least N = 10,000 (which
+   translates to 60,000 model runs). More complex models will be slower
+   to run and will also require more samples to calculate accurate
+   estimates of Sobol indices. Once you complete this process, pay
+   attention to the confidence bounds on your sensitivity indices to see
+   whether you need to run more samples.
+3. Run the parameter sets through your model. In the example above, the
+   Ishigami function could be evaluated through SALib since it is a
+   built in function. For your application, you will need to run these
+   parameter sets through the problem externally and save the output.
+   The output file should contain one row of output values for each
+   model run.
+4. Calculate the Sobol indices. Now, the Y will be a numpy array with
+   your external model output and you will need to include the parameter
+   samples as an additional argument.
+5. Finally, we interpet the results. If the confidence intervals of your
+   dominant indices are larger than roughly 10% of the value itself, you
+   may want to consider increasing your sample size as computation
+   permits. You should additionally read the references noted in Step 5
+   above to understand more about identifying important factors.
+
+
+.. only:: html
+    
+    **References**
+
+.. [1] Saltelli, A. (2002). "Making best use of model evaluations to compute sensitivity indices." Computer Physics Communications, 145(2):280-297, doi:`10.1016/S0010-4655(02)00280-1 <https://doi.org/10.1016/S0010-4655(02)00280-1>`_.
+
+.. [2] Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and S. Tarantola (2010). "Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index." Computer Physics Communications, 181(2):259-270, doi:`10.1016/j.cpc.2009.09.018 <https://doi.org/10.1016/j.cpc.2009.09.018>`_.
+
+.. [3] Sobol, I. M. (2001). "Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates." Mathematics and Computers in Simulation, 55(1-3):271-280, doi:`10.1016/S0378-4754(00)00270-6 <https://doi.org/10.1016/S0378-4754(00)00270-6>`_.
+
+.. [4] T.\  H.\  Andres, "Sampling methods and sensitivity analysis for large parameter sets," Journal of Statistical Computation and Simulation, vol. 57, no. 1–4, pp. 77–110, Apr. 1997, doi:`10.1080/00949659708811804 <https://doi.org/10.1080/00949659708811804>`_.
+
+.. [5] Y.\  Tang, P.\  Reed, T.\  Wagener, and K.\  van Werkhoven, "Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation," Hydrology and Earth System Sciences, vol. 11, no. 2, pp. 793–817, Feb. 2007, doi:`10.5194/hess-11-793-2007 <https://doi.org/10.5194/hess-11-793-2007>`_.
+
+.. [6] J.\  Nossent, P.\  Elsen, and W.\  Bauwens, "Sobol' sensitivity analysis of a complex environmental model," Environmental Modelling & Software, vol. 26, no. 12, pp. 1515–1525, Dec. 2011, doi:`10.1016/j.envsoft.2011.08.010 <https://doi.org/10.1016/j.envsoft.2011.08.010>`_.
diff --git a/dev/docs/html/_sources/A2.3_logistic.rst b/dev/docs/html/_sources/A2.3_logistic.rst
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+Logistic Regression Tutorial
+****************************
+
+.. note::
+
+    | Run the tutorial interactively:  `Logistic Regression Tutorial <nb_logistic_regression_>`_.
+    | Please be aware that notebooks can take a couple minutes to launch.
+    | To run the notebooks yourself, download the files `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks>`_ and use these `requirements <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt>`_.
+
+Tutorial: Logistic Regression for Factor Mapping
+================================================
+
+This tutorial replicates a scenario discovery analysis performed in
+`Hadjimichael et
+al. (2020) <https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020EF001503>`__.
+
+Background
+----------
+
+Planners in the the Upper Colorado River Basin (UCRB, shown in the
+figure below) are seeking to understand the vulnerability of water users
+to uncertainties stemming from climate change, population growth and
+water policy changes. The UCRB spans 25,682 km2 in western Colorado and
+is home to approximately 300,000 residents and 1,012 km2 of irrigated
+land. Several thousand irrigation ditches divert water from the main
+river and its tributaties for irrigation (shown as small black dots in
+the figure). Transmountain diversions of approximately 567,400,000 m3
+per year are exported for irrigation, industrial and municipal uses in
+northern and eastern Colorado, serving the major population centers of
+Denver and Colorado Springs. These diversions are carried through
+tunnels, shown as large black dots in the figure.
+
+.. image:: _static/basin_map.png
+
+An important planning consideration is the water rights of each user,
+defined by seniority across all water uses (irrigation diversions,
+transboundary diversions, power plants etc.) in the basin. To assess the
+vulnerability of users with varying degrees of water rights seniority,
+planners simulate the system across an ensemble of scenarios using the
+state of Colorado’s StateMod platform. The model simulates streamflow,
+diversions, instream demands, and reservoir operations.
+
+Hadjimichael et al. (2020) employ an exploratory analysis by simulating
+a large ensemble of plausible scenarios using StateMod and then
+identifying consequential decision-relevant combinations of uncertain
+factors, termed scenario discovery. Focusing on decision-relevant
+metrics (metrics that are important to the user, the scenario discovery
+is applied to the water shortages experienced by each individual user
+(i.e., not on a single basin-wide or sector-wide metric). For this
+training example, we’ll be performing scenario discovery for three
+different water users: two irrigation users and one municipal user.
+
+Let’s get started!
+------------------
+
+In this tutorial, we will be loading in data that has been produced in
+Hadjimichael et al. (2020). Before we start our analysis, we’ll load the
+relevant Python libraries, example data, and information for the three
+users.
+
+.. code:: ipython3
+
+    #import libraries 
+    import msdbook
+    import numpy as np
+    import pandas as pd
+    import matplotlib as mpl
+    import matplotlib.pyplot as plt
+    
+    # load example data from Hadjimichael et al. (2020)
+    msdbook.install_package_data()
+    
+    # Select the IDs for the three users that we will perform the analysis for
+    all_IDs = ['7000550','7200799','3704614'] 
+    usernames = ['Medium seniority irrigation',
+                 'Low seniority irrigation',
+                 'Transbasin municipal diversion']
+    nStructures = len(all_IDs)
+
+
+
+.. parsed-literal::
+
+    Downloading example data for msdbook version 0.1.5...
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/uncertain_params_bounds.txt
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_metric_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_vary_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_mth_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/solutions.resultfile
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/3704614_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/LHsamples_original_1000.txt
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/3704614_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/param_values.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_yr_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_mth_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7000550_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/collapse_days.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/hymod_params_256samples.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_vary_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7000550_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7200799_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_yr_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7200799_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/LeafCatch.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/hymod_simulations_256samples.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/Robustness.txt
+
+
+Step 1: Load Latin hypercube sample and set up problem
+------------------------------------------------------
+
+To examine regional vulnerability, we generate an ensemble of plausible
+future states of the world (SOWs) using Latin Hypercube Sampling. For
+this tutorial, we’ll load a file containing 1,000 samples across 14
+parameters. The sampled parameters encompass plausible changes to the
+future state of the basin, including changes to hydrology, water demands
+(irrigation, municipal & industry, transbasin), and institutional and
+environmental factors (environmental flows, reservoir storage, operation
+of the Shoshone Power Plant). These samples are taken from ranges
+identified in ``param_bounds``. Below we load in the 1000 samples, the
+range of values that the samples can take for each parameter, and the
+parameter names. More information on what each parameter constitutes can
+be found in Table 1 of Hadjimichael et al., 2020.
+
+.. code:: ipython3
+
+    #Identify the bounds for each of the 14 parameters 
+    param_bounds = msdbook.load_basin_param_bounds()
+    
+    #Load in the parameter samples 
+    LHsamples = msdbook.load_lhs_basin_sample()
+    
+    #Create an array of the parameter names
+    param_names=['Irrigation demand multiplier','Reservoir loss','Transbasin demand multiplier',
+                 'Municipal & industrial multiplier', 'Shoshone','Environmental flows',
+                 'Evaporation change','Mean dry flow','Dry flow variance',
+                 'Mean wet flow','Wet flow variance','Dry-dry probability',
+                 'Wet-wet probability', 'Snowmelt shift']
+
+
+Step 2: Define decision-relevant metrics for illustration
+---------------------------------------------------------
+
+Scenario discovery attempts to identify parametric regions that lead to
+‘success’ and ‘failure’. For this demonstration we’ll be defining
+‘success’ as states of the world where a shortage level doesn’t exceed
+its historical frequency.
+
+Step 3: Run the logistic regression
+-----------------------------------
+
+Logistic regression estimates the probability that a future SOW will be
+classified as a success or failure given a set of performance criteria.
+A logistic regression model is defined by:
+
+.. math:: ln \bigg (\frac{p_i}{1-p_i} \bigg ) = X^T_i \beta
+
+where :math:`p_i` is the probability the performance in the
+:math:`i^{th}` SOW will be classified as a success, :math:`X_i` is the
+vector of covariates describing the :math:`i^{th}` SOW, and
+:math:`\beta` is the vector of coefficients describing the relationship
+between the covariates and the response, which here will be estimated
+using maximum likelihood estimation.
+
+A logistic regression model was fit to the ensemble of SOWs using the
+performance criteria defined in step 2. Logistic regression modeling was
+conducted using the `Statsmodel
+Python <https://www.statsmodels.org/stable/index.html>`__ package. The
+data required for the full analysis is too large to include in this
+tutorial, but results can be found in the data file loaded below.
+
+The results files contain the occurence of different shortage frequency
+and magnitude combinations under the experiment, in increments of 10,
+between 0 and 100. These combinations (100 for each user) are
+alternative decision-relevant metrics that can be used for scenario
+discovery.
+
+.. code:: ipython3
+
+    # Set arrays for shortage frequencies and magnitudes
+    frequencies = np.arange(10, 110, 10)
+    magnitudes = np.arange(10, 110, 10)
+    realizations = 10
+    
+    # Load performance and pseudo r scores for each of the users
+    results = [msdbook.load_user_heatmap_array(all_IDs[i]) / 100 for i in range(len(all_IDs))]
+
+
+Step 4: Factor ranking
+----------------------
+
+To rank the importance of each uncertain factor, we utilize McFadden’s
+psuedo-R2, a measure that quantifies the improvement of the model when
+utilizing each given predictor as compared to prediction using the mean
+of the data set:
+
+.. math:: R^2_{McFadden}=1-\frac{ln \hat{L}(M_{full})}{ln \hat{L}(M_{intercept})}
+
+Here :math:`ln \hat{L}(M_{full})` is the log likelihood of the full
+model (including the predictor) and :math:`ln \hat{L}(M_{intercept})` is
+the log likelihood of the intercept model (which predicts the mean
+probability of success across all SOWs).
+
+Higher values of McFadden’s psuedo-R2 indicate higher factor importance
+(when the likelihood of the full model approaches one, the ratio of the
+likelihood of the full model compared to the intercept model will get
+very small).
+
+.. code:: ipython3
+
+    #Load the pseudo-R^2 scores
+    scores = [msdbook.load_user_pseudo_scores(all_IDs[i]) for i in range(len(all_IDs))]
+    
+    # Select indices of frequency and magnitudes that will be used for the visualization
+    freq = [1,0,0]
+    mag = [7,3,7]
+
+
+Step 5: Draw factor maps
+------------------------
+
+The McFadden’s psuedo-R2 scores files contain preliminary logistic
+regression results on parameter importance for each of these
+combinations. Using these psuedo-R2 scores we will identify the two most
+important factors for each metric which we’ll use to generate the final
+scenario discovery maps (note: there may be more than two important
+metrics for each user, but here we will demonstrate by mapping two).
+
+.. code:: ipython3
+
+    # setup figure
+    fig, axes = plt.subplots(3,1, figsize=(6,18), tight_layout=True)
+    fig.patch.set_facecolor('white')
+    
+    for i in range(len(axes.flat)):
+        
+        ax = axes.flat[i]
+        
+        allSOWsperformance = results[i]
+        all_pseudo_r_scores = scores[i]
+        
+        # construct dataframe
+        dta = pd.DataFrame(data=np.repeat(LHsamples, realizations, axis = 0), columns=param_names)
+        dta['Success'] = allSOWsperformance[freq[i],mag[i],:]
+        
+        pseudo_r_scores = all_pseudo_r_scores[str(frequencies[freq[i]])+'yrs_'+str(magnitudes[mag[i]])+'prc'].values
+        top_predictors = np.argsort(pseudo_r_scores)[::-1][:2] #Sort scores and pick top 2 predictors
+        
+        # define color map for dots representing SOWs in which the policy
+        # succeeds (light blue) and fails (dark red)
+        dot_cmap = mpl.colors.ListedColormap(np.array([[227,26,28],[166,206,227]])/255.0)
+        
+        # define color map for probability contours
+        contour_cmap = mpl.cm.get_cmap('RdBu')
+        
+        # define probability contours
+        contour_levels = np.arange(0.0, 1.05,0.1)
+        
+        # define base values of the predictors
+        SOW_values = np.array([1,1,1,1,0,0,1,1,1,1,1,0,0,0]) # default parameter values for base SOW
+        base = SOW_values[top_predictors]
+        ranges = param_bounds[top_predictors]
+        
+        # define grid of x (1st predictor), and y (2nd predictor) dimensions
+        # to plot contour map over
+        xgrid = np.arange(param_bounds[top_predictors[0]][0],
+                          param_bounds[top_predictors[0]][1], np.around((ranges[0][1]-ranges[0][0])/500,decimals=4))
+        ygrid = np.arange(param_bounds[top_predictors[1]][0],
+                          param_bounds[top_predictors[1]][1], np.around((ranges[1][1]-ranges[1][0])/500,decimals=4))
+        all_predictors = [ dta.columns.tolist()[i] for i in top_predictors]
+        dta['Interaction'] = dta[all_predictors[0]]*dta[all_predictors[1]]
+        
+        # logistic regression here
+        predictor_list = [all_predictors[i] for i in [0,1]]
+        result = msdbook.fit_logit(dta, predictor_list)
+        
+        # plot contour map
+        contourset = msdbook.plot_contour_map(ax, result, dta, contour_cmap,
+                                              dot_cmap, contour_levels, xgrid,
+                                              ygrid, all_predictors[0], all_predictors[1], base)
+        
+        ax.set_title(usernames[i])
+        
+    # set up colorbar
+    cbar_ax = fig.add_axes([0.98, 0.15, 0.05, 0.7])
+    cbar = fig.colorbar(contourset, cax=cbar_ax)
+    cbar_ax.set_ylabel('Probability of Success', fontsize=16)
+    cbar_ax.tick_params(axis='y', which='major', labelsize=12)
+
+
+
+.. parsed-literal::
+
+    /srv/conda/envs/notebook/lib/python3.7/site-packages/statsmodels/base/model.py:127: ValueWarning: unknown kwargs ['disp']
+      warnings.warn(msg, ValueWarning)
+
+
+.. parsed-literal::
+
+    Optimization terminated successfully.
+             Current function value: 0.378619
+             Iterations 8
+    Optimization terminated successfully.
+             Current function value: 0.397285
+             Iterations 8
+    Optimization terminated successfully.
+             Current function value: 0.377323
+             Iterations 8
+
+
+
+.. image:: _static/notebook_logistic_output_11_1.png
+
+
+The figure above demonstrates how different combinations of the
+uncertain factors lead to success or failure in different states of the
+world, which are denoted by the blue and red dots respectively. The
+probability of success and failure are further denoted by the contours
+in the figure. Several insights can be drawn from this figure.
+
+First, using metrics chosen to be decision-relevant (specific to each
+user) causes different factors to be identified as most important by
+this scenario-discovery exercise (the x- and y-axes for each of the
+subplots). In other words, depending on what the decision makers of this
+system want to prioritize they might choose to monitor different
+uncertain factors to track performance.
+
+Second, in the top panel, the two identified factors appear to also have
+an interactive effect on the metric used (shortages of a certain level
+and frequency in this example). In terms of scenario discovery, the
+Patient Rule Induction Method (PRIM) or Classification And Regression
+Trees (CART) would not be able to delineate this non-linear space and
+would therefore misclassify parameter combinations as ‘desirable’ when
+they were in fact undesirable, and vice versa.
+
+Lastly, logistic regression also produces contours of probability of
+success, i.e. different factor-value combinations are assigned different
+probabilities that a shortage level will be exceeded. This allows the
+decision makers to evaluate these insights while considering their risk
+aversion.
+
+Tips to Apply Scenario Discovery to Your Own Problem
+----------------------------------------------------
+
+In this tutorial, we demonstrated how to perform a scenario discovery
+analysis for three different users in the UCRB. The analysis allowed us
+to determine which parameters the users would be most affected by and to
+visualize how different ranges of these parameters lead to success and
+failure for different users. This framework can be applicable to any
+other application where it is of interest to characterize success and
+failure based on uncertain parameter ranges. In order to apply the same
+framework to your own problem:
+
+1. Choose sampling bounds for your parameters of interest, which will
+   represent uncertainties that characterize your system.
+2. Generate samples for these parameters (this can be done using the
+   ``saltelli.sample`` function or externally).
+3. Define what constitutes success and failure in your problem. In this
+   tutorial, success was defined based on not surpassing the historical
+   drought frequency. Choose a metric that is relevant to your problem
+   and decision-makers that might be involved. If your model involves an
+   optimization, you can also define metrics based on meeting certain
+   values of these objectives.
+4. Run the parameter sets through your model and calculate success and
+   failure based on your metrics and across different users if
+   applicable. This step will allow you to create the scatter plot part
+   of the final figure.
+5. If it is of interest, the contours on the figure can be created by
+   fitting the logistic regression model in a similiar manner as denoted
+   in Steps 3 and 5 of the tutorial.
+
+
diff --git a/dev/docs/html/_sources/A2.4_hymod.rst b/dev/docs/html/_sources/A2.4_hymod.rst
new file mode 100644
index 0000000..e847e8a
--- /dev/null
+++ b/dev/docs/html/_sources/A2.4_hymod.rst
@@ -0,0 +1,1242 @@
+HYMOD Dynamics Tutorial
+*************************
+
+.. note::
+    
+    | Run the tutorial interactively:  `HYMOD Notebook <nb_hymod_>`_.
+    | Please be aware that notebooks can take a couple minutes to launch.
+    | To run the notebooks yourself, download the files `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks>`_ and use these `requirements <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt>`_.
+
+Tutorial: Sensitivity Analysis of the HYMOD Model
+=================================================
+
+The purpose of this tutorial is to demonstrate the global sensitivity
+analysis concepts and tools established in the Section 3.1 of the main
+text of this eBook. This demonstration will highlight the central role
+of design of experiments (Section 3.3), when implementing global
+sensitivity analysis tools described in Section 3.4.
+
+We’ll explore these tools and concepts using the HYdrological MODel
+(HYMOD), a rainfall-runoff model developed and used for river flow
+forecasting. HYMOD was chosen for demonstration because its purpose is
+to abstract highly complex and non-linear systems. The methods
+demonstrated in thistutorial can be applied to numerical models that
+simulate other complex non-linear systems.
+
+This tutorial will first introduce HYMOD and use it to simulate
+streamflows in a river basin. Next, we’ll employ sensitivity analysis
+concepets described in Section 3 of the main text to examine how values
+of HYMOD’s parameters impact streamflow predictions. We’ll then explore
+how the effects of these parameters may change over time using
+time-varying sensitivtiy analysis. Finally, we’ll demonstrate concepts
+presented in Chapter 7 through two ensemble-based methods of uncertainty
+quantification - Generalized Likelihood Uncertainty Estimation (GLUE)
+and Pre-Calibration.
+
+The tutorial includes the following steps:
+
+1. Introduction to HYMOD
+^^^^^^^^^^^^^^^^^^^^^^^^
+
+`1.1 - Introduction to a simple hydrologic model (HYMOD) <#hymod>`__
+
+`1.2 - Input Data <#inputs>`__
+
+`1.3 - Running a basic simulation <#baseline>`__
+
+`1.4 - Model outputs <#outputs>`__
+
+2. Global Sensitivity Analysis
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+`2.1 - Design of Experiments <#sensitivity>`__
+
+`2.2 - Sensitivity analysis for one output <#sametrics>`__
+
+`2.3 - Sensitivity analysis across multiple outputs <#diffperformance>`__
+
+`2.4 - Time-varying sensitivity analysis <#time>`__
+
+1 - Introduction to HYMOD
+=========================
+
+.. _hymod:
+
+1.1 Overview
+
+HYMOD is a hydrologic model (rainfall-runoff model) that simulates key
+hydrologic fluxes such as infiltration, streamflow and
+evapotranspiration. The model was originally developed and used for
+river flow forecasting, but it has also been been used to explore
+different sensitivity analysis (e.g., `Herman et al.,
+2013 <https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/wrcr.20124>`__),
+uncertainty quantification (e.g., `Smith et al.,
+2008 <https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2006WR005205>`__),
+and optimization (e.g., `Ye et al.,
+2014 <https://www.sciencedirect.com/science/article/pii/S0022169414006362?casa_token=IRqE19Hkfa8AAAAA:_fXOqfwpzxMpchvu8_0njCe0Ok9H29Gyw2F46l9PzG9UVODDTUg6wIOSiyp6uybGevNVnZ7N>`__)
+concepts.
+
+HYMOD accepts two inputs - daily precepitation and daily potential
+evapotranspiration (PET)- and generates predicitons of daily streamflow.
+HYMOD abstracts the highly non-linear process of runoff routing by
+dividing the flow into two components: quick flow, representing
+precipitation that quickly runs off the surface of the watershed into
+the stream, and slow flow, that moves through the soil and takes much
+longer to arrive at the stream.
+
+To generate streamflow predictions, HYMOD first models vertical
+processes within the watershed to determine how much water infiltrates
+and evaporates from the soil at a given time step. It then determines
+how much water should be partitioned into quick flow and slow flow
+processes. Within each process it abstracts residence time (the time it
+takes a unit volume of water to move through the watershed and into the
+stream) using a series of “reservoirs” each with a calibrated residence
+time.
+
+HYMOD’s representation of hydrologic processes are shown Figure 1 below
+and controlled by the following parameters:
+
+:math:`H_{uz}`: the maximum water storage capacity of the soil (mm)
+
+:math:`B_{exp}`: parameters describing the degree of spatial variability
+within the basin between 0 and Huz
+
+:math:`Alp`: Fraction of runoff contributing to quick flow
+
+:math:`K_q`: Quick flow residence time of linear infinite reservoir (the
+Kq values of all three linear reservoirs are the same)
+
+:math:`K_s`: Slow flow residence time of linear infinite reservoir
+
+
+.. image:: _static/hymod_schematic-DAVE.png
+
+Vertical Processes
+~~~~~~~~~~~~~~~~~~
+
+HYMOD models the fraction of water that is stored in the soil
+:math:`(F(XH_{uz}))` using the following relationship:
+
+.. math:: F(XH_{uz}) = 1 - (1 - \frac{XH_{uz}}{H_{uz}})^{B}
+
+where :math:`XH_{uz}` is the water storage capacity of the soil;
+:math:`H_{uz}` is the parameter describing basin maximum water
+storage capacity (mm); and :math:`B` is the parameter describing the
+degree of spatial variability within the basin.
+
+The portion of precipitation that exceeds the water storage capacity is
+treated as runoff.
+
+Horizontal Processes
+~~~~~~~~~~~~~~~~~~~~
+
+To route runoff to streamflow, the excess runoff from the vertical
+processes is split into quick flow and slow flow. The proportion of
+runoff partitioned into quick flow and slow flow is determined by a
+parameter :math:`Alp`, which ranges between 0 and 1. Quick flow is
+routed through :math:`N` identical quick flow tanks :math:`Q1, Q2... QN`
+(shown above as :math:`N=3`). The rate at which runoff moves through the
+quick flow system is described by the residence time of the quick frlow
+tanks, :math:`Kq` (day). Slow flow is routed through a parallel slow
+flow tank and the rate at which slow flow is routed is described by the
+slow flow residences time, :math:`Ks` (day).
+
+Citation: Wagener, T., Boyle, D. P., Lees, M. J., Wheater, H. S., Gupta,
+H. V., & Sorooshian, S. (2001). A framework for development and
+application of hydrological models. Hydrology and Earth System Sciences,
+5(1), 13-26.
+
+.. _inputs:
+
+1.2 Input data
+
+The HYMOD model only requires precipitation and potential
+evapotranspiration as inputs. For this example, we’ll run HYMOD using
+data from the Leaf River, a humid catchment located north of Collins
+Mississippi that has been widely used to explore HYMOD. The dataset also
+includes daily observed runoff that we later use to evaluate the
+performace of each sensitvity analysis sample set.
+
+In the following section of code, we’ll load the necessary python
+libraries and read in the input file. For this exercise we’ll only use
+the first eleven years of data. The first five rows of the input dataset
+are printed to show what they look like:
+
+.. code:: ipython3
+
+    import msdbook
+
+    import numpy as np
+    import pandas as pd
+    import seaborn as sns
+
+    from sklearn import metrics
+    from matplotlib import pyplot as plt
+
+    # load example data
+    msdbook.install_package_data()
+
+    # load the Leaf River HYMOD input file
+    leaf_data = msdbook.load_hymod_input_file()
+
+    # extract the first eleven years of data
+    leaf_data = leaf_data.iloc[0:4015].copy()
+
+    print('Leaf River Data structure:')
+
+    # There are only three columns in the file including precipitation, potential evapotranspiration and  streamflow
+    leaf_data.head()
+
+
+.. parsed-literal::
+
+    Downloading example data for msdbook version 0.1.5...
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/uncertain_params_bounds.txt
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_metric_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_vary_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_mth_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/solutions.resultfile
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/3704614_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/LHsamples_original_1000.txt
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/3704614_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/param_values.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_yr_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_mth_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7000550_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/collapse_days.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/hymod_params_256samples.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_vary_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7000550_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7200799_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_yr_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7200799_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/LeafCatch.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/hymod_simulations_256samples.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/Robustness.txt
+    Leaf River Data structure:
+
+
+
+
+.. raw:: html
+
+    <div>
+    <style scoped>
+        .dataframe tbody tr th:only-of-type {
+            vertical-align: middle;
+        }
+
+        .dataframe tbody tr th {
+            vertical-align: top;
+        }
+
+        .dataframe thead th {
+            text-align: right;
+        }
+    </style>
+    <table border="1" class="dataframe">
+      <thead>
+        <tr style="text-align: right;">
+          <th></th>
+          <th>Precip</th>
+          <th>Pot_ET</th>
+          <th>Strmflw</th>
+        </tr>
+      </thead>
+      <tbody>
+        <tr>
+          <th>0</th>
+          <td>0.0</td>
+          <td>4.60</td>
+          <td>0.29</td>
+        </tr>
+        <tr>
+          <th>1</th>
+          <td>0.0</td>
+          <td>4.31</td>
+          <td>0.24</td>
+        </tr>
+        <tr>
+          <th>2</th>
+          <td>0.0</td>
+          <td>4.33</td>
+          <td>0.21</td>
+        </tr>
+        <tr>
+          <th>3</th>
+          <td>0.0</td>
+          <td>4.78</td>
+          <td>0.19</td>
+        </tr>
+        <tr>
+          <th>4</th>
+          <td>0.0</td>
+          <td>2.91</td>
+          <td>0.18</td>
+        </tr>
+      </tbody>
+    </table>
+    </div>
+
+
+
+To visualize catchment hydrology, streamflow and precipitation data are
+usually plotted together as a combined hydrograph (streamflow ) and
+hyetograph (rainfall, from Greek.hyetos, “rain”). Streamflow is plotted
+as a time series, while rainfall is shown as an inverted bar plot along
+the top of the graph. Streamflow labels are shown on the left y-axis,
+while rainfall labels are shown on the right y-axis.
+
+.. code:: ipython3
+
+    # make an axis for the hydrograph
+    fig, strmflw_ax = plt.subplots(figsize=[12,6])
+    strmflw_ax.set_ylim([0, 50])
+
+    #make a second y-axis for the hyetograph
+    precip_ax = strmflw_ax.twinx()
+    precip_ax.set_ylim([0, 200])
+    precip_ax.invert_yaxis()
+
+    precip = leaf_data['Precip']
+    strmflw_ax.plot(range(0, len(leaf_data['Precip'])), leaf_data['Strmflw'], color='lightcoral')
+    strmflw_ax.set_ylabel('Streamflow (mm/day)')
+
+    precip_ax.bar(range(0, len(leaf_data['Precip'])), leaf_data['Precip'], width=2)
+    precip_ax.set_ylabel('Rainfall (mm/day)')
+    precip_ax.legend(['Precip'], loc='center right')
+    strmflw_ax.legend(['Streamflow'],bbox_to_anchor=(1, 0.48))
+
+
+
+
+
+.. parsed-literal::
+
+    <matplotlib.legend.Legend at 0x7f53b95c6850>
+
+
+
+
+.. image:: _static/hymod1.png
+
+.. _baseline:
+
+1.3 Running a Baseline Model Simulation
+
+We’ll start our experiment by running HYMOD using its default
+parameters.
+
+.. code:: ipython3
+
+    # assign input parameters to generate a baseline simulated streamflow
+    Nq = 3  # Number of quickflow routing tanks
+    Kq = 0.5 # Quickflow routing tanks' rate parameter
+    Ks =  0.001 # Slowflow routing tank's rate parameter
+    Alp = 0.5 # Quick/slow split parameter
+    Huz = 100 # Maximum height of soil moisture accounting tank
+    B = 1.0 # Scaled distribution function shape parameter
+
+    # Note that the number of years is 11. One year of model warm-up and ten years are used for actual simulation
+    model = msdbook.hymod(Nq, Kq, Ks, Alp, Huz, B, leaf_data, ndays=4015)
+
+
+.. _outputs:
+
+1.4 Model Outputs
+
+Model outputs include actual evapotranspiration, quick and fast
+streamflow, and combined runoff. In this tutorial we focus on the total
+daily runoff, QQ (:math:`m^3/s`). We can use the following script to
+plot simulated streamflow against observed streamflow.
+
+.. tip:: View the source code used to create this plot here: `plot_observed_vs_simulated_streamflow <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-observed-vs-simulated-streamflow>`_
+
+.. code:: ipython3
+
+    ax = msdbook.plot_observed_vs_simulated_streamflow(df=leaf_data, hymod_dict=model)
+
+
+
+.. image:: _static/hymod2.png
+
+
+
+So how does our model perform? We can investigate model performance
+across several metrics:
+
+1: Mean Absolute Error (MAE); MAE conveys how the model performs on
+average across the 10 year simulation period, with smaller values
+indicating better performance. The absolute value is taken so that
+positive and negative errors do not cancel each other out.
+
+.. math:: MAE = \frac{1}{N}\sum_{t=0}^N\left\lvert Q_{sim,t}-Q_{obs,t}\right\rvert
+
+2: Root Mean Square Error (RMSE); RMSE is sum of square errors across
+the 10 year simulation period. RMSE is sensitive to large errors between
+the historical record and the simulated flows, and thus is useful for
+highlighting the model’s ability to capture of extreme flood events.
+
+.. math:: RMSE = \sqrt{\frac{1}{N}\sum_{t=1}^{N}(Q_{sim,t}-Q_{obs,t})^2}
+
+3: Log-Root Mean Square Error (Log(RMSE)) LOG(RMSE) focuses on model
+performance during low-flow events.
+
+.. math:: LOG(RMSE) = log(RMSE)
+
+.. code:: ipython3
+
+    mae = np.mean(abs(leaf_data['Strmflw'] - model['Q']))
+    mse = metrics.mean_squared_error(model['Q'], leaf_data['Strmflw'])
+    rmse = mse**(1/2)
+
+    print('MAE: ' + str(mae) + '\nRMSE: ' + str(mse) + '\nLOG(RMSE): ' + str(rmse))
+
+
+.. parsed-literal::
+
+    MAE: 1.0787471470460999
+    RMSE: 4.375695937555197
+    LOG(RMSE): 2.0918164206151544
+
+
+The error metrics show that HYMOD performs reasonably well, the MAE is
+around 1 :math:`m^3/s`, the RMSE is on the order of 10% of the largest
+observed streamflow and the LOG(RMSE) is fairly low.
+
+2- Global Sensitivity Analysis
+==============================
+
+.. _sensitivity:
+
+2.1 Experimental Design and Setup
+
+Now we’ll examine how sensitive streamflow simulations generated by
+HYMOD are to the model’s input parameters. We’ll perform global
+sensitivity analysis (see Section 3.1 of the main text) using the SALib
+Python library.
+
+.. code:: ipython3
+
+    from SALib.sample import saltelli
+    from SALib.analyze import sobol
+    from SALib.analyze import delta
+
+A first and critical step when conducting sensitivity analysis is
+determining the experimental design (see Design of Experiments, Section
+3.4 of the main text). Our experimental design involves defining the
+uncertainties that we’ll be examining, the output of interest, the
+ranges of each uncertainty that will be explored and the strategy for
+sampling the uncertainty space.
+
+For this experiment we’ll explore the five parameters highlighted in
+Figure 1. We’ll draw their ranges from existing literature on the model
+(note Jon H. paper). We’ll use a Sobol sampling an a quasi-random
+sampling with low sequences approach to sample the uncertainty space
+(Section 3.3.4).
+
+In this demonstration we’ll utilize Sobol Sensitivity Analysis, a
+variance based method (Section 3.4.5).
+
+To explore HYMOD’s behavoir, we’ll examine the sensitivity of four model
+ouputs to input parameters: 1) predicted flow, 2) Mean Absolute Error
+(compared with the calibaration data set), 3) Root Mean Square Error and
+4) Log Root Mean Square Error.
+
+This analysis will employ SALib, a Python implementation also utilized
+in the other SA tutorial (make this more formal).
+
+To start our analysis, we’ll create a dictionary that describes our
+model uncertainties and their ranges, this dictionary is named
+“problem_hymod” (SALib refers to these dictionaries as “problems”).
+
+.. code:: ipython3
+
+    problem_hymod = {
+        'num_vars': 5,
+        'names': ['Kq', 'Ks', 'Alp', 'Huz', 'B'],
+        'bounds': [[0.1, 1],  # Kq
+                   [0, 0.1],  # Ks
+                   [0, 1],    # Alp
+                   [0.1, 500],  # Huz
+                   [0, 1.9]]  # B
+    }
+
+After defining our uncertainites and ranges, we’ll use SALib to sample
+the uncertainty space and run the model for each of the sample sets. We
+will load a sample that has already been created ``param_values_hymod``
+for demonstration purposes. For HYMOD, literature recommends running at
+least N = 10,000 samples, to keep this demonstration easy to run
+however, we utilize only 256 sobol samples of uncertainties. To generate
+accurate approximations of second order sensitivity indicies SALib
+generates N*(2k+2) sets of samples, where N=256 and k=5 (number of
+uncertainties). For the math behind why this is needed, see (Saltelli,
+A., 2002. Making best use of model evaluations to compute sensitivity
+indices. Computer Physics Communications 145, 280–297.
+https://doi.org/10.1016/S0010-4655(02)00280-1).
+
+The actual model simulation takes an extended period, so we also load
+the simulation data from a previous run. The following demonstrates how
+to conduct this analysis:
+
+.. code:: python
+
+
+   # generate 256 samples.
+   param_values_hymod = saltelli.sample(problem_hymod, 256)
+
+   # dictionary to store outputs in
+   d_outputs = {}
+
+   # run simulation for each parameter sample
+   for i in range(0, len(param_values_hymod)):
+
+       # run model for each sensitivity analysis parameter sets
+       hymod_output = msdbook.hymod(Nq,
+                                    param_values_hymod[i, 0],
+                                    param_values_hymod[i, 1],
+                                    param_values_hymod[i, 2],
+                                    param_values_hymod[i, 3],
+                                    param_values_hymod[i, 4],
+                                    leaf_data,
+                                    ndays=4015)
+
+       # store the simulated total flow discharge
+       d_outputs[f"Q{i}"] = hymod_output["Q"]
+
+
+   Q_df_bw = pd.DataFrame(d_outputs)
+
+.. code:: ipython3
+
+    # load previously generated parameter values
+    param_values_hymod = msdbook.load_hymod_params()
+
+    # number of samples
+    n_samples = len(param_values_hymod)
+
+    # load previously generated hymod simulated outputs
+    Q_df_bw = msdbook.load_hymod_simulation()
+
+    # column names of each sample simulation number
+    sample_column_names = [i for i in Q_df_bw.columns if i[0] == 'Q']
+
+
+Running HYMOD - Model Warm-up
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+A hydrological model such as HYMOD usually includes ordinary
+differential equations that are sensitive to their initial condition.
+They also have components in their underlying formulation that have long
+memory such that prior time steps can affect their current simulations.
+For example, soil moisture or groundwater can hold water for a long time
+and therefore they are often considered to exhibit a long memory. This
+can affect the partitioning of water to runoff and infiltration, while
+also controlling the generation of base flow. Therefore, it is important
+to have a reasonable initial value for them. To achieve this,
+hydrologists usually extend their simulation period and after the
+simulations, they remove that extended time period that has unreasonable
+groundwater or surface water values. This time period is called the
+warm-up time period.
+
+Here we extended our simulation for one year (from 10 years to 11 years)
+and we removed the first year of simulation, therefore our warm-up
+period is one year.
+
+.. code:: ipython3
+
+    # exclude the first year of simulation from the simulations and reset the index
+    Q_df = Q_df_bw.iloc[365:4015].copy().reset_index(drop=True)
+
+    # exclude the first year of the input data and reset the index
+    leaf_data = leaf_data.iloc[365:4015].copy().reset_index(drop=True)
+
+
+Now that HYMOD has been warmed up, we’ll examine how HYMOD’s streamflow
+outputs vary under different sample sets, and compare them with the
+observed streamflow.
+
+.. code:: ipython3
+
+    # add date columns to our simulation data frame; for this data our start date is 1/1/2000
+    date_ts = pd.date_range(start='1/1/2000', periods=3650, freq='D')
+    Q_df['date'] = date_ts
+    Q_df['year'] = date_ts.year
+    Q_df['month'] = date_ts.month
+    Q_df['day'] = date_ts.day
+
+    # aggregate the simulated observed streamflow to monthly mean
+    df_sim_mth_mean = Q_df.groupby(['year', 'month'])[sample_column_names].mean()
+
+    # do the same for the observed data
+    date_ts = pd.date_range(start='1/1/2000', periods=len(leaf_data), freq='D')
+    leaf_data['date'] = date_ts
+    leaf_data['year'] = date_ts.year
+    leaf_data['month'] = date_ts.month
+    leaf_data['day'] = date_ts.day
+
+    # aggregate the daily observed streamflow to monthly mean
+    df_obs_mth_mean = leaf_data.groupby(['year', 'month']).mean()
+
+
+.. tip:: View the source code used to create this plot here: `plot_observed_vs_sensitivity_streamflow <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-observed-vs-sensitivity-streamflow>`_
+
+.. code:: ipython3
+
+    ax = msdbook.plot_observed_vs_sensitivity_streamflow(df_obs=df_obs_mth_mean,
+                                                         df_sim=df_sim_mth_mean)
+
+
+
+
+.. image:: _static/hymod3.png
+
+
+
+.. _sametrics:
+
+2.2 Sensitivity of streamflows to model parameters
+
+Now we’ll examine how each of HYMOD’s parameters impact the variance of
+simulated streamflows. Using SALib we’ll calculate the first order and
+total order sensitivity indicies of each model parameter. The first
+order sensitivity index measure’s the individual impact that a given
+parameter has on the variance of the simulated streamflows. The total
+order index measures the impact of a given parameter along with all
+interactions that other parameters have with the given parameter on
+simulated streamflows.
+
+We’ll start with an matrix, Y, which contains our simulated streamflows
+for every uncertainty sample. We’ll then use the sobol.analyze function
+from SALib to calculate the sensitivity indicies (Si). The arguments for
+this function are the problem dictionary defined in part 2.2 of this
+tutorial, and the matrix of simulated streamflows, Y.
+
+.. code:: ipython3
+
+    # overall aggregated indices
+    Y = Q_df[sample_column_names].mean().to_numpy()
+
+    # Perform analysis
+    Si = sobol.analyze(problem_hymod, Y)
+
+Now we can examine our results, we’ll print the first order and total
+order Si’s for each parameter, then visualize the results with bar plots
+
+.. code:: ipython3
+
+    print('First order indices = ', Si['S1'])
+
+    print('Total order indicies = ', Si['ST'])
+
+    sns.set_style('white')
+    fig = plt.figure(figsize=(8,4))
+    ax1 = fig.add_subplot(121)
+    ax1.bar(np.arange(5), Si['S1'])
+    ax1.set_xticklabels(['','Kq', 'Ks', 'Alp', 'Huz', 'B'])
+    ax1.set_ylabel('First order Si')
+    ax1.set_ylim([0,1])
+
+    ax2 = fig.add_subplot(122)
+    ax2.bar(np.arange(5), Si['ST'])
+    ax2.set_xticklabels(['','Kq', 'Ks', 'Alp', 'Huz', 'B'])
+    ax2.set_ylabel('Total order Si')
+    ax2.set_ylim([0,1])
+
+
+
+.. parsed-literal::
+
+    First order indices =  [9.55550001e-05 7.49249463e-04 5.62386413e-04 7.03327551e-01
+     2.53701895e-01]
+    Total order indicies =  [1.76174200e-06 1.63288175e-03 3.41378460e-04 6.88983864e-01
+     2.53922146e-01]
+
+
+.. parsed-literal::
+
+    /srv/conda/envs/notebook/lib/python3.7/site-packages/ipykernel_launcher.py:9: UserWarning: FixedFormatter should only be used together with FixedLocator
+      if __name__ == '__main__':
+    /srv/conda/envs/notebook/lib/python3.7/site-packages/ipykernel_launcher.py:15: UserWarning: FixedFormatter should only be used together with FixedLocator
+      from ipykernel import kernelapp as app
+
+
+
+
+.. parsed-literal::
+
+    (0.0, 1.0)
+
+
+
+
+.. image:: _static/hymod4.png
+
+
+
+Our findings indicate that in this instance, the streamflow estimate
+from HYMOD is highly sensitive to soil moisture parameters Huz and B and
+hardly affected by the routing parameters. Notably, there is very little
+interactions between parameters causing the total order indicies to be
+nearly identical to the first order indicies.
+
+
+.. _diffperformance:
+
+2.3 How do different performance metrics affect the results of our
+sensitivity analysis?
+
+Streamflow has many different properties. In this section, we discuss
+how the selection of metrics can lead to fundamentally different
+sensitivity analysis results. For example, one can only focus on
+aggregated streamflow metrics such as mean (what has been presented so
+far), or only on extreme events such as drought or floods.
+
+Here we compare three different metrics: 1- Mean error (ME) 2- Root Mean
+Square Error (RMSE) 3- Log-Root Mean Square Error (Log(RMSE))
+
+Each of these metrics focuses on a specific attribute of streamflow. For
+example, RMSE highlights the impacts of extreme flood events, while
+LOG(RMSE) focuses on model performance during low-flow events.
+
+.. code:: ipython3
+
+    # calculate error metrics
+    mae = Q_df[sample_column_names].apply(lambda x: abs(x-leaf_data["Strmflw"]), axis=0)
+    mse = Q_df[sample_column_names].apply(lambda x: metrics.mean_squared_error(x, leaf_data["Strmflw"]), axis=0)
+    rmse = mse**(1/2)
+
+    # add error metrics to a dictionary
+    d_metrics = {'MAE': mae.mean().values,
+                 'RMSE': rmse.values,
+                 'LOG[RMSE]': np.log10(rmse.values)}
+
+    # convert to a dataframe
+    df_metrics_SA = pd.DataFrame(d_metrics)
+
+
+We can use the following to calculate the SA indices for each metric and
+visualize it.
+
+.. code:: ipython3
+
+    df_metric_s1_result = pd.DataFrame(np.zeros((3, 5)), columns=['Kq', 'Ks', 'Alp', 'Huz', 'B'])
+    df_metric_sT_result = pd.DataFrame(np.zeros((3, 5)), columns=['Kq', 'Ks', 'Alp', 'Huz', 'B'])
+
+    # conduct sensitivity analysis for each metric
+    for index, i in enumerate(d_metrics.keys()):
+
+        # get the data as a numpy array for the target metric
+        Y = d_metrics[i]
+
+        # use the metric to conduct SA
+        Si = sobol.analyze(problem_hymod, Y, print_to_console=False)
+
+        # add the sensitivity indices to the output data frame
+        df_metric_s1_result.iloc[index, :] = Si['S1']
+        df_metric_sT_result.iloc[index, :] = Si['ST']
+
+.. code:: ipython3
+
+    # create seaborn heatmap with required labels
+    fig = plt.figure(figsize=(12,4))
+    ax1 = fig.add_subplot(121)
+    # labels for y-axis
+    y_axis_labels = ['Mean Absoulte Error', 'RSME', 'Log(RMSE)']
+
+    # plot heatmap
+    ax1 = sns.heatmap(df_metric_s1_result, yticklabels=y_axis_labels, annot=True,  cmap='inferno_r', cbar_kws={'label': 'Si'}, cbar=False)
+    ax1.figure.axes[-1].yaxis.label.set_size(14)
+    ax1.set_title('First Order Sensitivity')
+
+    ax2 = fig.add_subplot(122)
+    ax2 = sns.heatmap(df_metric_sT_result, yticklabels=y_axis_labels, annot=True,  cmap='inferno_r', cbar_kws={'label': 'Si'})
+    ax2.figure.axes[-1].yaxis.label.set_size(14)
+    ax2.set_title('Total Order Sensitivity')
+
+
+
+
+
+.. parsed-literal::
+
+    Text(0.5, 1.0, 'Total Order Sensitivity')
+
+
+
+
+.. image:: _static/hymod5.png
+
+
+
+The first order sensitivity indicies indicate that HYMOD’s sensitivity
+to its parameters is different depending on how its output is measured.
+Unsurprisingly, the mean absolute error is highly sensitive to the soil
+moisture accounting parameters Huz and B, just like the overall
+streamflow predictions above. However, when we examine RMSE and
+log(RMSE), the routing parameters Alp become sensitive, and the
+sensitivity to parameter B is reduced. As described above, RMSE and
+LOG(RMSE) respond to model performance in high-flow and low flow periods
+respectively. Our results indicate that for these flow regimes Alp, the
+parameter that governs the split between quick and slow flow is an
+important factor. While still the parameter with the highest most effect
+on all three measures, Huz is much less influential for RMSE and
+LOG(RMSE) than it is for MAE.
+
+The total order sensitivity indicies review a different, more complex
+story. While the MAE sensitivity is relatively governed by first order
+effects (like the streamflow predictions above), the RMSE and LOG(RMSE)
+error metrics show significant interactions. Alp has the highest total
+order sensitivity for RMSE and is eqal to Huz for Log(RMSE). Kq, which
+has a relatively low first order sensitivity index, shows strong
+contribution to variance when interactions are taken into account.
+
+Radial convergence plots are a helpful way to visualize the interactions
+between parameters. These plots array the model parameters in a circle
+and plot the first order, total order and second order Sobol sensitivity
+indices for each parameter. The first order sensitivity is shown as the
+size of a closed circle, the total order as the size of a larger open
+circle and the second order as the thickness of a line connecting two
+parameters. Below is an example of a radial convergence plot for the
+LOG(RMSE) measure. The plot indicates strong interactions between the
+Huz and Alp parameters, as well as Alp and Kq. There is also an
+interaction between Alp and Ks.
+
+.. code:: ipython3
+
+    import numpy as np
+    import itertools
+    import seaborn as sns
+    import math
+    sns.set_style('whitegrid', {'axes_linewidth': 0, 'axes.edgecolor': 'white'})
+
+    def is_significant(value, confidence_interval, threshold="conf"):
+        if threshold == "conf":
+            return value - abs(confidence_interval) > 0
+        else:
+            return value - abs(float(threshold)) > 0
+
+    def grouped_radial(SAresults, parameters, radSc=2.0, scaling=1, widthSc=0.5, STthick=1, varNameMult=1.3, colors=None, groups=None, gpNameMult=1.5, threshold="conf"):
+        # Derived from https://github.com/calvinwhealton/SensitivityAnalysisPlots
+        fig, ax = plt.subplots(1, 1)
+        color_map = {}
+
+        # initialize parameters and colors
+        if groups is None:
+
+            if colors is None:
+                colors = ["k"]
+
+            for i, parameter in enumerate(parameters):
+                color_map[parameter] = colors[i % len(colors)]
+        else:
+            if colors is None:
+                colors = sns.color_palette("deep", max(3, len(groups)))
+
+            for i, key in enumerate(groups.keys()):
+                #parameters.extend(groups[key])
+
+                for parameter in groups[key]:
+                    color_map[parameter] = colors[i % len(colors)]
+
+        n = len(parameters)
+        angles = radSc*math.pi*np.arange(0, n)/n
+        x = radSc*np.cos(angles)
+        y = radSc*np.sin(angles)
+
+        # plot second-order indices
+        for i, j in itertools.combinations(range(n), 2):
+            #key1 = parameters[i]
+            #key2 = parameters[j]
+
+            if is_significant(SAresults["S2"][i][j], SAresults["S2_conf"][i][j], threshold):
+                angle = math.atan((y[j]-y[i])/(x[j]-x[i]))
+
+                if y[j]-y[i] < 0:
+                    angle += math.pi
+
+                line_hw = scaling*(max(0, SAresults["S2"][i][j])**widthSc)/2
+
+                coords = np.empty((4, 2))
+                coords[0, 0] = x[i] - line_hw*math.sin(angle)
+                coords[1, 0] = x[i] + line_hw*math.sin(angle)
+                coords[2, 0] = x[j] + line_hw*math.sin(angle)
+                coords[3, 0] = x[j] - line_hw*math.sin(angle)
+                coords[0, 1] = y[i] + line_hw*math.cos(angle)
+                coords[1, 1] = y[i] - line_hw*math.cos(angle)
+                coords[2, 1] = y[j] - line_hw*math.cos(angle)
+                coords[3, 1] = y[j] + line_hw*math.cos(angle)
+
+                ax.add_artist(plt.Polygon(coords, color="0.75"))
+
+        # plot total order indices
+        for i, key in enumerate(parameters):
+            if is_significant(SAresults["ST"][i], SAresults["ST_conf"][i], threshold):
+                ax.add_artist(plt.Circle((x[i], y[i]), scaling*(SAresults["ST"][i]**widthSc)/2, color='w'))
+                ax.add_artist(plt.Circle((x[i], y[i]), scaling*(SAresults["ST"][i]**widthSc)/2, lw=STthick, color='0.4', fill=False))
+
+        # plot first-order indices
+        for i, key in enumerate(parameters):
+            if is_significant(SAresults["S1"][i], SAresults["S1_conf"][i], threshold):
+                ax.add_artist(plt.Circle((x[i], y[i]), scaling*(SAresults["S1"][i]**widthSc)/2, color='0.4'))
+
+        # add labels
+        for i, key in enumerate(parameters):
+            ax.text(varNameMult*x[i], varNameMult*y[i], key, ha='center', va='center',
+                    rotation=angles[i]*360/(2*math.pi) - 90,
+                    color=color_map[key])
+
+        if groups is not None:
+            for i, group in enumerate(groups.keys()):
+                print(group)
+                group_angle = np.mean([angles[j] for j in range(n) if parameters[j] in groups[group]])
+
+                ax.text(gpNameMult*radSc*math.cos(group_angle), gpNameMult*radSc*math.sin(group_angle), group, ha='center', va='center',
+                    rotation=group_angle*360/(2*math.pi) - 90,
+                    color=colors[i % len(colors)])
+
+        ax.set_facecolor('white')
+        ax.set_xticks([])
+        ax.set_yticks([])
+        plt.axis('equal')
+        plt.axis([-2*radSc, 2*radSc, -2*radSc, 2*radSc])
+        #plt.show()
+
+
+        return fig
+
+    # define groups for parameter uncertainties
+    groups={"Soil Moisture" : ["Huz", "B"],
+            "Routing" : ["Alp", "Kq", "Ks"]}
+
+
+    fig = grouped_radial(Si, ['Kq', 'Ks', 'Alp', 'Huz', 'B'], groups=groups, threshold=0.025)
+
+
+.. parsed-literal::
+
+    Soil Moisture
+    Routing
+
+
+
+.. image:: _static/hymod6.png
+
+
+2.4 Time-Varying Sensitivity Analysis
+
+.. _time:
+
+In section 2.5 we saw how performing sensitivity analysis on different
+measurements of model output can yeild in different results on the
+importance of each uncertain input. In this section we’ll examine how
+performing this analysis over time can yeild additional insight into the
+performance of HYMOD. We’ll first examine how model sensitivities vary
+by month, then examine how they change across each year of the
+simulation.
+
+For this demonstration, we’ll focus only on the monthly streamflow
+predictions generated by HYMOD.
+
+2.4.1 Sensitivity analysis indices for each month
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+.. code:: ipython3
+
+    # aggregate simulated streamflow data to monthly time series
+    df_sim_by_mth_mean = Q_df.groupby('month')[sample_column_names].mean()
+
+    # aggregate observed streamflow data to monthly time series
+    df_obs_by_mth_mean = leaf_data.groupby('month').mean()
+
+
+We can use the following to calculate the SA indices for each month and
+visualize it. Results are pre-loaded for efficiency.
+
+.. code:: python
+
+   # set up dataframes to store outputs
+   df_mth_s1 = pd.DataFrame(np.zeros((12,5)), columns=['Kq', 'Ks', 'Alp', 'Huz', 'B'])
+   df_mth_delta = df_mth_s1.copy()
+
+   # iterate through each month
+   for i in range(0, 12):
+
+       # generate the simulation data
+       Y = df_sim_by_mth_mean.iloc[i, :].to_numpy()
+
+       # run SA
+       Si = delta.analyze(problem_hymod, param_values_hymod, Y, print_to_console=False)
+
+       # add to output dataframes
+       df_mth_s1.iloc[i, :] = np.maximum(Si['S1'], 0)
+       df_mth_delta.iloc[i, :] = np.maximum(Si['delta'], 0)
+
+   # convert to arrays
+   arr_mth_s1 = df_mth_s1.values
+   arr_mth_delta = df_mth_delta.values
+
+First-order Indices
+^^^^^^^^^^^^^^^^^^^
+
+The following can be used to visualize the time-varying first-order
+indices. The first order represents the direct impacts of a specific
+parameter on model outputs.
+
+.. tip:: View the source code used to create this plot here: `plot_monthly_heatmap <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-monthly-heatmap>`_
+
+.. code:: ipython3
+
+    # load previously ran data
+    arr_mth_delta, arr_mth_s1 = msdbook.load_hymod_monthly_simulations()
+
+    # plot figure
+    ax, ax2 = msdbook.plot_monthly_heatmap(arr_sim=arr_mth_s1.T,
+                                           df_obs=df_obs_by_mth_mean,
+                                           title='First Order - Mean Monthly SA')
+
+
+
+
+.. image:: _static/hymod7.png
+
+
+
+This figure demonstrates the first order sensitivity indices when the
+streamflow data are aggregated by month. The purple line represents the
+observed monthly discharge. The figure indicates that the first order
+indices are highest for B and Huz across all months and lowest for Alp,
+Ks, and Kq. Note that in the months with the highest flow, Ks becomes an
+influential parameter.
+
+Total-order indices
+^^^^^^^^^^^^^^^^^^^
+
+We can also focus on the total order sensitivity index that includes
+first-order SA indices and interactions between parameters
+
+.. code:: ipython3
+
+    # plot figure
+    ax, ax2 = msdbook.plot_monthly_heatmap(arr_sim=arr_mth_delta.T,
+                                           df_obs=df_obs_by_mth_mean,
+                                           title='Total Order - Mean monthly SA')
+
+
+
+
+.. image:: _static/hymod8.png
+
+
+
+Notably, the total order sensitivity results are different than the
+first order sensitivity results, which indicates that interactions
+between the parameters (particularly in regards to routing parameters
+:math:`Kq`, :math:`Ks`, and :math:`Alp`) contribute to changes in HYMOD
+output.
+
+2.4.2 Annual sensitivity analysis indices
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+.. code:: ipython3
+
+    # group by year and get mean
+    df_sim_by_yr_mean = Q_df.groupby(['year'])[sample_column_names].mean()
+
+    # group input data and get mean
+    df_obs_by_yr_mean = leaf_data.groupby(['year']).mean()
+
+
+We can also calculate the sensitivity analysis indices for each
+individual year. This will allow us to understand if model control
+changes during different years. The following code first aggregates the
+outputs to annual time steps, and then calculates the SA indices.
+
+.. code:: python
+
+   # set up dataframes to store outputs
+   df_yr_s1 = pd.DataFrame(np.zeros((10, 5)), columns=['Kq', 'Ks', 'Alp', 'Huz', 'B'])
+   df_yr_delta = df_yr_s1.copy()
+
+   # iterate through each year
+   for i in range(0, 10):
+
+       # generate the simulation data
+       Y = df_sim_by_yr_mean.iloc[i, :].to_numpy()
+
+       # run SA
+       Si = delta.analyze(problem_hymod, param_values_hymod, Y, print_to_console=False)
+
+       # add to output dataframes
+       df_yr_s1.iloc[i, :] = np.maximum(Si['S1'], 0)
+       df_yr_delta.iloc[i, :] = np.maximum(Si['delta'], 0)
+
+   # convert to arrays
+   arr_yr_s1 = df_mth_s1.values
+   arr_yr_delta = df_mth_delta.values
+
+First-order indices
+^^^^^^^^^^^^^^^^^^^
+
+.. tip:: View the source code used to create this plot here: `plot_annual_heatmap <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-annual-heatmap>`_
+
+.. code:: ipython3
+
+    # load previously ran data
+    arr_yr_delta, arr_yr_s1 = msdbook.load_hymod_annual_simulations()
+
+    # plot figure
+    ax, ax2 = msdbook.plot_annual_heatmap(arr_sim=arr_yr_s1.T,
+                                          df_obs=df_obs_by_yr_mean,
+                                          title='First Order - Mean Annual SA')
+
+
+
+
+.. image:: _static/hymod9.png
+
+
+
+The first order sensitivities at the annual scale are not unlike the
+first order monthly sensitivities. Once again, sensitivities vary across
+year and Huz and B are the most consequential parameters.
+
+Total-order indices
+^^^^^^^^^^^^^^^^^^^
+
+.. code:: ipython3
+
+    # plot figure
+    ax, ax2 = msdbook.plot_annual_heatmap(arr_sim=arr_yr_delta.T,
+                                          df_obs=df_obs_by_yr_mean,
+                                          title='Total Order - Mean Annual SA and Observed flow')
+
+
+
+.. image:: _static/hymod10.png
+
+
+
+Our results indicate that sensitivity analysis indices vary in different
+years and now that interactions are included, the Kq, Ks, and Alp
+variables impact the sensitivity of the streamflow output.
+
+2.4.3 Monthly time-varying sensitivity analysis
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+Although time-varying sensitivity analysis (TVSA) at average monthly and
+average annual temporal resolutions is informative, TVSA is susceptible
+to the aggregation issue that we discussed earlier in section 3-2. To
+avoid that we can further discretize our time domain to zoom into
+individual months. This will provide us with even more information about
+model behavior and the sensitivity of different parameters in different
+states of the system. The block of code demonstrates how to implement
+the monthly TVSA.
+
+.. code:: python
+
+   # set up dataframes to store outputs
+   df_vary_s1 = pd.DataFrame(np.zeros((df_obs_mth_mean.shape[0], 5)),
+                             columns=['Kq', 'Ks', 'Alp', 'Huz', 'B'])
+
+   df_vary_delta = df_vary_s1.copy()
+
+   # iterate through each month
+   for i in range(0, df_obs_mth_mean.shape[0]):
+
+       # generate the simulation data
+       Y = df_sim_mth_mean.iloc[i, :].to_numpy()
+
+       # run SA
+       Si = delta.analyze(problem_hymod, param_values_hymod, Y, print_to_console=False)
+
+       # add to output dataframes
+       df_vary_s1.iloc[i, :] = np.maximum(Si['S1'], 0)
+       df_vary_delta.iloc[i, :] = np.maximum(Si['delta'], 0)
+
+   # convert to arrays
+   arr_vary_s1 = df_vary_s1.values
+   arr_vary_delta = df_vary_delta.values
+
+First-order indices
+^^^^^^^^^^^^^^^^^^^
+
+.. tip:: View the source code used to create this plot here: `plot_varying_heatmap <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-varying-heatmap>`_
+
+.. code:: ipython3
+
+    # load in previously ran data
+    arr_vary_delta, arr_vary_s1 = msdbook.load_hymod_varying_simulations()
+
+    # plot figure
+    ax, ax2 = msdbook.plot_varying_heatmap(arr_sim=arr_vary_s1.T,
+                                          df_obs=df_obs_mth_mean,
+                                          title='First Order - Time-Varying SA')
+
+
+
+.. image:: _static/hymod11.png
+
+
+
+Compared to the TVSA when streamflow was aggregated, this figure
+suggests that Kq is indeed a relevant parameter for influencing
+streamflow output when individual months are considered.
+
+Total order - time varying sensitivity analysis
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+.. code:: ipython3
+
+    # plot figure
+    ax, ax2 = msdbook.plot_varying_heatmap(arr_sim=arr_vary_delta.T,
+                                          df_obs=df_obs_mth_mean,
+                                          title='Total Order - Time-Varying SA')
+
+
+
+
+.. image:: _static/hymod12.png
+
+
+
+As above, the total order sensitivities further indicate the importance
+of Kq that is not apparent if aggregation is utilized.
+
+Tips to Apply this methodology to your own problem
+==================================================
+
+In this tutorial, we demonstrated how to use global sensitivtiy analysis
+to explore a complex, non-linear model. We showed how measuring
+sensitivity across multiple measures of model performance and temporal
+aggregations yeilding differing results about model
+sensitivity/behavoir. While these results may seem contraditory, they
+provide useful insight into the behavoir of HYMOD. Would we expect the
+same parameters to control high flow and low flow regimes within the
+model? Maybe, depending on the system, but also, maybe not. This
+analysis can provide insight into how the model responds to its input
+parameters, allowing us to compare the results to our expectaions. This
+may allow us to find problems with our intial assumptions, or call
+attention to model features that can be improved or expanded upon.
+Depending on the model and context, it may also yield insight into the
+workings of the underlying system.
+
+To run this tutorial on your own model you will need to:
+
+1. Design your experiment by choosing sampling bounds for your
+   parameters and setting up the problem dictionary as in step 2-2
+2. Choose the parameters of interest
+3. Generate samples using the saltelli.sample function. This step is
+   problem-dependent and note that the Sobol method can be
+   computationally intensive depending on the model being analyzed. More
+   complex models will be slower to run and will also require more
+   samples to calculate accurate estimates of Sobol indices. Once you
+   complete this process, pay attention to the confidence bounds on your
+   sensitivity indices to see whether you need to run more samples.
+4. Run the parameter sets through your model and record each of the
+   desired model outputs.
+5. Calculate the Sobol indices for each performance criteria. Now, the Y
+   will be a numpy array with your external model output and you will
+   need to include the parameter samples as an additional argument.
+6. Follow the procedure in step 2.6 to disaggregate performance across
+   time
diff --git a/dev/docs/html/_sources/A2.5_discovery.rst b/dev/docs/html/_sources/A2.5_discovery.rst
new file mode 100644
index 0000000..4487f45
--- /dev/null
+++ b/dev/docs/html/_sources/A2.5_discovery.rst
@@ -0,0 +1,384 @@
+Time-evolving scenario discovery for infrastructure pathways
+****************************************************************************************************
+
+.. note::
+
+    | Run the tutorial interactively:  `Scenario Discovery Notebook <nb_discovery_>`_.
+    | Please be aware that notebooks can take a couple minutes to launch.
+    | To run the notebooks yourself, download the files `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks>`_ and use these `requirements <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt>`_.
+
+
+
+Time-evolving scenario discovery for infrastructure pathways
+============================================================
+
+The purpose of this tutorial is to explore time-evolving vulnerability for systems that dynamically adapt to changing conditions. Using an example from water supply planning, we'll first examine how performance of a dynamic infrastructure pathway policy changes over time, then use factor mapping (main text Chapter 4.3) to understand which combinations of uncertaities generate vulnerability for two water utilities. Next, we'll perform factor prioritization (main text Chapter 4.3) to determine which uncertainties have the most influence on water supply performance. Finally, we'll provide an open platform to explore vulnerability across multiple measures of performance and different combinations of uncertainties.
+
+Background
+----------
+
+The Bedford-Greene metropolitan area (Figure 1) is a stylized water resources test case where two urban water utilities seek to develop an infrastructure and investment and management strategy to confront growing demands and changing climate. The utilities have agreed to jointly construct a new water treatment plant on Lake Classon, a large regional resource. Both utilities have also identified a set of individual infrastructure options to construct if necessary.
+
+.. figure:: _static/Map_small.png
+   :alt: Figure 1
+
+   Figure 1
+
+The utilities are formulating a cooperative and adaptive regional management strategy that uses a risk-of-failure (ROF) metric to trigger both short term drought mitigation actions (water use restrictions and treated transfers between utilities) and long-term infrastructure investment decisions (shown in Figure 2a). Both utilities have specified a set of risk triggers and developed a construction order for available infrastructure options.
+
+The utilities have run a Monte Carlo simulation to evaluate how these policies respond to a wide array of future States Of the World (SOWs). Each SOW represents a different combinations of thirteen uncertain system inputs including demand growth rates, changes to streamflows and financial variables. In this context, a fully specified SOW is composed of one sample of uncertain model inputs (e.g. one projection of demand growth rate coupled with one future streamflow scenario and one projection of future financial conditions). The water utilities used Latin Hypercube sampling (Chapter 3.3 of the main text) to develop an ensemble of 1,000 plausible future SOWs. The Monte Carlo simulation evaluates each candidate water supply infrastructure investment and management policy across all 1,000 SOWs, as shown in Figure 2b. For more details on the Monte Carlo sampling for this type  of analysis, see Trindade et al., (2019).
+
+The ROF-based policies respond to each SOW by generating a unique infrastructure pathway - a sequence of infrastructure investment decisions over time. Infrastructure pathways over a set of 1,000 future SOWs are shown in Figure 2c. Infrastructure options are arrayed along the vertical axis and the sequence of infrastructure investments triggered by the policy is plotted as pathways over time.  Since the adaptive rule system generates a unique infrastructure sequence for each scenario, Figure 2c summarizes the ensemble of pathways by clustering SOWs according to infrastructure intensity. Dark green lines represent SOWs where the utilities heavily invest in new infrastructure, light green lines represent SOWs with low infrastructure investment and medium shaded lines represent moderate investment. The shading behind each pathway represents the frequency that each infrastructure option was triggered over time across sampled scenarios
+
+.. figure:: _static/Policy_MonteCarlo_Pathways_small.png
+   :alt: Figure 2
+
+   Figure 2
+
+Evaluating Robustness over time
+-------------------------------
+
+The two water utilities are interested in mainting both supply
+reliability and financial stability across the broadest set of plausible
+future SOWs. To measure the performance of the infrastructure pathway
+policy, they’ve defined five critical performance criteria:
+
+-  Reliability > 99%
+-  Restriction Frequency < 20%
+-  Peak Financial Cost < 80% of annual revenue (a measure of debt
+   service spending)
+-  Worst-case drought management cost < 10% of annual revenue (a measure
+   of unexpected drought costs)
+-  Unit Cost of Expansion < 5 dollars/kgal
+
+To assess the robustness of the infrastructure pathway policy, the two
+utilities apply a satisficing metric, which measures the percentage of
+sampled SOWs where the pathway policy meets the peformance criteria:
+
+:math:`R =\frac{1}{N}\sum_{j=1}^{N}\Lambda_{\theta,j}`
+
+Where, :math:`\Lambda\_{\theta,j}=`
+
+.. math::
+
+   \begin{cases}
+   1, \quad \textrm{if}\ F(\theta)_{j}\leq \Phi_j \\
+   0, \quad \textrm{otherwise}
+   \end{cases}
+
+And :math:`\Phi` is a vector of performance criteria for utility
+:math:`j`, :math:`\theta` is the portfolio and :math:`N` is the total
+number of sampled SOWs.
+
+Below, we'll visualize how robustness for the two utilities evolves over the 45-year planning horizon. We'll assess robustness across three time periods, near-term (first 10 years), mid-term (22 years) and long term (45 years).
+
+We start by loading robustness values for the both utilities. These values are calculated by applying the robustness metric above across 2,000 simulated SOWs. To make this exercise computationally tractable, we've precomputed these values, which can be found in the files "short_term_robustness.csv", "mid_term_robustness.csv" and "long_term_robustness.csv". These values are calculated using the function "check_rdm_meet_criteria" within the helper functions.
+
+.. code:: ipython3
+
+    import numpy as np
+    from matplotlib import pyplot as plt
+    from functions.eBook_SD_helpers import check_rdm_meet_criteria, create_sd_input, plot_selected_tree_maps, get_factor_importances, open_exploration
+    import seaborn as sns
+
+    # load Deeply uncertain factors
+    rdm_factors = np.loadtxt('data/DU_Factors.csv', delimiter= ',')
+
+    sns.set()
+    short_term_robustness = np.loadtxt('data/short_term_robustness.csv', delimiter= ',')
+    mid_term_robustness = np.loadtxt('data/mid_term_robustness.csv', delimiter = ',')
+    long_term_robustness = np.loadtxt('data/long_term_robustness.csv', delimiter = ',')
+
+    # plot robustness over time
+    fig =plt.figure(figsize=(9,3))
+    plt.plot([10,22,45], [short_term_robustness[5]*100, mid_term_robustness[5]*100,long_term_robustness[5]*100], c='#B19CD9')
+    plt.plot([10, 22, 45], [short_term_robustness[11]*100, mid_term_robustness[11]*100, long_term_robustness[11]*100], c= '#43b284')
+    plt.scatter([10,22,45], [short_term_robustness[5]*100, mid_term_robustness[5]*100,long_term_robustness[5]*100], s=100, c='#B19CD9')
+    plt.scatter([10, 22, 45], [short_term_robustness[11]*100, mid_term_robustness[11]*100, long_term_robustness[11]*100], s=100, c='#43b284')
+    plt.xlabel('Time Horizon (yrs)')
+    plt.ylabel('Robustness (% SOWs)')
+    plt.legend(['Bedford', 'Greene'])
+    plt.title('Robustness Over Time')
+    plt.ylim([0, 107])
+
+
+
+
+
+.. parsed-literal::
+
+    (0.0, 107.0)
+
+
+
+
+.. image:: _static/discovery_4_1.png
+
+
+Exploring performance evolution
+-------------------------------
+
+The figure above reveals that the robustness of both water utilities degrades over time, with Bedford's robustness declining further than Greene. This suggests that the proposed pathway policy is likely insufficient to meet the long-term needs of the two utilities. But how is the current policy insufficient? To answer that question we examine the performance measures that fail to meet performance criteria for each utility across the three planning horizons.
+
+.. code:: ipython3
+
+    # Plot the type of vulnerability over time
+
+    ### Bedford ###
+    plot_robustness_1 = np.zeros([3,5])
+    # Determine the percentage of failure SOWs that violate each criterion (note some SOWS fail multiple criteria, so this may some to >1)
+    criteria = ['Reliability', 'Restriction Frequency', 'Peak Financial Cost', 'Worst-case drought\nManagement Cost', 'Stranded Assets']
+    plot_robustness_1[0,:] = (1 - short_term_robustness[0:5])/(1-short_term_robustness[5])
+    plot_robustness_1[1,:] = (1 - mid_term_robustness[0:5])/(1-mid_term_robustness[5])
+    plot_robustness_1[2,:] = (1 - long_term_robustness[0:5])/(1-long_term_robustness[5])
+
+    # Plot over time
+    fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(9,4))
+    axes[0].bar(np.arange(5), plot_robustness_1[0,:], color='#B19CD9')
+    axes[0].set_xticks(np.arange(5))
+    axes[0].set_xticklabels(criteria, rotation='vertical')
+    axes[0].set_ylim([0,1])
+    axes[0].set_title('10-year Horizon')
+    axes[0].set_ylabel('Fraction of failure SOWs')
+    axes[1].bar(np.arange(5), plot_robustness_1[1,:], color='#B19CD9')
+    axes[1].set_xticks(np.arange(5))
+    axes[1].set_xticklabels(criteria, rotation='vertical')
+    axes[1].set_ylim([0,1])
+    axes[1].set_title('22-year Horizon')
+    axes[2].bar(np.arange(5), plot_robustness_1[2,:], color='#B19CD9')
+    axes[2].set_xticks(np.arange(5))
+    axes[2].set_xticklabels(criteria, rotation='vertical')
+    axes[2].set_title('45-year Horizon')
+    axes[2].set_ylim([0,1])
+    fig.suptitle('Bedford')
+    plt.tight_layout()
+
+    ### Greene ###
+    # Determine the percentage of failure SOWs that violate each criterion (note some SOWS fail multiple criteria, so this may some to >1)
+    plot_robustness_2 = np.zeros([3, 5])
+    plot_robustness_2[0, :] = (1 - short_term_robustness[6:11]) / (1 - short_term_robustness[11])
+    plot_robustness_2[1, :] = (1 - mid_term_robustness[6:11]) / (1 - mid_term_robustness[11])
+    plot_robustness_2[2, :] = (1 - long_term_robustness[6:11]) / (1 - long_term_robustness[11])
+
+    fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(9, 4))
+    axes[0].bar(np.arange(5), plot_robustness_2[0, :], color='#43b284')
+    axes[0].set_xticks(np.arange(5))
+    axes[0].set_xticklabels(criteria, rotation='vertical')
+    axes[0].set_title('10-year Horizon')
+    axes[0].set_ylim([0,1])
+    axes[0].set_ylabel('Fraction of failure SOWs')
+    axes[1].bar(np.arange(5), plot_robustness_2[1, :], color='#43b284')
+    axes[1].set_xticks(np.arange(5))
+    axes[1].set_xticklabels(criteria, rotation='vertical')
+    axes[1].set_title('22-year Horizon')
+    axes[1].set_ylim([0,1])
+    axes[2].bar(np.arange(5), plot_robustness_2[2, :], color='#43b284')
+    axes[2].set_xticks(np.arange(5))
+    axes[2].set_xticklabels(criteria, rotation='vertical')
+    axes[2].set_title('45-year Horizon')
+    axes[2].set_ylim([0,1])
+    fig.suptitle('Greene')
+    plt.tight_layout()
+
+
+
+
+.. image:: _static/discovery_6_0.png
+
+
+
+.. image:: _static/discovery_6_1.png
+
+
+In the figures above, we observe that the vulnerability of both utilities changes in different ways. Early in the simulation period, Bedford is vulnerable to failures in reliability (though the robustness figure created in step B5.2 reveals that these failures are very rare). As the simulation period progresses, Bedford's vulnerability expands to include failures in restriction frequency and worst-case cost. These failures indicate that the utility has an overall inability to manage drought conditions and future conditions progress.
+
+Greene shows a very different evolution in vulnerability. Early in the simulation period, failures manifest in the restriction frequency objective, suggesting that the utility must rely on water use restrictions to maintain supply reliability. As the simulation progresses however, the vulnerability evolves. When evaluated across the 45-year planning horizon, a new failure modes emerges - financial failure manifesting in peak financial cost and stranded assets. This suggests that the proposed pathway policy may be over-investing in new infrastructure, straining the utility's budget with large debt payments that are unnecessary to maintain supply reliability.
+
+How do deep uncertainties generate vulnerability
+------------------------------------------------
+
+While the evolution of robustness provides insight into how the system
+evolves over time, it does not reveal *why* each utility is vulnerable.
+To examine how deep uncertainties generate vulnerability over time for
+the two utilities, we perform scenario discovery (factor mapping,
+Chapter 4.3). Here we’ll utilize gradient boosted trees to identify
+regions of the uncertainty space that cause the utilities to fail to
+meet performance criteria.
+
+.. code:: ipython3
+
+    # import the performance data across 2000 SOWs for three time periods
+    short_term_performance = np.loadtxt('data/short_term_performance.csv', delimiter= ',')
+    mid_term_performance = np.loadtxt('data/mid_term_performance.csv', delimiter = ',')
+    long_term_performance = np.loadtxt('data/long_term_performance.csv', delimiter = ',')
+
+    satisficing_criteria = [.98, .2, .8, .1, 5]
+
+    # transform into scenario discovery input
+    short_term_SD_input = create_sd_input(short_term_performance, satisficing_criteria)
+    mid_term_SD_input = create_sd_input(mid_term_performance, satisficing_criteria)
+    long_term_SD_input = create_sd_input(long_term_performance, satisficing_criteria)
+
+    # factor mapping Bedford
+    fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(9,3))
+    plot_selected_tree_maps(5, 'short_term', 0, 6, satisficing_criteria, 0, axes[0])
+    axes[0].set_title('10-year Horizon')
+    plot_selected_tree_maps(5, 'mid_term', 0, 6, satisficing_criteria, 0, axes[1])
+    axes[1].set_title('22-year Horizon')
+    plot_selected_tree_maps(5, 'long_term', 0, 1, satisficing_criteria, 0, axes[2])
+    axes[2].set_title('45-year Horizon')
+    fig.suptitle('Bedford Factor Maps')
+    plt.tight_layout()
+
+    # factor mapping Greene
+    fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(9,3))
+    plot_selected_tree_maps(11, 'short_term', 0, 8, satisficing_criteria, 0, axes[0])
+    axes[0].set_title('10-year Horizon')
+    plot_selected_tree_maps(11, 'mid_term', 0, 6, satisficing_criteria, 0, axes[1])
+    axes[1].set_title('22-year Horizon')
+    plot_selected_tree_maps(11, 'long_term', 0, 1, satisficing_criteria, 0, axes[2])
+    axes[2].set_title('45-year Horizon')
+    fig.suptitle('Greene Factor Maps')
+    plt.tight_layout()
+
+
+
+.. parsed-literal::
+
+    Factor map for Bedford
+    Factor map for Bedford
+    Factor map for Bedford
+    Factor map for Greene
+    Factor map for Greene
+    Factor map for Greene
+
+
+
+.. image:: _static/discovery_9_1.png
+
+
+
+.. image:: _static/discovery_9_2.png
+
+
+In the figures above, we learn more about how the vulnerability of the two utilities evolves over time. Bedford begins with very few possible failures but appears vulnerable to high demand growth scenarios under future scenarios with high demands. When evaluated across a 22-year planning horizon, Bedford is vulnerable when the near-term demand growth is high and water use restrictions are less effective than predicted. Under the full 45-year planning horizon, Bedford is vulnerable to sustained high levels of demand growth, failing if either near-term or mid-term demand growth exceeds expected levels.
+
+Greene's vulnerability evolves differently. It begins with vulnerability to high demand growth, but as the simulation progresses (and infrastructure is constructed), the utility becomes vulnerable to low-demand growth futures which cause the failures in financial criteria shown in section B.5.3. This indicates that the pathway policy over-builds in many SOWs, and becomes financially unstable if demand does not grow sufficiently to provide revenue to cover debt service payments.
+
+Which uncertainties have the most influence on time-evolving performance?
+-------------------------------------------------------------------------
+
+The factor maps generated in B.5.4 present the vulnerability generated by the two most important deep uncertainties as determined by Gradient Boosted Trees. Yet the factor prioritization shows that more than two uncertainties are influential to regional performance. Further, we can observe that individual uncertainties have different impacts on each performance obejctive, and these impacts may change over time. In the cells below, explore the impact of deep uncertainty by generating factor maps for different combinations of deep uncertain factors, objectives and time horizons.
+
+.. code:: ipython3
+
+    sns.set_style('white')
+    uncertainties = ['D1', 'D2', 'D3', 'BT', 'BM', 'DR', 'RE', 'EV', 'PM', 'CT', 'IA', 'IF', 'IP']
+    uncertainties = ['Near-term demand', 'Mid-term demand', 'Long-term demand', 'Bond Term', 'Bond Rate', 'Discount Rate', 'Restriction Effectiveness', 'Evaporation Rate', 'Permitting time', 'Construction time', 'Inflow Amplitude', 'Inflow Frequency', 'Inflow Period']
+
+    u1_st_FI = get_factor_importances(short_term_SD_input, rdm_factors, 250, 4, 5)
+    u1_mt_FI = get_factor_importances(mid_term_SD_input, rdm_factors, 250, 4, 5)
+    u1_lt_FI = get_factor_importances(long_term_SD_input, rdm_factors, 250, 4, 5)
+
+    u1_all = np.vstack([u1_st_FI,u1_mt_FI, u1_lt_FI])
+    u1_all = np.transpose(u1_all)
+
+    # factor ranking -- utility 2
+    u2_st_FI = get_factor_importances(short_term_SD_input, rdm_factors, 250, 4, 11)
+    u2_mt_FI = get_factor_importances(mid_term_SD_input, rdm_factors, 250, 4, 11)
+    u2_lt_FI = get_factor_importances(long_term_SD_input, rdm_factors, 250, 4, 11)
+    u2_all = np.vstack([u2_st_FI,u2_mt_FI, u2_lt_FI])
+    u2_all = np.transpose(u2_all)
+
+    fig, (ax, ax2, cax) = plt.subplots(ncols=3,figsize=(5,5),
+                      gridspec_kw={"width_ratios":[1,1, 0.1]})
+    fig.subplots_adjust(wspace=0.3)
+    im = ax.imshow(u1_all, cmap='Reds', vmin=0, vmax=.3)
+    ax.set_yticks(np.arange(13))
+    ax.set_yticklabels(uncertainties)
+    ax.set_xticks(np.arange(3))
+    ax.set_xlabel('Time Horizon')
+    ax.set_title('Bedford')
+
+    im1 = ax2.imshow(u2_all, cmap='Reds', vmin=0, vmax=.3)
+    ax2.set_yticks(np.arange(13))
+    ax2.set_yticklabels([])
+    ax2.set_xticks(np.arange(3))
+    ax2.set_xlabel('Time Horizon')
+    ax2.set_title('Greene')
+    fig.colorbar(im, cax=cax, label='Factor Importance')
+    plt.tight_layout()
+
+
+
+
+.. image:: _static/discovery_12_0.png
+
+
+
+
+The Figure above shows the factor importance as determined by gradient boosted trees for both utilities across the three planning horizons. While near-term demand growth is important for both utilities under all three planning horizons, the importance of other factors evolves over time. For example, restriction effectiveness plays an important role for Greene under the 22-year planning horizon but disappears under the 45-year planning horizon. In contrast, the bond interest rate is important for predicting success over the 45-year planning horizon, but does not appear important over the 10- or 22-year planning horizons. These findings highlight how assumptions about the planning period can have a large impact on modeling outcomes.
+
+Open exploration
+----------------
+
+In the cell below, use the function to explore how factor maps change
+for the two utilities based upon the uncertainties plotted, the
+objectives of interest and the time horizon.
+
+.. code:: ipython3
+
+    # specify the utility ("Bedford" or "Greene")
+    utility = "Bedford"
+
+    # specify which performance objectives to investigate (note that not all performance objectives have failures, which may result in a blank factor map)
+    # set this to one of the following: "Reliability", "Restriction Frequency", "Peak Financial Cost", "Worst Case Cost" or "Unit Cost"
+    objective = "Reliability"
+
+    # select uncertainties from the following list: 'D1', 'D2', 'D3', 'BT', 'BM', 'DR', 'RE', 'EV', 'PM', 'CT', 'IA', 'IF', 'IP'
+    uncertainty_1 = 'D1'
+    uncertainty_2 = 'D2'
+
+    # The code below will plot factor maps over the three planning horizons for the information above
+    fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(9,3))
+    open_exploration(utility, objective, 'short_term', uncertainty_1, uncertainty_2, axes[0])
+    open_exploration(utility, objective, 'mid_term', uncertainty_1, uncertainty_2, axes[1])
+    open_exploration(utility, objective, 'long_term', uncertainty_1, uncertainty_2, axes[2])
+    plt.tight_layout()
+
+
+
+.. parsed-literal::
+
+    Factor map for Bedford, reliability
+    Factor map for Bedford, reliability
+    Factor map for Bedford, reliability
+
+
+
+.. image:: _static/discovery_16_1.png
+
+
+
+
+Tips to apply this methodology to your own problem
+--------------------------------------------------
+
+In this tutorial, we demonstrated time-evolving scenario discovery for a
+cooperative water supply system. To apply this workflow to your own
+problem:
+
+1. Choose sampling bounds for your parameters of interest, which will
+   represent uncertainties that characterize your system.
+2. Generate samples for these parameters (this can be done using the
+   saltelli.sample function as in B.2 or done with another package).
+3. Define performance criteria for your problem
+4. Evaluate parameter sets through your model, and save performance
+   measures across multiple time horizons
+5. Draw from the supporting code for this tutorial to perform scneario
+   discovery and visualize results
+
+
+References
+--------------------------------------------------
+Trindade, B. C., Reed, P. M., & Characklis, G. W. (2019). Deeply uncertain pathways: Integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134, 103442.
diff --git a/dev/docs/html/_sources/A2.6_hmm.rst b/dev/docs/html/_sources/A2.6_hmm.rst
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+A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios Tutorial
+****************************************************************************************************
+
+.. note::
+
+    | Run the tutorial interactively:  `HMM Notebook <nb_hmm_>`_.
+    | Please be aware that notebooks can take a couple minutes to launch.
+    | To run the notebooks yourself, download the files `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks>`_ and use these `requirements <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt>`_.
+
+
+A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios
+============================================================================
+
+In this notebook, we will be covering the basics of fitting a Hidden
+Markov Model-based synthetic streamflow generator for a single site in
+the Upper Colorado River Basin. First, we will characterize the observed
+historical flow in the basin from 1909-2013. Then, we will fit a
+synthetic streamflow generator to the observed flows in the basin in
+order to create stationary synthetic flows. Finally, we will create a
+non-stationary version of the generator to create flows that could be
+representative of plausible future climate in the region. We ultimately
+show how to place the synthetically generated flows in the context of
+physically-informed CMIP5 projections to compare the two methods.
+
+Background
+----------
+
+In the Western United States (US), and particularly the Colorado River
+Basin, a recent study used tree-ring reconstructions to suggest that the
+megadrought that has been occurring in the Southwest over the past 22
+years is the region’s worst drought since about 800 AD (Williams et al.,
+2022). The study’s lead author, UCLA climatologist Park Williams,
+suggested that had the sequence of wet-dry years occurred without
+anthropogenic forcing, the 2000s would have likely still been dry, but
+not on the same level as the worst of the last millennium’s
+megadroughts.
+
+The recent trend of warming and reduced soil moisture in the Southwest
+US is highly challenging from a water systems planning and management
+perspective for the Colorado River Basin. Given the wide recognition
+that the river is over-allocated, the most recent drought highlights the
+difficulty of sustaining the flow requirements as dictated by the
+Colorado Compact. Thus, there has been an increasing focus in
+exploratory modeling efforts to clarify how vulnerable water systems in
+this region are to plausible drought streamflow scenarios for the
+future. In this tutorial, we’ll discuss how to create these scenarios
+using a Hidden Markov Model (HMM)- based streamflow synthetic generator.
+As discussed in `Section
+2.1 <https://uc-ebook.org/docs/html/2_diagnostic_modeling_overview_and_perspectives.html#overview-of-model-diagnostics>`__
+and
+`4.2 <https://uc-ebook.org/docs/html/4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-dynamics-what-is-controlling-model-behaviors-of-interest>`__
+of the eBook, future climate conditions in the basin represent a deep
+uncertainty that can lead to highly consequential water scarcity
+outcomes. It is advantageous to create a model such as the HMM-based
+generator in order to facilitate the creation of many ensembles of
+streamflow that can ultimately be used to force regional water systems
+models to understand how variability and drought extremes affect
+regional water shortages, operations, and policies.
+
+.. raw:: html
+
+  <figure>
+  <center><img src="_static/1-lede_pix_-_usbr-powell-8517152024_aa66437c2e_o-cropped.jpg" style="width:50%"></center>
+  <figcaption style="text-align:center"> Lake Powell shows persistent effects from drought (Source: U.S. Bureau of Reclamation) </figcaption>
+  </figure>
+
+Let’s Get Started!
+==================
+
+Observed Record
+---------------
+
+First, let’s take a look at the observed data from 1909-2013 for a
+specific site. In this example, we use the outlet gauge of the Upper
+Colorado River (USGS Gauge 09163500 at the Colorado-Utah state line).
+Below, we create a plot of the annual streamflow.
+
+.. code:: ipython3
+
+    # Import libraries
+    import matplotlib as mpl
+    import matplotlib.pyplot as plt
+    import matplotlib.patches as patches
+    import numpy as np
+    import pandas as pd
+    from random import random
+    from SALib.sample import latin
+    from scipy import stats as ss
+    import statistics 
+    import statsmodels.api as sm
+    
+    
+    # Import helper functions from local package 
+    from functions import fitmodel
+    from functions import plotstates
+    from functions import plotdist
+
+
+.. code:: ipython3
+
+    # Read in annual historical data
+    AnnualQ = pd.read_csv('data/uc_historical.csv')
+    AnnualQ['Year'] = list(range(1909, 2014))
+    
+    # Plot a line graph
+    fig, ax = plt.subplots(figsize=(12, 8))
+    ax.plot(AnnualQ.iloc[:, 1],
+            AnnualQ.iloc[:, 0],
+            color='#005F73',
+            label='Annual')
+    
+    # Add labels and title
+    ax.set_title("Upper Colorado Annual Flow")
+    ax.set_xlabel("Year", fontsize=16)
+    ax.set_ylabel("Annual Flow (cubic feet per year)", fontsize=16)
+    plt.xticks(fontsize=12)
+    plt.yticks(fontsize=12)
+    mpl.rc('legend', fontsize=16)
+    legend = plt.legend(loc="upper right") 
+    plt.show()
+    plt.close()
+
+
+
+
+.. image:: _static/hmm_9_0.png
+
+
+Let’s calculate an 11-year rolling mean of the same data to get a sense
+of long-term trends.
+
+.. code:: ipython3
+
+    fig, ax = plt.subplots(figsize=(12, 8))
+    
+    # Plot the original line graph
+    plt.plot(AnnualQ.iloc[:,1],
+             AnnualQ.iloc[:,0],
+             color='#005F73',
+             label='Annual')
+    
+    # Plot an 11-year rolling mean 
+    plt.plot(AnnualQ.iloc[:, 1].rolling(11).mean(),
+             AnnualQ.iloc[:, 0].rolling(11).mean(),
+             color='#183A2E',
+             label='11-Year Rolling Mean')
+    
+    # Add labels and title
+    plt.title("Upper Colorado Annual Flow")
+    ax.set_xlabel("Year",fontsize=16)
+    ax.set_ylabel("Annual Flow (cubic feet per year)", fontsize=16)
+    plt.xticks(fontsize=12)
+    plt.yticks(fontsize=12)
+    mpl.rc('legend', fontsize=16)
+    legend = plt.legend() 
+    plt.show()
+    plt.close()
+
+
+
+
+.. image:: _static/hmm_11_0.png
+
+
+The Colorado Compact, which prescribes flows between the Upper and Lower
+Colorado Basins, was negotiated using data prior to 1922, a time period
+revealed by the above figure to be one of the consistently wetter
+periods on record. It’s clear today that since the 1980s, the Southwest
+US has been experiencing aridification (Overpeck et al., 2020) and that
+this observed record alone isn’t an accurate representation of what
+future climate might look like in this region.
+
+Let’s get a little more specific and formally quantify decadal droughts
+that have occurred in the observed period. We use a metric proposed in
+Ault et al. (2014). The authors define a decadal drought as when the
+11-year rolling mean falls below a threshold that is 1/2 a standard
+deviation below the overall mean of the record. We can then highlight
+the block of years that fall in a decadal drought using yellow
+rectangles below.
+
+.. code:: ipython3
+
+    # Define drought threshold 
+    std = statistics.stdev(AnnualQ.iloc[:, 0])
+    threshold = np.mean(AnnualQ.iloc[:, 0] - (0.5 * std))
+    
+    # Find where the rolling mean dip below the threshold?
+    drought_instances = [i for i, v in enumerate(AnnualQ.iloc[:,0].rolling(11).mean()) if v < threshold]
+    drought_years = AnnualQ.iloc[:, 1].rolling(11).mean()[drought_instances]
+    
+    # Add labels and title
+    fig, ax = plt.subplots(figsize=(12, 8))
+    ax.plot(AnnualQ.iloc[:,1],
+            AnnualQ.iloc[:,0],
+            color='#005F73',
+            label='Annual') 
+    
+    ax.plot(AnnualQ.iloc[:,1].rolling(11,center=True).mean(),
+            AnnualQ.iloc[:,0].rolling(11,center=True).mean(),
+            color='#183A2E',
+            label='11-Year Rolling Mean')
+    
+    ax.axhline(y=threshold, 
+               color='black', 
+               linestyle='--',
+               label='Drought Threshold')
+    
+    # Visualize the drought periods as yellow rectangles
+    for i in drought_years:
+        
+        # Plot a box centered around those values and with 5 years on either side.
+        rect = patches.Rectangle((i-5,0), 11,2e7, linewidth=1, edgecolor='#EFE2BE', facecolor='#EFE2BE')
+        
+        # Add the patch to the Axes
+        ax.add_patch(rect)
+        
+        
+    plt.title("Upper Colorado Annual Flow")
+    ax.set_xlabel("Year", fontsize=16)
+    ax.set_ylabel("Annual Flow (cubic feet per year)", fontsize=16)
+    plt.xticks(fontsize=12)
+    plt.yticks(fontsize=12)
+    mpl.rc('legend', fontsize=16)
+    legend = plt.legend() 
+    plt.show()
+    plt.close()
+
+
+
+
+.. image:: _static/hmm_14_0.png
+
+
+By this metric, the Upper Colorado Basin region has experienced two
+decadal droughts over the last century.
+
+Synthetic Stationary Generator to Better Quantify Natural Variability
+---------------------------------------------------------------------
+
+It is important to remember that the streamflow that we have observed in
+the region over the last century is only one instance of the hydrology
+that could occur since the atmosphere is an inherently stochastic
+system. Thus, we require a tool that will allow us to see multiple
+plausible realizations of the streamflow record to understand the
+internal variability that characterizes the historical period. One
+observed realization of historical streamflow is limited in its ability
+to capture rare extremes; plausible (but not observed) alternative
+instances of streamflow records can help to fill this gap. The tool that
+we use to develop synthetic flows for the region is a Gaussian Hidden
+Markov Model (HMM). If a system follows a Markov process, it switches
+between a number of “hidden states” dictated by a transition matrix.
+Each state has its own Gaussian probability distribution (defined by a
+mean and standard deviation) and one can draw from this distribution to
+create synthetic flows that fit the properties of the historical
+distribution. HMMs are an attractive choice for this region because they
+can simulate persistence (i.e., long duration droughts), which is a
+characteristic of the region’s hydro-climatology. The figure below shows
+an example of a 2-state Gaussian HMM that we will be fitting for this
+example.
+
+.. raw:: html
+
+  <figure>
+      <center><img src="_static/HMM_example.png" style="width:75%"></center>
+  <figcaption style="text-align:center"> Two-state Gaussian HMM with mean and standard deviation parameters</figcaption>
+  </figure>
+
+Below is the code that fits the HMM model to the last 2/3 of the
+historical record of log annual flows at the CO-UT stateline gauge and
+creates an alternative trace of 105 years. A subset of the dataset is
+chosen in order to minimize overfitting and to retain a set of data for
+validation of the model. When we fit our model, we utilize the
+Baum-Welch algorithm (a special version of the expectation-maximization
+algorithm) to find the optimal parameters that maximize the likelihood
+of seeing the observed flows. Ultimately, the algorithm will return a
+mean and standard deviation associated with each state (mus and sigmas
+defined below) and a 2x2 transition probability matrix that captures the
+likelihood of transitioning between states (P). We can also retrieve the
+annual hidden states across the observed series, also known as the
+Viterbi sequence of states, which classifies each year in a “wet” or
+“dry” state.
+
+.. code:: ipython3
+
+    # Number of years for alternative trace
+    n_years = 105
+    
+    # Import historical data that it used to fit HMM model 
+    AnnualQ_h = pd.read_csv('data/uc_historical.csv')
+    
+    # Fit the model and pull out relevant parameters and samples 
+    logQ = np.log(AnnualQ_h)
+    hidden_states, mus, sigmas, P, logProb, samples, model = fitmodel.fitHMM(logQ, n_years)
+
+
+We’ve fit our HMM, but what does the model look like? Let’s plot the
+annual time series of hidden states, or the Viterbi sequence. In the
+code, above, we have defined that the drier state is always represented
+by state 0. Thus, we know that hidden_states = 0 corresponds to the dry
+state and hidden_states = 1 to the wet state.
+
+.. code:: ipython3
+
+    # Plot Vitebi sequence 
+    plotstates.plotTimeSeries(np.log(AnnualQ.iloc[:,0]), hidden_states, 'Annual Flow (cube feet per year)')
+
+
+
+
+.. image:: _static/hmm_21_0.png
+
+
+In the figure above, we see that the years with the higher log flows
+tend to be classified in a “wet” state and the opposite is true of the
+“dry” state. We can also print the transition matrix, which shows the
+likelihood of transitioning between states. Note that the system has a
+high likelihood of persisting in the same state.
+
+.. code:: ipython3
+
+    print(model.transmat_)
+
+
+.. parsed-literal::
+
+    [[0.65095026 0.34904974]
+     [0.3205531  0.6794469 ]]
+
+
+Let’s also plot the distribution of log annual flows associated with the
+wet and dry states.
+
+.. code:: ipython3
+
+    # Plot wet and dry state distributions
+    plotdist.plotDistribution(logQ, mus, sigmas, P)
+
+
+
+
+.. image:: _static/hmm_25_0.png
+
+
+The wet state distribution is characterized by a greater mean flow, but
+note that there is significant overlap in the tails of the distributions
+below which demonstrates why years with similiar flows can be classified
+in different states.
+
+Now let’s see what the drought dynamics look like in the synthetic
+scenario that we created using the same definition that we had used for
+the historical period.
+
+.. code:: ipython3
+
+    # Retrieve samples and back-transform out of log space
+    AnnualQ_s = np.exp(samples[0])
+    AnnualQ_s = pd.DataFrame(AnnualQ_s)
+    AnnualQ_s['Year'] = list(range(1909, 2014))
+    
+    # Define drought threshold 
+    std=statistics.stdev(AnnualQ_s.iloc[:, 0])
+    threshold=np.mean(AnnualQ_s.iloc[:, 0] - (0.5 * std))
+    
+    # Where does the rolling mean dip below the threshold
+    drought_instances = [i for i,v in enumerate(AnnualQ_s.iloc[:, 0].rolling(11).mean()) if v < threshold]
+    drought_years = AnnualQ_s.iloc[:, 1].rolling(11).mean()[drought_instances]
+    
+    #Visualize the streamflow scenario
+    fig, ax = plt.subplots(figsize=(12, 8))
+    
+    #Plot the original line graph
+    ax.plot(AnnualQ_s.iloc[:,1],
+            AnnualQ_s.iloc[:,0],
+            color='#005F73',
+            label='Annual')
+    
+    #Plot a 11-year rolling mean 
+    ax.plot(AnnualQ_s.iloc[:,1],
+            AnnualQ_s.iloc[:,0].rolling(11, center=True).mean(),
+            color='#183A2E',
+            label='11-Year Rolling Mean')
+    
+    # Add labels and title
+    ax.axhline(y=threshold, 
+               color='black', 
+               linestyle='--',
+               label='Drought Threshold')
+    
+    
+    for i in drought_years:
+        
+        #Plot a box centered around those values and with 5 years on either side.
+        rect = patches.Rectangle((i - 5,
+                                  0), 
+                                  11,
+                                  2e7, 
+                                  linewidth=1, 
+                                  edgecolor='#EFE2BE', 
+                                  facecolor='#EFE2BE')
+        
+        # Add the patch to the Axes
+        ax.add_patch(rect)
+        
+        
+    plt.title("Upper Colorado Annual Flow (Synthetic Stationary)",fontsize=16)
+    plt.xlabel("Year", fontsize=16)
+    plt.ylabel("Annual Flow (cubic feet per year)", fontsize=16)
+    mpl.rc('legend', fontsize=16)
+    plt.legend() 
+    plt.xticks(fontsize=12)
+    plt.yticks(fontsize=12)
+    plt.show()
+    plt.close()
+
+
+
+
+.. image:: _static/hmm_28_0.png
+
+
+You can sample from the model and create more 105-year traces and note
+how the location and number of decadal droughts changes. This
+demonstrates how different the historical record can look just within
+the range of natural variability. It’s also important to remember that
+when droughts occur can also define the ultimate effect of the drought
+(i.e. is it a time when there is a large population growth or a time
+when humans can adapt by conserving or building more infrastructure?). A
+hydrologic drought need not manifest into an agricultural or operational
+drought of the same magnitude if stored surface water is available.
+
+We externally run the HMM many times to create a dataset of 100
+instances of the 105-year traces and 1000 instances of the 105-year
+traces that are available in the package
+(“synthetic_stationary_small_sample_100.csv”,“synthetic_stationary_large_sample_1000”).
+The shaded green lines correspond to the flow duration curves (FDCs) for
+the generated streamflow traces in comparison with the FDC of the
+historical record in beige.
+
+.. raw:: html
+
+  <figure>
+      <center><img src="_static/stationary_synthetic_FDC.png" style="width:75%"></center>
+  <figcaption style="text-align:center">Generated streamflow traces in comparison with the FDC of the historical record.</figcaption>
+  </figure>
+
+As expected, the stationary synthetic FDCs envelope the historical FDC
+and particularly, the synthetic traces offer many more instances of low
+flow conditions that could lead to more extreme drought conditions than
+what has been observed historically. It is also useful to check for
+convergence of samples and to determine how many samples are needed to
+fully represent internal variability. Above we see that the extension to
+1000 instances of 105-year traces fills out regions of the FDC,
+including creating some more extreme drought conditions, but that
+additional samples will likely not fill out the FDC substantially more.
+
+Non-Stationary Synthetic Generator to Impose Climate Changes
+------------------------------------------------------------
+
+Now, we create flows under non-stationary conditions to get a better
+understanding of what flows can look like under climate changes. In
+order to create flows under non-stationary conditions, we can toggle the
+parameters of the HMM model in order to create systematic changes to the
+model that can represent a changing climate. The HMM has 6 parameters
+that define it. When we fit the historical model, the parameters that
+are fit represent a baseline parameter value. In this non-stationary
+generator, we define a range to sample these parameters from.
+
++-----------------------+---------------+-------------+-------------+
+| Parameter             | Current Value | Lower Bound | Upper Bound |
++=======================+===============+=============+=============+
+| Log-Space Wet State   | 1.00          | 0.98        | 1.02        |
+| Mean Multiplier       |               |             |             |
++-----------------------+---------------+-------------+-------------+
+| Log-Space Dry State   | 1.00          | 0.98        | 1.02        |
+| Mean Multiplier       |               |             |             |
++-----------------------+---------------+-------------+-------------+
+| Log-Space Wet State   | 1.00          | 0.75        | 1.25        |
+| Standard Deviation    |               |             |             |
+| Multiplier            |               |             |             |
++-----------------------+---------------+-------------+-------------+
+| Log-Space Dry State   | 1.00          | 0.75        | 1.25        |
+| Standard Deviation    |               |             |             |
+| Multiplier            |               |             |             |
++-----------------------+---------------+-------------+-------------+
+| Change in Dry-Dry     | 0.00          | -0.30       | +0.30       |
+| Transition            |               |             |             |
+| Probability           |               |             |             |
++-----------------------+---------------+-------------+-------------+
+| Change in Wet-Wet     | 0.00          | -0.30       | +0.30       |
+| Transition            |               |             |             |
+| Probability           |               |             |             |
++-----------------------+---------------+-------------+-------------+
+
+Now let’s sample 1000 times from these bounds to create 1000 new
+parameterizations of the model. Here we use SALib and the Latin
+Hypercube sample function.
+
+.. code:: ipython3
+
+    # Create problem structure with parameters that we want to sample 
+    problem = {
+        'num_vars': 6,
+        'names': ['wet_mu', 'dry_mu', 'wet_std','dry_std','dry_tp',"wet_tp"],
+        'bounds': [[0.98, 1.02],
+                   [0.98, 1.02],
+                   [0.75,1.25],
+                   [0.75,1.25],
+                   [-0.3,0.3],
+                   [-0.3,0.3]]
+    }
+    
+    # generate 1000 parameterizations
+    n_samples = 1000
+    
+    # set random seed for reproducibility
+    seed_value = 123
+    
+    # Generate our samples 
+    LHsamples = latin.sample(problem, n_samples, seed_value)
+
+
+Now let’s look at what some of the traces look like in our
+non-stationary generator. Let’s choose a random instance from the
+1000-member space and adjust the parameters accordingly.
+
+.. code:: ipython3
+
+    # Define static parameters 
+    n_years = 105
+    
+    # Sample parameter; Adjust to any sample number from 0-999 
+    sample = 215 
+    
+    # Create empty arrays to store the new Gaussian HMM parameters for each SOW
+    Pnew = np.empty([2,2])
+    piNew = np.empty([2])
+    musNew_HMM = np.empty([2])
+    sigmasNew_HMM = np.empty([2])
+    logAnnualQ_s = np.empty([n_years]) 
+    
+    # Calculate new transition matrix and stationary distribution of SOW at last node as well as new means and standard deviations
+    Pnew[0, 0] = max(0.0, min(1.0, P[0, 0] + LHsamples[sample][4]))
+    Pnew[1, 1] = max(0.0, min(1.0, P[1, 1] + LHsamples[sample][5]))
+    Pnew[0, 1] = 1 - Pnew[0, 0]
+    Pnew[1, 0] = 1 - Pnew[1, 1]
+    eigenvals, eigenvecs = np.linalg.eig(np.transpose(Pnew))
+    one_eigval = np.argmin(np.abs(eigenvals - 1))
+    piNew = np.divide(np.dot(np.transpose(Pnew), eigenvecs[:, one_eigval]), 
+                      np.sum(np.dot(np.transpose(Pnew), eigenvecs[:,one_eigval])))
+    
+    musNew_HMM[0] = mus[0] * LHsamples[sample][1]
+    musNew_HMM[1] = mus[1] * LHsamples[sample][0]
+    sigmasNew_HMM[0] = sigmas[0] * LHsamples[sample][3]
+    sigmasNew_HMM[1] = sigmas[1] * LHsamples[sample][2]
+    
+    # Generate first state and log-space annual flow at last node
+    states = np.empty([n_years])
+    if random() <= piNew[0]:
+        states[0] = 0
+        logAnnualQ_s[0] = ss.norm.rvs(musNew_HMM[0], sigmasNew_HMM[0])
+    else:
+        states[0] = 1
+        logAnnualQ_s[0] = ss.norm.rvs(musNew_HMM[1], sigmasNew_HMM[1])
+    
+    # Generate remaining state trajectory and log space flows at last node
+    for j in range(1, n_years):
+        if random() <= Pnew[int(states[j-1]), int(states[j-1])]:
+            states[j] = states[j-1]
+        else:
+            states[j] = 1 - states[j-1]
+    
+        if states[j] == 0:
+            logAnnualQ_s[j] = ss.norm.rvs(musNew_HMM[0], sigmasNew_HMM[0])
+        else:
+            logAnnualQ_s[j] = ss.norm.rvs(musNew_HMM[1], sigmasNew_HMM[1])
+    
+    # Convert log-space flows to real-space flows
+    AnnualQ_s = np.exp(logAnnualQ_s)-1
+
+
+Now let’s see what this synthetic trace looks like.
+
+.. code:: ipython3
+
+    # Retrieve samples and back-transform out of log space
+    AnnualQ_s = pd.DataFrame(AnnualQ_s)
+    AnnualQ_s['Year'] = list(range(1909, 2014))
+    
+    # Define drought threshold 
+    std = statistics.stdev(AnnualQ_s.iloc[:, 0])
+    threshold = np.mean(AnnualQ_s.iloc[:, 0] - (0.5 * std))
+    
+    # Where does the rolling mean dip below the threshold
+    drought_instances = [i for i, v in enumerate(AnnualQ_s.iloc[:, 0].rolling(11).mean()) if v < threshold]
+    drought_years = AnnualQ_s.iloc[:, 1].rolling(11).mean()[drought_instances]
+    
+    # Visualize the streamflow scenario
+    fig, ax = plt.subplots(figsize=(12, 8))
+    
+    # Plot the original line graph
+    ax.plot(AnnualQ_s.iloc[:,1],
+            AnnualQ_s.iloc[:,0],
+            color='#005F73',
+            label='Annual')
+    
+    # Plot a 11-year rolling mean 
+    ax.plot(AnnualQ_s.iloc[:, 1],
+            AnnualQ_s.iloc[:, 0].rolling(11, center=True).mean(),
+            color='#183A2E',
+            label='11-Year Rolling Mean')
+    
+    # Add labels and title
+    ax.axhline(y=threshold, 
+               color='black', 
+               linestyle='--',
+               label='Drought Threshold')
+    
+    
+    for i in drought_years:
+        
+        # Plot a box centered around those values and with 5 years on either side.
+        rect = patches.Rectangle((i - 5,0), 
+                                 11,
+                                 2e7, 
+                                 linewidth=1, 
+                                 edgecolor='#EFE2BE', 
+                                 facecolor='#EFE2BE')
+        
+        # Add the patch to the Axes
+        ax.add_patch(rect)
+        
+        
+    plt.title("Annual Flow (Synthetic Non-Stationary)", fontsize=16)
+    plt.xlabel("Year", fontsize=16)
+    plt.ylabel("Annual Flow (cubic feet per year)", fontsize=16)
+    plt.xticks(fontsize=12)
+    plt.yticks(fontsize=12)
+    mpl.rc('legend', fontsize=16)
+    legend = plt.legend(loc="upper right") 
+    plt.show()
+    plt.close()
+    
+
+
+
+
+.. image:: _static/hmm_40_0.png
+
+
+Above is the example trace from the new non-stationary model. You may
+see fewer or more decadal drought instances. We can further summarize
+overall decadal drought characteristics across the samples. Let’s plot a
+histogram of the total number of times we go below the drought threshold
+across these realizations.
+
+.. code:: ipython3
+
+    decadal_drought_occurence=np.empty([1000]) 
+    
+    for y in range(1000):
+        
+        # Create empty arrays to store the new Gaussian HMM parameters for each SOW
+        Pnew = np.empty([2, 2])
+        piNew = np.empty([2])
+        musNew_HMM = np.empty([2])
+        sigmasNew_HMM = np.empty([2])
+        logAnnualQ_s = np.empty([n_years]) 
+        
+        # Calculate new transition matrix and stationary distribution of SOW at last node
+        # as well as new means and standard deviations
+    
+        Pnew[0, 0] = max(0.0,min(1.0, P[0, 0] + LHsamples[y][4]))
+        Pnew[1, 1] = max(0.0,min(1.0, P[1, 1] + LHsamples[y][5]))
+        Pnew[0, 1] = 1 - Pnew[0, 0]
+        Pnew[1, 0] = 1 - Pnew[1, 1]
+        eigenvals, eigenvecs = np.linalg.eig(np.transpose(Pnew))
+        one_eigval = np.argmin(np.abs(eigenvals - 1))
+        piNew = np.divide(np.dot(np.transpose(Pnew), eigenvecs[:, one_eigval]),
+                          np.sum(np.dot(np.transpose(Pnew), eigenvecs[:, one_eigval])))
+    
+        musNew_HMM[0] = mus[0] * LHsamples[y][1]
+        musNew_HMM[1] = mus[1] * LHsamples[y][0]
+        sigmasNew_HMM[0] = sigmas[0] * LHsamples[y][3]
+        sigmasNew_HMM[1] = sigmas[1] * LHsamples[y][2]
+    
+        # Generate first state and log-space annual flow at last node
+        states = np.empty([n_years])
+        if random() <= piNew[0]:
+            states[0] = 0
+            logAnnualQ_s[0] = ss.norm.rvs(musNew_HMM[0], sigmasNew_HMM[0])
+        else:
+            states[0] = 1
+            logAnnualQ_s[0] = ss.norm.rvs(musNew_HMM[1], sigmasNew_HMM[1])
+    
+        # generate remaining state trajectory and log space flows at last node
+        for j in range(1, n_years):
+            if random() <= Pnew[int(states[j-1]), int(states[j-1])]:
+                states[j] = states[j-1]
+            else:
+                states[j] = 1 - states[j-1]
+    
+            if states[j] == 0:
+                logAnnualQ_s[j] = ss.norm.rvs(musNew_HMM[0], sigmasNew_HMM[0])
+            else:
+                logAnnualQ_s[j] = ss.norm.rvs(musNew_HMM[1], sigmasNew_HMM[1])
+    
+        # Convert log-space flows to real-space flows
+        AnnualQ_s = np.exp(logAnnualQ_s) - 1
+        AnnualQ_s = pd.DataFrame(AnnualQ_s)
+        AnnualQ_s['Year'] = list(range(1909, 2014))
+    
+        # Define drought threshold 
+        std = statistics.stdev(AnnualQ_s.iloc[:, 0])
+        threshold = np.mean(AnnualQ_s.iloc[:, 0] - (0.5 * std))
+    
+        # Where does the rolling mean dip below the threshold
+        drought_instances = [i for i, v in enumerate(AnnualQ_s.iloc[:, 0].rolling(11).mean()) if v < threshold]
+        decadal_drought_occurence[y] = len(drought_instances)
+        
+
+.. code:: ipython3
+
+    fig, ax = plt.subplots(figsize=(12, 8))
+    ax.hist(decadal_drought_occurence,label='Non-Stationary generator',color="#005F73")
+    ax.set_xlabel('Number of Instances of Decadal Drought',fontsize=16)
+    ax.set_ylabel('Frequency',fontsize=16)
+    ax.axvline(x=2, color='r', linestyle='-',label='Observed')
+    mpl.rc('legend', fontsize = 16)
+    plt.xticks(fontsize = 12)
+    plt.yticks(fontsize = 12)
+    plt.show()
+    plt.close()
+
+
+
+
+.. image:: _static/hmm_43_0.png
+
+
+Note how many more instances of the decadal droughts we are creating
+with the non-stationary generator than our observed 105-year trace which
+creates a rich space in which we can test our models. Just as we did
+with the stationary generator, we can externally run the non-stationary
+generator to create 10,000 instances of the 105-year traces that are
+available in the package
+(“synthetic_nonstationary_large_sample_10000.csv”). The shaded green and
+blue lines correspond to the FDCs for the stationary and non-stationary
+generated streamflow traces in comparison with the FDC of the historical
+record in beige. Note how the non-stationary generator produces even
+more drought extremes than the stationary non-synthetic traces.
+
+.. raw:: html
+
+  <figure>
+      <center><img src="_static/nonstationary_synthetic_FDC.png" style="width:75%"></center>
+  <figcaption style="text-align:center">Generated streamflow traces in comparison with the FDC of the historical record.</figcaption>
+  </figure>
+
+Placing CMIP5 Projections in the Context of Non-Stationary Flows
+----------------------------------------------------------------
+
+We have broadened the drought conditions that we are creating which that
+can be very useful to understand how our water systems model performs
+under potentially extreme scenarios. However, it’s useful to compare our
+bottom-up synthetically generated flows in the context of global
+physically-driven CMIP5 projections to get a better understanding of how
+the two approaches compare. We first aquire 97 CMIP5 projections from
+the Colorado River Water Availability Study (CWCB, 2012). In each of
+these projections, monthly precipitation factor changes and temperature
+delta changes were computed between mean projected 2035–2065 climate
+statistics and mean historical climate statistics from 1950–2013. These
+97 different combinations of 12 monthly precipitation multipliers and 12
+monthly temperature delta shifts were applied to historical
+precipitation and temperature time series from 1950–2013. The resulting
+climate time series were run through a Variable Infiltration Capacity
+(VIC) model of the UCRB, resulting in 97 time series of projected future
+streamflows at the Colorado‐Utah state line.
+
+We fit an HMM to each trace of projected streamflow and get a set of
+corresponding HMM parameters. Then we take the ratio between these
+parameters and the baseline HMM parameters that we calculated earlier in
+the notebook in order to calculate the multipliers associated with each
+CMIP5 projection. This is all done externally, so we import the
+resulting multipliers in the next line.
+
+.. code:: ipython3
+
+    # Read in CMIP5 and paleo multipliers
+    CMIP5_multipliers = pd.read_csv('data/CMIP5_SOWs.txt', header=None, sep=" ")
+
+
+Let’s plot a response surface that will allow us to see how combinations
+of HMM parameters tend to influence decadal drought. In order to get a
+continuous surface, we’ll fit a non-linear regression to the parameter
+values and then predict the decadal drought over a set of grid points.
+We fit the response surface for two parameters that should have an
+affect on decadal drought: the dry distribution mean and the dry-dry
+transition probabilites.
+
+.. code:: ipython3
+
+    # Choose two parameters to fit the response surface for
+    mu_dry=[i[1] for i in LHsamples]
+    tp_dry=[i[4] for i in LHsamples]
+    
+    # Create an interpolation grid
+    xgrid = np.arange(np.min(mu_dry),
+                      np.max(mu_dry), 
+                      (np.max(mu_dry) - np.min(mu_dry)) / 100)
+    
+    ygrid = np.arange(np.min(tp_dry),
+                      np.max(tp_dry), 
+                      (np.max(tp_dry) - np.min(tp_dry)) / 100)
+    
+    # Fit regression 
+    d = {'Dry_Tp': tp_dry, 
+         'Dry_Mu': mu_dry,
+         'Drought_Occurrence':decadal_drought_occurence}
+    
+    df = pd.DataFrame(d)
+    df['Intercept'] = np.ones(np.shape(df)[0])
+    df['Interaction'] = df['Dry_Tp'] * df['Dry_Mu']
+    cols = ['Intercept'] + ['Dry_Mu'] + ['Dry_Tp'] + ['Interaction']
+    ols = sm.OLS(df['Drought_Occurrence'], df[cols])
+    result = ols.fit()
+    
+    # Calculate drought occurence for each grid point
+    X, Y = np.meshgrid(xgrid, ygrid)
+    x = X.flatten()
+    y = Y.flatten()
+    grid = np.column_stack([np.ones(len(x)), x, y, x * y])
+    z = result.predict(grid)
+    z[z < 0.0] = 0.0 # replace negative shortage predictions with 0
+
+
+Let’s plot our results:
+
+.. code:: ipython3
+
+    # Set color gradient for response surface 
+    drought_map = mpl.cm.get_cmap('RdBu_r')
+    
+    # Reshape our predicted drought occurrence and define bounds of colors 
+    Z = np.reshape(z, np.shape(X))
+    vmin = np.min([np.min(z), np.min(df['Drought_Occurrence'].values)])
+    vmax = 15
+    norm = mpl.colors.Normalize(vmin, vmax)
+    
+    # Plot response surface and CMIP5 projections
+    fig, ax = plt.subplots(figsize=(12, 8))
+    ax.contourf(X, Y, Z, cmap=drought_map, norm=norm)
+    ax.scatter(CMIP5_multipliers.iloc[:,7],
+               CMIP5_multipliers.iloc[:,12], 
+               c='#ffffb3', 
+               edgecolor='none', 
+               s=30)
+    cbar = ax.figure.colorbar(mpl.cm.ScalarMappable(norm=norm, cmap=drought_map), ax=ax)
+    ax.set_xlim(np.nanmin(X), np.nanmax(X))
+    ax.set_ylim(np.nanmin(Y), np.nanmax(Y))
+    ax.set_xlabel('Dry State Mu', fontsize=14)
+    ax.set_ylabel('Dry-Dry Transition Probability', fontsize=14)
+    ax.tick_params(axis='both', labelsize=14)
+    cbar.ax.set_ylabel('Decadal Drought Occurrence', rotation=-90, fontsize=14, labelpad=15)
+    cbar.ax.tick_params(axis='y',labelsize=14)
+    plt.show()
+    plt.close()
+    
+
+
+
+
+.. image:: _static/hmm_53_0.png
+
+
+We see the influence of the dry state mean and dry-dry transition
+parameters. We’re likely to see more decadal droughts when we (1)
+increase the dry-dry transition probability, which inherently will
+increase persistence of the dry state, and (2) when we make the dry
+state log mean drier. Note that the CMIP5 scenarios tend to span the
+extent of the dry mean sample space, but are less representative of the
+dry transition probability sample space, which suggests that the types
+of hydrological droughts represented in the projections tend to only be
+wetter to slightly drier than our baseline. Both methods of producing
+these scenarios are valid, though studies have suggested that
+globally-resolved GCMs may be inappropriate to represent regional
+extremes. Ultimately, if your goal is to produce a variety of ensembles
+that are characterized by many different drought characteristics, you
+will likely find that a generator approach will serve this purpose
+better.
+
+Tips to Create an HMM-Based Generator for your System
+-----------------------------------------------------
+
+In this tutorial, we demonstrated how to fit an HMM-based generator for
+a single gauge located in the Upper Colorado River Basin. In order to
+apply this methodology to your problem, you will need to first ask:
+
+(1) Is this model appropriate for my location of interest? We have
+    applied this style of generator to locations where persistent wet
+    and dry states are characteristic, which tends to be in the Western
+    US. Ultimately the best way to judge if an HMM is useful for your
+    application is to fit the model and explore the resulting
+    distributions. Are there two (or more) distinct states that emerge?
+    If not, then your location may not exhibit the type of persistence
+    that an HMM-based generator is useful for. You can consider
+    exploring other styles of generators such as the Kirsch-Nowak
+    generator (Kirsch et al., 2013).
+
+(2) Do I have the right datasets? We use annual data for our location of
+    interest. In this notebook, the HMM is fit to log annual flows.
+    Ultimately, it can be disaggregated to daily flows (using a
+    reference historical daily dataset) to be useful in water resources
+    operational applications. You could also disaggregate to a finer
+    resolution than daily if the historical dataset exists.
+
+If you meet these requirements, feel free to proceed through fitting the
+model using the code available in the notebook. Be sure to consider the
+appropirate number of samples to generate (both in a stationary and
+non-stationary case). Make sure that you test multiple sample sizes and
+continue to increase your sample size until you converge to a consistent
+representation of extremes. What is the appropriate number of LHS
+samples of the parameters to use? In this experiment we used 1,000
+samples of parameters due to extensive stability tests described in
+Quinn et al. (2020).
+
+Finally, to learn more about this test case refer to Hadmichael et
+al. (2020a) and Hadmichael et al. (2020b). For another study on
+synthetic drought generation to support vulnerability assessments in the
+Research Triangle region of North Carolina, please refer to Herman et
+al. (2016)
+
+References
+----------
+
+Ault, T. R., Cole, J. E., Overpeck, J. T., Pederson, G. T., & Meko, D.
+M. (2014). Assessing the risk of persistent drought using climate model
+simulations and paleoclimate data. Journal of Climate, 27(20),
+7529-7549.
+
+CWCB (2012).Colorado River Water Availability Study Phase I Report.
+Colorado Water Conservation Board
+
+Hadjimichael, A., Quinn, J., Wilson, E., Reed, P., Basdekas, L., Yates,
+D., & Garrison, M. (2020a). Defining robustness, vulnerabilities, and
+consequential scenarios for diverse stakeholder interests in
+institutionally complex river basins. Earth’s Future, 8(7),
+e2020EF001503.
+
+Hadjimichael, A., Quinn, J., & Reed, P. (2020). Advancing diagnostic
+model evaluation to better understand water shortage mechanisms in
+institutionally complex river basins. Water Resources Research, 56(10),
+e2020WR028079.
+
+Herman, J. D., Zeff, H. B., Lamontagne, J. R., Reed, P. M., &
+Characklis, G. W. (2016). Synthetic drought scenario generation to
+support bottom-up water supply vulnerability assessments. Journal of
+Water Resources Planning and Management, (11), 04016050.
+
+Kirsch, B. R., Characklis, G. W., & Zeff, H. B. (2013). Evaluating the
+impact of alternative hydro-climate scenarios on transfer agreements:
+Practical improvement for generating synthetic streamflows. Journal of
+Water Resources Planning and Management, 139(4), 396-406.
+
+Overpeck, J.T. & Udall, B. (2020) “Climate change and the aridification
+of North America.” Proceedings of the national academy of sciences
+117.22 11856-11858.
+
+Quinn, J. D., Hadjimichael, A., Reed,P. M., & Steinschneider, S. (2020).
+Canexploratory modeling of water scarcity vulnerabilities and robustness
+bescenario neutral?Earth’s Future,8,e2020EF001650.
+https://doi.org/10.1029/2020EF001650Received
+
+Williams, A. P., Cook, B. I., & Smerdon, J. E. (2022). Rapid
+intensification of the emerging southwestern North American megadrought
+in 2020–2021. Nature Climate Change, 12(3), 232-234.
diff --git a/dev/docs/html/_sources/A2_Jupyter_Notebooks.rst b/dev/docs/html/_sources/A2_Jupyter_Notebooks.rst
new file mode 100644
index 0000000..cfa77c0
--- /dev/null
+++ b/dev/docs/html/_sources/A2_Jupyter_Notebooks.rst
@@ -0,0 +1,12 @@
+.. _A2_Jupyter_Notebooks:
+
+**************************
+Jupyter Notebook Tutorials
+**************************
+
+.. include:: A2.1_fishgame.rst
+.. include:: A2.2_saltelli.rst
+.. include:: A2.3_logistic.rst
+.. include:: A2.4_hymod.rst
+.. include:: A2.5_discovery.rst
+.. include:: A2.6_hmm.rst
diff --git a/dev/docs/html/_sources/A3_plotting_code.rst b/dev/docs/html/_sources/A3_plotting_code.rst
new file mode 100644
index 0000000..f3b3499
--- /dev/null
+++ b/dev/docs/html/_sources/A3_plotting_code.rst
@@ -0,0 +1,549 @@
+Plotting Code Samples
+*********************
+
+hymod.ipynb
+---------------------------------------
+
+The following are the plotting functions as described in the ``hymod.ipynb`` Jupyter notebook tutorial.
+
+The following are the necessary package imports to run these functions:
+
+.. code-block:: python
+
+    import numpy as np
+    import seaborn as sns
+    import matplotlib.pyplot as plt
+
+    from matplotlib.lines import Line2D
+
+
+**plot_observed_vs_simulated_streamflow()**
+___________________________________________
+
+.. code-block:: python
+
+    def plot_observed_vs_simulated_streamflow(df, hymod_dict, figsize=[12, 6]):
+        """Plot observed versus simulated streamflow.
+
+        :param df:              Dataframe of hymod input data including columns for precip, potential evapotranspiration,
+                                and streamflow
+
+        :param hymod_dict:      A dictionary of hymod outputs
+        :type hymod_dict:       dict
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        # set plot style
+        plt.style.use('seaborn-white')
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # plot observed streamflow
+        ax.plot(range(0, len(df['Strmflw'])), df['Strmflw'], color='pink')
+
+        # plot simulated streamflow
+        ax.plot(range(0, len(df['Strmflw'])), hymod_dict['Q'], color='black')
+
+        # set axis labels
+        ax.set_ylabel('Streamflow($m^3/s$)')
+        ax.set_xlabel('Days')
+
+        # set plot title
+        plt.title('Observed vs. Simulated Streamflow')
+
+        return ax
+
+
+**plot_observed_vs_sensitivity_streamflow()**
+_____________________________________________
+
+.. code-block:: python
+
+    def plot_observed_vs_sensitivity_streamflow(df_obs, df_sim, figsize=[10, 4]):
+        """Plot observed streamflow versus simulations generated from sensitivity analysis.
+
+        :param df_obs:          Dataframe of mean monthly hymod input data including columns for precip,
+                                potential evapotranspiration, and streamflow
+
+        :param df_sim:          Dataframe of mean monthly simulation data from sensitivity analysis
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        month_list = range(len(df_sim))
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # set labels
+        ax.set_xlabel('Days')
+        ax.set_ylabel('Flow Discharge (m^3/s)')
+
+        # plots all simulated streamflow cases under different sample sets
+        for i in df_sim.columns:
+            plt.plot(month_list, df_sim[i], color="pink", alpha=0.2)
+
+        # plot observed streamflow
+        plt.plot(month_list, df_obs['Strmflw'], color="black")
+
+        plt.title('Observed vs. Sensitivity Analysis Outputs')
+
+        return ax
+
+
+**plot_monthly_heatmap()**
+__________________________
+
+.. code-block:: python
+
+    def plot_monthly_heatmap(arr_sim, df_obs, title='', figsize=[14, 6]):
+        """Plot a sensitivity metric overlain by observed flow.
+
+        :param arr_sim:         Numpy array of simulated metrics
+
+        :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis
+
+        :param title:           Title of plot
+        :type title:            str
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # plot heatmap
+        sns.heatmap(arr_sim,
+                    ax=ax,
+                    yticklabels=['Kq', 'Ks', 'Alp', 'Huz', 'B'],
+                    cmap=sns.color_palette("ch:s=-.2,r=.6"))
+
+        # setup overlay axis
+        ax2 = ax.twinx()
+
+        # plot line
+        ax2.plot(np.arange(0.5, 12.5), df_obs['Strmflw'], color='slateblue')
+
+        # plot points on line
+        ax2.plot(np.arange(0.5, 12.5), df_obs['Strmflw'], color='slateblue', marker='o')
+
+        # set axis limits and labels
+        ax.set_ylim(0, 5)
+        ax.set_xlim(0, 12)
+        ax.set_xticklabels(['jan', 'feb', 'mar', 'apr', 'may', 'jun', 'jul', 'aug', 'sep', 'oct', 'nov', 'dec'])
+        ax2.set_ylabel('Flow Discharge($m^3/s$)')
+
+        plt.title(title)
+
+        plt.show()
+
+        return ax, ax2
+
+
+**plot_annual_heatmap()**
+__________________________
+
+.. code-block:: python
+
+    def plot_annual_heatmap(arr_sim, df_obs, title='', figsize=[14,5]):
+        """Plot a sensitivity metric overlain by observed flow..
+
+        :param arr_sim:         Numpy array of simulated metrics
+
+        :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis
+
+        :param title:           Title of plot
+        :type title:            str
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # plot heatmap
+        sns.heatmap(arr_sim, ax=ax, cmap=sns.color_palette("YlOrBr"))
+
+        # setup overlay axis
+        ax2 = ax.twinx()
+
+        # plot line
+        ax2.plot(np.arange(0.5, 10.5), df_obs['Strmflw'], color='slateblue')
+
+        # plot points on line
+        ax2.plot(np.arange(0.5, 10.5), df_obs['Strmflw'], color='slateblue', marker='o')
+
+        # set up axis lables and limits
+        ax.set_ylim(0, 5)
+        ax.set_xlim(0, 10)
+        ax.set_yticklabels(['Kq', 'Ks', 'Alp', 'Huz', 'B'])
+        ax.set_xticklabels(range(2000, 2010))
+        ax2.set_ylabel('Flow Discharge($m^3/s$)')
+
+        plt.title(title)
+
+        return ax, ax2
+
+
+**plot_varying_heatmap()**
+___________________________
+
+.. code-block:: python
+
+    def plot_varying_heatmap(arr_sim, df_obs, title='', figsize=[14,5]):
+        """Plot a sensitivity metric overlain by observed flow..
+
+        :param arr_sim:         Numpy array of simulated metrics
+
+        :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis
+
+        :param title:           Title of plot
+        :type title:            str
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # plot heatmap
+        sns.heatmap(arr_sim,
+                    ax=ax,
+                    yticklabels=['Kq', 'Ks', 'Alp', 'Huz', 'B'],
+                    cmap=sns.light_palette("seagreen", as_cmap=True))
+
+        n_years = df_obs.shape[0]
+
+        # setup overlay axis
+        ax2 = ax.twinx()
+
+        # plot line
+        ax2.plot(range(0, n_years), df_obs['Strmflw'], color='slateblue')
+
+        # plot points on line
+        ax2.plot(range(0, n_years), df_obs['Strmflw'], color='slateblue', marker='o')
+
+        # set up axis lables and limits
+        ax.set_ylim(0, 5)
+        ax.set_xlim(-0.5, 119.5)
+        ax2.set_ylabel('Flow Discharge')
+        ax.set_xlabel('Number of Months')
+
+        plt.title(title)
+
+        return ax, ax2
+
+
+**plot_precalibration_flow()**
+_______________________________
+
+.. code-block:: python
+
+    def plot_precalibration_flow(df_sim, df_obs, figsize=[10, 4]):
+        """Plot flow discharge provided by the ensemble of parameters sets from Pre-Calibration versus the observed
+        flow data.
+
+        :param df_sim:          Dataframe of simulated metrics
+
+        :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # set axis labels
+        ax.set_xlabel('Days')
+        ax.set_ylabel('Flow Discharge')
+
+        # plot pre-calibration results
+        for i in range(df_sim.shape[1]):
+            plt.plot(range(len(df_sim)), df_sim.iloc[:, i],  color="lightgreen", alpha=0.2)
+
+        # plot observed
+        plt.plot(range(len(df_sim)), df_obs['Strmflw'],  color="black")
+
+        plt.title('Observed vs. Pre-Calibration Outputs')
+
+        # customize legend
+        custom_lines = [Line2D([0], [0],  color="lightgreen", lw=4),
+                        Line2D([0], [0], color="black", lw=4)]
+        plt.legend(custom_lines, ['Pre-Calibration', 'Observed'])
+
+        return ax
+
+
+**plot_precalibration_glue()**
+_______________________________
+
+.. code-block:: python
+
+    def plot_precalibration_glue(df_precal, df_glue, df_obs, figsize=[10, 4]):
+        """Plot flow discharge provided by the ensemble of parameters sets from Pre-Calibration versus the observed
+        flow data.
+
+        :param df_sim:          Dataframe of simulated metrics
+
+        :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # set axis labels
+        ax.set_xlabel('Days')
+        ax.set_ylabel('Flow Discharge')
+
+        # plot pre-calibration results
+        for i in range(df_precal.shape[1]):
+            plt.plot(range(len(df_precal)), df_precal.iloc[:, i],  color="lightgreen", alpha=0.2)
+
+        # plot glue
+        for i in range(df_glue.shape[1]):
+            plt.plot(range(len(df_glue)), df_glue.iloc[:, i], color="lightblue", alpha=0.2)
+
+        # plot observed
+        plt.plot(range(len(df_precal)), df_obs['Strmflw'],  color="black")
+
+        plt.title('Observed vs. Sensitivity Analysis Outputs across GLUE/Pre-Calibration')
+
+        # customize legend
+        custom_lines = [Line2D([0], [0],  color="lightgreen", lw=4),
+                        Line2D([0], [0], color="lightblue", lw=4),
+                        Line2D([0], [0], color="black", lw=4)]
+        plt.legend(custom_lines, ['Pre-Calibration', 'GLUE', 'Observed'])
+
+        return ax
+
+
+fishery_dynamics.ipynb
+---------------------------------------
+
+The following are the plotting functions as described in the ``fishery_dynamics.ipynb`` Jupyter notebook tutorial.
+
+The following are the necessary package imports to run these functions:
+
+.. code-block:: python
+
+    import numpy as np
+    import matplotlib.pyplot as plt
+
+    from matplotlib import patheffects as pe
+
+
+**plot_objective_performance()**
+_________________________________
+
+.. code-block:: python
+
+    def plot_objective_performance(objective_performance, profit_solution, robust_solution, figsize=(18, 9)):
+        """Plot the identified solutions with regards to their objective performance
+        in a parallel axis plot
+
+        :param objective_performance:               Objective performance array
+        :param profit_solution:                     Profitable solutions array
+        :param robust_solution:                     Robust solutions array
+        :param figsize:                             Figure size
+        :type figsize:                              tuple
+
+        """
+
+        # create the figure object
+        fig = plt.figure(figsize=figsize)
+
+        # set up subplot axis object
+        ax = fig.add_subplot(1, 1, 1)
+
+        # labels where constraint is always 0
+        objs_labels = ['Net present\nvalue (NPV)',
+                       'Prey population deficit',
+                       'Longest duration\nof low harvest',
+                       'Worst harvest instance',
+                       'Variance of harvest',
+                       'Duration of predator\npopulation collapse']
+
+        # normalization across objectives
+        mins = objective_performance.min(axis=0)
+        maxs = objective_performance.max(axis=0)
+        norm_reference = objective_performance.copy()
+
+        for i in range(5):
+            mm = objective_performance[:, i].min()
+            mx = objective_performance[:, i].max()
+            if mm != mx:
+                norm_reference[:, i] = (objective_performance[:, i] - mm) / (mx - mm)
+            else:
+                norm_reference[:, i] = 1
+
+        # colormap from matplotlib
+        cmap = plt.cm.get_cmap("Blues")
+
+        # plot all solutions
+        for i in range(len(norm_reference[:, 0])):
+            ys = np.append(norm_reference[i, :], 1.0)
+            xs = range(len(ys))
+            ax.plot(xs, ys, c=cmap(ys[0]), linewidth=2)
+
+        # to highlight robust solutions
+        ys = np.append(norm_reference[profit_solution, :], 1.0)  # Most profitable
+        xs = range(len(ys))
+        l1 = ax.plot(xs[0:6],
+                     ys[0:6],
+                     c=cmap(ys[0]),
+                     linewidth=3,
+                     label='Most robust in NPV',
+                     path_effects=[pe.Stroke(linewidth=6, foreground='darkgoldenrod'), pe.Normal()])
+
+        ys = np.append(norm_reference[robust_solution, :], 1.0)  # Most robust in all criteria
+        xs = range(len(ys))
+        l2 = ax.plot(xs[0:6],
+                     ys[0:6],
+                     c=cmap(ys[0]),
+                     linewidth=3,
+                     label='Most robust across criteria',
+                     path_effects=[pe.Stroke(linewidth=6, foreground='gold'), pe.Normal()])
+
+        # build colorbar
+        sm = plt.cm.ScalarMappable(cmap=cmap)
+        sm.set_array([objective_performance[:, 0].min(), objective_performance[:, 0].max()])
+        cbar = fig.colorbar(sm)
+        cbar.ax.set_ylabel("\nNet present value (NPV)")
+
+        # tick values
+        minvalues = ["{0:.3f}".format(mins[0]),
+                     "{0:.3f}".format(-mins[1]),
+                     str(-mins[2]),
+                     "{0:.3f}".format(-mins[3]),
+                     "{0:.2f}".format(-mins[4]),
+                     str(0)]
+
+        maxvalues = ["{0:.2f}".format(maxs[0]),
+                     "{0:.3f}".format(-maxs[1]),
+                     str(-maxs[2]),
+                     "{0:.2f}".format(maxs[3]),
+                     "{0:.2f}".format(-maxs[4]),
+                     str(0)]
+
+        ax.set_ylabel("Preference ->", size=12)
+        ax.set_yticks([])
+        ax.set_xticks([0, 1, 2, 3, 4, 5])
+        ax.set_xticklabels([minvalues[i] + '\n' + objs_labels[i] for i in range(len(objs_labels))])
+
+        # make a twin axis for toplabels
+        ax1 = ax.twiny()
+        ax1.set_yticks([])
+        ax1.set_xticks([0, 1, 2, 3, 4, 5])
+        ax1.set_xticklabels([maxvalues[i] for i in range(len(maxs) + 1)])
+
+        return ax, ax1
+
+
+**plot_factor_performance()**
+_________________________________
+
+.. code-block:: python
+
+    def plot_factor_performance(param_values, collapse_days, b, m, a):
+        """Visualize the performance of our policies in three-dimensional
+        parametric space.
+
+        :param param_values:                Saltelli sample array
+        :param collapse_days:               Simulation array
+        :param b:                           b parameter boundary interval
+        :param m:                           m parameter boundary interval
+        :param a:                           a parameter boundary interval
+
+        """
+
+        # set colormap
+        cmap = plt.cm.get_cmap("RdBu_r")
+
+        # build figure object
+        fig = plt.figure(figsize=plt.figaspect(0.5), dpi=600, constrained_layout=True)
+
+        # set up scalable colormap
+        sm = plt.cm.ScalarMappable(cmap=cmap)
+
+        # set up subplot for profit maximizing policy
+        ax1 = fig.add_subplot(1, 2, 1, projection='3d')
+
+        # add point data for profit plot
+        sows = ax1.scatter(param_values[:,1],
+                           param_values[:,6],
+                           param_values[:,0],
+                           c=collapse_days[:,0],
+                           cmap=cmap,
+                           s=0.5)
+
+        # add surface data for boundary separating successful and failed states of the world
+        pts_ineq = ax1.plot_surface(b, m, a, color='black', alpha=0.25, zorder=1)
+
+        # add reference point to plot
+        pt_ref = ax1.scatter(0.5, 0.7, 0.005, c='black', s=50, zorder=0)
+
+        # set up plot aesthetics and labels
+        ax1.set_xlabel("b")
+        ax1.set_ylabel("m")
+        ax1.set_zlabel("a")
+        ax1.set_zlim([0.0, 2.0])
+        ax1.set_xlim([0.0, 1.0])
+        ax1.set_ylim([0.0, 1.5])
+        ax1.xaxis.set_view_interval(0,  0.5)
+        ax1.set_facecolor('white')
+        ax1.view_init(12, -17)
+        ax1.set_title('Profit maximizing policy')
+
+        # set up subplot for robust policy
+        ax2 = fig.add_subplot(1, 2, 2, projection='3d')
+
+        # add point data for robust plot
+        sows = ax2.scatter(param_values[:,1],
+                           param_values[:,6],
+                           param_values[:,0],
+                           c=collapse_days[:,1],
+                           cmap=cmap,
+                           s=0.5)
+
+        # add surface data for boundary separating successful and failed states of the world
+        pts_ineq = ax2.plot_surface(b, m, a, color='black', alpha=0.25, zorder=1)
+
+        # add reference point to plot
+        pt_ref = ax2.scatter(0.5, 0.7, 0.005, c='black', s=50, zorder=0)
+
+        # set up plot aesthetics and labels
+        ax2.set_xlabel("b")
+        ax2.set_ylabel("m")
+        ax2.set_zlabel("a")
+        ax2.set_zlim([0.0, 2.0])
+        ax2.set_xlim([0.0, 1.0])
+        ax2.set_ylim([0.0, 1.5])
+        ax2.xaxis.set_view_interval(0, 0.5)
+        ax2.set_facecolor('white')
+        ax2.view_init(12, -17)
+        ax2.set_title('Robust policy')
+
+        # set up colorbar
+        sm.set_array([collapse_days.min(), collapse_days.max()])
+        cbar = fig.colorbar(sm)
+        cbar.set_label('Days with predator collapse')
+
+        return ax1, ax2
diff --git a/dev/docs/html/_sources/R.Bibliography.rst b/dev/docs/html/_sources/R.Bibliography.rst
new file mode 100644
index 0000000..60cc211
--- /dev/null
+++ b/dev/docs/html/_sources/R.Bibliography.rst
@@ -0,0 +1,6 @@
+============
+Bibliography
+============
+
+.. bibliography::
+    :style: unsrt
diff --git a/dev/docs/html/_sources/acknowledgements.rst b/dev/docs/html/_sources/acknowledgements.rst
new file mode 100644
index 0000000..1f34009
--- /dev/null
+++ b/dev/docs/html/_sources/acknowledgements.rst
@@ -0,0 +1,25 @@
+
+Acknowledgements
+################
+
+This e-book was developed by the `Integrated Multisector, Multiscale Modeling (IM3) project <https://im3.pnnl.gov>`_, supported by the `U.S. Department of Energy <https://www.energy.gov/>`_, `Office of Science <https://www.energy.gov/science/office-science>`_, as part of research in the `MultiSector Dynamics <https://climatemodeling.science.energy.gov/program/multisector-dynamics>`_, Earth and Environmental System Modeling Program.
+
+The authors would like to thank Casey Burleyson (PNNL) and Isaac Thompson (PNNL) for their insightful feedback on drafts of the text and tutorials.
+
+Open source under license `CC BY-NC-ND-4.0 <https://creativecommons.org/licenses/by-nc-nd/4.0/>`_.
+
+Portions of cover image by `V.Cid7413 <https://commons.wikimedia.org/wiki/File:The_Amazing_Earth.jpg>`_, distributed under a `CC-BY-4.0 <https://creativecommons.org/licenses/by/4.0>`_ license.
+
+Open Source Disclaimer:
+
+    This material was prepared as an account of work sponsored by an agency of the United States Government.  Neither the United States Government nor the United States Department of Energy, nor Battelle, nor any of their employees, nor any jurisdiction or organization that has cooperated in the development of these materials, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness or any information, apparatus, product, software, or process disclosed, or represents that its use would not infringe privately owned rights.
+    Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or Battelle Memorial Institute. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
+
+| PACIFIC NORTHWEST NATIONAL LABORATORY
+| operated by
+| BATTELLE
+| for the
+| UNITED STATES DEPARTMENT OF ENERGY
+| under Contract DE-AC05-76RL01830
+
+Copyright (c) 2022-present; Battelle Memorial Institute
diff --git a/dev/docs/html/_sources/citation.rst b/dev/docs/html/_sources/citation.rst
new file mode 100644
index 0000000..63ee0ef
--- /dev/null
+++ b/dev/docs/html/_sources/citation.rst
@@ -0,0 +1,29 @@
+Suggested Citation
+###################
+
+    Reed, P.M., Hadjimichael, A., Malek, K., Karimi, T., Vernon, C.R., Srikrishnan, V., Gupta, R.S., Gold, D.F., Lee, B., Keller, K., Thurber, T.B, & Rice, J.S. (2022). Addressing Uncertainty in Multisector Dynamics Research [Book]. Zenodo. https://doi.org/10.5281/zenodo.6110623
+
+.. only:: html
+
+    .. raw:: html
+
+        <div style="display:flex;flex-direction:row;align-items:center;justify-content:start;margin:1em 0;">
+            <a
+                class="btn btn-light"
+                style="margin-right:1em;"
+                role="button"
+                href="data:text/plain;base64,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"
+                download="uc-ebook.bib"
+            >
+                <i style="margin-right:0.5em;" class="fas fa-download"></i>Download citation (.bib)
+            </a>
+            <a
+                class="btn btn-light"
+                style="margin-right:1em;"
+                role="button"
+                href="data:text/plain;base64,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"
+                download="uc-ebook.ris"
+            >
+                <i style="margin-right:0.5em;" class="fas fa-download"></i>Download citation (.ris)
+            </a>
+        </div>
\ No newline at end of file
diff --git a/dev/docs/html/_sources/code_of_conduct.rst b/dev/docs/html/_sources/code_of_conduct.rst
new file mode 100644
index 0000000..56672b8
--- /dev/null
+++ b/dev/docs/html/_sources/code_of_conduct.rst
@@ -0,0 +1,58 @@
+Code of Conduct
+===============
+
+Our Pledge
+----------
+
+We as members, contributors, and leaders pledge to make participation in our
+community a harassment-free experience for everyone, regardless of age, body
+size, visible or invisible disability, ethnicity, sex characteristics, gender
+identity and expression, level of experience, education, socio-economic status,
+nationality, personal appearance, race, religion, or sexual identity and
+orientation.
+
+We pledge to act and interact in ways that contribute to an open, welcoming,
+diverse, inclusive, and healthy community.
+
+Our Standards
+-------------
+
+Examples of behavior that contributes to a positive environment for our
+community include:
+
+- Demonstrating empathy and kindness toward other people
+- Being respectful of differing opinions, viewpoints, and experiences
+- Giving and gracefully accepting constructive feedback
+- Accepting responsibility and apologizing to those affected by our mistakes,
+  and learning from the experience
+- Focusing on what is best not just for us as individuals, but for the overall
+  community
+
+Examples of unacceptable behavior include:
+
+- The use of sexualized language or imagery, and sexual attention or
+  advances of any kind
+- Trolling, insulting or derogatory comments, and personal or political attacks
+- Public or private harassment
+- Publishing others' private information, such as a physical or email
+  address, without their explicit permission
+- Other conduct which could reasonably be considered inappropriate in a
+  professional setting
+
+Scope
+-----
+
+This Code of Conduct applies within all community spaces, and also applies
+when an individual is officially representing the community in public spaces.
+Examples of representing our community include using an official e-mail
+address, posting via an official social media account, or acting as an appointed
+representative at an online or offline event.
+
+Attribution
+-----------
+
+This Code of Conduct is adapted from the Contributor Covenant, version 2.0,
+available at https://www.contributor-covenant.org/version/2/0/code_of_conduct.html.
+
+For answers to common questions about this code of conduct, see the FAQ at
+https://www.contributor-covenant.org/faq. Translations are available at https://www.contributor-covenant.org/translations.
diff --git a/dev/docs/html/_sources/contributing.rst b/dev/docs/html/_sources/contributing.rst
new file mode 100644
index 0000000..e1e1227
--- /dev/null
+++ b/dev/docs/html/_sources/contributing.rst
@@ -0,0 +1,18 @@
+******************
+Contribution Guide
+******************
+
+Our eBook is a living product that we hope continues to grow over time to stay relevant with methodological and technological advancements in the field of uncertainty quantification and MultiSector Dynamics at large. We extend an invitation to you the reader to contribute to one of our interactive Jupyter notebook tutorials. If you feel you have a contribution that would be relevant and generalizable to the MSD community, please submit a proposal idea `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/issues/new?assignees=thurber%2C+crvernon&labels=triage&projects=&template=contribution_proposal.yml&title=Contribution+Proposal>`_. Your proposal will be reviewing by our team and feedback and/or a decision will be provided shortly. Any contribution accepted will receive it's own DOI and citation so that you may provide an independent reference to your work as you see fit.
+
+Please consider the following requirements for contribution:
+
+- All elements of your contribution MUST be fully open-source and can be distributed with an `Open Source Initiative approved license <https://opensource.org/licenses/>`_ with an understanding that they may be used in community demonstrations and other activities. Author citations will be present in the notebook for any contributed work to ensure the author(s) receive full credit for their contribution.
+- Any data or code reused in your submission must be correctly cited giving credit to the original authors. 
+- The notebook provided must be written in English and able to be a stand-alone product that needs no further explanation past what is written in the notebook to make use of it.
+- The provided work is not merely a regurgitation of an existing tutorial or demonstration but represents a novel contribution.
+- All contributions and communication thereof must abide by our `code of conduct <https://uc-ebook.org/docs/html/code_of_conduct.html>`_.
+
+
+If you feel your work meets the criteria above, please submit a proposal issue `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/issues/new?assignees=thurber%2C+crvernon&labels=triage&projects=&template=contribution_proposal.yml&title=Contribution+Proposal>`_. If your proposal is approved, please create a pull request with the submission `template <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/.github/PULL_REQUEST_TEMPLATE/contribution_checklist.md>`_ copied into the pull request description and filled out. We will then review your pull request and provide feedback. Once your contribution has been deemed ready to deploy, we will generate a DOI for your work, launch it to our MSD-LIVE set of interactive notebooks, and feature the contribution in the index of our eBook.
+
+Please feel free to reach out with any further questions.
diff --git a/dev/docs/html/_sources/examples.rst b/dev/docs/html/_sources/examples.rst
new file mode 100644
index 0000000..ca7125a
--- /dev/null
+++ b/dev/docs/html/_sources/examples.rst
@@ -0,0 +1,44 @@
+========
+Examples
+========
+
+
+Using mathjax
+===================
+
+The **first-order sensitivity index** indicates the percent of model output variance contributed by a factor individually (i.e., the effect of varying *x*\ :sub:`i`\ alone) and is obtained using the following (Saltelli, 2002a; Sobol, 2001):
+
+.. math::
+
+  S_i^1 = \frac{V_{x_i} [E_{x\sim_i} (x_i)]}{V(y)}
+
+with *E* and *V* denoting the expected value and the variance, respectively.
+
+Building a codeblock
+=======================
+
+.. code-block:: python
+  :linenos:
+
+   import ebook
+
+   ebook.plot_experimental_design()
+
+Testing out a note
+==================
+
+.. note::
+  Keep track of the latest in programming solutions on the `Water Programming <https://waterprogramming.wordpress.com/>`_ blog!
+
+
+Insert a figure with a caption
+==============================
+
+Be sure to add your figure into 'msd_uncertainty_ebook/docs/source/_static'
+
+  .. figure:: _static/im3.png
+      :alt: IM3 logo
+      :width: 100px
+      :align: center
+
+      This is my caption.
diff --git a/dev/docs/html/_sources/index.rst b/dev/docs/html/_sources/index.rst
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--- /dev/null
+++ b/dev/docs/html/_sources/index.rst
@@ -0,0 +1,134 @@
+.. msd_uncertainty_ebook documentation master file, created by
+   sphinx-quickstart on Wed May 26 22:27:12 2021.
+   You can adapt this file completely to your liking, but it should at least
+   contain the root `toctree` directive.
+
+
+=======================================================
+Addressing Uncertainty in MultiSector Dynamics Research
+=======================================================
+
+.. only:: html
+
+    .. epigraph::
+
+        Patrick M. Reed, Antonia Hadjimichael, Keyvan Malek,
+        Tina Karimi, Chris R. Vernon, Vivek Srikrishnan, Rohini S. Gupta,
+        David F. Gold, Ben Lee, Klaus Keller, Travis B. Thurber, Jennie S. Rice
+
+    A practical guide to sensitivity analysis and diagnostic model evaluation techniques for confronting the computational and conceptual challenges of multi-model, transdisciplinary workflows.
+
+    **Topics in the Book**
+
+    * `Preface <preface.html>`_
+    * `Introduction <1_introduction.html>`_
+    * `An overview of diagnostic modeling and perspectives on model evaluation <2_diagnostic_modeling_overview_and_perspectives.html>`_
+    * `A framework for the basic methods and concepts used in sensitivity analysis <3_sensitivity_analysis_the_basics.html>`_
+    * `Technical applications supporting diagnostic model evaluation and exploration <4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html>`_
+
+    **Interactive Tutorials**
+
+    * `Factor Discovery <nb_fishery_dynamics_>`_
+    * `Model Calibration <nb_hymod_>`_
+    * `Sobol Sensitivity Analysis <nb_saltelli_sobol_>`_
+    * `Factor Mapping using Logistic Regression <nb_logistic_regression_>`_
+    * `Time-evolving scenario discovery for infrastructure pathways <nb_discovery_>`_
+    * `A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios <nb_hmm_>`_
+
+    .. tip::
+
+        .. raw:: html
+
+            <p>
+                Use the sidebar on the left to quickly navigate the eBook!
+                <br/>
+                Click or tap the <i style="padding: 0 0.25rem;" class="fas fa-bars"></i> icon to show and hide the sidebar.
+            </p>
+
+    .. admonition:: Info
+
+        .. raw:: html
+
+            <p>
+                Report a typo or just pass along something you like about the book by <a target="_blank" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/issues/new?assignees=thurber%2C+crvernon&labels=documentation%2C+triage&template=custom.md&title=Publication+Feedback">opening an issue</a> on GitHub!
+                <br/>
+                Click or tap the <i style="padding: 0 0.25rem;" class="fab fa-github"></i> icon to find this link again.
+            </p>
+
+.. raw:: latex
+
+    \frontmatter
+    \sphinxmaketitle
+
+.. toctree::
+    :hidden:
+    :glob:
+
+    preface
+    citation
+    acknowledgements
+    code_of_conduct
+    contributing
+
+.. raw:: latex
+
+    \sphinxtableofcontents
+
+.. raw:: latex
+
+    \mainmatter
+
+.. toctree::
+    :hidden:
+    :includehidden:
+    :numbered: 4
+    :maxdepth: 4
+    :caption: Contents
+    :name: mastertoc
+    :glob:
+
+    1_introduction
+    2_diagnostic_modeling_overview_and_perspectives
+    3_sensitivity_analysis_the_basics
+    4_sensitivity_analysis_diagnostic_and_exploratory_modeling
+    5_conclusion
+
+.. raw:: latex
+
+    \appendix
+
+.. appendix::
+    :hidden:
+    :numbered: 4
+    :maxdepth: 4
+    :caption: Appendices
+    :glob:
+
+    A1_Uncertainty_Quantification
+    A2_Jupyter_Notebooks
+    A3_plotting_code
+
+.. raw:: latex
+
+    \backmatter
+
+.. toctree::
+    :hidden:
+    :maxdepth: 1
+    :caption: Glossary
+    :glob:
+
+    6_glossary
+
+
+.. raw:: latex
+
+    \backmatter
+
+.. toctree::
+    :hidden:
+    :maxdepth: 1
+    :caption: References
+    :glob:
+
+    R.Bibliography
diff --git a/dev/docs/html/_sources/modules.rst b/dev/docs/html/_sources/modules.rst
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--- /dev/null
+++ b/dev/docs/html/_sources/modules.rst
@@ -0,0 +1,6 @@
+nanites
+=======
+
+.. toctree::
+   :maxdepth: 4
+
diff --git a/dev/docs/html/_sources/preface.rst b/dev/docs/html/_sources/preface.rst
new file mode 100644
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--- /dev/null
+++ b/dev/docs/html/_sources/preface.rst
@@ -0,0 +1,34 @@
+
+.. _preface:
+
+*******
+Preface
+*******
+
+This online book is meant to provide an open science “living” resource on uncertainty characterization methods for the MultiSector Dynamics (MSD) community and other technical communities confronting sustainability, climate, and energy transition challenges. The last decade has seen rapid growth in science efforts seeking to address the interconnected nature of these challenges across scales, sectors, and systems. Accompanying these advances is the growing realization that the deep integration of research from many disciplinary fields is non-trivial and raises important questions. How and why models are developed seems to have an obvious answer (“to gain understanding”). But what does it actually mean to gain understanding? What if a small change in a model or its data fundamentally changes our perceptions of what we thought we understood? What controls the outcomes of our model(s)? How do we understand the implications of model coupling, such as when one model is on the receiving end of several other models that are considered “input data”? 
+
+The often quoted “All models are wrong, but some are useful.” (George Box) is a bit of a conflation trap, often used to excuse known weaknesses in complex models as just an unavoidable outcome of being a modeler. In fact, the quote actually refers to a specific class of small-scale statistical models within an application context that assures a much higher degree of understanding and data quality control than is typical for the coupled human-natural systems applications in the MSD area. Moreover, Box was actually warning readers to avoid overparameterization and emphasizing the need to better understand what underlying factors cause your model to be wrong :cite:p:`box_1976`. 
+
+So, in short, there is a tension when attaining better performance by means of increasing the complexity of a model or model-based workflow. Box highlights that a modeler requires a clear diagnostic understanding of this performance-complexity tradeoff. If we move from small-scale models simulating readily-observed phenomena to the MSD context, things get quite a bit more complicated. How can we provide robust insights for unseen futures that emerge across a myriad of human and natural systems? Sometimes even asking, “what is a model?” or “what is data?” is complicated (e.g., data assimilated weather products, satellite-based signals translated through retrieval algorithms, demographic changes, resource demands, etc.). This MSD guidance text seeks to help readers navigate these challenges. It is meant to serve as an evolving resource that helps the MSD community learn how to better address uncertainty while working with complex chains of models bridging sectors, scales, and systems. It is not intended to be an exhaustive resource, but instead should be seen as a guided tour through state-of-the-science methods in uncertainty characterization, including global sensitivity analysis and exploratory modeling, to provide insights into complex human-natural systems interactions. 
+
+To aid readers in navigating the text, the key goals for each chapter are summarized below.
+
+:numref:`introduction` uses the `Integrated Multisector Multiscale Modeling <https://im3.pnnl.gov>`_ project as a living lab to encapsulate the challenges that emerge in bridging disciplines to make consequential model-based insights while acknowledging the tremendous array of uncertainties that shape them.
+
+:numref:`2_diagnostic_modeling` helps the reader to better understand the importance of using diagnostic modeling to interrogate why uncertain model behaviors may emerge. The chapter also aids readers to better understand the diverse disciplinary perspectives that exist on how best to pursue consequential model-based discoveries.
+
+:numref:`3_sensitivity_analysis_the_basics` is a technical tools-focused primer for readers on the key elements of uncertainty characterization that includes ensemble-based design of experiments, quantitative methods for computing global sensitivities, and a summary of existing software packages.
+
+:numref:`4_sensitivity_analysis` narrates for readers how and why the tools from the previous chapter can be applied in a range of tasks from diagnosing model performance to formal exploratory modeling methods for making consequential model-based discoveries.
+
+The supplemental appendices provided in the text are also important resources for readers. They provide a glossary to help bridge terminology challenges, a brief summary of uncertainty quantification tools for more advanced readers, and a suite of Jupyter notebook tutorials that provide hands-on training tied to the contents of :numref:`3_sensitivity_analysis_the_basics` and :numref:`4_sensitivity_analysis`.
+
+This text was written with a number of different audiences in mind. 
+
+Technical experts in uncertainty may find this to be a helpful and unique resource bridging a number of perspectives that have not been combined in prior books (e.g., formal model diagnostics, global sensitivity analysis, and exploratory modeling under deep uncertainty). 
+
+Readers from different sector-specific and disciplinary-specific backgrounds can use this text to better understand potential differences and opportunities in how to make model-based insights.
+
+Academic or junior researchers can utilize this freely available text for training and teaching resources that include hands-on coding experiences.
+
+This text itself represents our strong commitment to open science and will evolve as a living resource as the communities of researchers provide feedback, innovations, and future tools.
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+ *
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+  "INPUT",
+  "SELECT",
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+const _ready = (callback) => {
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+    Documentation.LOCALE = catalog.locale;
+  },
+
+  /**
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+    document.querySelectorAll("input[name=q]")[0]?.focus();
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+    document.addEventListener("keydown", (event) => {
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+      if (!event.shiftKey) {
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+            const prevLink = document.querySelector('link[rel="prev"]');
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+              window.location.href = prevLink.href;
+              event.preventDefault();
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+            break;
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+
+            const nextLink = document.querySelector('link[rel="next"]');
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+              window.location.href = nextLink.href;
+              event.preventDefault();
+            }
+            break;
+        }
+      }
+
+      // some keyboard layouts may need Shift to get /
+      switch (event.key) {
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+          if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break;
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+_ready(Documentation.init);
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diff --git a/dev/docs/html/_static/im3.png b/dev/docs/html/_static/im3.png
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diff --git a/dev/docs/html/_static/images/logo_binder.svg b/dev/docs/html/_static/images/logo_binder.svg
new file mode 100644
index 0000000..45fecf7
--- /dev/null
+++ b/dev/docs/html/_static/images/logo_binder.svg
@@ -0,0 +1,19 @@
+<?xml version="1.0" encoding="utf-8"?>
+<!-- Generator: Adobe Illustrator 23.0.1, SVG Export Plug-In . SVG Version: 6.00 Build 0)  -->
+<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px"
+	 viewBox="0 0 44.4 44.4" style="enable-background:new 0 0 44.4 44.4;" xml:space="preserve">
+<style type="text/css">
+	.st0{fill:none;stroke:#F5A252;stroke-width:5;stroke-miterlimit:10;}
+	.st1{fill:none;stroke:#579ACA;stroke-width:5;stroke-miterlimit:10;}
+	.st2{fill:none;stroke:#E66581;stroke-width:5;stroke-miterlimit:10;}
+</style>
+<title>logo</title>
+<g>
+	<path class="st0" d="M33.9,6.4c3.6,3.9,3.4,9.9-0.5,13.5s-9.9,3.4-13.5-0.5s-3.4-9.9,0.5-13.5l0,0C24.2,2.4,30.2,2.6,33.9,6.4z"/>
+	<path class="st1" d="M35.1,27.3c2.6,4.6,1.1,10.4-3.5,13c-4.6,2.6-10.4,1.1-13-3.5s-1.1-10.4,3.5-13l0,0
+		C26.6,21.2,32.4,22.7,35.1,27.3z"/>
+	<path class="st2" d="M25.9,17.8c2.6,4.6,1.1,10.4-3.5,13s-10.4,1.1-13-3.5s-1.1-10.4,3.5-13l0,0C17.5,11.7,23.3,13.2,25.9,17.8z"/>
+	<path class="st1" d="M19.2,26.4c3.1-4.3,9.1-5.2,13.3-2.1c1.1,0.8,2,1.8,2.7,3"/>
+	<path class="st0" d="M19.9,19.4c-3.6-3.9-3.4-9.9,0.5-13.5s9.9-3.4,13.5,0.5"/>
+</g>
+</svg>
diff --git a/dev/docs/html/_static/images/logo_colab.png b/dev/docs/html/_static/images/logo_colab.png
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diff --git a/dev/docs/html/_static/images/logo_deepnote.svg b/dev/docs/html/_static/images/logo_deepnote.svg
new file mode 100644
index 0000000..fa77ebf
--- /dev/null
+++ b/dev/docs/html/_static/images/logo_deepnote.svg
@@ -0,0 +1 @@
+<svg viewBox="0 0 128 128" fill="none" xmlns="http://www.w3.org/2000/svg"><path fill-rule="evenodd" clip-rule="evenodd" d="M0 128h52.512l29.539-11.077-11.077-43.487-34.051 3.693L0 128Z" fill="#0076D4"/><path fill-rule="evenodd" clip-rule="evenodd" d="M52.513 128s16.6-8.759 19.673-24.277c3.072-15.517-12.091-26.594-35.263-26.594 0-.41 20.343-28.718 20.343-28.718l49.4 1.435L95.71 107.7l-20.452 15.978L52.513 128Z" fill="#002868"/><path fill-rule="evenodd" clip-rule="evenodd" d="M0 60.718 41.025.001s1.006.01 3.282 0c16.082-.068 81.23 3.12 81.23 60.368 0 65.352-73.025 67.631-73.025 67.631s30.495-5.839 30.495-34.816c0-28.978-27.541-32.466-45.264-32.466H0Z" fill="#00A9FF"/></svg>
diff --git a/dev/docs/html/_static/images/logo_jupyterhub.svg b/dev/docs/html/_static/images/logo_jupyterhub.svg
new file mode 100644
index 0000000..60cfe9f
--- /dev/null
+++ b/dev/docs/html/_static/images/logo_jupyterhub.svg
@@ -0,0 +1 @@
+<svg id="Layer_1" data-name="Layer 1" xmlns="http://www.w3.org/2000/svg" width="38.73" height="50" viewBox="0 0 38.73 50"><defs><style>.cls-1{fill:#767677;}.cls-2{fill:#f37726;}.cls-3{fill:#9e9e9e;}.cls-4{fill:#616262;}.cls-5{font-size:17.07px;fill:#fff;font-family:Roboto-Regular, Roboto;}</style></defs><title>logo_jupyterhub</title><g id="Canvas"><path id="path7_fill" data-name="path7 fill" class="cls-1" d="M39.51,3.53a3,3,0,0,1-1.7,2.9A3,3,0,0,1,34.48,6a3,3,0,0,1-.82-3.26,3,3,0,0,1,1.05-1.41A3,3,0,0,1,37.52.86a2.88,2.88,0,0,1,1,.6,3,3,0,0,1,.7.93,3.18,3.18,0,0,1,.28,1.14Z" transform="translate(-1.87 -0.69)"/><path id="path8_fill" data-name="path8 fill" class="cls-2" d="M21.91,38.39c-8,0-15.06-2.87-18.7-7.12a19.93,19.93,0,0,0,37.39,0C37,35.52,30,38.39,21.91,38.39Z" transform="translate(-1.87 -0.69)"/><path id="path9_fill" data-name="path9 fill" class="cls-2" d="M21.91,10.78c8,0,15.05,2.87,18.69,7.12a19.93,19.93,0,0,0-37.39,0C6.85,13.64,13.86,10.78,21.91,10.78Z" transform="translate(-1.87 -0.69)"/><path id="path10_fill" data-name="path10 fill" class="cls-3" d="M10.88,46.66a3.86,3.86,0,0,1-.52,2.15,3.81,3.81,0,0,1-1.62,1.51,3.93,3.93,0,0,1-2.19.34,3.79,3.79,0,0,1-2-.94,3.73,3.73,0,0,1-1.14-1.9,3.79,3.79,0,0,1,.1-2.21,3.86,3.86,0,0,1,1.33-1.78,3.92,3.92,0,0,1,3.54-.53,3.85,3.85,0,0,1,2.14,1.93,3.74,3.74,0,0,1,.37,1.43Z" transform="translate(-1.87 -0.69)"/><path id="path11_fill" data-name="path11 fill" class="cls-4" d="M4.12,9.81A2.18,2.18,0,0,1,2.9,9.48a2.23,2.23,0,0,1-.84-1A2.26,2.26,0,0,1,1.9,7.26a2.13,2.13,0,0,1,.56-1.13,2.18,2.18,0,0,1,2.36-.56,2.13,2.13,0,0,1,1,.76,2.18,2.18,0,0,1,.42,1.2A2.22,2.22,0,0,1,4.12,9.81Z" transform="translate(-1.87 -0.69)"/></g><text class="cls-5" transform="translate(5.24 30.01)">Hub</text></svg>
diff --git a/dev/docs/html/_static/language_data.js b/dev/docs/html/_static/language_data.js
new file mode 100644
index 0000000..367b8ed
--- /dev/null
+++ b/dev/docs/html/_static/language_data.js
@@ -0,0 +1,199 @@
+/*
+ * language_data.js
+ * ~~~~~~~~~~~~~~~~
+ *
+ * This script contains the language-specific data used by searchtools.js,
+ * namely the list of stopwords, stemmer, scorer and splitter.
+ *
+ * :copyright: Copyright 2007-2024 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+
+var stopwords = ["a", "and", "are", "as", "at", "be", "but", "by", "for", "if", "in", "into", "is", "it", "near", "no", "not", "of", "on", "or", "such", "that", "the", "their", "then", "there", "these", "they", "this", "to", "was", "will", "with"];
+
+
+/* Non-minified version is copied as a separate JS file, if available */
+
+/**
+ * Porter Stemmer
+ */
+var Stemmer = function() {
+
+  var step2list = {
+    ational: 'ate',
+    tional: 'tion',
+    enci: 'ence',
+    anci: 'ance',
+    izer: 'ize',
+    bli: 'ble',
+    alli: 'al',
+    entli: 'ent',
+    eli: 'e',
+    ousli: 'ous',
+    ization: 'ize',
+    ation: 'ate',
+    ator: 'ate',
+    alism: 'al',
+    iveness: 'ive',
+    fulness: 'ful',
+    ousness: 'ous',
+    aliti: 'al',
+    iviti: 'ive',
+    biliti: 'ble',
+    logi: 'log'
+  };
+
+  var step3list = {
+    icate: 'ic',
+    ative: '',
+    alize: 'al',
+    iciti: 'ic',
+    ical: 'ic',
+    ful: '',
+    ness: ''
+  };
+
+  var c = "[^aeiou]";          // consonant
+  var v = "[aeiouy]";          // vowel
+  var C = c + "[^aeiouy]*";    // consonant sequence
+  var V = v + "[aeiou]*";      // vowel sequence
+
+  var mgr0 = "^(" + C + ")?" + V + C;                      // [C]VC... is m>0
+  var meq1 = "^(" + C + ")?" + V + C + "(" + V + ")?$";    // [C]VC[V] is m=1
+  var mgr1 = "^(" + C + ")?" + V + C + V + C;              // [C]VCVC... is m>1
+  var s_v   = "^(" + C + ")?" + v;                         // vowel in stem
+
+  this.stemWord = function (w) {
+    var stem;
+    var suffix;
+    var firstch;
+    var origword = w;
+
+    if (w.length < 3)
+      return w;
+
+    var re;
+    var re2;
+    var re3;
+    var re4;
+
+    firstch = w.substr(0,1);
+    if (firstch == "y")
+      w = firstch.toUpperCase() + w.substr(1);
+
+    // Step 1a
+    re = /^(.+?)(ss|i)es$/;
+    re2 = /^(.+?)([^s])s$/;
+
+    if (re.test(w))
+      w = w.replace(re,"$1$2");
+    else if (re2.test(w))
+      w = w.replace(re2,"$1$2");
+
+    // Step 1b
+    re = /^(.+?)eed$/;
+    re2 = /^(.+?)(ed|ing)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      re = new RegExp(mgr0);
+      if (re.test(fp[1])) {
+        re = /.$/;
+        w = w.replace(re,"");
+      }
+    }
+    else if (re2.test(w)) {
+      var fp = re2.exec(w);
+      stem = fp[1];
+      re2 = new RegExp(s_v);
+      if (re2.test(stem)) {
+        w = stem;
+        re2 = /(at|bl|iz)$/;
+        re3 = new RegExp("([^aeiouylsz])\\1$");
+        re4 = new RegExp("^" + C + v + "[^aeiouwxy]$");
+        if (re2.test(w))
+          w = w + "e";
+        else if (re3.test(w)) {
+          re = /.$/;
+          w = w.replace(re,"");
+        }
+        else if (re4.test(w))
+          w = w + "e";
+      }
+    }
+
+    // Step 1c
+    re = /^(.+?)y$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(s_v);
+      if (re.test(stem))
+        w = stem + "i";
+    }
+
+    // Step 2
+    re = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      suffix = fp[2];
+      re = new RegExp(mgr0);
+      if (re.test(stem))
+        w = stem + step2list[suffix];
+    }
+
+    // Step 3
+    re = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      suffix = fp[2];
+      re = new RegExp(mgr0);
+      if (re.test(stem))
+        w = stem + step3list[suffix];
+    }
+
+    // Step 4
+    re = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/;
+    re2 = /^(.+?)(s|t)(ion)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(mgr1);
+      if (re.test(stem))
+        w = stem;
+    }
+    else if (re2.test(w)) {
+      var fp = re2.exec(w);
+      stem = fp[1] + fp[2];
+      re2 = new RegExp(mgr1);
+      if (re2.test(stem))
+        w = stem;
+    }
+
+    // Step 5
+    re = /^(.+?)e$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(mgr1);
+      re2 = new RegExp(meq1);
+      re3 = new RegExp("^" + C + v + "[^aeiouwxy]$");
+      if (re.test(stem) || (re2.test(stem) && !(re3.test(stem))))
+        w = stem;
+    }
+    re = /ll$/;
+    re2 = new RegExp(mgr1);
+    if (re.test(w) && re2.test(w)) {
+      re = /.$/;
+      w = w.replace(re,"");
+    }
+
+    // and turn initial Y back to y
+    if (firstch == "y")
+      w = firstch.toLowerCase() + w.substr(1);
+    return w;
+  }
+}
+
diff --git a/dev/docs/html/_static/locales/ar/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/ar/LC_MESSAGES/booktheme.mo
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index 0000000..15541a6
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diff --git a/dev/docs/html/_static/locales/ar/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/ar/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..34d404c
--- /dev/null
+++ b/dev/docs/html/_static/locales/ar/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ar\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "طباعة إلى PDF"
+
+msgid "Theme by the"
+msgstr "موضوع بواسطة"
+
+msgid "Download source file"
+msgstr "تنزيل ملف المصدر"
+
+msgid "open issue"
+msgstr "قضية مفتوحة"
+
+msgid "Contents"
+msgstr "محتويات"
+
+msgid "previous page"
+msgstr "الصفحة السابقة"
+
+msgid "Download notebook file"
+msgstr "تنزيل ملف دفتر الملاحظات"
+
+msgid "Copyright"
+msgstr "حقوق النشر"
+
+msgid "Download this page"
+msgstr "قم بتنزيل هذه الصفحة"
+
+msgid "Source repository"
+msgstr "مستودع المصدر"
+
+msgid "By"
+msgstr "بواسطة"
+
+msgid "repository"
+msgstr "مخزن"
+
+msgid "Last updated on"
+msgstr "آخر تحديث في"
+
+msgid "Toggle navigation"
+msgstr "تبديل التنقل"
+
+msgid "Sphinx Book Theme"
+msgstr "موضوع كتاب أبو الهول"
+
+msgid "suggest edit"
+msgstr "أقترح تحرير"
+
+msgid "Open an issue"
+msgstr "افتح قضية"
+
+msgid "Launch"
+msgstr "إطلاق"
+
+msgid "Fullscreen mode"
+msgstr "وضع ملء الشاشة"
+
+msgid "Edit this page"
+msgstr "قم بتحرير هذه الصفحة"
+
+msgid "By the"
+msgstr "بواسطة"
+
+msgid "next page"
+msgstr "الصفحة التالية"
diff --git a/dev/docs/html/_static/locales/bg/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/bg/LC_MESSAGES/booktheme.mo
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index 0000000..da95120
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diff --git a/dev/docs/html/_static/locales/bg/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/bg/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..7420c19
--- /dev/null
+++ b/dev/docs/html/_static/locales/bg/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: bg\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Печат в PDF"
+
+msgid "Theme by the"
+msgstr "Тема от"
+
+msgid "Download source file"
+msgstr "Изтеглете изходния файл"
+
+msgid "open issue"
+msgstr "отворен брой"
+
+msgid "Contents"
+msgstr "Съдържание"
+
+msgid "previous page"
+msgstr "предишна страница"
+
+msgid "Download notebook file"
+msgstr "Изтеглете файла на бележника"
+
+msgid "Copyright"
+msgstr "Авторско право"
+
+msgid "Download this page"
+msgstr "Изтеглете тази страница"
+
+msgid "Source repository"
+msgstr "Хранилище на източника"
+
+msgid "By"
+msgstr "От"
+
+msgid "repository"
+msgstr "хранилище"
+
+msgid "Last updated on"
+msgstr "Последна актуализация на"
+
+msgid "Toggle navigation"
+msgstr "Превключване на навигацията"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема на книгата Sphinx"
+
+msgid "suggest edit"
+msgstr "предложи редактиране"
+
+msgid "Open an issue"
+msgstr "Отворете проблем"
+
+msgid "Launch"
+msgstr "Стартиране"
+
+msgid "Fullscreen mode"
+msgstr "Режим на цял екран"
+
+msgid "Edit this page"
+msgstr "Редактирайте тази страница"
+
+msgid "By the"
+msgstr "По"
+
+msgid "next page"
+msgstr "Следваща страница"
diff --git a/dev/docs/html/_static/locales/bn/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/bn/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..6b96639
Binary files /dev/null and b/dev/docs/html/_static/locales/bn/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/bn/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/bn/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..63a07c3
--- /dev/null
+++ b/dev/docs/html/_static/locales/bn/LC_MESSAGES/booktheme.po
@@ -0,0 +1,63 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: bn\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "পিডিএফ প্রিন্ট করুন"
+
+msgid "Theme by the"
+msgstr "থিম দ্বারা"
+
+msgid "Download source file"
+msgstr "উত্স ফাইল ডাউনলোড করুন"
+
+msgid "open issue"
+msgstr "খোলা সমস্যা"
+
+msgid "previous page"
+msgstr "আগের পৃষ্ঠা"
+
+msgid "Download notebook file"
+msgstr "নোটবুক ফাইল ডাউনলোড করুন"
+
+msgid "Copyright"
+msgstr "কপিরাইট"
+
+msgid "Download this page"
+msgstr "এই পৃষ্ঠাটি ডাউনলোড করুন"
+
+msgid "Source repository"
+msgstr "উত্স সংগ্রহস্থল"
+
+msgid "By"
+msgstr "দ্বারা"
+
+msgid "Last updated on"
+msgstr "সর্বশেষ আপডেট"
+
+msgid "Toggle navigation"
+msgstr "নেভিগেশন টগল করুন"
+
+msgid "Sphinx Book Theme"
+msgstr "স্পিনিক্স বুক থিম"
+
+msgid "Open an issue"
+msgstr "একটি সমস্যা খুলুন"
+
+msgid "Launch"
+msgstr "শুরু করা"
+
+msgid "Edit this page"
+msgstr "এই পৃষ্ঠাটি সম্পাদনা করুন"
+
+msgid "By the"
+msgstr "দ্বারা"
+
+msgid "next page"
+msgstr "পরবর্তী পৃষ্ঠা"
diff --git a/dev/docs/html/_static/locales/ca/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/ca/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..a4dd30e
Binary files /dev/null and b/dev/docs/html/_static/locales/ca/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/ca/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/ca/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..8fb358b
--- /dev/null
+++ b/dev/docs/html/_static/locales/ca/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ca\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Imprimeix a PDF"
+
+msgid "Theme by the"
+msgstr "Tema del"
+
+msgid "Download source file"
+msgstr "Baixeu el fitxer font"
+
+msgid "open issue"
+msgstr "número obert"
+
+msgid "previous page"
+msgstr "Pàgina anterior"
+
+msgid "Download notebook file"
+msgstr "Descarregar fitxer de quadern"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Download this page"
+msgstr "Descarregueu aquesta pàgina"
+
+msgid "Source repository"
+msgstr "Dipòsit de fonts"
+
+msgid "By"
+msgstr "Per"
+
+msgid "Last updated on"
+msgstr "Darrera actualització el"
+
+msgid "Toggle navigation"
+msgstr "Commuta la navegació"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema del llibre Esfinx"
+
+msgid "suggest edit"
+msgstr "suggerir edició"
+
+msgid "Open an issue"
+msgstr "Obriu un número"
+
+msgid "Launch"
+msgstr "Llançament"
+
+msgid "Edit this page"
+msgstr "Editeu aquesta pàgina"
+
+msgid "By the"
+msgstr "Per la"
+
+msgid "next page"
+msgstr "pàgina següent"
diff --git a/dev/docs/html/_static/locales/cs/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/cs/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..c39e01a
Binary files /dev/null and b/dev/docs/html/_static/locales/cs/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/cs/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/cs/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..c6ef469
--- /dev/null
+++ b/dev/docs/html/_static/locales/cs/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: cs\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Tisk do PDF"
+
+msgid "Theme by the"
+msgstr "Téma od"
+
+msgid "Download source file"
+msgstr "Stáhněte si zdrojový soubor"
+
+msgid "open issue"
+msgstr "otevřené číslo"
+
+msgid "Contents"
+msgstr "Obsah"
+
+msgid "previous page"
+msgstr "předchozí stránka"
+
+msgid "Download notebook file"
+msgstr "Stáhnout soubor poznámkového bloku"
+
+msgid "Copyright"
+msgstr "autorská práva"
+
+msgid "Download this page"
+msgstr "Stáhněte si tuto stránku"
+
+msgid "Source repository"
+msgstr "Zdrojové úložiště"
+
+msgid "By"
+msgstr "Podle"
+
+msgid "repository"
+msgstr "úložiště"
+
+msgid "Last updated on"
+msgstr "Naposledy aktualizováno"
+
+msgid "Toggle navigation"
+msgstr "Přepnout navigaci"
+
+msgid "Sphinx Book Theme"
+msgstr "Téma knihy Sfinga"
+
+msgid "suggest edit"
+msgstr "navrhnout úpravy"
+
+msgid "Open an issue"
+msgstr "Otevřete problém"
+
+msgid "Launch"
+msgstr "Zahájení"
+
+msgid "Fullscreen mode"
+msgstr "Režim celé obrazovky"
+
+msgid "Edit this page"
+msgstr "Upravit tuto stránku"
+
+msgid "By the"
+msgstr "Podle"
+
+msgid "next page"
+msgstr "další strana"
diff --git a/dev/docs/html/_static/locales/da/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/da/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..f43157d
Binary files /dev/null and b/dev/docs/html/_static/locales/da/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/da/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/da/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..306a38e
--- /dev/null
+++ b/dev/docs/html/_static/locales/da/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: da\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Udskriv til PDF"
+
+msgid "Theme by the"
+msgstr "Tema af"
+
+msgid "Download source file"
+msgstr "Download kildefil"
+
+msgid "open issue"
+msgstr "åbent nummer"
+
+msgid "Contents"
+msgstr "Indhold"
+
+msgid "previous page"
+msgstr "forrige side"
+
+msgid "Download notebook file"
+msgstr "Download notesbog-fil"
+
+msgid "Copyright"
+msgstr "ophavsret"
+
+msgid "Download this page"
+msgstr "Download denne side"
+
+msgid "Source repository"
+msgstr "Kildelager"
+
+msgid "By"
+msgstr "Ved"
+
+msgid "repository"
+msgstr "lager"
+
+msgid "Last updated on"
+msgstr "Sidst opdateret den"
+
+msgid "Toggle navigation"
+msgstr "Skift navigation"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx bogtema"
+
+msgid "suggest edit"
+msgstr "foreslå redigering"
+
+msgid "Open an issue"
+msgstr "Åbn et problem"
+
+msgid "Launch"
+msgstr "Start"
+
+msgid "Fullscreen mode"
+msgstr "Fuldskærmstilstand"
+
+msgid "Edit this page"
+msgstr "Rediger denne side"
+
+msgid "By the"
+msgstr "Ved"
+
+msgid "next page"
+msgstr "Næste side"
diff --git a/dev/docs/html/_static/locales/de/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/de/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..648b565
Binary files /dev/null and b/dev/docs/html/_static/locales/de/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/de/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/de/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..4925360
--- /dev/null
+++ b/dev/docs/html/_static/locales/de/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: de\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "In PDF drucken"
+
+msgid "Theme by the"
+msgstr "Thema von der"
+
+msgid "Download source file"
+msgstr "Quelldatei herunterladen"
+
+msgid "open issue"
+msgstr "offenes Thema"
+
+msgid "Contents"
+msgstr "Inhalt"
+
+msgid "previous page"
+msgstr "vorherige Seite"
+
+msgid "Download notebook file"
+msgstr "Notebook-Datei herunterladen"
+
+msgid "Copyright"
+msgstr "Urheberrechte ©"
+
+msgid "Download this page"
+msgstr "Laden Sie diese Seite herunter"
+
+msgid "Source repository"
+msgstr "Quell-Repository"
+
+msgid "By"
+msgstr "Durch"
+
+msgid "repository"
+msgstr "Repository"
+
+msgid "Last updated on"
+msgstr "Zuletzt aktualisiert am"
+
+msgid "Toggle navigation"
+msgstr "Navigation umschalten"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx-Buch-Thema"
+
+msgid "suggest edit"
+msgstr "vorschlagen zu bearbeiten"
+
+msgid "Open an issue"
+msgstr "Öffnen Sie ein Problem"
+
+msgid "Launch"
+msgstr "Starten"
+
+msgid "Fullscreen mode"
+msgstr "Vollbildmodus"
+
+msgid "Edit this page"
+msgstr "Bearbeite diese Seite"
+
+msgid "By the"
+msgstr "Bis zum"
+
+msgid "next page"
+msgstr "Nächste Seite"
diff --git a/dev/docs/html/_static/locales/el/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/el/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..fca6e93
Binary files /dev/null and b/dev/docs/html/_static/locales/el/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/el/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/el/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..3e01acb
--- /dev/null
+++ b/dev/docs/html/_static/locales/el/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: el\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Εκτύπωση σε PDF"
+
+msgid "Theme by the"
+msgstr "Θέμα από το"
+
+msgid "Download source file"
+msgstr "Λήψη αρχείου προέλευσης"
+
+msgid "open issue"
+msgstr "ανοιχτό ζήτημα"
+
+msgid "Contents"
+msgstr "Περιεχόμενα"
+
+msgid "previous page"
+msgstr "προηγούμενη σελίδα"
+
+msgid "Download notebook file"
+msgstr "Λήψη αρχείου σημειωματάριου"
+
+msgid "Copyright"
+msgstr "Πνευματική ιδιοκτησία"
+
+msgid "Download this page"
+msgstr "Λήψη αυτής της σελίδας"
+
+msgid "Source repository"
+msgstr "Αποθήκη πηγής"
+
+msgid "By"
+msgstr "Με"
+
+msgid "repository"
+msgstr "αποθήκη"
+
+msgid "Last updated on"
+msgstr "Τελευταία ενημέρωση στις"
+
+msgid "Toggle navigation"
+msgstr "Εναλλαγή πλοήγησης"
+
+msgid "Sphinx Book Theme"
+msgstr "Θέμα βιβλίου Sphinx"
+
+msgid "suggest edit"
+msgstr "προτείνω επεξεργασία"
+
+msgid "Open an issue"
+msgstr "Ανοίξτε ένα ζήτημα"
+
+msgid "Launch"
+msgstr "Εκτόξευση"
+
+msgid "Fullscreen mode"
+msgstr "ΛΕΙΤΟΥΡΓΙΑ ΠΛΗΡΟΥΣ ΟΘΟΝΗΣ"
+
+msgid "Edit this page"
+msgstr "Επεξεργαστείτε αυτήν τη σελίδα"
+
+msgid "By the"
+msgstr "Από το"
+
+msgid "next page"
+msgstr "επόμενη σελίδα"
diff --git a/dev/docs/html/_static/locales/eo/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/eo/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..d1072bb
Binary files /dev/null and b/dev/docs/html/_static/locales/eo/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/eo/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/eo/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..f7ed226
--- /dev/null
+++ b/dev/docs/html/_static/locales/eo/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: eo\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Presi al PDF"
+
+msgid "Theme by the"
+msgstr "Temo de la"
+
+msgid "Download source file"
+msgstr "Elŝutu fontodosieron"
+
+msgid "open issue"
+msgstr "malferma numero"
+
+msgid "Contents"
+msgstr "Enhavo"
+
+msgid "previous page"
+msgstr "antaŭa paĝo"
+
+msgid "Download notebook file"
+msgstr "Elŝutu kajeran dosieron"
+
+msgid "Copyright"
+msgstr "Kopirajto"
+
+msgid "Download this page"
+msgstr "Elŝutu ĉi tiun paĝon"
+
+msgid "Source repository"
+msgstr "Fonto-deponejo"
+
+msgid "By"
+msgstr "De"
+
+msgid "repository"
+msgstr "deponejo"
+
+msgid "Last updated on"
+msgstr "Laste ĝisdatigita la"
+
+msgid "Toggle navigation"
+msgstr "Ŝalti navigadon"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinksa Libro-Temo"
+
+msgid "suggest edit"
+msgstr "sugesti redaktadon"
+
+msgid "Open an issue"
+msgstr "Malfermu numeron"
+
+msgid "Launch"
+msgstr "Lanĉo"
+
+msgid "Fullscreen mode"
+msgstr "Plenekrana reĝimo"
+
+msgid "Edit this page"
+msgstr "Redaktu ĉi tiun paĝon"
+
+msgid "By the"
+msgstr "Per la"
+
+msgid "next page"
+msgstr "sekva paĝo"
diff --git a/dev/docs/html/_static/locales/es/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/es/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..ba2ee4d
Binary files /dev/null and b/dev/docs/html/_static/locales/es/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/es/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/es/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..5e0029e
--- /dev/null
+++ b/dev/docs/html/_static/locales/es/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: es\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Imprimir en PDF"
+
+msgid "Theme by the"
+msgstr "Tema por el"
+
+msgid "Download source file"
+msgstr "Descargar archivo fuente"
+
+msgid "open issue"
+msgstr "Tema abierto"
+
+msgid "Contents"
+msgstr "Contenido"
+
+msgid "previous page"
+msgstr "pagina anterior"
+
+msgid "Download notebook file"
+msgstr "Descargar archivo de cuaderno"
+
+msgid "Copyright"
+msgstr "Derechos de autor"
+
+msgid "Download this page"
+msgstr "Descarga esta pagina"
+
+msgid "Source repository"
+msgstr "Repositorio de origen"
+
+msgid "By"
+msgstr "Por"
+
+msgid "repository"
+msgstr "repositorio"
+
+msgid "Last updated on"
+msgstr "Ultima actualización en"
+
+msgid "Toggle navigation"
+msgstr "Navegación de palanca"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema del libro de la esfinge"
+
+msgid "suggest edit"
+msgstr "sugerir editar"
+
+msgid "Open an issue"
+msgstr "Abrir un problema"
+
+msgid "Launch"
+msgstr "Lanzamiento"
+
+msgid "Fullscreen mode"
+msgstr "Modo de pantalla completa"
+
+msgid "Edit this page"
+msgstr "Edita esta página"
+
+msgid "By the"
+msgstr "Por el"
+
+msgid "next page"
+msgstr "siguiente página"
diff --git a/dev/docs/html/_static/locales/et/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/et/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..983b823
Binary files /dev/null and b/dev/docs/html/_static/locales/et/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/et/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/et/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..8680982
--- /dev/null
+++ b/dev/docs/html/_static/locales/et/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: et\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Prindi PDF-i"
+
+msgid "Theme by the"
+msgstr "Teema"
+
+msgid "Download source file"
+msgstr "Laadige alla lähtefail"
+
+msgid "open issue"
+msgstr "avatud küsimus"
+
+msgid "Contents"
+msgstr "Sisu"
+
+msgid "previous page"
+msgstr "eelmine leht"
+
+msgid "Download notebook file"
+msgstr "Laadige sülearvuti fail alla"
+
+msgid "Copyright"
+msgstr "Autoriõigus"
+
+msgid "Download this page"
+msgstr "Laadige see leht alla"
+
+msgid "Source repository"
+msgstr "Allikahoidla"
+
+msgid "By"
+msgstr "Kõrval"
+
+msgid "repository"
+msgstr "hoidla"
+
+msgid "Last updated on"
+msgstr "Viimati uuendatud"
+
+msgid "Toggle navigation"
+msgstr "Lülita navigeerimine sisse"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinksiraamatu teema"
+
+msgid "suggest edit"
+msgstr "soovita muuta"
+
+msgid "Open an issue"
+msgstr "Avage probleem"
+
+msgid "Launch"
+msgstr "Käivitage"
+
+msgid "Fullscreen mode"
+msgstr "Täisekraanirežiim"
+
+msgid "Edit this page"
+msgstr "Muutke seda lehte"
+
+msgid "By the"
+msgstr "Autor"
+
+msgid "next page"
+msgstr "järgmine leht"
diff --git a/dev/docs/html/_static/locales/fi/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/fi/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..d8ac054
Binary files /dev/null and b/dev/docs/html/_static/locales/fi/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/fi/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/fi/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..34dac21
--- /dev/null
+++ b/dev/docs/html/_static/locales/fi/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: fi\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Tulosta PDF-tiedostoon"
+
+msgid "Theme by the"
+msgstr "Teeman tekijä"
+
+msgid "Download source file"
+msgstr "Lataa lähdetiedosto"
+
+msgid "open issue"
+msgstr "avoin ongelma"
+
+msgid "Contents"
+msgstr "Sisällys"
+
+msgid "previous page"
+msgstr "Edellinen sivu"
+
+msgid "Download notebook file"
+msgstr "Lataa muistikirjatiedosto"
+
+msgid "Copyright"
+msgstr "Tekijänoikeus"
+
+msgid "Download this page"
+msgstr "Lataa tämä sivu"
+
+msgid "Source repository"
+msgstr "Lähteen arkisto"
+
+msgid "By"
+msgstr "Tekijä"
+
+msgid "repository"
+msgstr "arkisto"
+
+msgid "Last updated on"
+msgstr "Viimeksi päivitetty"
+
+msgid "Toggle navigation"
+msgstr "Vaihda navigointia"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx-kirjan teema"
+
+msgid "suggest edit"
+msgstr "ehdottaa muokkausta"
+
+msgid "Open an issue"
+msgstr "Avaa ongelma"
+
+msgid "Launch"
+msgstr "Tuoda markkinoille"
+
+msgid "Fullscreen mode"
+msgstr "Koko näytön tila"
+
+msgid "Edit this page"
+msgstr "Muokkaa tätä sivua"
+
+msgid "By the"
+msgstr "Mukaan"
+
+msgid "next page"
+msgstr "seuraava sivu"
diff --git a/dev/docs/html/_static/locales/fr/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/fr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..f663d39
Binary files /dev/null and b/dev/docs/html/_static/locales/fr/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/fr/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/fr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..8991a1b
--- /dev/null
+++ b/dev/docs/html/_static/locales/fr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: fr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Imprimer au format PDF"
+
+msgid "Theme by the"
+msgstr "Thème par le"
+
+msgid "Download source file"
+msgstr "Télécharger le fichier source"
+
+msgid "open issue"
+msgstr "signaler un problème"
+
+msgid "Contents"
+msgstr "Contenu"
+
+msgid "previous page"
+msgstr "page précédente"
+
+msgid "Download notebook file"
+msgstr "Télécharger le fichier notebook"
+
+msgid "Copyright"
+msgstr "droits d'auteur"
+
+msgid "Download this page"
+msgstr "Téléchargez cette page"
+
+msgid "Source repository"
+msgstr "Dépôt source"
+
+msgid "By"
+msgstr "Par"
+
+msgid "repository"
+msgstr "dépôt"
+
+msgid "Last updated on"
+msgstr "Dernière mise à jour le"
+
+msgid "Toggle navigation"
+msgstr "Basculer la navigation"
+
+msgid "Sphinx Book Theme"
+msgstr "Thème du livre Sphinx"
+
+msgid "suggest edit"
+msgstr "suggestion de modification"
+
+msgid "Open an issue"
+msgstr "Ouvrez un problème"
+
+msgid "Launch"
+msgstr "lancement"
+
+msgid "Fullscreen mode"
+msgstr "Mode plein écran"
+
+msgid "Edit this page"
+msgstr "Modifier cette page"
+
+msgid "By the"
+msgstr "Par le"
+
+msgid "next page"
+msgstr "page suivante"
diff --git a/dev/docs/html/_static/locales/hr/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/hr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..eca4a1a
Binary files /dev/null and b/dev/docs/html/_static/locales/hr/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/hr/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/hr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..42c4233
--- /dev/null
+++ b/dev/docs/html/_static/locales/hr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: hr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Ispis u PDF"
+
+msgid "Theme by the"
+msgstr "Tema autora"
+
+msgid "Download source file"
+msgstr "Preuzmi izvornu datoteku"
+
+msgid "open issue"
+msgstr "otvoreno izdanje"
+
+msgid "Contents"
+msgstr "Sadržaj"
+
+msgid "previous page"
+msgstr "Prethodna stranica"
+
+msgid "Download notebook file"
+msgstr "Preuzmi datoteku bilježnice"
+
+msgid "Copyright"
+msgstr "Autorska prava"
+
+msgid "Download this page"
+msgstr "Preuzmite ovu stranicu"
+
+msgid "Source repository"
+msgstr "Izvorno spremište"
+
+msgid "By"
+msgstr "Po"
+
+msgid "repository"
+msgstr "spremište"
+
+msgid "Last updated on"
+msgstr "Posljednje ažuriranje:"
+
+msgid "Toggle navigation"
+msgstr "Uključi / isključi navigaciju"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema knjige Sphinx"
+
+msgid "suggest edit"
+msgstr "predloži uređivanje"
+
+msgid "Open an issue"
+msgstr "Otvorite izdanje"
+
+msgid "Launch"
+msgstr "Pokrenite"
+
+msgid "Fullscreen mode"
+msgstr "Način preko cijelog zaslona"
+
+msgid "Edit this page"
+msgstr "Uredite ovu stranicu"
+
+msgid "By the"
+msgstr "Od strane"
+
+msgid "next page"
+msgstr "sljedeća stranica"
diff --git a/dev/docs/html/_static/locales/id/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/id/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..d07a06a
Binary files /dev/null and b/dev/docs/html/_static/locales/id/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/id/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/id/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..b8d8d89
--- /dev/null
+++ b/dev/docs/html/_static/locales/id/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: id\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Cetak ke PDF"
+
+msgid "Theme by the"
+msgstr "Tema oleh"
+
+msgid "Download source file"
+msgstr "Unduh file sumber"
+
+msgid "open issue"
+msgstr "masalah terbuka"
+
+msgid "Contents"
+msgstr "Isi"
+
+msgid "previous page"
+msgstr "halaman sebelumnya"
+
+msgid "Download notebook file"
+msgstr "Unduh file notebook"
+
+msgid "Copyright"
+msgstr "hak cipta"
+
+msgid "Download this page"
+msgstr "Unduh halaman ini"
+
+msgid "Source repository"
+msgstr "Repositori sumber"
+
+msgid "By"
+msgstr "Oleh"
+
+msgid "repository"
+msgstr "gudang"
+
+msgid "Last updated on"
+msgstr "Terakhir diperbarui saat"
+
+msgid "Toggle navigation"
+msgstr "Alihkan navigasi"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema Buku Sphinx"
+
+msgid "suggest edit"
+msgstr "menyarankan edit"
+
+msgid "Open an issue"
+msgstr "Buka masalah"
+
+msgid "Launch"
+msgstr "Meluncurkan"
+
+msgid "Fullscreen mode"
+msgstr "Mode layar penuh"
+
+msgid "Edit this page"
+msgstr "Edit halaman ini"
+
+msgid "By the"
+msgstr "Oleh"
+
+msgid "next page"
+msgstr "halaman selanjutnya"
diff --git a/dev/docs/html/_static/locales/it/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/it/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..53ba476
Binary files /dev/null and b/dev/docs/html/_static/locales/it/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/it/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/it/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..36fca59
--- /dev/null
+++ b/dev/docs/html/_static/locales/it/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: it\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Stampa in PDF"
+
+msgid "Theme by the"
+msgstr "Tema di"
+
+msgid "Download source file"
+msgstr "Scarica il file sorgente"
+
+msgid "open issue"
+msgstr "questione aperta"
+
+msgid "Contents"
+msgstr "Contenuti"
+
+msgid "previous page"
+msgstr "pagina precedente"
+
+msgid "Download notebook file"
+msgstr "Scarica il file del taccuino"
+
+msgid "Copyright"
+msgstr "Diritto d'autore"
+
+msgid "Download this page"
+msgstr "Scarica questa pagina"
+
+msgid "Source repository"
+msgstr "Repository di origine"
+
+msgid "By"
+msgstr "Di"
+
+msgid "repository"
+msgstr "repository"
+
+msgid "Last updated on"
+msgstr "Ultimo aggiornamento il"
+
+msgid "Toggle navigation"
+msgstr "Attiva / disattiva la navigazione"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema del libro della Sfinge"
+
+msgid "suggest edit"
+msgstr "suggerisci modifica"
+
+msgid "Open an issue"
+msgstr "Apri un problema"
+
+msgid "Launch"
+msgstr "Lanciare"
+
+msgid "Fullscreen mode"
+msgstr "Modalità schermo intero"
+
+msgid "Edit this page"
+msgstr "Modifica questa pagina"
+
+msgid "By the"
+msgstr "Dal"
+
+msgid "next page"
+msgstr "pagina successiva"
diff --git a/dev/docs/html/_static/locales/iw/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/iw/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..a45c657
Binary files /dev/null and b/dev/docs/html/_static/locales/iw/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/iw/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/iw/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..dede9cb
--- /dev/null
+++ b/dev/docs/html/_static/locales/iw/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: iw\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "הדפס לקובץ PDF"
+
+msgid "Theme by the"
+msgstr "נושא מאת"
+
+msgid "Download source file"
+msgstr "הורד את קובץ המקור"
+
+msgid "open issue"
+msgstr "בעיה פתוחה"
+
+msgid "Contents"
+msgstr "תוכן"
+
+msgid "previous page"
+msgstr "עמוד קודם"
+
+msgid "Download notebook file"
+msgstr "הורד קובץ מחברת"
+
+msgid "Copyright"
+msgstr "זכויות יוצרים"
+
+msgid "Download this page"
+msgstr "הורד דף זה"
+
+msgid "Source repository"
+msgstr "מאגר המקורות"
+
+msgid "By"
+msgstr "על ידי"
+
+msgid "repository"
+msgstr "מאגר"
+
+msgid "Last updated on"
+msgstr "עודכן לאחרונה ב"
+
+msgid "Toggle navigation"
+msgstr "החלף ניווט"
+
+msgid "Sphinx Book Theme"
+msgstr "נושא ספר ספינקס"
+
+msgid "suggest edit"
+msgstr "מציע לערוך"
+
+msgid "Open an issue"
+msgstr "פתח גיליון"
+
+msgid "Launch"
+msgstr "לְהַשִׁיק"
+
+msgid "Fullscreen mode"
+msgstr "מצב מסך מלא"
+
+msgid "Edit this page"
+msgstr "ערוך דף זה"
+
+msgid "By the"
+msgstr "דרך"
+
+msgid "next page"
+msgstr "עמוד הבא"
diff --git a/dev/docs/html/_static/locales/ja/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/ja/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..1cefd29
Binary files /dev/null and b/dev/docs/html/_static/locales/ja/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/ja/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/ja/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..2615f0d
--- /dev/null
+++ b/dev/docs/html/_static/locales/ja/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ja\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "PDFに印刷"
+
+msgid "Theme by the"
+msgstr "のテーマ"
+
+msgid "Download source file"
+msgstr "ソースファイルをダウンロード"
+
+msgid "open issue"
+msgstr "未解決の問題"
+
+msgid "Contents"
+msgstr "目次"
+
+msgid "previous page"
+msgstr "前のページ"
+
+msgid "Download notebook file"
+msgstr "ノートブックファイルをダウンロード"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Download this page"
+msgstr "このページをダウンロード"
+
+msgid "Source repository"
+msgstr "ソースリポジトリ"
+
+msgid "By"
+msgstr "著者"
+
+msgid "repository"
+msgstr "リポジトリ"
+
+msgid "Last updated on"
+msgstr "最終更新日"
+
+msgid "Toggle navigation"
+msgstr "ナビゲーションを切り替え"
+
+msgid "Sphinx Book Theme"
+msgstr "スフィンクスの本のテーマ"
+
+msgid "suggest edit"
+msgstr "編集を提案する"
+
+msgid "Open an issue"
+msgstr "問題を報告"
+
+msgid "Launch"
+msgstr "起動"
+
+msgid "Fullscreen mode"
+msgstr "全画面モード"
+
+msgid "Edit this page"
+msgstr "このページを編集"
+
+msgid "By the"
+msgstr "によって"
+
+msgid "next page"
+msgstr "次のページ"
diff --git a/dev/docs/html/_static/locales/ko/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/ko/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..06c7ec9
Binary files /dev/null and b/dev/docs/html/_static/locales/ko/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/ko/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/ko/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..c9e13a4
--- /dev/null
+++ b/dev/docs/html/_static/locales/ko/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ko\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "PDF로 인쇄"
+
+msgid "Theme by the"
+msgstr "테마별"
+
+msgid "Download source file"
+msgstr "소스 파일 다운로드"
+
+msgid "open issue"
+msgstr "열린 문제"
+
+msgid "Contents"
+msgstr "내용"
+
+msgid "previous page"
+msgstr "이전 페이지"
+
+msgid "Download notebook file"
+msgstr "노트북 파일 다운로드"
+
+msgid "Copyright"
+msgstr "저작권"
+
+msgid "Download this page"
+msgstr "이 페이지 다운로드"
+
+msgid "Source repository"
+msgstr "소스 저장소"
+
+msgid "By"
+msgstr "으로"
+
+msgid "repository"
+msgstr "저장소"
+
+msgid "Last updated on"
+msgstr "마지막 업데이트"
+
+msgid "Toggle navigation"
+msgstr "탐색 전환"
+
+msgid "Sphinx Book Theme"
+msgstr "스핑크스 도서 테마"
+
+msgid "suggest edit"
+msgstr "편집 제안"
+
+msgid "Open an issue"
+msgstr "이슈 열기"
+
+msgid "Launch"
+msgstr "시작하다"
+
+msgid "Fullscreen mode"
+msgstr "전체 화면으로보기"
+
+msgid "Edit this page"
+msgstr "이 페이지 편집"
+
+msgid "By the"
+msgstr "에 의해"
+
+msgid "next page"
+msgstr "다음 페이지"
diff --git a/dev/docs/html/_static/locales/lt/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/lt/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..4468ba0
Binary files /dev/null and b/dev/docs/html/_static/locales/lt/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/lt/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/lt/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..35eabd9
--- /dev/null
+++ b/dev/docs/html/_static/locales/lt/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: lt\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Spausdinti į PDF"
+
+msgid "Theme by the"
+msgstr "Tema"
+
+msgid "Download source file"
+msgstr "Atsisiųsti šaltinio failą"
+
+msgid "open issue"
+msgstr "atviras klausimas"
+
+msgid "Contents"
+msgstr "Turinys"
+
+msgid "previous page"
+msgstr "Ankstesnis puslapis"
+
+msgid "Download notebook file"
+msgstr "Atsisiųsti nešiojamojo kompiuterio failą"
+
+msgid "Copyright"
+msgstr "Autorių teisės"
+
+msgid "Download this page"
+msgstr "Atsisiųskite šį puslapį"
+
+msgid "Source repository"
+msgstr "Šaltinio saugykla"
+
+msgid "By"
+msgstr "Iki"
+
+msgid "repository"
+msgstr "saugykla"
+
+msgid "Last updated on"
+msgstr "Paskutinį kartą atnaujinta"
+
+msgid "Toggle navigation"
+msgstr "Perjungti naršymą"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinkso knygos tema"
+
+msgid "suggest edit"
+msgstr "pasiūlyti redaguoti"
+
+msgid "Open an issue"
+msgstr "Atidarykite problemą"
+
+msgid "Launch"
+msgstr "Paleiskite"
+
+msgid "Fullscreen mode"
+msgstr "Pilno ekrano režimas"
+
+msgid "Edit this page"
+msgstr "Redaguoti šį puslapį"
+
+msgid "By the"
+msgstr "Prie"
+
+msgid "next page"
+msgstr "Kitas puslapis"
diff --git a/dev/docs/html/_static/locales/lv/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/lv/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..74aa4d8
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diff --git a/dev/docs/html/_static/locales/lv/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/lv/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..ee1bd08
--- /dev/null
+++ b/dev/docs/html/_static/locales/lv/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: lv\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Drukāt PDF formātā"
+
+msgid "Theme by the"
+msgstr "Autora tēma"
+
+msgid "Download source file"
+msgstr "Lejupielādēt avota failu"
+
+msgid "open issue"
+msgstr "atklāts jautājums"
+
+msgid "Contents"
+msgstr "Saturs"
+
+msgid "previous page"
+msgstr "iepriekšējā lapa"
+
+msgid "Download notebook file"
+msgstr "Lejupielādēt piezīmju grāmatiņu"
+
+msgid "Copyright"
+msgstr "Autortiesības"
+
+msgid "Download this page"
+msgstr "Lejupielādējiet šo lapu"
+
+msgid "Source repository"
+msgstr "Avota krātuve"
+
+msgid "By"
+msgstr "Autors"
+
+msgid "repository"
+msgstr "krātuve"
+
+msgid "Last updated on"
+msgstr "Pēdējoreiz atjaunināts"
+
+msgid "Toggle navigation"
+msgstr "Pārslēgt navigāciju"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinksa grāmatas tēma"
+
+msgid "suggest edit"
+msgstr "ieteikt rediģēt"
+
+msgid "Open an issue"
+msgstr "Atveriet problēmu"
+
+msgid "Launch"
+msgstr "Uzsākt"
+
+msgid "Fullscreen mode"
+msgstr "Pilnekrāna režīms"
+
+msgid "Edit this page"
+msgstr "Rediģēt šo lapu"
+
+msgid "By the"
+msgstr "Ar"
+
+msgid "next page"
+msgstr "nākamā lapaspuse"
diff --git a/dev/docs/html/_static/locales/ml/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/ml/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..2736e8f
Binary files /dev/null and b/dev/docs/html/_static/locales/ml/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/ml/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/ml/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..d471277
--- /dev/null
+++ b/dev/docs/html/_static/locales/ml/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ml\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "PDF- ലേക്ക് പ്രിന്റുചെയ്യുക"
+
+msgid "Theme by the"
+msgstr "പ്രമേയം"
+
+msgid "Download source file"
+msgstr "ഉറവിട ഫയൽ ഡൗൺലോഡുചെയ്യുക"
+
+msgid "open issue"
+msgstr "തുറന്ന പ്രശ്നം"
+
+msgid "previous page"
+msgstr "മുൻപത്തെ താൾ"
+
+msgid "Download notebook file"
+msgstr "നോട്ട്ബുക്ക് ഫയൽ ഡൺലോഡ് ചെയ്യുക"
+
+msgid "Copyright"
+msgstr "പകർപ്പവകാശം"
+
+msgid "Download this page"
+msgstr "ഈ പേജ് ഡൗൺലോഡുചെയ്യുക"
+
+msgid "Source repository"
+msgstr "ഉറവിട ശേഖരം"
+
+msgid "By"
+msgstr "എഴുതിയത്"
+
+msgid "Last updated on"
+msgstr "അവസാനം അപ്‌ഡേറ്റുചെയ്‌തത്"
+
+msgid "Toggle navigation"
+msgstr "നാവിഗേഷൻ ടോഗിൾ ചെയ്യുക"
+
+msgid "Sphinx Book Theme"
+msgstr "സ്ഫിങ്ക്സ് പുസ്തക തീം"
+
+msgid "suggest edit"
+msgstr "എഡിറ്റുചെയ്യാൻ നിർദ്ദേശിക്കുക"
+
+msgid "Open an issue"
+msgstr "ഒരു പ്രശ്നം തുറക്കുക"
+
+msgid "Launch"
+msgstr "സമാരംഭിക്കുക"
+
+msgid "Edit this page"
+msgstr "ഈ പേജ് എഡിറ്റുചെയ്യുക"
+
+msgid "By the"
+msgstr "എഴുതിയത്"
+
+msgid "next page"
+msgstr "അടുത്ത പേജ്"
diff --git a/dev/docs/html/_static/locales/mr/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/mr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..fe53010
Binary files /dev/null and b/dev/docs/html/_static/locales/mr/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/mr/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/mr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..f3694ac
--- /dev/null
+++ b/dev/docs/html/_static/locales/mr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: mr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "पीडीएफवर मुद्रित करा"
+
+msgid "Theme by the"
+msgstr "द्वारा थीम"
+
+msgid "Download source file"
+msgstr "स्त्रोत फाइल डाउनलोड करा"
+
+msgid "open issue"
+msgstr "खुला मुद्दा"
+
+msgid "previous page"
+msgstr "मागील पान"
+
+msgid "Download notebook file"
+msgstr "नोटबुक फाईल डाउनलोड करा"
+
+msgid "Copyright"
+msgstr "कॉपीराइट"
+
+msgid "Download this page"
+msgstr "हे पृष्ठ डाउनलोड करा"
+
+msgid "Source repository"
+msgstr "स्त्रोत भांडार"
+
+msgid "By"
+msgstr "द्वारा"
+
+msgid "Last updated on"
+msgstr "अखेरचे अद्यतनित"
+
+msgid "Toggle navigation"
+msgstr "नेव्हिगेशन टॉगल करा"
+
+msgid "Sphinx Book Theme"
+msgstr "स्फिंक्स बुक थीम"
+
+msgid "suggest edit"
+msgstr "संपादन सुचवा"
+
+msgid "Open an issue"
+msgstr "एक मुद्दा उघडा"
+
+msgid "Launch"
+msgstr "लाँच करा"
+
+msgid "Edit this page"
+msgstr "हे पृष्ठ संपादित करा"
+
+msgid "By the"
+msgstr "द्वारा"
+
+msgid "next page"
+msgstr "पुढील पृष्ठ"
diff --git a/dev/docs/html/_static/locales/ms/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/ms/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..f02603f
Binary files /dev/null and b/dev/docs/html/_static/locales/ms/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/ms/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/ms/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..65b7c60
--- /dev/null
+++ b/dev/docs/html/_static/locales/ms/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ms\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Cetak ke PDF"
+
+msgid "Theme by the"
+msgstr "Tema oleh"
+
+msgid "Download source file"
+msgstr "Muat turun fail sumber"
+
+msgid "open issue"
+msgstr "isu terbuka"
+
+msgid "previous page"
+msgstr "halaman sebelumnya"
+
+msgid "Download notebook file"
+msgstr "Muat turun fail buku nota"
+
+msgid "Copyright"
+msgstr "hak cipta"
+
+msgid "Download this page"
+msgstr "Muat turun halaman ini"
+
+msgid "Source repository"
+msgstr "Repositori sumber"
+
+msgid "By"
+msgstr "Oleh"
+
+msgid "Last updated on"
+msgstr "Terakhir dikemas kini pada"
+
+msgid "Toggle navigation"
+msgstr "Togol navigasi"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema Buku Sphinx"
+
+msgid "suggest edit"
+msgstr "cadangkan edit"
+
+msgid "Open an issue"
+msgstr "Buka masalah"
+
+msgid "Launch"
+msgstr "Lancarkan"
+
+msgid "Edit this page"
+msgstr "Edit halaman ini"
+
+msgid "By the"
+msgstr "Oleh"
+
+msgid "next page"
+msgstr "muka surat seterusnya"
diff --git a/dev/docs/html/_static/locales/nl/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/nl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..e59e7ec
Binary files /dev/null and b/dev/docs/html/_static/locales/nl/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/nl/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/nl/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..71bd1cd
--- /dev/null
+++ b/dev/docs/html/_static/locales/nl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: nl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Afdrukken naar pdf"
+
+msgid "Theme by the"
+msgstr "Thema door de"
+
+msgid "Download source file"
+msgstr "Download het bronbestand"
+
+msgid "open issue"
+msgstr "open probleem"
+
+msgid "Contents"
+msgstr "Inhoud"
+
+msgid "previous page"
+msgstr "vorige pagina"
+
+msgid "Download notebook file"
+msgstr "Download notebookbestand"
+
+msgid "Copyright"
+msgstr "auteursrechten"
+
+msgid "Download this page"
+msgstr "Download deze pagina"
+
+msgid "Source repository"
+msgstr "Bronopslagplaats"
+
+msgid "By"
+msgstr "Door"
+
+msgid "repository"
+msgstr "repository"
+
+msgid "Last updated on"
+msgstr "Laatst geupdate op"
+
+msgid "Toggle navigation"
+msgstr "Schakel navigatie"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx-boekthema"
+
+msgid "suggest edit"
+msgstr "suggereren bewerken"
+
+msgid "Open an issue"
+msgstr "Open een probleem"
+
+msgid "Launch"
+msgstr "Lancering"
+
+msgid "Fullscreen mode"
+msgstr "Volledig scherm"
+
+msgid "Edit this page"
+msgstr "bewerk deze pagina"
+
+msgid "By the"
+msgstr "Door de"
+
+msgid "next page"
+msgstr "volgende bladzijde"
diff --git a/dev/docs/html/_static/locales/no/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/no/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..6cd15c8
Binary files /dev/null and b/dev/docs/html/_static/locales/no/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/no/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/no/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..b21346a
--- /dev/null
+++ b/dev/docs/html/_static/locales/no/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: no\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Skriv ut til PDF"
+
+msgid "Theme by the"
+msgstr "Tema av"
+
+msgid "Download source file"
+msgstr "Last ned kildefilen"
+
+msgid "open issue"
+msgstr "åpent nummer"
+
+msgid "Contents"
+msgstr "Innhold"
+
+msgid "previous page"
+msgstr "forrige side"
+
+msgid "Download notebook file"
+msgstr "Last ned notatbokfilen"
+
+msgid "Copyright"
+msgstr "opphavsrett"
+
+msgid "Download this page"
+msgstr "Last ned denne siden"
+
+msgid "Source repository"
+msgstr "Kildedepot"
+
+msgid "By"
+msgstr "Av"
+
+msgid "repository"
+msgstr "oppbevaringssted"
+
+msgid "Last updated on"
+msgstr "Sist oppdatert den"
+
+msgid "Toggle navigation"
+msgstr "Bytt navigasjon"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx boktema"
+
+msgid "suggest edit"
+msgstr "foreslå redigering"
+
+msgid "Open an issue"
+msgstr "Åpne et problem"
+
+msgid "Launch"
+msgstr "Start"
+
+msgid "Fullscreen mode"
+msgstr "Fullskjerm-modus"
+
+msgid "Edit this page"
+msgstr "Rediger denne siden"
+
+msgid "By the"
+msgstr "Ved"
+
+msgid "next page"
+msgstr "neste side"
diff --git a/dev/docs/html/_static/locales/pl/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/pl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..9ebb584
Binary files /dev/null and b/dev/docs/html/_static/locales/pl/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/pl/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/pl/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..1b7233f
--- /dev/null
+++ b/dev/docs/html/_static/locales/pl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: pl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Drukuj do PDF"
+
+msgid "Theme by the"
+msgstr "Motyw autorstwa"
+
+msgid "Download source file"
+msgstr "Pobierz plik źródłowy"
+
+msgid "open issue"
+msgstr "otwarty problem"
+
+msgid "Contents"
+msgstr "Zawartość"
+
+msgid "previous page"
+msgstr "Poprzednia strona"
+
+msgid "Download notebook file"
+msgstr "Pobierz plik notatnika"
+
+msgid "Copyright"
+msgstr "prawa autorskie"
+
+msgid "Download this page"
+msgstr "Pobierz tę stronę"
+
+msgid "Source repository"
+msgstr "Repozytorium źródłowe"
+
+msgid "By"
+msgstr "Przez"
+
+msgid "repository"
+msgstr "magazyn"
+
+msgid "Last updated on"
+msgstr "Ostatnia aktualizacja"
+
+msgid "Toggle navigation"
+msgstr "Przełącz nawigację"
+
+msgid "Sphinx Book Theme"
+msgstr "Motyw książki Sphinx"
+
+msgid "suggest edit"
+msgstr "zaproponuj edycję"
+
+msgid "Open an issue"
+msgstr "Otwórz problem"
+
+msgid "Launch"
+msgstr "Uruchomić"
+
+msgid "Fullscreen mode"
+msgstr "Pełny ekran"
+
+msgid "Edit this page"
+msgstr "Edytuj tę strone"
+
+msgid "By the"
+msgstr "Przez"
+
+msgid "next page"
+msgstr "Następna strona"
diff --git a/dev/docs/html/_static/locales/pt/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/pt/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..d0ddb87
Binary files /dev/null and b/dev/docs/html/_static/locales/pt/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/pt/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/pt/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..1b27314
--- /dev/null
+++ b/dev/docs/html/_static/locales/pt/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: pt\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Imprimir em PDF"
+
+msgid "Theme by the"
+msgstr "Tema por"
+
+msgid "Download source file"
+msgstr "Baixar arquivo fonte"
+
+msgid "open issue"
+msgstr "questão aberta"
+
+msgid "Contents"
+msgstr "Conteúdo"
+
+msgid "previous page"
+msgstr "página anterior"
+
+msgid "Download notebook file"
+msgstr "Baixar arquivo de notebook"
+
+msgid "Copyright"
+msgstr "direito autoral"
+
+msgid "Download this page"
+msgstr "Baixe esta página"
+
+msgid "Source repository"
+msgstr "Repositório fonte"
+
+msgid "By"
+msgstr "De"
+
+msgid "repository"
+msgstr "repositório"
+
+msgid "Last updated on"
+msgstr "Última atualização em"
+
+msgid "Toggle navigation"
+msgstr "Alternar de navegação"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema do livro Sphinx"
+
+msgid "suggest edit"
+msgstr "sugerir edição"
+
+msgid "Open an issue"
+msgstr "Abra um problema"
+
+msgid "Launch"
+msgstr "Lançamento"
+
+msgid "Fullscreen mode"
+msgstr "Modo tela cheia"
+
+msgid "Edit this page"
+msgstr "Edite essa página"
+
+msgid "By the"
+msgstr "Pelo"
+
+msgid "next page"
+msgstr "próxima página"
diff --git a/dev/docs/html/_static/locales/ro/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/ro/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..3c36ab1
Binary files /dev/null and b/dev/docs/html/_static/locales/ro/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/ro/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/ro/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..1783ad2
--- /dev/null
+++ b/dev/docs/html/_static/locales/ro/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ro\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Imprimați în PDF"
+
+msgid "Theme by the"
+msgstr "Tema de"
+
+msgid "Download source file"
+msgstr "Descărcați fișierul sursă"
+
+msgid "open issue"
+msgstr "problema deschisă"
+
+msgid "Contents"
+msgstr "Cuprins"
+
+msgid "previous page"
+msgstr "pagina anterioară"
+
+msgid "Download notebook file"
+msgstr "Descărcați fișierul notebook"
+
+msgid "Copyright"
+msgstr "Drepturi de autor"
+
+msgid "Download this page"
+msgstr "Descarcă această pagină"
+
+msgid "Source repository"
+msgstr "Depozit sursă"
+
+msgid "By"
+msgstr "De"
+
+msgid "repository"
+msgstr "repertoriu"
+
+msgid "Last updated on"
+msgstr "Ultima actualizare la"
+
+msgid "Toggle navigation"
+msgstr "Comutare navigare"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema Sphinx Book"
+
+msgid "suggest edit"
+msgstr "sugerează editare"
+
+msgid "Open an issue"
+msgstr "Deschideți o problemă"
+
+msgid "Launch"
+msgstr "Lansa"
+
+msgid "Fullscreen mode"
+msgstr "Modul ecran întreg"
+
+msgid "Edit this page"
+msgstr "Editați această pagină"
+
+msgid "By the"
+msgstr "Langa"
+
+msgid "next page"
+msgstr "pagina următoare"
diff --git a/dev/docs/html/_static/locales/ru/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/ru/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..6b8ca41
Binary files /dev/null and b/dev/docs/html/_static/locales/ru/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/ru/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/ru/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..b1176b7
--- /dev/null
+++ b/dev/docs/html/_static/locales/ru/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ru\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Распечатать в PDF"
+
+msgid "Theme by the"
+msgstr "Тема от"
+
+msgid "Download source file"
+msgstr "Скачать исходный файл"
+
+msgid "open issue"
+msgstr "открытый вопрос"
+
+msgid "Contents"
+msgstr "Содержание"
+
+msgid "previous page"
+msgstr "Предыдущая страница"
+
+msgid "Download notebook file"
+msgstr "Скачать файл записной книжки"
+
+msgid "Copyright"
+msgstr "авторское право"
+
+msgid "Download this page"
+msgstr "Загрузите эту страницу"
+
+msgid "Source repository"
+msgstr "Исходный репозиторий"
+
+msgid "By"
+msgstr "По"
+
+msgid "repository"
+msgstr "хранилище"
+
+msgid "Last updated on"
+msgstr "Последнее обновление"
+
+msgid "Toggle navigation"
+msgstr "Переключить навигацию"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема книги Сфинкс"
+
+msgid "suggest edit"
+msgstr "предложить редактировать"
+
+msgid "Open an issue"
+msgstr "Открыть вопрос"
+
+msgid "Launch"
+msgstr "Запуск"
+
+msgid "Fullscreen mode"
+msgstr "Полноэкранный режим"
+
+msgid "Edit this page"
+msgstr "Редактировать эту страницу"
+
+msgid "By the"
+msgstr "Посредством"
+
+msgid "next page"
+msgstr "Следующая страница"
diff --git a/dev/docs/html/_static/locales/sk/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/sk/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..59bd0dd
Binary files /dev/null and b/dev/docs/html/_static/locales/sk/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/sk/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/sk/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..6501288
--- /dev/null
+++ b/dev/docs/html/_static/locales/sk/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sk\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Tlač do PDF"
+
+msgid "Theme by the"
+msgstr "Téma od"
+
+msgid "Download source file"
+msgstr "Stiahnite si zdrojový súbor"
+
+msgid "open issue"
+msgstr "otvorené vydanie"
+
+msgid "Contents"
+msgstr "Obsah"
+
+msgid "previous page"
+msgstr "predchádzajúca strana"
+
+msgid "Download notebook file"
+msgstr "Stiahnite si zošit"
+
+msgid "Copyright"
+msgstr "Autorské práva"
+
+msgid "Download this page"
+msgstr "Stiahnite si túto stránku"
+
+msgid "Source repository"
+msgstr "Zdrojové úložisko"
+
+msgid "By"
+msgstr "Autor:"
+
+msgid "repository"
+msgstr "Úložisko"
+
+msgid "Last updated on"
+msgstr "Posledná aktualizácia dňa"
+
+msgid "Toggle navigation"
+msgstr "Prepnúť navigáciu"
+
+msgid "Sphinx Book Theme"
+msgstr "Téma knihy Sfinga"
+
+msgid "suggest edit"
+msgstr "navrhnúť úpravu"
+
+msgid "Open an issue"
+msgstr "Otvorte problém"
+
+msgid "Launch"
+msgstr "Spustiť"
+
+msgid "Fullscreen mode"
+msgstr "Režim celej obrazovky"
+
+msgid "Edit this page"
+msgstr "Upraviť túto stránku"
+
+msgid "By the"
+msgstr "Podľa"
+
+msgid "next page"
+msgstr "ďalšia strana"
diff --git a/dev/docs/html/_static/locales/sl/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/sl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..87bf26d
Binary files /dev/null and b/dev/docs/html/_static/locales/sl/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/sl/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/sl/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..3c7e3a8
--- /dev/null
+++ b/dev/docs/html/_static/locales/sl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Natisni v PDF"
+
+msgid "Theme by the"
+msgstr "Tema avtorja"
+
+msgid "Download source file"
+msgstr "Prenesite izvorno datoteko"
+
+msgid "open issue"
+msgstr "odprto vprašanje"
+
+msgid "Contents"
+msgstr "Vsebina"
+
+msgid "previous page"
+msgstr "Prejšnja stran"
+
+msgid "Download notebook file"
+msgstr "Prenesite datoteko zvezka"
+
+msgid "Copyright"
+msgstr "avtorske pravice"
+
+msgid "Download this page"
+msgstr "Prenesite to stran"
+
+msgid "Source repository"
+msgstr "Izvorno skladišče"
+
+msgid "By"
+msgstr "Avtor"
+
+msgid "repository"
+msgstr "odlagališče"
+
+msgid "Last updated on"
+msgstr "Nazadnje posodobljeno dne"
+
+msgid "Toggle navigation"
+msgstr "Preklopi navigacijo"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema knjige Sphinx"
+
+msgid "suggest edit"
+msgstr "predlagajte urejanje"
+
+msgid "Open an issue"
+msgstr "Odprite številko"
+
+msgid "Launch"
+msgstr "Kosilo"
+
+msgid "Fullscreen mode"
+msgstr "Celozaslonski način"
+
+msgid "Edit this page"
+msgstr "Uredite to stran"
+
+msgid "By the"
+msgstr "Avtor"
+
+msgid "next page"
+msgstr "Naslednja stran"
diff --git a/dev/docs/html/_static/locales/sr/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/sr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..ec740f4
Binary files /dev/null and b/dev/docs/html/_static/locales/sr/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/sr/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/sr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..773b8ad
--- /dev/null
+++ b/dev/docs/html/_static/locales/sr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Испис у ПДФ"
+
+msgid "Theme by the"
+msgstr "Тхеме би"
+
+msgid "Download source file"
+msgstr "Преузми изворну датотеку"
+
+msgid "open issue"
+msgstr "отворено издање"
+
+msgid "Contents"
+msgstr "Садржај"
+
+msgid "previous page"
+msgstr "Претходна страница"
+
+msgid "Download notebook file"
+msgstr "Преузмите датотеку бележнице"
+
+msgid "Copyright"
+msgstr "Ауторско право"
+
+msgid "Download this page"
+msgstr "Преузмите ову страницу"
+
+msgid "Source repository"
+msgstr "Изворно спремиште"
+
+msgid "By"
+msgstr "Од стране"
+
+msgid "repository"
+msgstr "спремиште"
+
+msgid "Last updated on"
+msgstr "Последње ажурирање"
+
+msgid "Toggle navigation"
+msgstr "Укључи / искључи навигацију"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема књиге Спхинк"
+
+msgid "suggest edit"
+msgstr "предложи уређивање"
+
+msgid "Open an issue"
+msgstr "Отворите издање"
+
+msgid "Launch"
+msgstr "Лансирање"
+
+msgid "Fullscreen mode"
+msgstr "Режим целог екрана"
+
+msgid "Edit this page"
+msgstr "Уредите ову страницу"
+
+msgid "By the"
+msgstr "Од"
+
+msgid "next page"
+msgstr "Следећа страна"
diff --git a/dev/docs/html/_static/locales/sv/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/sv/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..b07dc76
Binary files /dev/null and b/dev/docs/html/_static/locales/sv/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/sv/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/sv/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..bcac54c
--- /dev/null
+++ b/dev/docs/html/_static/locales/sv/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sv\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Skriv ut till PDF"
+
+msgid "Theme by the"
+msgstr "Tema av"
+
+msgid "Download source file"
+msgstr "Ladda ner källfil"
+
+msgid "open issue"
+msgstr "öppna problemrapport"
+
+msgid "Contents"
+msgstr "Innehåll"
+
+msgid "previous page"
+msgstr "föregående sida"
+
+msgid "Download notebook file"
+msgstr "Ladda ner notebook-fil"
+
+msgid "Copyright"
+msgstr "Upphovsrätt"
+
+msgid "Download this page"
+msgstr "Ladda ner den här sidan"
+
+msgid "Source repository"
+msgstr "Källkodsrepositorium"
+
+msgid "By"
+msgstr "Av"
+
+msgid "repository"
+msgstr "repositorium"
+
+msgid "Last updated on"
+msgstr "Senast uppdaterad den"
+
+msgid "Toggle navigation"
+msgstr "Växla navigering"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx Boktema"
+
+msgid "suggest edit"
+msgstr "föreslå ändring"
+
+msgid "Open an issue"
+msgstr "Öppna en problemrapport"
+
+msgid "Launch"
+msgstr "Öppna"
+
+msgid "Fullscreen mode"
+msgstr "Fullskärmsläge"
+
+msgid "Edit this page"
+msgstr "Redigera den här sidan"
+
+msgid "By the"
+msgstr "Av den"
+
+msgid "next page"
+msgstr "nästa sida"
diff --git a/dev/docs/html/_static/locales/ta/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/ta/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..29f52e1
Binary files /dev/null and b/dev/docs/html/_static/locales/ta/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/ta/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/ta/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..b48bdfa
--- /dev/null
+++ b/dev/docs/html/_static/locales/ta/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ta\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "PDF இல் அச்சிடுக"
+
+msgid "Theme by the"
+msgstr "வழங்கிய தீம்"
+
+msgid "Download source file"
+msgstr "மூல கோப்பைப் பதிவிறக்குக"
+
+msgid "open issue"
+msgstr "திறந்த பிரச்சினை"
+
+msgid "previous page"
+msgstr "முந்தைய பக்கம்"
+
+msgid "Download notebook file"
+msgstr "நோட்புக் கோப்பைப் பதிவிறக்கவும்"
+
+msgid "Copyright"
+msgstr "பதிப்புரிமை"
+
+msgid "Download this page"
+msgstr "இந்தப் பக்கத்தைப் பதிவிறக்கவும்"
+
+msgid "Source repository"
+msgstr "மூல களஞ்சியம்"
+
+msgid "By"
+msgstr "வழங்கியவர்"
+
+msgid "Last updated on"
+msgstr "கடைசியாக புதுப்பிக்கப்பட்டது"
+
+msgid "Toggle navigation"
+msgstr "வழிசெலுத்தலை நிலைமாற்று"
+
+msgid "Sphinx Book Theme"
+msgstr "ஸ்பிங்க்ஸ் புத்தக தீம்"
+
+msgid "suggest edit"
+msgstr "திருத்த பரிந்துரைக்கவும்"
+
+msgid "Open an issue"
+msgstr "சிக்கலைத் திறக்கவும்"
+
+msgid "Launch"
+msgstr "தொடங்க"
+
+msgid "Edit this page"
+msgstr "இந்தப் பக்கத்தைத் திருத்தவும்"
+
+msgid "By the"
+msgstr "மூலம்"
+
+msgid "next page"
+msgstr "அடுத்த பக்கம்"
diff --git a/dev/docs/html/_static/locales/te/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/te/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..0a5f4b4
Binary files /dev/null and b/dev/docs/html/_static/locales/te/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/te/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/te/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..952278f
--- /dev/null
+++ b/dev/docs/html/_static/locales/te/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: te\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "PDF కి ముద్రించండి"
+
+msgid "Theme by the"
+msgstr "ద్వారా థీమ్"
+
+msgid "Download source file"
+msgstr "మూల ఫైల్‌ను డౌన్‌లోడ్ చేయండి"
+
+msgid "open issue"
+msgstr "ఓపెన్ ఇష్యూ"
+
+msgid "previous page"
+msgstr "ముందు పేజి"
+
+msgid "Download notebook file"
+msgstr "నోట్బుక్ ఫైల్ను డౌన్లోడ్ చేయండి"
+
+msgid "Copyright"
+msgstr "కాపీరైట్"
+
+msgid "Download this page"
+msgstr "ఈ పేజీని డౌన్‌లోడ్ చేయండి"
+
+msgid "Source repository"
+msgstr "మూల రిపోజిటరీ"
+
+msgid "By"
+msgstr "ద్వారా"
+
+msgid "Last updated on"
+msgstr "చివరిగా నవీకరించబడింది"
+
+msgid "Toggle navigation"
+msgstr "నావిగేషన్‌ను టోగుల్ చేయండి"
+
+msgid "Sphinx Book Theme"
+msgstr "సింహిక పుస్తక థీమ్"
+
+msgid "suggest edit"
+msgstr "సవరించమని సూచించండి"
+
+msgid "Open an issue"
+msgstr "సమస్యను తెరవండి"
+
+msgid "Launch"
+msgstr "ప్రారంభించండి"
+
+msgid "Edit this page"
+msgstr "ఈ పేజీని సవరించండి"
+
+msgid "By the"
+msgstr "ద్వారా"
+
+msgid "next page"
+msgstr "తరువాతి పేజీ"
diff --git a/dev/docs/html/_static/locales/tg/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/tg/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..b21c6c6
Binary files /dev/null and b/dev/docs/html/_static/locales/tg/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/tg/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/tg/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..c33dc42
--- /dev/null
+++ b/dev/docs/html/_static/locales/tg/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: tg\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Чоп ба PDF"
+
+msgid "Theme by the"
+msgstr "Мавзӯъи аз"
+
+msgid "Download source file"
+msgstr "Файли манбаъро зеркашӣ кунед"
+
+msgid "open issue"
+msgstr "барориши кушод"
+
+msgid "Contents"
+msgstr "Мундариҷа"
+
+msgid "previous page"
+msgstr "саҳифаи қаблӣ"
+
+msgid "Download notebook file"
+msgstr "Файли дафтарро зеркашӣ кунед"
+
+msgid "Copyright"
+msgstr "Ҳуқуқи муаллиф"
+
+msgid "Download this page"
+msgstr "Ин саҳифаро зеркашӣ кунед"
+
+msgid "Source repository"
+msgstr "Анбори манбаъ"
+
+msgid "By"
+msgstr "Бо"
+
+msgid "repository"
+msgstr "анбор"
+
+msgid "Last updated on"
+msgstr "Last навсозӣ дар"
+
+msgid "Toggle navigation"
+msgstr "Гузаришро иваз кунед"
+
+msgid "Sphinx Book Theme"
+msgstr "Сфинкс Мавзӯи китоб"
+
+msgid "suggest edit"
+msgstr "пешниҳод вироиш"
+
+msgid "Open an issue"
+msgstr "Масъаларо кушоед"
+
+msgid "Launch"
+msgstr "Оғоз"
+
+msgid "Fullscreen mode"
+msgstr "Ҳолати экрани пурра"
+
+msgid "Edit this page"
+msgstr "Ин саҳифаро таҳрир кунед"
+
+msgid "By the"
+msgstr "Бо"
+
+msgid "next page"
+msgstr "саҳифаи оянда"
diff --git a/dev/docs/html/_static/locales/th/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/th/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..abede98
Binary files /dev/null and b/dev/docs/html/_static/locales/th/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/th/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/th/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..9d24294
--- /dev/null
+++ b/dev/docs/html/_static/locales/th/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: th\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "พิมพ์เป็น PDF"
+
+msgid "Theme by the"
+msgstr "ธีมโดย"
+
+msgid "Download source file"
+msgstr "ดาวน์โหลดไฟล์ต้นฉบับ"
+
+msgid "open issue"
+msgstr "เปิดปัญหา"
+
+msgid "Contents"
+msgstr "สารบัญ"
+
+msgid "previous page"
+msgstr "หน้าที่แล้ว"
+
+msgid "Download notebook file"
+msgstr "ดาวน์โหลดไฟล์สมุดบันทึก"
+
+msgid "Copyright"
+msgstr "ลิขสิทธิ์"
+
+msgid "Download this page"
+msgstr "ดาวน์โหลดหน้านี้"
+
+msgid "Source repository"
+msgstr "ที่เก็บซอร์ส"
+
+msgid "By"
+msgstr "โดย"
+
+msgid "repository"
+msgstr "ที่เก็บ"
+
+msgid "Last updated on"
+msgstr "ปรับปรุงล่าสุดเมื่อ"
+
+msgid "Toggle navigation"
+msgstr "ไม่ต้องสลับช่องทาง"
+
+msgid "Sphinx Book Theme"
+msgstr "ธีมหนังสือสฟิงซ์"
+
+msgid "suggest edit"
+msgstr "แนะนำแก้ไข"
+
+msgid "Open an issue"
+msgstr "เปิดปัญหา"
+
+msgid "Launch"
+msgstr "เปิด"
+
+msgid "Fullscreen mode"
+msgstr "โหมดเต็มหน้าจอ"
+
+msgid "Edit this page"
+msgstr "แก้ไขหน้านี้"
+
+msgid "By the"
+msgstr "โดย"
+
+msgid "next page"
+msgstr "หน้าต่อไป"
diff --git a/dev/docs/html/_static/locales/tl/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/tl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..8df1b73
Binary files /dev/null and b/dev/docs/html/_static/locales/tl/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/tl/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/tl/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..20e0d07
--- /dev/null
+++ b/dev/docs/html/_static/locales/tl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: tl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "I-print sa PDF"
+
+msgid "Theme by the"
+msgstr "Tema ng"
+
+msgid "Download source file"
+msgstr "Mag-download ng file ng pinagmulan"
+
+msgid "open issue"
+msgstr "bukas na isyu"
+
+msgid "previous page"
+msgstr "Nakaraang pahina"
+
+msgid "Download notebook file"
+msgstr "Mag-download ng file ng notebook"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Download this page"
+msgstr "I-download ang pahinang ito"
+
+msgid "Source repository"
+msgstr "Pinagmulan ng imbakan"
+
+msgid "By"
+msgstr "Ni"
+
+msgid "Last updated on"
+msgstr "Huling na-update noong"
+
+msgid "Toggle navigation"
+msgstr "I-toggle ang pag-navigate"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema ng Sphinx Book"
+
+msgid "suggest edit"
+msgstr "iminumungkahi i-edit"
+
+msgid "Open an issue"
+msgstr "Magbukas ng isyu"
+
+msgid "Launch"
+msgstr "Ilunsad"
+
+msgid "Edit this page"
+msgstr "I-edit ang pahinang ito"
+
+msgid "By the"
+msgstr "Sa pamamagitan ng"
+
+msgid "next page"
+msgstr "Susunod na pahina"
diff --git a/dev/docs/html/_static/locales/tr/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/tr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..029ae18
Binary files /dev/null and b/dev/docs/html/_static/locales/tr/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/tr/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/tr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..a77eb02
--- /dev/null
+++ b/dev/docs/html/_static/locales/tr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: tr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "PDF olarak yazdır"
+
+msgid "Theme by the"
+msgstr "Tarafından tema"
+
+msgid "Download source file"
+msgstr "Kaynak dosyayı indirin"
+
+msgid "open issue"
+msgstr "Açık konu"
+
+msgid "Contents"
+msgstr "İçindekiler"
+
+msgid "previous page"
+msgstr "önceki sayfa"
+
+msgid "Download notebook file"
+msgstr "Defter dosyasını indirin"
+
+msgid "Copyright"
+msgstr "Telif hakkı"
+
+msgid "Download this page"
+msgstr "Bu sayfayı indirin"
+
+msgid "Source repository"
+msgstr "Kaynak kod deposu"
+
+msgid "By"
+msgstr "Tarafından"
+
+msgid "repository"
+msgstr "depo"
+
+msgid "Last updated on"
+msgstr "Son güncelleme tarihi"
+
+msgid "Toggle navigation"
+msgstr "Gezinmeyi değiştir"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfenks Kitap Teması"
+
+msgid "suggest edit"
+msgstr "düzenleme öner"
+
+msgid "Open an issue"
+msgstr "Bir sorunu açın"
+
+msgid "Launch"
+msgstr "Başlatmak"
+
+msgid "Fullscreen mode"
+msgstr "Tam ekran modu"
+
+msgid "Edit this page"
+msgstr "Bu sayfayı düzenle"
+
+msgid "By the"
+msgstr "Tarafından"
+
+msgid "next page"
+msgstr "sonraki Sayfa"
diff --git a/dev/docs/html/_static/locales/uk/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/uk/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..16ab789
Binary files /dev/null and b/dev/docs/html/_static/locales/uk/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/uk/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/uk/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..993dd07
--- /dev/null
+++ b/dev/docs/html/_static/locales/uk/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: uk\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Друк у форматі PDF"
+
+msgid "Theme by the"
+msgstr "Тема від"
+
+msgid "Download source file"
+msgstr "Завантажити вихідний файл"
+
+msgid "open issue"
+msgstr "відкритий випуск"
+
+msgid "Contents"
+msgstr "Зміст"
+
+msgid "previous page"
+msgstr "Попередня сторінка"
+
+msgid "Download notebook file"
+msgstr "Завантажте файл блокнота"
+
+msgid "Copyright"
+msgstr "Авторське право"
+
+msgid "Download this page"
+msgstr "Завантажте цю сторінку"
+
+msgid "Source repository"
+msgstr "Джерело сховища"
+
+msgid "By"
+msgstr "Автор"
+
+msgid "repository"
+msgstr "сховище"
+
+msgid "Last updated on"
+msgstr "Останнє оновлення:"
+
+msgid "Toggle navigation"
+msgstr "Переключити навігацію"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема книги \"Сфінкс\""
+
+msgid "suggest edit"
+msgstr "запропонувати редагувати"
+
+msgid "Open an issue"
+msgstr "Відкрийте випуск"
+
+msgid "Launch"
+msgstr "Запуск"
+
+msgid "Fullscreen mode"
+msgstr "Повноекранний режим"
+
+msgid "Edit this page"
+msgstr "Редагувати цю сторінку"
+
+msgid "By the"
+msgstr "По"
+
+msgid "next page"
+msgstr "Наступна сторінка"
diff --git a/dev/docs/html/_static/locales/ur/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/ur/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..de8c84b
Binary files /dev/null and b/dev/docs/html/_static/locales/ur/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/ur/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/ur/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..2f77426
--- /dev/null
+++ b/dev/docs/html/_static/locales/ur/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ur\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "پی ڈی ایف پرنٹ کریں"
+
+msgid "Theme by the"
+msgstr "کے ذریعہ تھیم"
+
+msgid "Download source file"
+msgstr "سورس فائل ڈاؤن لوڈ کریں"
+
+msgid "open issue"
+msgstr "کھلا مسئلہ"
+
+msgid "previous page"
+msgstr "سابقہ ​​صفحہ"
+
+msgid "Download notebook file"
+msgstr "نوٹ بک فائل ڈاؤن لوڈ کریں"
+
+msgid "Copyright"
+msgstr "کاپی رائٹ"
+
+msgid "Download this page"
+msgstr "اس صفحے کو ڈاؤن لوڈ کریں"
+
+msgid "Source repository"
+msgstr "ماخذ ذخیرہ"
+
+msgid "By"
+msgstr "بذریعہ"
+
+msgid "Last updated on"
+msgstr "آخری بار تازہ کاری ہوئی"
+
+msgid "Toggle navigation"
+msgstr "نیویگیشن ٹوگل کریں"
+
+msgid "Sphinx Book Theme"
+msgstr "سپنکس بک تھیم"
+
+msgid "suggest edit"
+msgstr "ترمیم کی تجویز کریں"
+
+msgid "Open an issue"
+msgstr "ایک مسئلہ کھولیں"
+
+msgid "Launch"
+msgstr "لانچ کریں"
+
+msgid "Edit this page"
+msgstr "اس صفحے میں ترمیم کریں"
+
+msgid "By the"
+msgstr "کی طرف"
+
+msgid "next page"
+msgstr "اگلا صفحہ"
diff --git a/dev/docs/html/_static/locales/vi/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/vi/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..2bb3255
Binary files /dev/null and b/dev/docs/html/_static/locales/vi/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/vi/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/vi/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..33159f3
--- /dev/null
+++ b/dev/docs/html/_static/locales/vi/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: vi\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "In sang PDF"
+
+msgid "Theme by the"
+msgstr "Chủ đề của"
+
+msgid "Download source file"
+msgstr "Tải xuống tệp nguồn"
+
+msgid "open issue"
+msgstr "vấn đề mở"
+
+msgid "Contents"
+msgstr "Nội dung"
+
+msgid "previous page"
+msgstr "trang trước"
+
+msgid "Download notebook file"
+msgstr "Tải xuống tệp sổ tay"
+
+msgid "Copyright"
+msgstr "Bản quyền"
+
+msgid "Download this page"
+msgstr "Tải xuống trang này"
+
+msgid "Source repository"
+msgstr "Kho nguồn"
+
+msgid "By"
+msgstr "Bởi"
+
+msgid "repository"
+msgstr "kho"
+
+msgid "Last updated on"
+msgstr "Cập nhật lần cuối vào"
+
+msgid "Toggle navigation"
+msgstr "Chuyển đổi điều hướng thành"
+
+msgid "Sphinx Book Theme"
+msgstr "Chủ đề sách nhân sư"
+
+msgid "suggest edit"
+msgstr "đề nghị chỉnh sửa"
+
+msgid "Open an issue"
+msgstr "Mở một vấn đề"
+
+msgid "Launch"
+msgstr "Phóng"
+
+msgid "Fullscreen mode"
+msgstr "Chế độ toàn màn hình"
+
+msgid "Edit this page"
+msgstr "chỉnh sửa trang này"
+
+msgid "By the"
+msgstr "Bằng"
+
+msgid "next page"
+msgstr "Trang tiếp theo"
diff --git a/dev/docs/html/_static/locales/zh_CN/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/zh_CN/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..0e3235d
Binary files /dev/null and b/dev/docs/html/_static/locales/zh_CN/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/zh_CN/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/zh_CN/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..2e519ef
--- /dev/null
+++ b/dev/docs/html/_static/locales/zh_CN/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: zh_CN\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "列印成 PDF"
+
+msgid "Theme by the"
+msgstr "主题作者："
+
+msgid "Download source file"
+msgstr "下载源文件"
+
+msgid "open issue"
+msgstr "创建议题"
+
+msgid "Contents"
+msgstr "目录"
+
+msgid "previous page"
+msgstr "上一页"
+
+msgid "Download notebook file"
+msgstr "下载笔记本文件"
+
+msgid "Copyright"
+msgstr "版权"
+
+msgid "Download this page"
+msgstr "下载此页面"
+
+msgid "Source repository"
+msgstr "源码库"
+
+msgid "By"
+msgstr "作者："
+
+msgid "repository"
+msgstr "仓库"
+
+msgid "Last updated on"
+msgstr "上次更新时间："
+
+msgid "Toggle navigation"
+msgstr "显示或隐藏导航栏"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx Book 主题"
+
+msgid "suggest edit"
+msgstr "提出修改建议"
+
+msgid "Open an issue"
+msgstr "创建议题"
+
+msgid "Launch"
+msgstr "启动"
+
+msgid "Fullscreen mode"
+msgstr "全屏模式"
+
+msgid "Edit this page"
+msgstr "编辑此页面"
+
+msgid "By the"
+msgstr "作者："
+
+msgid "next page"
+msgstr "下一页"
diff --git a/dev/docs/html/_static/locales/zh_TW/LC_MESSAGES/booktheme.mo b/dev/docs/html/_static/locales/zh_TW/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..9116fa9
Binary files /dev/null and b/dev/docs/html/_static/locales/zh_TW/LC_MESSAGES/booktheme.mo differ
diff --git a/dev/docs/html/_static/locales/zh_TW/LC_MESSAGES/booktheme.po b/dev/docs/html/_static/locales/zh_TW/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..beecb07
--- /dev/null
+++ b/dev/docs/html/_static/locales/zh_TW/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: zh_TW\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "列印成 PDF"
+
+msgid "Theme by the"
+msgstr "佈景主題作者："
+
+msgid "Download source file"
+msgstr "下載原始檔"
+
+msgid "open issue"
+msgstr "公開的問題"
+
+msgid "Contents"
+msgstr "目錄"
+
+msgid "previous page"
+msgstr "上一頁"
+
+msgid "Download notebook file"
+msgstr "下載 Notebook 檔案"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Download this page"
+msgstr "下載此頁面"
+
+msgid "Source repository"
+msgstr "來源儲存庫"
+
+msgid "By"
+msgstr "作者："
+
+msgid "repository"
+msgstr "儲存庫"
+
+msgid "Last updated on"
+msgstr "最後更新時間："
+
+msgid "Toggle navigation"
+msgstr "顯示或隱藏導覽列"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx Book 佈景主題"
+
+msgid "suggest edit"
+msgstr "提出修改建議"
+
+msgid "Open an issue"
+msgstr "開啟議題"
+
+msgid "Launch"
+msgstr "啟動"
+
+msgid "Fullscreen mode"
+msgstr "全螢幕模式"
+
+msgid "Edit this page"
+msgstr "編輯此頁面"
+
+msgid "By the"
+msgstr "作者："
+
+msgid "next page"
+msgstr "下一頁"
diff --git a/dev/docs/html/_static/minus.png b/dev/docs/html/_static/minus.png
new file mode 100644
index 0000000..d96755f
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diff --git a/dev/docs/html/_static/nbsphinx-broken-thumbnail.svg b/dev/docs/html/_static/nbsphinx-broken-thumbnail.svg
new file mode 100644
index 0000000..4919ca8
--- /dev/null
+++ b/dev/docs/html/_static/nbsphinx-broken-thumbnail.svg
@@ -0,0 +1,9 @@
+<svg xmlns="http://www.w3.org/2000/svg" width="100" height="100">
+  <style>
+    svg { fill: lightcoral; }
+    @media (prefers-color-scheme: dark) {
+      svg { fill: crimson; }
+    }
+  </style>
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new file mode 100644
index 0000000..a3fb27c
--- /dev/null
+++ b/dev/docs/html/_static/nbsphinx-code-cells.css
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+div.nboutput.container div.output_area pre,
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+div.nboutput.container div.output_area .highlight {
+    border: none;
+    padding: 0;
+    margin: 0;
+    box-shadow: none;
+}
+
+div.nbinput.container > div[class*=highlight],
+div.nboutput.container > div[class*=highlight] {
+    margin: 0;
+}
+
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+div.nboutput.container div.prompt * {
+    background: none;
+}
+
+div.nboutput.container div.output_area .highlight,
+div.nboutput.container div.output_area pre {
+    background: unset;
+}
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+div.nboutput.container div.output_area div.highlight {
+    color: unset;  /* override Pygments text color */
+}
+
+/* avoid gaps between output lines */
+div.nboutput.container div[class*=highlight] pre {
+    line-height: normal;
+}
+
+/* input/output containers */
+div.nbinput.container,
+div.nboutput.container {
+    display: -webkit-flex;
+    display: flex;
+    align-items: flex-start;
+    margin: 0;
+    width: 100%;
+}
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+}
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+    width: 4.5ex;
+    padding-top: 5px;
+    position: relative;
+    user-select: none;
+}
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+div.nboutput.container div.prompt > div {
+    position: absolute;
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+    margin-right: 0.3ex;
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+        text-align: left;
+        padding: 0.4em;
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+    div.nboutput.container div.prompt.empty {
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+    }
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+    div.nbinput.container div.prompt > div,
+    div.nboutput.container div.prompt > div {
+        position: unset;
+    }
+}
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+div.nbinput.container div.prompt pre,
+div.nboutput.container div.prompt pre {
+    overflow: hidden;
+    white-space: pre;
+}
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+div.nbinput.container div.input_area,
+div.nboutput.container div.output_area {
+    -webkit-flex: 1;
+    flex: 1;
+    overflow: auto;
+}
+@media (max-width: 540px) {
+    div.nbinput.container div.input_area,
+    div.nboutput.container div.output_area {
+        width: 100%;
+    }
+}
+
+/* input area */
+div.nbinput.container div.input_area {
+    border: 1px solid #e0e0e0;
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+    /*background: #f5f5f5;*/
+}
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+div.nboutput.container div.output_area div[class*=highlight] > pre,
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+div.nboutput.container div.output_area > div.output_javascript,
+div.nboutput.container div.output_area:not(.rendered_html) > img{
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+    margin: 0;
+}
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+  vertical-align: middle;
+  padding: 0.5em 0.5em;
+  line-height: normal;
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+    margin-bottom: 1em;
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+    border: 1px dotted currentColor;
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+    text-align: center;
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+html[data-theme="dark"] .highlight span.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+html[data-theme="dark"] .highlight td.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
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+html[data-theme="dark"] .highlight .err { color: #ffa07a } /* Error */
+html[data-theme="dark"] .highlight .k { color: #dcc6e0 } /* Keyword */
+html[data-theme="dark"] .highlight .l { color: #ffd900 } /* Literal */
+html[data-theme="dark"] .highlight .n { color: #f8f8f2 } /* Name */
+html[data-theme="dark"] .highlight .o { color: #abe338 } /* Operator */
+html[data-theme="dark"] .highlight .p { color: #f8f8f2 } /* Punctuation */
+html[data-theme="dark"] .highlight .ch { color: #ffd900 } /* Comment.Hashbang */
+html[data-theme="dark"] .highlight .cm { color: #ffd900 } /* Comment.Multiline */
+html[data-theme="dark"] .highlight .cp { color: #ffd900 } /* Comment.Preproc */
+html[data-theme="dark"] .highlight .cpf { color: #ffd900 } /* Comment.PreprocFile */
+html[data-theme="dark"] .highlight .c1 { color: #ffd900 } /* Comment.Single */
+html[data-theme="dark"] .highlight .cs { color: #ffd900 } /* Comment.Special */
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+html[data-theme="dark"] .highlight .sh { color: #abe338 } /* Literal.String.Heredoc */
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+html[data-theme="dark"] .highlight .sx { color: #abe338 } /* Literal.String.Other */
+html[data-theme="dark"] .highlight .sr { color: #ffa07a } /* Literal.String.Regex */
+html[data-theme="dark"] .highlight .s1 { color: #abe338 } /* Literal.String.Single */
+html[data-theme="dark"] .highlight .ss { color: #00e0e0 } /* Literal.String.Symbol */
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new file mode 100644
index 0000000..6cbf559
--- /dev/null
+++ b/dev/docs/html/_static/sbt-webpack-macros.html
@@ -0,0 +1,11 @@
+<!--
+  All these macros are auto-generated and must **NOT** be edited by hand.
+  See the webpack.config.js file, to learn more about how this is generated.
+-->
+{% macro head_pre_bootstrap() %}
+  <link href="{{ pathto('_static/styles/sphinx-book-theme.css', 1) }}" rel="stylesheet">
+{% endmacro %}
+
+{% macro body_post() %}
+  <script src="{{ pathto('_static/scripts/sphinx-book-theme.js', 1) }}"></script>
+{% endmacro %}
diff --git a/dev/docs/html/_static/scripts/bootstrap.js b/dev/docs/html/_static/scripts/bootstrap.js
new file mode 100644
index 0000000..c8178de
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je(this._element,t)}_keydown(t){if(/input|textarea/i.test(t.target.tagName))return;const e=oi[t.key];e&&(t.preventDefault(),this._slide(this._directionToOrder(e)))}_getItemIndex(t){return this._getItems().indexOf(t)}_setActiveIndicatorElement(t){if(!this._indicatorsElement)return;const e=we.findOne(ii,this._indicatorsElement);e.classList.remove(ei),e.removeAttribute("aria-current");const i=we.findOne(`[data-bs-slide-to="${t}"]`,this._indicatorsElement);i&&(i.classList.add(ei),i.setAttribute("aria-current","true"))}_updateInterval(){const t=this._activeElement||this._getActive();if(!t)return;const e=Number.parseInt(t.getAttribute("data-bs-interval"),10);this._config.interval=e||this._config.defaultInterval}_slide(t,e=null){if(this._isSliding)return;const i=this._getActive(),n=t===ze,s=e||Gt(this._getItems(),i,n,this._config.wrap);if(s===i)return;const 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Kt()?t===Re?qe:Ve:t===Re?Ve:qe}static jQueryInterface(t){return this.each((function(){const e=li.getOrCreateInstance(this,t);if("number"!=typeof t){if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t]()}}else e.to(t)}))}}fe.on(document,Ze,"[data-bs-slide], [data-bs-slide-to]",(function(t){const e=we.getElementFromSelector(this);if(!e||!e.classList.contains(ti))return;t.preventDefault();const i=li.getOrCreateInstance(e),n=this.getAttribute("data-bs-slide-to");return n?(i.to(n),void i._maybeEnableCycle()):"next"===_e.getDataAttribute(this,"slide")?(i.next(),void i._maybeEnableCycle()):(i.prev(),void i._maybeEnableCycle())})),fe.on(window,Je,(()=>{const t=we.find('[data-bs-ride="carousel"]');for(const e of t)li.getOrCreateInstance(e)})),Qt(li);const ci=".bs.collapse",hi=`show${ci}`,di=`shown${ci}`,ui=`hide${ci}`,fi=`hidden${ci}`,pi=`click${ci}.data-api`,mi="show",gi="collapse",_i="collapsing",bi=`:scope .${gi} .${gi}`,vi='[data-bs-toggle="collapse"]',yi={parent:null,toggle:!0},wi={parent:"(null|element)",toggle:"boolean"};class Ei extends ve{constructor(t,e){super(t,e),this._isTransitioning=!1,this._triggerArray=[];const i=we.find(vi);for(const t of i){const e=we.getSelectorFromElement(t),i=we.find(e).filter((t=>t===this._element));null!==e&&i.length&&this._triggerArray.push(t)}this._initializeChildren(),this._config.parent||this._addAriaAndCollapsedClass(this._triggerArray,this._isShown()),this._config.toggle&&this.toggle()}static get Default(){return yi}static get DefaultType(){return wi}static get NAME(){return"collapse"}toggle(){this._isShown()?this.hide():this.show()}show(){if(this._isTransitioning||this._isShown())return;let t=[];if(this._config.parent&&(t=this._getFirstLevelChildren(".collapse.show, .collapse.collapsing").filter((t=>t!==this._element)).map((t=>Ei.getOrCreateInstance(t,{toggle:!1})))),t.length&&t[0]._isTransitioning)return;if(fe.trigger(this._element,hi).defaultPrevented)return;for(const e of t)e.hide();const e=this._getDimension();this._element.classList.remove(gi),this._element.classList.add(_i),this._element.style[e]=0,this._addAriaAndCollapsedClass(this._triggerArray,!0),this._isTransitioning=!0;const i=`scroll${e[0].toUpperCase()+e.slice(1)}`;this._queueCallback((()=>{this._isTransitioning=!1,this._element.classList.remove(_i),this._element.classList.add(gi,mi),this._element.style[e]="",fe.trigger(this._element,di)}),this._element,!0),this._element.style[e]=`${this._element[i]}px`}hide(){if(this._isTransitioning||!this._isShown())return;if(fe.trigger(this._element,ui).defaultPrevented)return;const t=this._getDimension();this._element.style[t]=`${this._element.getBoundingClientRect()[t]}px`,qt(this._element),this._element.classList.add(_i),this._element.classList.remove(gi,mi);for(const t of this._triggerArray){const e=we.getElementFromSelector(t);e&&!this._isShown(e)&&this._addAriaAndCollapsedClass([t],!1)}this._isTransitioning=!0,this._element.style[t]="",this._queueCallback((()=>{this._isTransitioning=!1,this._element.classList.remove(_i),this._element.classList.add(gi),fe.trigger(this._element,fi)}),this._element,!0)}_isShown(t=this._element){return t.classList.contains(mi)}_configAfterMerge(t){return t.toggle=Boolean(t.toggle),t.parent=Ht(t.parent),t}_getDimension(){return this._element.classList.contains("collapse-horizontal")?"width":"height"}_initializeChildren(){if(!this._config.parent)return;const t=this._getFirstLevelChildren(vi);for(const e of t){const t=we.getElementFromSelector(e);t&&this._addAriaAndCollapsedClass([e],this._isShown(t))}}_getFirstLevelChildren(t){const e=we.find(bi,this._config.parent);return we.find(t,this._config.parent).filter((t=>!e.includes(t)))}_addAriaAndCollapsedClass(t,e){if(t.length)for(const i of t)i.classList.toggle("collapsed",!e),i.setAttribute("aria-expanded",e)}static jQueryInterface(t){const e={};return"string"==typeof t&&/show|hide/.test(t)&&(e.toggle=!1),this.each((function(){const i=Ei.getOrCreateInstance(this,e);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t]()}}))}}fe.on(document,pi,vi,(function(t){("A"===t.target.tagName||t.delegateTarget&&"A"===t.delegateTarget.tagName)&&t.preventDefault();for(const t of we.getMultipleElementsFromSelector(this))Ei.getOrCreateInstance(t,{toggle:!1}).toggle()})),Qt(Ei);const Ai="dropdown",Ti=".bs.dropdown",Ci=".data-api",Oi="ArrowUp",xi="ArrowDown",ki=`hide${Ti}`,Li=`hidden${Ti}`,Si=`show${Ti}`,Di=`shown${Ti}`,$i=`click${Ti}${Ci}`,Ii=`keydown${Ti}${Ci}`,Ni=`keyup${Ti}${Ci}`,Pi="show",Mi='[data-bs-toggle="dropdown"]:not(.disabled):not(:disabled)',ji=`${Mi}.${Pi}`,Fi=".dropdown-menu",Hi=Kt()?"top-end":"top-start",Bi=Kt()?"top-start":"top-end",Wi=Kt()?"bottom-end":"bottom-start",zi=Kt()?"bottom-start":"bottom-end",Ri=Kt()?"left-start":"right-start",qi=Kt()?"right-start":"left-start",Vi={autoClose:!0,boundary:"clippingParents",display:"dynamic",offset:[0,2],popperConfig:null,reference:"toggle"},Yi={autoClose:"(boolean|string)",boundary:"(string|element)",display:"string",offset:"(array|string|function)",popperConfig:"(null|object|function)",reference:"(string|element|object)"};class Ki extends ve{constructor(t,e){super(t,e),this._popper=null,this._parent=this._element.parentNode,this._menu=we.next(this._element,Fi)[0]||we.prev(this._element,Fi)[0]||we.findOne(Fi,this._parent),this._inNavbar=this._detectNavbar()}static get Default(){return Vi}static get DefaultType(){return Yi}static get NAME(){return Ai}toggle(){return this._isShown()?this.hide():this.show()}show(){if(Wt(this._element)||this._isShown())return;const t={relatedTarget:this._element};if(!fe.trigger(this._element,Si,t).defaultPrevented){if(this._createPopper(),"ontouchstart"in document.documentElement&&!this._parent.closest(".navbar-nav"))for(const t of[].concat(...document.body.children))fe.on(t,"mouseover",Rt);this._element.focus(),this._element.setAttribute("aria-expanded",!0),this._menu.classList.add(Pi),this._element.classList.add(Pi),fe.trigger(this._element,Di,t)}}hide(){if(Wt(this._element)||!this._isShown())return;const t={relatedTarget:this._element};this._completeHide(t)}dispose(){this._popper&&this._popper.destroy(),super.dispose()}update(){this._inNavbar=this._detectNavbar(),this._popper&&this._popper.update()}_completeHide(t){if(!fe.trigger(this._element,ki,t).defaultPrevented){if("ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))fe.off(t,"mouseover",Rt);this._popper&&this._popper.destroy(),this._menu.classList.remove(Pi),this._element.classList.remove(Pi),this._element.setAttribute("aria-expanded","false"),_e.removeDataAttribute(this._menu,"popper"),fe.trigger(this._element,Li,t)}}_getConfig(t){if("object"==typeof(t=super._getConfig(t)).reference&&!Ft(t.reference)&&"function"!=typeof t.reference.getBoundingClientRect)throw new TypeError(`${Ai.toUpperCase()}: Option "reference" provided type "object" without a required "getBoundingClientRect" method.`);return t}_createPopper(){if(void 0===e)throw new TypeError("Bootstrap's dropdowns require Popper (https://popper.js.org)");let t=this._element;"parent"===this._config.reference?t=this._parent:Ft(this._config.reference)?t=Ht(this._config.reference):"object"==typeof this._config.reference&&(t=this._config.reference);const i=this._getPopperConfig();this._popper=Dt(t,this._menu,i)}_isShown(){return this._menu.classList.contains(Pi)}_getPlacement(){const t=this._parent;if(t.classList.contains("dropend"))return Ri;if(t.classList.contains("dropstart"))return qi;if(t.classList.contains("dropup-center"))return"top";if(t.classList.contains("dropdown-center"))return"bottom";const e="end"===getComputedStyle(this._menu).getPropertyValue("--bs-position").trim();return t.classList.contains("dropup")?e?Bi:Hi:e?zi:Wi}_detectNavbar(){return null!==this._element.closest(".navbar")}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_getPopperConfig(){const t={placement:this._getPlacement(),modifiers:[{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"offset",options:{offset:this._getOffset()}}]};return(this._inNavbar||"static"===this._config.display)&&(_e.setDataAttribute(this._menu,"popper","static"),t.modifiers=[{name:"applyStyles",enabled:!1}]),{...t,...Xt(this._config.popperConfig,[t])}}_selectMenuItem({key:t,target:e}){const i=we.find(".dropdown-menu .dropdown-item:not(.disabled):not(:disabled)",this._menu).filter((t=>Bt(t)));i.length&&Gt(i,e,t===xi,!i.includes(e)).focus()}static jQueryInterface(t){return this.each((function(){const e=Ki.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}static clearMenus(t){if(2===t.button||"keyup"===t.type&&"Tab"!==t.key)return;const e=we.find(ji);for(const i of e){const e=Ki.getInstance(i);if(!e||!1===e._config.autoClose)continue;const n=t.composedPath(),s=n.includes(e._menu);if(n.includes(e._element)||"inside"===e._config.autoClose&&!s||"outside"===e._config.autoClose&&s)continue;if(e._menu.contains(t.target)&&("keyup"===t.type&&"Tab"===t.key||/input|select|option|textarea|form/i.test(t.target.tagName)))continue;const o={relatedTarget:e._element};"click"===t.type&&(o.clickEvent=t),e._completeHide(o)}}static dataApiKeydownHandler(t){const e=/input|textarea/i.test(t.target.tagName),i="Escape"===t.key,n=[Oi,xi].includes(t.key);if(!n&&!i)return;if(e&&!i)return;t.preventDefault();const s=this.matches(Mi)?this:we.prev(this,Mi)[0]||we.next(this,Mi)[0]||we.findOne(Mi,t.delegateTarget.parentNode),o=Ki.getOrCreateInstance(s);if(n)return t.stopPropagation(),o.show(),void o._selectMenuItem(t);o._isShown()&&(t.stopPropagation(),o.hide(),s.focus())}}fe.on(document,Ii,Mi,Ki.dataApiKeydownHandler),fe.on(document,Ii,Fi,Ki.dataApiKeydownHandler),fe.on(document,$i,Ki.clearMenus),fe.on(document,Ni,Ki.clearMenus),fe.on(document,$i,Mi,(function(t){t.preventDefault(),Ki.getOrCreateInstance(this).toggle()})),Qt(Ki);const Qi="backdrop",Xi="show",Ui=`mousedown.bs.${Qi}`,Gi={className:"modal-backdrop",clickCallback:null,isAnimated:!1,isVisible:!0,rootElement:"body"},Ji={className:"string",clickCallback:"(function|null)",isAnimated:"boolean",isVisible:"boolean",rootElement:"(element|string)"};class Zi extends be{constructor(t){super(),this._config=this._getConfig(t),this._isAppended=!1,this._element=null}static get Default(){return Gi}static get DefaultType(){return Ji}static get NAME(){return Qi}show(t){if(!this._config.isVisible)return void Xt(t);this._append();const e=this._getElement();this._config.isAnimated&&qt(e),e.classList.add(Xi),this._emulateAnimation((()=>{Xt(t)}))}hide(t){this._config.isVisible?(this._getElement().classList.remove(Xi),this._emulateAnimation((()=>{this.dispose(),Xt(t)}))):Xt(t)}dispose(){this._isAppended&&(fe.off(this._element,Ui),this._element.remove(),this._isAppended=!1)}_getElement(){if(!this._element){const t=document.createElement("div");t.className=this._config.className,this._config.isAnimated&&t.classList.add("fade"),this._element=t}return this._element}_configAfterMerge(t){return t.rootElement=Ht(t.rootElement),t}_append(){if(this._isAppended)return;const t=this._getElement();this._config.rootElement.append(t),fe.on(t,Ui,(()=>{Xt(this._config.clickCallback)})),this._isAppended=!0}_emulateAnimation(t){Ut(t,this._getElement(),this._config.isAnimated)}}const tn=".bs.focustrap",en=`focusin${tn}`,nn=`keydown.tab${tn}`,sn="backward",on={autofocus:!0,trapElement:null},rn={autofocus:"boolean",trapElement:"element"};class an extends be{constructor(t){super(),this._config=this._getConfig(t),this._isActive=!1,this._lastTabNavDirection=null}static get Default(){return on}static get DefaultType(){return rn}static get NAME(){return"focustrap"}activate(){this._isActive||(this._config.autofocus&&this._config.trapElement.focus(),fe.off(document,tn),fe.on(document,en,(t=>this._handleFocusin(t))),fe.on(document,nn,(t=>this._handleKeydown(t))),this._isActive=!0)}deactivate(){this._isActive&&(this._isActive=!1,fe.off(document,tn))}_handleFocusin(t){const{trapElement:e}=this._config;if(t.target===document||t.target===e||e.contains(t.target))return;const i=we.focusableChildren(e);0===i.length?e.focus():this._lastTabNavDirection===sn?i[i.length-1].focus():i[0].focus()}_handleKeydown(t){"Tab"===t.key&&(this._lastTabNavDirection=t.shiftKey?sn:"forward")}}const ln=".fixed-top, .fixed-bottom, .is-fixed, .sticky-top",cn=".sticky-top",hn="padding-right",dn="margin-right";class un{constructor(){this._element=document.body}getWidth(){const t=document.documentElement.clientWidth;return Math.abs(window.innerWidth-t)}hide(){const t=this.getWidth();this._disableOverFlow(),this._setElementAttributes(this._element,hn,(e=>e+t)),this._setElementAttributes(ln,hn,(e=>e+t)),this._setElementAttributes(cn,dn,(e=>e-t))}reset(){this._resetElementAttributes(this._element,"overflow"),this._resetElementAttributes(this._element,hn),this._resetElementAttributes(ln,hn),this._resetElementAttributes(cn,dn)}isOverflowing(){return this.getWidth()>0}_disableOverFlow(){this._saveInitialAttribute(this._element,"overflow"),this._element.style.overflow="hidden"}_setElementAttributes(t,e,i){const n=this.getWidth();this._applyManipulationCallback(t,(t=>{if(t!==this._element&&window.innerWidth>t.clientWidth+n)return;this._saveInitialAttribute(t,e);const s=window.getComputedStyle(t).getPropertyValue(e);t.style.setProperty(e,`${i(Number.parseFloat(s))}px`)}))}_saveInitialAttribute(t,e){const i=t.style.getPropertyValue(e);i&&_e.setDataAttribute(t,e,i)}_resetElementAttributes(t,e){this._applyManipulationCallback(t,(t=>{const i=_e.getDataAttribute(t,e);null!==i?(_e.removeDataAttribute(t,e),t.style.setProperty(e,i)):t.style.removeProperty(e)}))}_applyManipulationCallback(t,e){if(Ft(t))e(t);else for(const i of we.find(t,this._element))e(i)}}const fn=".bs.modal",pn=`hide${fn}`,mn=`hidePrevented${fn}`,gn=`hidden${fn}`,_n=`show${fn}`,bn=`shown${fn}`,vn=`resize${fn}`,yn=`click.dismiss${fn}`,wn=`mousedown.dismiss${fn}`,En=`keydown.dismiss${fn}`,An=`click${fn}.data-api`,Tn="modal-open",Cn="show",On="modal-static",xn={backdrop:!0,focus:!0,keyboard:!0},kn={backdrop:"(boolean|string)",focus:"boolean",keyboard:"boolean"};class Ln extends ve{constructor(t,e){super(t,e),this._dialog=we.findOne(".modal-dialog",this._element),this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._isShown=!1,this._isTransitioning=!1,this._scrollBar=new un,this._addEventListeners()}static get Default(){return xn}static get DefaultType(){return kn}static get NAME(){return"modal"}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||this._isTransitioning||fe.trigger(this._element,_n,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._isTransitioning=!0,this._scrollBar.hide(),document.body.classList.add(Tn),this._adjustDialog(),this._backdrop.show((()=>this._showElement(t))))}hide(){this._isShown&&!this._isTransitioning&&(fe.trigger(this._element,pn).defaultPrevented||(this._isShown=!1,this._isTransitioning=!0,this._focustrap.deactivate(),this._element.classList.remove(Cn),this._queueCallback((()=>this._hideModal()),this._element,this._isAnimated())))}dispose(){fe.off(window,fn),fe.off(this._dialog,fn),this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}handleUpdate(){this._adjustDialog()}_initializeBackDrop(){return new Zi({isVisible:Boolean(this._config.backdrop),isAnimated:this._isAnimated()})}_initializeFocusTrap(){return new an({trapElement:this._element})}_showElement(t){document.body.contains(this._element)||document.body.append(this._element),this._element.style.display="block",this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.scrollTop=0;const e=we.findOne(".modal-body",this._dialog);e&&(e.scrollTop=0),qt(this._element),this._element.classList.add(Cn),this._queueCallback((()=>{this._config.focus&&this._focustrap.activate(),this._isTransitioning=!1,fe.trigger(this._element,bn,{relatedTarget:t})}),this._dialog,this._isAnimated())}_addEventListeners(){fe.on(this._element,En,(t=>{"Escape"===t.key&&(this._config.keyboard?this.hide():this._triggerBackdropTransition())})),fe.on(window,vn,(()=>{this._isShown&&!this._isTransitioning&&this._adjustDialog()})),fe.on(this._element,wn,(t=>{fe.one(this._element,yn,(e=>{this._element===t.target&&this._element===e.target&&("static"!==this._config.backdrop?this._config.backdrop&&this.hide():this._triggerBackdropTransition())}))}))}_hideModal(){this._element.style.display="none",this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._isTransitioning=!1,this._backdrop.hide((()=>{document.body.classList.remove(Tn),this._resetAdjustments(),this._scrollBar.reset(),fe.trigger(this._element,gn)}))}_isAnimated(){return this._element.classList.contains("fade")}_triggerBackdropTransition(){if(fe.trigger(this._element,mn).defaultPrevented)return;const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._element.style.overflowY;"hidden"===e||this._element.classList.contains(On)||(t||(this._element.style.overflowY="hidden"),this._element.classList.add(On),this._queueCallback((()=>{this._element.classList.remove(On),this._queueCallback((()=>{this._element.style.overflowY=e}),this._dialog)}),this._dialog),this._element.focus())}_adjustDialog(){const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._scrollBar.getWidth(),i=e>0;if(i&&!t){const t=Kt()?"paddingLeft":"paddingRight";this._element.style[t]=`${e}px`}if(!i&&t){const t=Kt()?"paddingRight":"paddingLeft";this._element.style[t]=`${e}px`}}_resetAdjustments(){this._element.style.paddingLeft="",this._element.style.paddingRight=""}static jQueryInterface(t,e){return this.each((function(){const i=Ln.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t](e)}}))}}fe.on(document,An,'[data-bs-toggle="modal"]',(function(t){const e=we.getElementFromSelector(this);["A","AREA"].includes(this.tagName)&&t.preventDefault(),fe.one(e,_n,(t=>{t.defaultPrevented||fe.one(e,gn,(()=>{Bt(this)&&this.focus()}))}));const i=we.findOne(".modal.show");i&&Ln.getInstance(i).hide(),Ln.getOrCreateInstance(e).toggle(this)})),Ee(Ln),Qt(Ln);const Sn=".bs.offcanvas",Dn=".data-api",$n=`load${Sn}${Dn}`,In="show",Nn="showing",Pn="hiding",Mn=".offcanvas.show",jn=`show${Sn}`,Fn=`shown${Sn}`,Hn=`hide${Sn}`,Bn=`hidePrevented${Sn}`,Wn=`hidden${Sn}`,zn=`resize${Sn}`,Rn=`click${Sn}${Dn}`,qn=`keydown.dismiss${Sn}`,Vn={backdrop:!0,keyboard:!0,scroll:!1},Yn={backdrop:"(boolean|string)",keyboard:"boolean",scroll:"boolean"};class Kn extends ve{constructor(t,e){super(t,e),this._isShown=!1,this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._addEventListeners()}static get Default(){return Vn}static get DefaultType(){return Yn}static get NAME(){return"offcanvas"}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||fe.trigger(this._element,jn,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._backdrop.show(),this._config.scroll||(new un).hide(),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.classList.add(Nn),this._queueCallback((()=>{this._config.scroll&&!this._config.backdrop||this._focustrap.activate(),this._element.classList.add(In),this._element.classList.remove(Nn),fe.trigger(this._element,Fn,{relatedTarget:t})}),this._element,!0))}hide(){this._isShown&&(fe.trigger(this._element,Hn).defaultPrevented||(this._focustrap.deactivate(),this._element.blur(),this._isShown=!1,this._element.classList.add(Pn),this._backdrop.hide(),this._queueCallback((()=>{this._element.classList.remove(In,Pn),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._config.scroll||(new un).reset(),fe.trigger(this._element,Wn)}),this._element,!0)))}dispose(){this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}_initializeBackDrop(){const t=Boolean(this._config.backdrop);return new Zi({className:"offcanvas-backdrop",isVisible:t,isAnimated:!0,rootElement:this._element.parentNode,clickCallback:t?()=>{"static"!==this._config.backdrop?this.hide():fe.trigger(this._element,Bn)}:null})}_initializeFocusTrap(){return new an({trapElement:this._element})}_addEventListeners(){fe.on(this._element,qn,(t=>{"Escape"===t.key&&(this._config.keyboard?this.hide():fe.trigger(this._element,Bn))}))}static jQueryInterface(t){return this.each((function(){const e=Kn.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}fe.on(document,Rn,'[data-bs-toggle="offcanvas"]',(function(t){const e=we.getElementFromSelector(this);if(["A","AREA"].includes(this.tagName)&&t.preventDefault(),Wt(this))return;fe.one(e,Wn,(()=>{Bt(this)&&this.focus()}));const i=we.findOne(Mn);i&&i!==e&&Kn.getInstance(i).hide(),Kn.getOrCreateInstance(e).toggle(this)})),fe.on(window,$n,(()=>{for(const t of we.find(Mn))Kn.getOrCreateInstance(t).show()})),fe.on(window,zn,(()=>{for(const t of we.find("[aria-modal][class*=show][class*=offcanvas-]"))"fixed"!==getComputedStyle(t).position&&Kn.getOrCreateInstance(t).hide()})),Ee(Kn),Qt(Kn);const Qn={"*":["class","dir","id","lang","role",/^aria-[\w-]*$/i],a:["target","href","title","rel"],area:[],b:[],br:[],col:[],code:[],dd:[],div:[],dl:[],dt:[],em:[],hr:[],h1:[],h2:[],h3:[],h4:[],h5:[],h6:[],i:[],img:["src","srcset","alt","title","width","height"],li:[],ol:[],p:[],pre:[],s:[],small:[],span:[],sub:[],sup:[],strong:[],u:[],ul:[]},Xn=new Set(["background","cite","href","itemtype","longdesc","poster","src","xlink:href"]),Un=/^(?!javascript:)(?:[a-z0-9+.-]+:|[^&:/?#]*(?:[/?#]|$))/i,Gn=(t,e)=>{const i=t.nodeName.toLowerCase();return e.includes(i)?!Xn.has(i)||Boolean(Un.test(t.nodeValue)):e.filter((t=>t instanceof RegExp)).some((t=>t.test(i)))},Jn={allowList:Qn,content:{},extraClass:"",html:!1,sanitize:!0,sanitizeFn:null,template:"<div></div>"},Zn={allowList:"object",content:"object",extraClass:"(string|function)",html:"boolean",sanitize:"boolean",sanitizeFn:"(null|function)",template:"string"},ts={entry:"(string|element|function|null)",selector:"(string|element)"};class es extends be{constructor(t){super(),this._config=this._getConfig(t)}static get Default(){return Jn}static get DefaultType(){return Zn}static get NAME(){return"TemplateFactory"}getContent(){return Object.values(this._config.content).map((t=>this._resolvePossibleFunction(t))).filter(Boolean)}hasContent(){return this.getContent().length>0}changeContent(t){return this._checkContent(t),this._config.content={...this._config.content,...t},this}toHtml(){const t=document.createElement("div");t.innerHTML=this._maybeSanitize(this._config.template);for(const[e,i]of Object.entries(this._config.content))this._setContent(t,i,e);const e=t.children[0],i=this._resolvePossibleFunction(this._config.extraClass);return i&&e.classList.add(...i.split(" ")),e}_typeCheckConfig(t){super._typeCheckConfig(t),this._checkContent(t.content)}_checkContent(t){for(const[e,i]of Object.entries(t))super._typeCheckConfig({selector:e,entry:i},ts)}_setContent(t,e,i){const n=we.findOne(i,t);n&&((e=this._resolvePossibleFunction(e))?Ft(e)?this._putElementInTemplate(Ht(e),n):this._config.html?n.innerHTML=this._maybeSanitize(e):n.textContent=e:n.remove())}_maybeSanitize(t){return this._config.sanitize?function(t,e,i){if(!t.length)return t;if(i&&"function"==typeof i)return i(t);const n=(new window.DOMParser).parseFromString(t,"text/html"),s=[].concat(...n.body.querySelectorAll("*"));for(const t of s){const i=t.nodeName.toLowerCase();if(!Object.keys(e).includes(i)){t.remove();continue}const n=[].concat(...t.attributes),s=[].concat(e["*"]||[],e[i]||[]);for(const e of n)Gn(e,s)||t.removeAttribute(e.nodeName)}return n.body.innerHTML}(t,this._config.allowList,this._config.sanitizeFn):t}_resolvePossibleFunction(t){return Xt(t,[this])}_putElementInTemplate(t,e){if(this._config.html)return e.innerHTML="",void e.append(t);e.textContent=t.textContent}}const is=new Set(["sanitize","allowList","sanitizeFn"]),ns="fade",ss="show",os=".tooltip-inner",rs=".modal",as="hide.bs.modal",ls="hover",cs="focus",hs={AUTO:"auto",TOP:"top",RIGHT:Kt()?"left":"right",BOTTOM:"bottom",LEFT:Kt()?"right":"left"},ds={allowList:Qn,animation:!0,boundary:"clippingParents",container:!1,customClass:"",delay:0,fallbackPlacements:["top","right","bottom","left"],html:!1,offset:[0,6],placement:"top",popperConfig:null,sanitize:!0,sanitizeFn:null,selector:!1,template:'<div class="tooltip" role="tooltip"><div class="tooltip-arrow"></div><div class="tooltip-inner"></div></div>',title:"",trigger:"hover focus"},us={allowList:"object",animation:"boolean",boundary:"(string|element)",container:"(string|element|boolean)",customClass:"(string|function)",delay:"(number|object)",fallbackPlacements:"array",html:"boolean",offset:"(array|string|function)",placement:"(string|function)",popperConfig:"(null|object|function)",sanitize:"boolean",sanitizeFn:"(null|function)",selector:"(string|boolean)",template:"string",title:"(string|element|function)",trigger:"string"};class fs extends ve{constructor(t,i){if(void 0===e)throw new TypeError("Bootstrap's tooltips require Popper (https://popper.js.org)");super(t,i),this._isEnabled=!0,this._timeout=0,this._isHovered=null,this._activeTrigger={},this._popper=null,this._templateFactory=null,this._newContent=null,this.tip=null,this._setListeners(),this._config.selector||this._fixTitle()}static get Default(){return ds}static get DefaultType(){return us}static get NAME(){return"tooltip"}enable(){this._isEnabled=!0}disable(){this._isEnabled=!1}toggleEnabled(){this._isEnabled=!this._isEnabled}toggle(){this._isEnabled&&(this._activeTrigger.click=!this._activeTrigger.click,this._isShown()?this._leave():this._enter())}dispose(){clearTimeout(this._timeout),fe.off(this._element.closest(rs),as,this._hideModalHandler),this._element.getAttribute("data-bs-original-title")&&this._element.setAttribute("title",this._element.getAttribute("data-bs-original-title")),this._disposePopper(),super.dispose()}show(){if("none"===this._element.style.display)throw new Error("Please use show on visible elements");if(!this._isWithContent()||!this._isEnabled)return;const t=fe.trigger(this._element,this.constructor.eventName("show")),e=(zt(this._element)||this._element.ownerDocument.documentElement).contains(this._element);if(t.defaultPrevented||!e)return;this._disposePopper();const i=this._getTipElement();this._element.setAttribute("aria-describedby",i.getAttribute("id"));const{container:n}=this._config;if(this._element.ownerDocument.documentElement.contains(this.tip)||(n.append(i),fe.trigger(this._element,this.constructor.eventName("inserted"))),this._popper=this._createPopper(i),i.classList.add(ss),"ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))fe.on(t,"mouseover",Rt);this._queueCallback((()=>{fe.trigger(this._element,this.constructor.eventName("shown")),!1===this._isHovered&&this._leave(),this._isHovered=!1}),this.tip,this._isAnimated())}hide(){if(this._isShown()&&!fe.trigger(this._element,this.constructor.eventName("hide")).defaultPrevented){if(this._getTipElement().classList.remove(ss),"ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))fe.off(t,"mouseover",Rt);this._activeTrigger.click=!1,this._activeTrigger[cs]=!1,this._activeTrigger[ls]=!1,this._isHovered=null,this._queueCallback((()=>{this._isWithActiveTrigger()||(this._isHovered||this._disposePopper(),this._element.removeAttribute("aria-describedby"),fe.trigger(this._element,this.constructor.eventName("hidden")))}),this.tip,this._isAnimated())}}update(){this._popper&&this._popper.update()}_isWithContent(){return Boolean(this._getTitle())}_getTipElement(){return this.tip||(this.tip=this._createTipElement(this._newContent||this._getContentForTemplate())),this.tip}_createTipElement(t){const e=this._getTemplateFactory(t).toHtml();if(!e)return null;e.classList.remove(ns,ss),e.classList.add(`bs-${this.constructor.NAME}-auto`);const i=(t=>{do{t+=Math.floor(1e6*Math.random())}while(document.getElementById(t));return t})(this.constructor.NAME).toString();return e.setAttribute("id",i),this._isAnimated()&&e.classList.add(ns),e}setContent(t){this._newContent=t,this._isShown()&&(this._disposePopper(),this.show())}_getTemplateFactory(t){return this._templateFactory?this._templateFactory.changeContent(t):this._templateFactory=new es({...this._config,content:t,extraClass:this._resolvePossibleFunction(this._config.customClass)}),this._templateFactory}_getContentForTemplate(){return{[os]:this._getTitle()}}_getTitle(){return this._resolvePossibleFunction(this._config.title)||this._element.getAttribute("data-bs-original-title")}_initializeOnDelegatedTarget(t){return this.constructor.getOrCreateInstance(t.delegateTarget,this._getDelegateConfig())}_isAnimated(){return this._config.animation||this.tip&&this.tip.classList.contains(ns)}_isShown(){return this.tip&&this.tip.classList.contains(ss)}_createPopper(t){const e=Xt(this._config.placement,[this,t,this._element]),i=hs[e.toUpperCase()];return Dt(this._element,t,this._getPopperConfig(i))}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_resolvePossibleFunction(t){return Xt(t,[this._element])}_getPopperConfig(t){const e={placement:t,modifiers:[{name:"flip",options:{fallbackPlacements:this._config.fallbackPlacements}},{name:"offset",options:{offset:this._getOffset()}},{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"arrow",options:{element:`.${this.constructor.NAME}-arrow`}},{name:"preSetPlacement",enabled:!0,phase:"beforeMain",fn:t=>{this._getTipElement().setAttribute("data-popper-placement",t.state.placement)}}]};return{...e,...Xt(this._config.popperConfig,[e])}}_setListeners(){const t=this._config.trigger.split(" ");for(const e of t)if("click"===e)fe.on(this._element,this.constructor.eventName("click"),this._config.selector,(t=>{this._initializeOnDelegatedTarget(t).toggle()}));else if("manual"!==e){const t=e===ls?this.constructor.eventName("mouseenter"):this.constructor.eventName("focusin"),i=e===ls?this.constructor.eventName("mouseleave"):this.constructor.eventName("focusout");fe.on(this._element,t,this._config.selector,(t=>{const e=this._initializeOnDelegatedTarget(t);e._activeTrigger["focusin"===t.type?cs:ls]=!0,e._enter()})),fe.on(this._element,i,this._config.selector,(t=>{const e=this._initializeOnDelegatedTarget(t);e._activeTrigger["focusout"===t.type?cs:ls]=e._element.contains(t.relatedTarget),e._leave()}))}this._hideModalHandler=()=>{this._element&&this.hide()},fe.on(this._element.closest(rs),as,this._hideModalHandler)}_fixTitle(){const t=this._element.getAttribute("title");t&&(this._element.getAttribute("aria-label")||this._element.textContent.trim()||this._element.setAttribute("aria-label",t),this._element.setAttribute("data-bs-original-title",t),this._element.removeAttribute("title"))}_enter(){this._isShown()||this._isHovered?this._isHovered=!0:(this._isHovered=!0,this._setTimeout((()=>{this._isHovered&&this.show()}),this._config.delay.show))}_leave(){this._isWithActiveTrigger()||(this._isHovered=!1,this._setTimeout((()=>{this._isHovered||this.hide()}),this._config.delay.hide))}_setTimeout(t,e){clearTimeout(this._timeout),this._timeout=setTimeout(t,e)}_isWithActiveTrigger(){return Object.values(this._activeTrigger).includes(!0)}_getConfig(t){const e=_e.getDataAttributes(this._element);for(const t of Object.keys(e))is.has(t)&&delete e[t];return t={...e,..."object"==typeof t&&t?t:{}},t=this._mergeConfigObj(t),t=this._configAfterMerge(t),this._typeCheckConfig(t),t}_configAfterMerge(t){return t.container=!1===t.container?document.body:Ht(t.container),"number"==typeof t.delay&&(t.delay={show:t.delay,hide:t.delay}),"number"==typeof t.title&&(t.title=t.title.toString()),"number"==typeof t.content&&(t.content=t.content.toString()),t}_getDelegateConfig(){const t={};for(const[e,i]of Object.entries(this._config))this.constructor.Default[e]!==i&&(t[e]=i);return t.selector=!1,t.trigger="manual",t}_disposePopper(){this._popper&&(this._popper.destroy(),this._popper=null),this.tip&&(this.tip.remove(),this.tip=null)}static jQueryInterface(t){return this.each((function(){const e=fs.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}Qt(fs);const ps=".popover-header",ms=".popover-body",gs={...fs.Default,content:"",offset:[0,8],placement:"right",template:'<div class="popover" role="tooltip"><div class="popover-arrow"></div><h3 class="popover-header"></h3><div class="popover-body"></div></div>',trigger:"click"},_s={...fs.DefaultType,content:"(null|string|element|function)"};class bs extends fs{static get Default(){return gs}static get DefaultType(){return _s}static get NAME(){return"popover"}_isWithContent(){return this._getTitle()||this._getContent()}_getContentForTemplate(){return{[ps]:this._getTitle(),[ms]:this._getContent()}}_getContent(){return this._resolvePossibleFunction(this._config.content)}static jQueryInterface(t){return this.each((function(){const e=bs.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}Qt(bs);const vs=".bs.scrollspy",ys=`activate${vs}`,ws=`click${vs}`,Es=`load${vs}.data-api`,As="active",Ts="[href]",Cs=".nav-link",Os=`${Cs}, .nav-item > ${Cs}, .list-group-item`,xs={offset:null,rootMargin:"0px 0px -25%",smoothScroll:!1,target:null,threshold:[.1,.5,1]},ks={offset:"(number|null)",rootMargin:"string",smoothScroll:"boolean",target:"element",threshold:"array"};class Ls extends ve{constructor(t,e){super(t,e),this._targetLinks=new Map,this._observableSections=new Map,this._rootElement="visible"===getComputedStyle(this._element).overflowY?null:this._element,this._activeTarget=null,this._observer=null,this._previousScrollData={visibleEntryTop:0,parentScrollTop:0},this.refresh()}static get Default(){return xs}static get DefaultType(){return ks}static get NAME(){return"scrollspy"}refresh(){this._initializeTargetsAndObservables(),this._maybeEnableSmoothScroll(),this._observer?this._observer.disconnect():this._observer=this._getNewObserver();for(const t of this._observableSections.values())this._observer.observe(t)}dispose(){this._observer.disconnect(),super.dispose()}_configAfterMerge(t){return t.target=Ht(t.target)||document.body,t.rootMargin=t.offset?`${t.offset}px 0px -30%`:t.rootMargin,"string"==typeof t.threshold&&(t.threshold=t.threshold.split(",").map((t=>Number.parseFloat(t)))),t}_maybeEnableSmoothScroll(){this._config.smoothScroll&&(fe.off(this._config.target,ws),fe.on(this._config.target,ws,Ts,(t=>{const e=this._observableSections.get(t.target.hash);if(e){t.preventDefault();const i=this._rootElement||window,n=e.offsetTop-this._element.offsetTop;if(i.scrollTo)return void i.scrollTo({top:n,behavior:"smooth"});i.scrollTop=n}})))}_getNewObserver(){const t={root:this._rootElement,threshold:this._config.threshold,rootMargin:this._config.rootMargin};return new IntersectionObserver((t=>this._observerCallback(t)),t)}_observerCallback(t){const e=t=>this._targetLinks.get(`#${t.target.id}`),i=t=>{this._previousScrollData.visibleEntryTop=t.target.offsetTop,this._process(e(t))},n=(this._rootElement||document.documentElement).scrollTop,s=n>=this._previousScrollData.parentScrollTop;this._previousScrollData.parentScrollTop=n;for(const o of t){if(!o.isIntersecting){this._activeTarget=null,this._clearActiveClass(e(o));continue}const t=o.target.offsetTop>=this._previousScrollData.visibleEntryTop;if(s&&t){if(i(o),!n)return}else s||t||i(o)}}_initializeTargetsAndObservables(){this._targetLinks=new Map,this._observableSections=new Map;const t=we.find(Ts,this._config.target);for(const e of t){if(!e.hash||Wt(e))continue;const t=we.findOne(decodeURI(e.hash),this._element);Bt(t)&&(this._targetLinks.set(decodeURI(e.hash),e),this._observableSections.set(e.hash,t))}}_process(t){this._activeTarget!==t&&(this._clearActiveClass(this._config.target),this._activeTarget=t,t.classList.add(As),this._activateParents(t),fe.trigger(this._element,ys,{relatedTarget:t}))}_activateParents(t){if(t.classList.contains("dropdown-item"))we.findOne(".dropdown-toggle",t.closest(".dropdown")).classList.add(As);else for(const e of we.parents(t,".nav, .list-group"))for(const t of we.prev(e,Os))t.classList.add(As)}_clearActiveClass(t){t.classList.remove(As);const e=we.find(`${Ts}.${As}`,t);for(const t of e)t.classList.remove(As)}static jQueryInterface(t){return this.each((function(){const e=Ls.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t]()}}))}}fe.on(window,Es,(()=>{for(const t of we.find('[data-bs-spy="scroll"]'))Ls.getOrCreateInstance(t)})),Qt(Ls);const Ss=".bs.tab",Ds=`hide${Ss}`,$s=`hidden${Ss}`,Is=`show${Ss}`,Ns=`shown${Ss}`,Ps=`click${Ss}`,Ms=`keydown${Ss}`,js=`load${Ss}`,Fs="ArrowLeft",Hs="ArrowRight",Bs="ArrowUp",Ws="ArrowDown",zs="Home",Rs="End",qs="active",Vs="fade",Ys="show",Ks=".dropdown-toggle",Qs=`:not(${Ks})`,Xs='[data-bs-toggle="tab"], [data-bs-toggle="pill"], [data-bs-toggle="list"]',Us=`.nav-link${Qs}, .list-group-item${Qs}, [role="tab"]${Qs}, ${Xs}`,Gs=`.${qs}[data-bs-toggle="tab"], .${qs}[data-bs-toggle="pill"], .${qs}[data-bs-toggle="list"]`;class Js extends ve{constructor(t){super(t),this._parent=this._element.closest('.list-group, .nav, [role="tablist"]'),this._parent&&(this._setInitialAttributes(this._parent,this._getChildren()),fe.on(this._element,Ms,(t=>this._keydown(t))))}static get NAME(){return"tab"}show(){const t=this._element;if(this._elemIsActive(t))return;const e=this._getActiveElem(),i=e?fe.trigger(e,Ds,{relatedTarget:t}):null;fe.trigger(t,Is,{relatedTarget:e}).defaultPrevented||i&&i.defaultPrevented||(this._deactivate(e,t),this._activate(t,e))}_activate(t,e){t&&(t.classList.add(qs),this._activate(we.getElementFromSelector(t)),this._queueCallback((()=>{"tab"===t.getAttribute("role")?(t.removeAttribute("tabindex"),t.setAttribute("aria-selected",!0),this._toggleDropDown(t,!0),fe.trigger(t,Ns,{relatedTarget:e})):t.classList.add(Ys)}),t,t.classList.contains(Vs)))}_deactivate(t,e){t&&(t.classList.remove(qs),t.blur(),this._deactivate(we.getElementFromSelector(t)),this._queueCallback((()=>{"tab"===t.getAttribute("role")?(t.setAttribute("aria-selected",!1),t.setAttribute("tabindex","-1"),this._toggleDropDown(t,!1),fe.trigger(t,$s,{relatedTarget:e})):t.classList.remove(Ys)}),t,t.classList.contains(Vs)))}_keydown(t){if(![Fs,Hs,Bs,Ws,zs,Rs].includes(t.key))return;t.stopPropagation(),t.preventDefault();const e=this._getChildren().filter((t=>!Wt(t)));let i;if([zs,Rs].includes(t.key))i=e[t.key===zs?0:e.length-1];else{const n=[Hs,Ws].includes(t.key);i=Gt(e,t.target,n,!0)}i&&(i.focus({preventScroll:!0}),Js.getOrCreateInstance(i).show())}_getChildren(){return we.find(Us,this._parent)}_getActiveElem(){return this._getChildren().find((t=>this._elemIsActive(t)))||null}_setInitialAttributes(t,e){this._setAttributeIfNotExists(t,"role","tablist");for(const t of e)this._setInitialAttributesOnChild(t)}_setInitialAttributesOnChild(t){t=this._getInnerElement(t);const e=this._elemIsActive(t),i=this._getOuterElement(t);t.setAttribute("aria-selected",e),i!==t&&this._setAttributeIfNotExists(i,"role","presentation"),e||t.setAttribute("tabindex","-1"),this._setAttributeIfNotExists(t,"role","tab"),this._setInitialAttributesOnTargetPanel(t)}_setInitialAttributesOnTargetPanel(t){const e=we.getElementFromSelector(t);e&&(this._setAttributeIfNotExists(e,"role","tabpanel"),t.id&&this._setAttributeIfNotExists(e,"aria-labelledby",`${t.id}`))}_toggleDropDown(t,e){const i=this._getOuterElement(t);if(!i.classList.contains("dropdown"))return;const n=(t,n)=>{const s=we.findOne(t,i);s&&s.classList.toggle(n,e)};n(Ks,qs),n(".dropdown-menu",Ys),i.setAttribute("aria-expanded",e)}_setAttributeIfNotExists(t,e,i){t.hasAttribute(e)||t.setAttribute(e,i)}_elemIsActive(t){return t.classList.contains(qs)}_getInnerElement(t){return t.matches(Us)?t:we.findOne(Us,t)}_getOuterElement(t){return t.closest(".nav-item, .list-group-item")||t}static jQueryInterface(t){return this.each((function(){const e=Js.getOrCreateInstance(this);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t]()}}))}}fe.on(document,Ps,Xs,(function(t){["A","AREA"].includes(this.tagName)&&t.preventDefault(),Wt(this)||Js.getOrCreateInstance(this).show()})),fe.on(window,js,(()=>{for(const t of we.find(Gs))Js.getOrCreateInstance(t)})),Qt(Js);const Zs=".bs.toast",to=`mouseover${Zs}`,eo=`mouseout${Zs}`,io=`focusin${Zs}`,no=`focusout${Zs}`,so=`hide${Zs}`,oo=`hidden${Zs}`,ro=`show${Zs}`,ao=`shown${Zs}`,lo="hide",co="show",ho="showing",uo={animation:"boolean",autohide:"boolean",delay:"number"},fo={animation:!0,autohide:!0,delay:5e3};class po extends ve{constructor(t,e){super(t,e),this._timeout=null,this._hasMouseInteraction=!1,this._hasKeyboardInteraction=!1,this._setListeners()}static get Default(){return fo}static get DefaultType(){return uo}static get NAME(){return"toast"}show(){fe.trigger(this._element,ro).defaultPrevented||(this._clearTimeout(),this._config.animation&&this._element.classList.add("fade"),this._element.classList.remove(lo),qt(this._element),this._element.classList.add(co,ho),this._queueCallback((()=>{this._element.classList.remove(ho),fe.trigger(this._element,ao),this._maybeScheduleHide()}),this._element,this._config.animation))}hide(){this.isShown()&&(fe.trigger(this._element,so).defaultPrevented||(this._element.classList.add(ho),this._queueCallback((()=>{this._element.classList.add(lo),this._element.classList.remove(ho,co),fe.trigger(this._element,oo)}),this._element,this._config.animation)))}dispose(){this._clearTimeout(),this.isShown()&&this._element.classList.remove(co),super.dispose()}isShown(){return 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+//# sourceMappingURL=bootstrap.js.map
\ No newline at end of file
diff --git a/dev/docs/html/_static/scripts/bootstrap.js.LICENSE.txt b/dev/docs/html/_static/scripts/bootstrap.js.LICENSE.txt
new file mode 100644
index 0000000..28755c2
--- /dev/null
+++ b/dev/docs/html/_static/scripts/bootstrap.js.LICENSE.txt
@@ -0,0 +1,5 @@
+/*!
+  * Bootstrap v5.3.3 (https://getbootstrap.com/)
+  * Copyright 2011-2024 The Bootstrap Authors (https://github.com/twbs/bootstrap/graphs/contributors)
+  * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)
+  */
diff --git a/dev/docs/html/_static/scripts/bootstrap.js.map b/dev/docs/html/_static/scripts/bootstrap.js.map
new file mode 100644
index 0000000..e9e8158
--- /dev/null
+++ b/dev/docs/html/_static/scripts/bootstrap.js.map
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(element.nodeName || '').toLowerCase() : null;\n}","export default function getWindow(node) {\n  if (node == null) {\n    return window;\n  }\n\n  if (node.toString() !== '[object Window]') {\n    var ownerDocument = node.ownerDocument;\n    return ownerDocument ? ownerDocument.defaultView || window : window;\n  }\n\n  return node;\n}","import getWindow from \"./getWindow.js\";\n\nfunction isElement(node) {\n  var OwnElement = getWindow(node).Element;\n  return node instanceof OwnElement || node instanceof Element;\n}\n\nfunction isHTMLElement(node) {\n  var OwnElement = getWindow(node).HTMLElement;\n  return node instanceof OwnElement || node instanceof HTMLElement;\n}\n\nfunction isShadowRoot(node) {\n  // IE 11 has no ShadowRoot\n  if (typeof ShadowRoot === 'undefined') {\n    return false;\n  }\n\n  var OwnElement = getWindow(node).ShadowRoot;\n  return node instanceof OwnElement || node instanceof ShadowRoot;\n}\n\nexport { isElement, isHTMLElement, isShadowRoot };","import getNodeName from \"../dom-utils/getNodeName.js\";\nimport { isHTMLElement } from \"../dom-utils/instanceOf.js\"; // This modifier takes the styles prepared by the `computeStyles` modifier\n// and applies them to the HTMLElements such as popper and arrow\n\nfunction applyStyles(_ref) {\n  var state = _ref.state;\n  Object.keys(state.elements).forEach(function (name) {\n    var style = state.styles[name] || {};\n    var attributes = state.attributes[name] || {};\n    var element = state.elements[name]; // arrow is optional + virtual elements\n\n    if (!isHTMLElement(element) || !getNodeName(element)) {\n      return;\n    } // Flow doesn't support to extend this property, but it's the most\n    // effective way to apply styles to an HTMLElement\n    // $FlowFixMe[cannot-write]\n\n\n    Object.assign(element.style, style);\n    Object.keys(attributes).forEach(function (name) {\n      var value = attributes[name];\n\n      if (value === false) {\n        element.removeAttribute(name);\n      } else {\n        element.setAttribute(name, value === true ? '' : value);\n      }\n    });\n  });\n}\n\nfunction effect(_ref2) {\n  var state = _ref2.state;\n  var initialStyles = {\n    popper: {\n      position: state.options.strategy,\n      left: '0',\n      top: '0',\n      margin: '0'\n    },\n    arrow: {\n      position: 'absolute'\n    },\n    reference: {}\n  };\n  Object.assign(state.elements.popper.style, initialStyles.popper);\n  state.styles = initialStyles;\n\n  if (state.elements.arrow) {\n    Object.assign(state.elements.arrow.style, initialStyles.arrow);\n  }\n\n  return function () {\n    Object.keys(state.elements).forEach(function (name) {\n      var element = state.elements[name];\n      var attributes = state.attributes[name] || {};\n      var styleProperties = Object.keys(state.styles.hasOwnProperty(name) ? state.styles[name] : initialStyles[name]); // Set all values to an empty string to unset them\n\n      var style = styleProperties.reduce(function (style, property) {\n        style[property] = '';\n        return style;\n      }, {}); // arrow is optional + virtual elements\n\n      if (!isHTMLElement(element) || !getNodeName(element)) {\n        return;\n      }\n\n      Object.assign(element.style, style);\n      Object.keys(attributes).forEach(function (attribute) {\n        element.removeAttribute(attribute);\n      });\n    });\n  };\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'applyStyles',\n  enabled: true,\n  phase: 'write',\n  fn: applyStyles,\n  effect: effect,\n  requires: ['computeStyles']\n};","import { auto } from \"../enums.js\";\nexport default function getBasePlacement(placement) {\n  return placement.split('-')[0];\n}","export var max = Math.max;\nexport var min = Math.min;\nexport var round = Math.round;","export default function getUAString() {\n  var uaData = navigator.userAgentData;\n\n  if (uaData != null && uaData.brands && Array.isArray(uaData.brands)) {\n    return uaData.brands.map(function (item) {\n      return item.brand + \"/\" + item.version;\n    }).join(' ');\n  }\n\n  return navigator.userAgent;\n}","import getUAString from \"../utils/userAgent.js\";\nexport default function isLayoutViewport() {\n  return !/^((?!chrome|android).)*safari/i.test(getUAString());\n}","import { isElement, isHTMLElement } from \"./instanceOf.js\";\nimport { round } from \"../utils/math.js\";\nimport getWindow from \"./getWindow.js\";\nimport isLayoutViewport from \"./isLayoutViewport.js\";\nexport default function getBoundingClientRect(element, includeScale, isFixedStrategy) {\n  if (includeScale === void 0) {\n    includeScale = false;\n  }\n\n  if (isFixedStrategy === void 0) {\n    isFixedStrategy = false;\n  }\n\n  var clientRect = element.getBoundingClientRect();\n  var scaleX = 1;\n  var scaleY = 1;\n\n  if (includeScale && isHTMLElement(element)) {\n    scaleX = element.offsetWidth > 0 ? round(clientRect.width) / element.offsetWidth || 1 : 1;\n    scaleY = element.offsetHeight > 0 ? round(clientRect.height) / element.offsetHeight || 1 : 1;\n  }\n\n  var _ref = isElement(element) ? getWindow(element) : window,\n      visualViewport = _ref.visualViewport;\n\n  var addVisualOffsets = !isLayoutViewport() && isFixedStrategy;\n  var x = (clientRect.left + (addVisualOffsets && visualViewport ? visualViewport.offsetLeft : 0)) / scaleX;\n  var y = (clientRect.top + (addVisualOffsets && visualViewport ? visualViewport.offsetTop : 0)) / scaleY;\n  var width = clientRect.width / scaleX;\n  var height = clientRect.height / scaleY;\n  return {\n    width: width,\n    height: height,\n    top: y,\n    right: x + width,\n    bottom: y + height,\n    left: x,\n    x: x,\n    y: y\n  };\n}","import getBoundingClientRect from \"./getBoundingClientRect.js\"; // Returns the layout rect of an element relative to its offsetParent. Layout\n// means it doesn't take into account transforms.\n\nexport default function getLayoutRect(element) {\n  var clientRect = getBoundingClientRect(element); // Use the clientRect sizes if it's not been transformed.\n  // Fixes https://github.com/popperjs/popper-core/issues/1223\n\n  var width = element.offsetWidth;\n  var height = element.offsetHeight;\n\n  if (Math.abs(clientRect.width - width) <= 1) {\n    width = clientRect.width;\n  }\n\n  if (Math.abs(clientRect.height - height) <= 1) {\n    height = clientRect.height;\n  }\n\n  return {\n    x: element.offsetLeft,\n    y: element.offsetTop,\n    width: width,\n    height: height\n  };\n}","import { isShadowRoot } from \"./instanceOf.js\";\nexport default function contains(parent, child) {\n  var rootNode = child.getRootNode && child.getRootNode(); // First, attempt with faster native method\n\n  if (parent.contains(child)) {\n    return true;\n  } // then fallback to custom implementation with Shadow DOM support\n  else if (rootNode && isShadowRoot(rootNode)) {\n      var next = child;\n\n      do {\n        if (next && parent.isSameNode(next)) {\n          return true;\n        } // $FlowFixMe[prop-missing]: need a better way to handle this...\n\n\n        next = next.parentNode || next.host;\n      } while (next);\n    } // Give up, the result is false\n\n\n  return false;\n}","import getWindow from \"./getWindow.js\";\nexport default function getComputedStyle(element) {\n  return getWindow(element).getComputedStyle(element);\n}","import getNodeName from \"./getNodeName.js\";\nexport default function isTableElement(element) {\n  return ['table', 'td', 'th'].indexOf(getNodeName(element)) >= 0;\n}","import { isElement } from \"./instanceOf.js\";\nexport default function getDocumentElement(element) {\n  // $FlowFixMe[incompatible-return]: assume body is always available\n  return ((isElement(element) ? element.ownerDocument : // $FlowFixMe[prop-missing]\n  element.document) || window.document).documentElement;\n}","import getNodeName from \"./getNodeName.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport { isShadowRoot } from \"./instanceOf.js\";\nexport default function getParentNode(element) {\n  if (getNodeName(element) === 'html') {\n    return element;\n  }\n\n  return (// this is a quicker (but less type safe) way to save quite some bytes from the bundle\n    // $FlowFixMe[incompatible-return]\n    // $FlowFixMe[prop-missing]\n    element.assignedSlot || // step into the shadow DOM of the parent of a slotted node\n    element.parentNode || ( // DOM Element detected\n    isShadowRoot(element) ? element.host : null) || // ShadowRoot detected\n    // $FlowFixMe[incompatible-call]: HTMLElement is a Node\n    getDocumentElement(element) // fallback\n\n  );\n}","import getWindow from \"./getWindow.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport getComputedStyle from \"./getComputedStyle.js\";\nimport { isHTMLElement, isShadowRoot } from \"./instanceOf.js\";\nimport isTableElement from \"./isTableElement.js\";\nimport getParentNode from \"./getParentNode.js\";\nimport getUAString from \"../utils/userAgent.js\";\n\nfunction getTrueOffsetParent(element) {\n  if (!isHTMLElement(element) || // https://github.com/popperjs/popper-core/issues/837\n  getComputedStyle(element).position === 'fixed') {\n    return null;\n  }\n\n  return element.offsetParent;\n} // `.offsetParent` reports `null` for fixed elements, while absolute elements\n// return the containing block\n\n\nfunction getContainingBlock(element) {\n  var isFirefox = /firefox/i.test(getUAString());\n  var isIE = /Trident/i.test(getUAString());\n\n  if (isIE && isHTMLElement(element)) {\n    // In IE 9, 10 and 11 fixed elements containing block is always established by the viewport\n    var elementCss = getComputedStyle(element);\n\n    if (elementCss.position === 'fixed') {\n      return null;\n    }\n  }\n\n  var currentNode = getParentNode(element);\n\n  if (isShadowRoot(currentNode)) {\n    currentNode = currentNode.host;\n  }\n\n  while (isHTMLElement(currentNode) && ['html', 'body'].indexOf(getNodeName(currentNode)) < 0) {\n    var css = getComputedStyle(currentNode); // This is non-exhaustive but covers the most common CSS properties that\n    // create a containing block.\n    // https://developer.mozilla.org/en-US/docs/Web/CSS/Containing_block#identifying_the_containing_block\n\n    if (css.transform !== 'none' || css.perspective !== 'none' || css.contain === 'paint' || ['transform', 'perspective'].indexOf(css.willChange) !== -1 || isFirefox && css.willChange === 'filter' || isFirefox && css.filter && css.filter !== 'none') {\n      return currentNode;\n    } else {\n      currentNode = currentNode.parentNode;\n    }\n  }\n\n  return null;\n} // Gets the closest ancestor positioned element. Handles some edge cases,\n// such as table ancestors and cross browser bugs.\n\n\nexport default function getOffsetParent(element) {\n  var window = getWindow(element);\n  var offsetParent = getTrueOffsetParent(element);\n\n  while (offsetParent && isTableElement(offsetParent) && getComputedStyle(offsetParent).position === 'static') {\n    offsetParent = getTrueOffsetParent(offsetParent);\n  }\n\n  if (offsetParent && (getNodeName(offsetParent) === 'html' || getNodeName(offsetParent) === 'body' && getComputedStyle(offsetParent).position === 'static')) {\n    return window;\n  }\n\n  return offsetParent || getContainingBlock(element) || window;\n}","export default function getMainAxisFromPlacement(placement) {\n  return ['top', 'bottom'].indexOf(placement) >= 0 ? 'x' : 'y';\n}","import { max as mathMax, min as mathMin } from \"./math.js\";\nexport function within(min, value, max) {\n  return mathMax(min, mathMin(value, max));\n}\nexport function withinMaxClamp(min, value, max) {\n  var v = within(min, value, max);\n  return v > max ? max : v;\n}","import getFreshSideObject from \"./getFreshSideObject.js\";\nexport default function mergePaddingObject(paddingObject) {\n  return Object.assign({}, getFreshSideObject(), paddingObject);\n}","export default function getFreshSideObject() {\n  return {\n    top: 0,\n    right: 0,\n    bottom: 0,\n    left: 0\n  };\n}","export default function expandToHashMap(value, keys) {\n  return keys.reduce(function (hashMap, key) {\n    hashMap[key] = value;\n    return hashMap;\n  }, {});\n}","import getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getLayoutRect from \"../dom-utils/getLayoutRect.js\";\nimport contains from \"../dom-utils/contains.js\";\nimport getOffsetParent from \"../dom-utils/getOffsetParent.js\";\nimport getMainAxisFromPlacement from \"../utils/getMainAxisFromPlacement.js\";\nimport { within } from \"../utils/within.js\";\nimport mergePaddingObject from \"../utils/mergePaddingObject.js\";\nimport expandToHashMap from \"../utils/expandToHashMap.js\";\nimport { left, right, basePlacements, top, bottom } from \"../enums.js\"; // eslint-disable-next-line import/no-unused-modules\n\nvar toPaddingObject = function toPaddingObject(padding, state) {\n  padding = typeof padding === 'function' ? padding(Object.assign({}, state.rects, {\n    placement: state.placement\n  })) : padding;\n  return mergePaddingObject(typeof padding !== 'number' ? padding : expandToHashMap(padding, basePlacements));\n};\n\nfunction arrow(_ref) {\n  var _state$modifiersData$;\n\n  var state = _ref.state,\n      name = _ref.name,\n      options = _ref.options;\n  var arrowElement = state.elements.arrow;\n  var popperOffsets = state.modifiersData.popperOffsets;\n  var basePlacement = getBasePlacement(state.placement);\n  var axis = getMainAxisFromPlacement(basePlacement);\n  var isVertical = [left, right].indexOf(basePlacement) >= 0;\n  var len = isVertical ? 'height' : 'width';\n\n  if (!arrowElement || !popperOffsets) {\n    return;\n  }\n\n  var paddingObject = toPaddingObject(options.padding, state);\n  var arrowRect = getLayoutRect(arrowElement);\n  var minProp = axis === 'y' ? top : left;\n  var maxProp = axis === 'y' ? bottom : right;\n  var endDiff = state.rects.reference[len] + state.rects.reference[axis] - popperOffsets[axis] - state.rects.popper[len];\n  var startDiff = popperOffsets[axis] - state.rects.reference[axis];\n  var arrowOffsetParent = getOffsetParent(arrowElement);\n  var clientSize = arrowOffsetParent ? axis === 'y' ? arrowOffsetParent.clientHeight || 0 : arrowOffsetParent.clientWidth || 0 : 0;\n  var centerToReference = endDiff / 2 - startDiff / 2; // Make sure the arrow doesn't overflow the popper if the center point is\n  // outside of the popper bounds\n\n  var min = paddingObject[minProp];\n  var max = clientSize - arrowRect[len] - paddingObject[maxProp];\n  var center = clientSize / 2 - arrowRect[len] / 2 + centerToReference;\n  var offset = within(min, center, max); // Prevents breaking syntax highlighting...\n\n  var axisProp = axis;\n  state.modifiersData[name] = (_state$modifiersData$ = {}, _state$modifiersData$[axisProp] = offset, _state$modifiersData$.centerOffset = offset - center, _state$modifiersData$);\n}\n\nfunction effect(_ref2) {\n  var state = _ref2.state,\n      options = _ref2.options;\n  var _options$element = options.element,\n      arrowElement = _options$element === void 0 ? '[data-popper-arrow]' : _options$element;\n\n  if (arrowElement == null) {\n    return;\n  } // CSS selector\n\n\n  if (typeof arrowElement === 'string') {\n    arrowElement = state.elements.popper.querySelector(arrowElement);\n\n    if (!arrowElement) {\n      return;\n    }\n  }\n\n  if (!contains(state.elements.popper, arrowElement)) {\n    return;\n  }\n\n  state.elements.arrow = arrowElement;\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'arrow',\n  enabled: true,\n  phase: 'main',\n  fn: arrow,\n  effect: effect,\n  requires: ['popperOffsets'],\n  requiresIfExists: ['preventOverflow']\n};","export default function getVariation(placement) {\n  return placement.split('-')[1];\n}","import { top, left, right, bottom, end } from \"../enums.js\";\nimport getOffsetParent from \"../dom-utils/getOffsetParent.js\";\nimport getWindow from \"../dom-utils/getWindow.js\";\nimport getDocumentElement from \"../dom-utils/getDocumentElement.js\";\nimport getComputedStyle from \"../dom-utils/getComputedStyle.js\";\nimport getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getVariation from \"../utils/getVariation.js\";\nimport { round } from \"../utils/math.js\"; // eslint-disable-next-line import/no-unused-modules\n\nvar unsetSides = {\n  top: 'auto',\n  right: 'auto',\n  bottom: 'auto',\n  left: 'auto'\n}; // Round the offsets to the nearest suitable subpixel based on the DPR.\n// Zooming can change the DPR, but it seems to report a value that will\n// cleanly divide the values into the appropriate subpixels.\n\nfunction roundOffsetsByDPR(_ref, win) {\n  var x = _ref.x,\n      y = _ref.y;\n  var dpr = win.devicePixelRatio || 1;\n  return {\n    x: round(x * dpr) / dpr || 0,\n    y: round(y * dpr) / dpr || 0\n  };\n}\n\nexport function mapToStyles(_ref2) {\n  var _Object$assign2;\n\n  var popper = _ref2.popper,\n      popperRect = _ref2.popperRect,\n      placement = _ref2.placement,\n      variation = _ref2.variation,\n      offsets = _ref2.offsets,\n      position = _ref2.position,\n      gpuAcceleration = _ref2.gpuAcceleration,\n      adaptive = _ref2.adaptive,\n      roundOffsets = _ref2.roundOffsets,\n      isFixed = _ref2.isFixed;\n  var _offsets$x = offsets.x,\n      x = _offsets$x === void 0 ? 0 : _offsets$x,\n      _offsets$y = offsets.y,\n      y = _offsets$y === void 0 ? 0 : _offsets$y;\n\n  var _ref3 = typeof roundOffsets === 'function' ? roundOffsets({\n    x: x,\n    y: y\n  }) : {\n    x: x,\n    y: y\n  };\n\n  x = _ref3.x;\n  y = _ref3.y;\n  var hasX = offsets.hasOwnProperty('x');\n  var hasY = offsets.hasOwnProperty('y');\n  var sideX = left;\n  var sideY = top;\n  var win = window;\n\n  if (adaptive) {\n    var offsetParent = getOffsetParent(popper);\n    var heightProp = 'clientHeight';\n    var widthProp = 'clientWidth';\n\n    if (offsetParent === getWindow(popper)) {\n      offsetParent = getDocumentElement(popper);\n\n      if (getComputedStyle(offsetParent).position !== 'static' && position === 'absolute') {\n        heightProp = 'scrollHeight';\n        widthProp = 'scrollWidth';\n      }\n    } // $FlowFixMe[incompatible-cast]: force type refinement, we compare offsetParent with window above, but Flow doesn't detect it\n\n\n    offsetParent = offsetParent;\n\n    if (placement === top || (placement === left || placement === right) && variation === end) {\n      sideY = bottom;\n      var offsetY = isFixed && offsetParent === win && win.visualViewport ? win.visualViewport.height : // $FlowFixMe[prop-missing]\n      offsetParent[heightProp];\n      y -= offsetY - popperRect.height;\n      y *= gpuAcceleration ? 1 : -1;\n    }\n\n    if (placement === left || (placement === top || placement === bottom) && variation === end) {\n      sideX = right;\n      var offsetX = isFixed && offsetParent === win && win.visualViewport ? win.visualViewport.width : // $FlowFixMe[prop-missing]\n      offsetParent[widthProp];\n      x -= offsetX - popperRect.width;\n      x *= gpuAcceleration ? 1 : -1;\n    }\n  }\n\n  var commonStyles = Object.assign({\n    position: position\n  }, adaptive && unsetSides);\n\n  var _ref4 = roundOffsets === true ? roundOffsetsByDPR({\n    x: x,\n    y: y\n  }, getWindow(popper)) : {\n    x: x,\n    y: y\n  };\n\n  x = _ref4.x;\n  y = _ref4.y;\n\n  if (gpuAcceleration) {\n    var _Object$assign;\n\n    return Object.assign({}, commonStyles, (_Object$assign = {}, _Object$assign[sideY] = hasY ? '0' : '', _Object$assign[sideX] = hasX ? '0' : '', _Object$assign.transform = (win.devicePixelRatio || 1) <= 1 ? \"translate(\" + x + \"px, \" + y + \"px)\" : \"translate3d(\" + x + \"px, \" + y + \"px, 0)\", _Object$assign));\n  }\n\n  return Object.assign({}, commonStyles, (_Object$assign2 = {}, _Object$assign2[sideY] = hasY ? y + \"px\" : '', _Object$assign2[sideX] = hasX ? x + \"px\" : '', _Object$assign2.transform = '', _Object$assign2));\n}\n\nfunction computeStyles(_ref5) {\n  var state = _ref5.state,\n      options = _ref5.options;\n  var _options$gpuAccelerat = options.gpuAcceleration,\n      gpuAcceleration = _options$gpuAccelerat === void 0 ? true : _options$gpuAccelerat,\n      _options$adaptive = options.adaptive,\n      adaptive = _options$adaptive === void 0 ? true : _options$adaptive,\n      _options$roundOffsets = options.roundOffsets,\n      roundOffsets = _options$roundOffsets === void 0 ? true : _options$roundOffsets;\n  var commonStyles = {\n    placement: getBasePlacement(state.placement),\n    variation: getVariation(state.placement),\n    popper: state.elements.popper,\n    popperRect: state.rects.popper,\n    gpuAcceleration: gpuAcceleration,\n    isFixed: state.options.strategy === 'fixed'\n  };\n\n  if (state.modifiersData.popperOffsets != null) {\n    state.styles.popper = Object.assign({}, state.styles.popper, mapToStyles(Object.assign({}, commonStyles, {\n      offsets: state.modifiersData.popperOffsets,\n      position: state.options.strategy,\n      adaptive: adaptive,\n      roundOffsets: roundOffsets\n    })));\n  }\n\n  if (state.modifiersData.arrow != null) {\n    state.styles.arrow = Object.assign({}, state.styles.arrow, mapToStyles(Object.assign({}, commonStyles, {\n      offsets: state.modifiersData.arrow,\n      position: 'absolute',\n      adaptive: false,\n      roundOffsets: roundOffsets\n    })));\n  }\n\n  state.attributes.popper = Object.assign({}, state.attributes.popper, {\n    'data-popper-placement': state.placement\n  });\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'computeStyles',\n  enabled: true,\n  phase: 'beforeWrite',\n  fn: computeStyles,\n  data: {}\n};","import getWindow from \"../dom-utils/getWindow.js\"; // eslint-disable-next-line import/no-unused-modules\n\nvar passive = {\n  passive: true\n};\n\nfunction effect(_ref) {\n  var state = _ref.state,\n      instance = _ref.instance,\n      options = _ref.options;\n  var _options$scroll = options.scroll,\n      scroll = _options$scroll === void 0 ? true : _options$scroll,\n      _options$resize = options.resize,\n      resize = _options$resize === void 0 ? true : _options$resize;\n  var window = getWindow(state.elements.popper);\n  var scrollParents = [].concat(state.scrollParents.reference, state.scrollParents.popper);\n\n  if (scroll) {\n    scrollParents.forEach(function (scrollParent) {\n      scrollParent.addEventListener('scroll', instance.update, passive);\n    });\n  }\n\n  if (resize) {\n    window.addEventListener('resize', instance.update, passive);\n  }\n\n  return function () {\n    if (scroll) {\n      scrollParents.forEach(function (scrollParent) {\n        scrollParent.removeEventListener('scroll', instance.update, passive);\n      });\n    }\n\n    if (resize) {\n      window.removeEventListener('resize', instance.update, passive);\n    }\n  };\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'eventListeners',\n  enabled: true,\n  phase: 'write',\n  fn: function fn() {},\n  effect: effect,\n  data: {}\n};","var hash = {\n  left: 'right',\n  right: 'left',\n  bottom: 'top',\n  top: 'bottom'\n};\nexport default function getOppositePlacement(placement) {\n  return placement.replace(/left|right|bottom|top/g, function (matched) {\n    return hash[matched];\n  });\n}","var hash = {\n  start: 'end',\n  end: 'start'\n};\nexport default function getOppositeVariationPlacement(placement) {\n  return placement.replace(/start|end/g, function (matched) {\n    return hash[matched];\n  });\n}","import getWindow from \"./getWindow.js\";\nexport default function getWindowScroll(node) {\n  var win = getWindow(node);\n  var scrollLeft = win.pageXOffset;\n  var scrollTop = win.pageYOffset;\n  return {\n    scrollLeft: scrollLeft,\n    scrollTop: scrollTop\n  };\n}","import getBoundingClientRect from \"./getBoundingClientRect.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport getWindowScroll from \"./getWindowScroll.js\";\nexport default function getWindowScrollBarX(element) {\n  // If <html> has a CSS width greater than the viewport, then this will be\n  // incorrect for RTL.\n  // Popper 1 is broken in this case and never had a bug report so let's assume\n  // it's not an issue. I don't think anyone ever specifies width on <html>\n  // anyway.\n  // Browsers where the left scrollbar doesn't cause an issue report `0` for\n  // this (e.g. Edge 2019, IE11, Safari)\n  return getBoundingClientRect(getDocumentElement(element)).left + getWindowScroll(element).scrollLeft;\n}","import getComputedStyle from \"./getComputedStyle.js\";\nexport default function isScrollParent(element) {\n  // Firefox wants us to check `-x` and `-y` variations as well\n  var _getComputedStyle = getComputedStyle(element),\n      overflow = _getComputedStyle.overflow,\n      overflowX = _getComputedStyle.overflowX,\n      overflowY = _getComputedStyle.overflowY;\n\n  return /auto|scroll|overlay|hidden/.test(overflow + overflowY + overflowX);\n}","import getParentNode from \"./getParentNode.js\";\nimport isScrollParent from \"./isScrollParent.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport { isHTMLElement } from \"./instanceOf.js\";\nexport default function getScrollParent(node) {\n  if (['html', 'body', '#document'].indexOf(getNodeName(node)) >= 0) {\n    // $FlowFixMe[incompatible-return]: assume body is always available\n    return node.ownerDocument.body;\n  }\n\n  if (isHTMLElement(node) && isScrollParent(node)) {\n    return node;\n  }\n\n  return getScrollParent(getParentNode(node));\n}","import getScrollParent from \"./getScrollParent.js\";\nimport getParentNode from \"./getParentNode.js\";\nimport getWindow from \"./getWindow.js\";\nimport isScrollParent from \"./isScrollParent.js\";\n/*\ngiven a DOM element, return the list of all scroll parents, up the list of ancesors\nuntil we get to the top window object. This list is what we attach scroll listeners\nto, because if any of these parent elements scroll, we'll need to re-calculate the\nreference element's position.\n*/\n\nexport default function listScrollParents(element, list) {\n  var _element$ownerDocumen;\n\n  if (list === void 0) {\n    list = [];\n  }\n\n  var scrollParent = getScrollParent(element);\n  var isBody = scrollParent === ((_element$ownerDocumen = element.ownerDocument) == null ? void 0 : _element$ownerDocumen.body);\n  var win = getWindow(scrollParent);\n  var target = isBody ? [win].concat(win.visualViewport || [], isScrollParent(scrollParent) ? scrollParent : []) : scrollParent;\n  var updatedList = list.concat(target);\n  return isBody ? updatedList : // $FlowFixMe[incompatible-call]: isBody tells us target will be an HTMLElement here\n  updatedList.concat(listScrollParents(getParentNode(target)));\n}","export default function rectToClientRect(rect) {\n  return Object.assign({}, rect, {\n    left: rect.x,\n    top: rect.y,\n    right: rect.x + rect.width,\n    bottom: rect.y + rect.height\n  });\n}","import { viewport } from \"../enums.js\";\nimport getViewportRect from \"./getViewportRect.js\";\nimport getDocumentRect from \"./getDocumentRect.js\";\nimport listScrollParents from \"./listScrollParents.js\";\nimport getOffsetParent from \"./getOffsetParent.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport getComputedStyle from \"./getComputedStyle.js\";\nimport { isElement, isHTMLElement } from \"./instanceOf.js\";\nimport getBoundingClientRect from \"./getBoundingClientRect.js\";\nimport getParentNode from \"./getParentNode.js\";\nimport contains from \"./contains.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport rectToClientRect from \"../utils/rectToClientRect.js\";\nimport { max, min } from \"../utils/math.js\";\n\nfunction getInnerBoundingClientRect(element, strategy) {\n  var rect = getBoundingClientRect(element, false, strategy === 'fixed');\n  rect.top = rect.top + element.clientTop;\n  rect.left = rect.left + element.clientLeft;\n  rect.bottom = rect.top + element.clientHeight;\n  rect.right = rect.left + element.clientWidth;\n  rect.width = element.clientWidth;\n  rect.height = element.clientHeight;\n  rect.x = rect.left;\n  rect.y = rect.top;\n  return rect;\n}\n\nfunction getClientRectFromMixedType(element, clippingParent, strategy) {\n  return clippingParent === viewport ? rectToClientRect(getViewportRect(element, strategy)) : isElement(clippingParent) ? getInnerBoundingClientRect(clippingParent, strategy) : rectToClientRect(getDocumentRect(getDocumentElement(element)));\n} // A \"clipping parent\" is an overflowable container with the characteristic of\n// clipping (or hiding) overflowing elements with a position different from\n// `initial`\n\n\nfunction getClippingParents(element) {\n  var clippingParents = listScrollParents(getParentNode(element));\n  var canEscapeClipping = ['absolute', 'fixed'].indexOf(getComputedStyle(element).position) >= 0;\n  var clipperElement = canEscapeClipping && isHTMLElement(element) ? getOffsetParent(element) : element;\n\n  if (!isElement(clipperElement)) {\n    return [];\n  } // $FlowFixMe[incompatible-return]: https://github.com/facebook/flow/issues/1414\n\n\n  return clippingParents.filter(function (clippingParent) {\n    return isElement(clippingParent) && contains(clippingParent, clipperElement) && getNodeName(clippingParent) !== 'body';\n  });\n} // Gets the maximum area that the element is visible in due to any number of\n// clipping parents\n\n\nexport default function getClippingRect(element, boundary, rootBoundary, strategy) {\n  var mainClippingParents = boundary === 'clippingParents' ? getClippingParents(element) : [].concat(boundary);\n  var clippingParents = [].concat(mainClippingParents, [rootBoundary]);\n  var firstClippingParent = clippingParents[0];\n  var clippingRect = clippingParents.reduce(function (accRect, clippingParent) {\n    var rect = getClientRectFromMixedType(element, clippingParent, strategy);\n    accRect.top = max(rect.top, accRect.top);\n    accRect.right = min(rect.right, accRect.right);\n    accRect.bottom = min(rect.bottom, accRect.bottom);\n    accRect.left = max(rect.left, accRect.left);\n    return accRect;\n  }, getClientRectFromMixedType(element, firstClippingParent, strategy));\n  clippingRect.width = clippingRect.right - clippingRect.left;\n  clippingRect.height = clippingRect.bottom - clippingRect.top;\n  clippingRect.x = clippingRect.left;\n  clippingRect.y = clippingRect.top;\n  return clippingRect;\n}","import getWindow from \"./getWindow.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport getWindowScrollBarX from \"./getWindowScrollBarX.js\";\nimport isLayoutViewport from \"./isLayoutViewport.js\";\nexport default function getViewportRect(element, strategy) {\n  var win = getWindow(element);\n  var html = getDocumentElement(element);\n  var visualViewport = win.visualViewport;\n  var width = html.clientWidth;\n  var height = html.clientHeight;\n  var x = 0;\n  var y = 0;\n\n  if (visualViewport) {\n    width = visualViewport.width;\n    height = visualViewport.height;\n    var layoutViewport = isLayoutViewport();\n\n    if (layoutViewport || !layoutViewport && strategy === 'fixed') {\n      x = visualViewport.offsetLeft;\n      y = visualViewport.offsetTop;\n    }\n  }\n\n  return {\n    width: width,\n    height: height,\n    x: x + getWindowScrollBarX(element),\n    y: y\n  };\n}","import getDocumentElement from \"./getDocumentElement.js\";\nimport getComputedStyle from \"./getComputedStyle.js\";\nimport getWindowScrollBarX from \"./getWindowScrollBarX.js\";\nimport getWindowScroll from \"./getWindowScroll.js\";\nimport { max } from \"../utils/math.js\"; // Gets the entire size of the scrollable document area, even extending outside\n// of the `<html>` and `<body>` rect bounds if horizontally scrollable\n\nexport default function getDocumentRect(element) {\n  var _element$ownerDocumen;\n\n  var html = getDocumentElement(element);\n  var winScroll = getWindowScroll(element);\n  var body = (_element$ownerDocumen = element.ownerDocument) == null ? void 0 : _element$ownerDocumen.body;\n  var width = max(html.scrollWidth, html.clientWidth, body ? body.scrollWidth : 0, body ? body.clientWidth : 0);\n  var height = max(html.scrollHeight, html.clientHeight, body ? body.scrollHeight : 0, body ? body.clientHeight : 0);\n  var x = -winScroll.scrollLeft + getWindowScrollBarX(element);\n  var y = -winScroll.scrollTop;\n\n  if (getComputedStyle(body || html).direction === 'rtl') {\n    x += max(html.clientWidth, body ? body.clientWidth : 0) - width;\n  }\n\n  return {\n    width: width,\n    height: height,\n    x: x,\n    y: y\n  };\n}","import getBasePlacement from \"./getBasePlacement.js\";\nimport getVariation from \"./getVariation.js\";\nimport getMainAxisFromPlacement from \"./getMainAxisFromPlacement.js\";\nimport { top, right, bottom, left, start, end } from \"../enums.js\";\nexport default function computeOffsets(_ref) {\n  var reference = _ref.reference,\n      element = _ref.element,\n      placement = _ref.placement;\n  var basePlacement = placement ? getBasePlacement(placement) : null;\n  var variation = placement ? getVariation(placement) : null;\n  var commonX = reference.x + reference.width / 2 - element.width / 2;\n  var commonY = reference.y + reference.height / 2 - element.height / 2;\n  var offsets;\n\n  switch (basePlacement) {\n    case top:\n      offsets = {\n        x: commonX,\n        y: reference.y - element.height\n      };\n      break;\n\n    case bottom:\n      offsets = {\n        x: commonX,\n        y: reference.y + reference.height\n      };\n      break;\n\n    case right:\n      offsets = {\n        x: reference.x + reference.width,\n        y: commonY\n      };\n      break;\n\n    case left:\n      offsets = {\n        x: reference.x - element.width,\n        y: commonY\n      };\n      break;\n\n    default:\n      offsets = {\n        x: reference.x,\n        y: reference.y\n      };\n  }\n\n  var mainAxis = basePlacement ? getMainAxisFromPlacement(basePlacement) : null;\n\n  if (mainAxis != null) {\n    var len = mainAxis === 'y' ? 'height' : 'width';\n\n    switch (variation) {\n      case start:\n        offsets[mainAxis] = offsets[mainAxis] - (reference[len] / 2 - element[len] / 2);\n        break;\n\n      case end:\n        offsets[mainAxis] = offsets[mainAxis] + (reference[len] / 2 - element[len] / 2);\n        break;\n\n      default:\n    }\n  }\n\n  return offsets;\n}","import getClippingRect from \"../dom-utils/getClippingRect.js\";\nimport getDocumentElement from \"../dom-utils/getDocumentElement.js\";\nimport getBoundingClientRect from \"../dom-utils/getBoundingClientRect.js\";\nimport computeOffsets from \"./computeOffsets.js\";\nimport rectToClientRect from \"./rectToClientRect.js\";\nimport { clippingParents, reference, popper, bottom, top, right, basePlacements, viewport } from \"../enums.js\";\nimport { isElement } from \"../dom-utils/instanceOf.js\";\nimport mergePaddingObject from \"./mergePaddingObject.js\";\nimport expandToHashMap from \"./expandToHashMap.js\"; // eslint-disable-next-line import/no-unused-modules\n\nexport default function detectOverflow(state, options) {\n  if (options === void 0) {\n    options = {};\n  }\n\n  var _options = options,\n      _options$placement = _options.placement,\n      placement = _options$placement === void 0 ? state.placement : _options$placement,\n      _options$strategy = _options.strategy,\n      strategy = _options$strategy === void 0 ? state.strategy : _options$strategy,\n      _options$boundary = _options.boundary,\n      boundary = _options$boundary === void 0 ? clippingParents : _options$boundary,\n      _options$rootBoundary = _options.rootBoundary,\n      rootBoundary = _options$rootBoundary === void 0 ? viewport : _options$rootBoundary,\n      _options$elementConte = _options.elementContext,\n      elementContext = _options$elementConte === void 0 ? popper : _options$elementConte,\n      _options$altBoundary = _options.altBoundary,\n      altBoundary = _options$altBoundary === void 0 ? false : _options$altBoundary,\n      _options$padding = _options.padding,\n      padding = _options$padding === void 0 ? 0 : _options$padding;\n  var paddingObject = mergePaddingObject(typeof padding !== 'number' ? padding : expandToHashMap(padding, basePlacements));\n  var altContext = elementContext === popper ? reference : popper;\n  var popperRect = state.rects.popper;\n  var element = state.elements[altBoundary ? altContext : elementContext];\n  var clippingClientRect = getClippingRect(isElement(element) ? element : element.contextElement || getDocumentElement(state.elements.popper), boundary, rootBoundary, strategy);\n  var referenceClientRect = getBoundingClientRect(state.elements.reference);\n  var popperOffsets = computeOffsets({\n    reference: referenceClientRect,\n    element: popperRect,\n    strategy: 'absolute',\n    placement: placement\n  });\n  var popperClientRect = rectToClientRect(Object.assign({}, popperRect, popperOffsets));\n  var elementClientRect = elementContext === popper ? popperClientRect : referenceClientRect; // positive = overflowing the clipping rect\n  // 0 or negative = within the clipping rect\n\n  var overflowOffsets = {\n    top: clippingClientRect.top - elementClientRect.top + paddingObject.top,\n    bottom: elementClientRect.bottom - clippingClientRect.bottom + paddingObject.bottom,\n    left: clippingClientRect.left - elementClientRect.left + paddingObject.left,\n    right: elementClientRect.right - clippingClientRect.right + paddingObject.right\n  };\n  var offsetData = state.modifiersData.offset; // Offsets can be applied only to the popper element\n\n  if (elementContext === popper && offsetData) {\n    var offset = offsetData[placement];\n    Object.keys(overflowOffsets).forEach(function (key) {\n      var multiply = [right, bottom].indexOf(key) >= 0 ? 1 : -1;\n      var axis = [top, bottom].indexOf(key) >= 0 ? 'y' : 'x';\n      overflowOffsets[key] += offset[axis] * multiply;\n    });\n  }\n\n  return overflowOffsets;\n}","import getOppositePlacement from \"../utils/getOppositePlacement.js\";\nimport getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getOppositeVariationPlacement from \"../utils/getOppositeVariationPlacement.js\";\nimport detectOverflow from \"../utils/detectOverflow.js\";\nimport computeAutoPlacement from \"../utils/computeAutoPlacement.js\";\nimport { bottom, top, start, right, left, auto } from \"../enums.js\";\nimport getVariation from \"../utils/getVariation.js\"; // eslint-disable-next-line import/no-unused-modules\n\nfunction getExpandedFallbackPlacements(placement) {\n  if (getBasePlacement(placement) === auto) {\n    return [];\n  }\n\n  var oppositePlacement = getOppositePlacement(placement);\n  return [getOppositeVariationPlacement(placement), oppositePlacement, getOppositeVariationPlacement(oppositePlacement)];\n}\n\nfunction flip(_ref) {\n  var state = _ref.state,\n      options = _ref.options,\n      name = _ref.name;\n\n  if (state.modifiersData[name]._skip) {\n    return;\n  }\n\n  var _options$mainAxis = options.mainAxis,\n      checkMainAxis = _options$mainAxis === void 0 ? true : _options$mainAxis,\n      _options$altAxis = options.altAxis,\n      checkAltAxis = _options$altAxis === void 0 ? true : _options$altAxis,\n      specifiedFallbackPlacements = options.fallbackPlacements,\n      padding = options.padding,\n      boundary = options.boundary,\n      rootBoundary = options.rootBoundary,\n      altBoundary = options.altBoundary,\n      _options$flipVariatio = options.flipVariations,\n      flipVariations = _options$flipVariatio === void 0 ? true : _options$flipVariatio,\n      allowedAutoPlacements = options.allowedAutoPlacements;\n  var preferredPlacement = state.options.placement;\n  var basePlacement = getBasePlacement(preferredPlacement);\n  var isBasePlacement = basePlacement === preferredPlacement;\n  var fallbackPlacements = specifiedFallbackPlacements || (isBasePlacement || !flipVariations ? [getOppositePlacement(preferredPlacement)] : getExpandedFallbackPlacements(preferredPlacement));\n  var placements = [preferredPlacement].concat(fallbackPlacements).reduce(function (acc, placement) {\n    return acc.concat(getBasePlacement(placement) === auto ? computeAutoPlacement(state, {\n      placement: placement,\n      boundary: boundary,\n      rootBoundary: rootBoundary,\n      padding: padding,\n      flipVariations: flipVariations,\n      allowedAutoPlacements: allowedAutoPlacements\n    }) : placement);\n  }, []);\n  var referenceRect = state.rects.reference;\n  var popperRect = state.rects.popper;\n  var checksMap = new Map();\n  var makeFallbackChecks = true;\n  var firstFittingPlacement = placements[0];\n\n  for (var i = 0; i < placements.length; i++) {\n    var placement = placements[i];\n\n    var _basePlacement = getBasePlacement(placement);\n\n    var isStartVariation = getVariation(placement) === start;\n    var isVertical = [top, bottom].indexOf(_basePlacement) >= 0;\n    var len = isVertical ? 'width' : 'height';\n    var overflow = detectOverflow(state, {\n      placement: placement,\n      boundary: boundary,\n      rootBoundary: rootBoundary,\n      altBoundary: altBoundary,\n      padding: padding\n    });\n    var mainVariationSide = isVertical ? isStartVariation ? right : left : isStartVariation ? bottom : top;\n\n    if (referenceRect[len] > popperRect[len]) {\n      mainVariationSide = getOppositePlacement(mainVariationSide);\n    }\n\n    var altVariationSide = getOppositePlacement(mainVariationSide);\n    var checks = [];\n\n    if (checkMainAxis) {\n      checks.push(overflow[_basePlacement] <= 0);\n    }\n\n    if (checkAltAxis) {\n      checks.push(overflow[mainVariationSide] <= 0, overflow[altVariationSide] <= 0);\n    }\n\n    if (checks.every(function (check) {\n      return check;\n    })) {\n      firstFittingPlacement = placement;\n      makeFallbackChecks = false;\n      break;\n    }\n\n    checksMap.set(placement, checks);\n  }\n\n  if (makeFallbackChecks) {\n    // `2` may be desired in some cases – research later\n    var numberOfChecks = flipVariations ? 3 : 1;\n\n    var _loop = function _loop(_i) {\n      var fittingPlacement = placements.find(function (placement) {\n        var checks = checksMap.get(placement);\n\n        if (checks) {\n          return checks.slice(0, _i).every(function (check) {\n            return check;\n          });\n        }\n      });\n\n      if (fittingPlacement) {\n        firstFittingPlacement = fittingPlacement;\n        return \"break\";\n      }\n    };\n\n    for (var _i = numberOfChecks; _i > 0; _i--) {\n      var _ret = _loop(_i);\n\n      if (_ret === \"break\") break;\n    }\n  }\n\n  if (state.placement !== firstFittingPlacement) {\n    state.modifiersData[name]._skip = true;\n    state.placement = firstFittingPlacement;\n    state.reset = true;\n  }\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'flip',\n  enabled: true,\n  phase: 'main',\n  fn: flip,\n  requiresIfExists: ['offset'],\n  data: {\n    _skip: false\n  }\n};","import getVariation from \"./getVariation.js\";\nimport { variationPlacements, basePlacements, placements as allPlacements } from \"../enums.js\";\nimport detectOverflow from \"./detectOverflow.js\";\nimport getBasePlacement from \"./getBasePlacement.js\";\nexport default function computeAutoPlacement(state, options) {\n  if (options === void 0) {\n    options = {};\n  }\n\n  var _options = options,\n      placement = _options.placement,\n      boundary = _options.boundary,\n      rootBoundary = _options.rootBoundary,\n      padding = _options.padding,\n      flipVariations = _options.flipVariations,\n      _options$allowedAutoP = _options.allowedAutoPlacements,\n      allowedAutoPlacements = _options$allowedAutoP === void 0 ? allPlacements : _options$allowedAutoP;\n  var variation = getVariation(placement);\n  var placements = variation ? flipVariations ? variationPlacements : variationPlacements.filter(function (placement) {\n    return getVariation(placement) === variation;\n  }) : basePlacements;\n  var allowedPlacements = placements.filter(function (placement) {\n    return allowedAutoPlacements.indexOf(placement) >= 0;\n  });\n\n  if (allowedPlacements.length === 0) {\n    allowedPlacements = placements;\n  } // $FlowFixMe[incompatible-type]: Flow seems to have problems with two array unions...\n\n\n  var overflows = allowedPlacements.reduce(function (acc, placement) {\n    acc[placement] = detectOverflow(state, {\n      placement: placement,\n      boundary: boundary,\n      rootBoundary: rootBoundary,\n      padding: padding\n    })[getBasePlacement(placement)];\n    return acc;\n  }, {});\n  return Object.keys(overflows).sort(function (a, b) {\n    return overflows[a] - overflows[b];\n  });\n}","import { top, bottom, left, right } from \"../enums.js\";\nimport detectOverflow from \"../utils/detectOverflow.js\";\n\nfunction getSideOffsets(overflow, rect, preventedOffsets) {\n  if (preventedOffsets === void 0) {\n    preventedOffsets = {\n      x: 0,\n      y: 0\n    };\n  }\n\n  return {\n    top: overflow.top - rect.height - preventedOffsets.y,\n    right: overflow.right - rect.width + preventedOffsets.x,\n    bottom: overflow.bottom - rect.height + preventedOffsets.y,\n    left: overflow.left - rect.width - preventedOffsets.x\n  };\n}\n\nfunction isAnySideFullyClipped(overflow) {\n  return [top, right, bottom, left].some(function (side) {\n    return overflow[side] >= 0;\n  });\n}\n\nfunction hide(_ref) {\n  var state = _ref.state,\n      name = _ref.name;\n  var referenceRect = state.rects.reference;\n  var popperRect = state.rects.popper;\n  var preventedOffsets = state.modifiersData.preventOverflow;\n  var referenceOverflow = detectOverflow(state, {\n    elementContext: 'reference'\n  });\n  var popperAltOverflow = detectOverflow(state, {\n    altBoundary: true\n  });\n  var referenceClippingOffsets = getSideOffsets(referenceOverflow, referenceRect);\n  var popperEscapeOffsets = getSideOffsets(popperAltOverflow, popperRect, preventedOffsets);\n  var isReferenceHidden = isAnySideFullyClipped(referenceClippingOffsets);\n  var hasPopperEscaped = isAnySideFullyClipped(popperEscapeOffsets);\n  state.modifiersData[name] = {\n    referenceClippingOffsets: referenceClippingOffsets,\n    popperEscapeOffsets: popperEscapeOffsets,\n    isReferenceHidden: isReferenceHidden,\n    hasPopperEscaped: hasPopperEscaped\n  };\n  state.attributes.popper = Object.assign({}, state.attributes.popper, {\n    'data-popper-reference-hidden': isReferenceHidden,\n    'data-popper-escaped': hasPopperEscaped\n  });\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'hide',\n  enabled: true,\n  phase: 'main',\n  requiresIfExists: ['preventOverflow'],\n  fn: hide\n};","import getBasePlacement from \"../utils/getBasePlacement.js\";\nimport { top, left, right, placements } from \"../enums.js\"; // eslint-disable-next-line import/no-unused-modules\n\nexport function distanceAndSkiddingToXY(placement, rects, offset) {\n  var basePlacement = getBasePlacement(placement);\n  var invertDistance = [left, top].indexOf(basePlacement) >= 0 ? -1 : 1;\n\n  var _ref = typeof offset === 'function' ? offset(Object.assign({}, rects, {\n    placement: placement\n  })) : offset,\n      skidding = _ref[0],\n      distance = _ref[1];\n\n  skidding = skidding || 0;\n  distance = (distance || 0) * invertDistance;\n  return [left, right].indexOf(basePlacement) >= 0 ? {\n    x: distance,\n    y: skidding\n  } : {\n    x: skidding,\n    y: distance\n  };\n}\n\nfunction offset(_ref2) {\n  var state = _ref2.state,\n      options = _ref2.options,\n      name = _ref2.name;\n  var _options$offset = options.offset,\n      offset = _options$offset === void 0 ? [0, 0] : _options$offset;\n  var data = placements.reduce(function (acc, placement) {\n    acc[placement] = distanceAndSkiddingToXY(placement, state.rects, offset);\n    return acc;\n  }, {});\n  var _data$state$placement = data[state.placement],\n      x = _data$state$placement.x,\n      y = _data$state$placement.y;\n\n  if (state.modifiersData.popperOffsets != null) {\n    state.modifiersData.popperOffsets.x += x;\n    state.modifiersData.popperOffsets.y += y;\n  }\n\n  state.modifiersData[name] = data;\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'offset',\n  enabled: true,\n  phase: 'main',\n  requires: ['popperOffsets'],\n  fn: offset\n};","import computeOffsets from \"../utils/computeOffsets.js\";\n\nfunction popperOffsets(_ref) {\n  var state = _ref.state,\n      name = _ref.name;\n  // Offsets are the actual position the popper needs to have to be\n  // properly positioned near its reference element\n  // This is the most basic placement, and will be adjusted by\n  // the modifiers in the next step\n  state.modifiersData[name] = computeOffsets({\n    reference: state.rects.reference,\n    element: state.rects.popper,\n    strategy: 'absolute',\n    placement: state.placement\n  });\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'popperOffsets',\n  enabled: true,\n  phase: 'read',\n  fn: popperOffsets,\n  data: {}\n};","import { top, left, right, bottom, start } from \"../enums.js\";\nimport getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getMainAxisFromPlacement from \"../utils/getMainAxisFromPlacement.js\";\nimport getAltAxis from \"../utils/getAltAxis.js\";\nimport { within, withinMaxClamp } from \"../utils/within.js\";\nimport getLayoutRect from \"../dom-utils/getLayoutRect.js\";\nimport getOffsetParent from \"../dom-utils/getOffsetParent.js\";\nimport detectOverflow from \"../utils/detectOverflow.js\";\nimport getVariation from \"../utils/getVariation.js\";\nimport getFreshSideObject from \"../utils/getFreshSideObject.js\";\nimport { min as mathMin, max as mathMax } from \"../utils/math.js\";\n\nfunction preventOverflow(_ref) {\n  var state = _ref.state,\n      options = _ref.options,\n      name = _ref.name;\n  var _options$mainAxis = options.mainAxis,\n      checkMainAxis = _options$mainAxis === void 0 ? true : _options$mainAxis,\n      _options$altAxis = options.altAxis,\n      checkAltAxis = _options$altAxis === void 0 ? false : _options$altAxis,\n      boundary = options.boundary,\n      rootBoundary = options.rootBoundary,\n      altBoundary = options.altBoundary,\n      padding = options.padding,\n      _options$tether = options.tether,\n      tether = _options$tether === void 0 ? true : _options$tether,\n      _options$tetherOffset = options.tetherOffset,\n      tetherOffset = _options$tetherOffset === void 0 ? 0 : _options$tetherOffset;\n  var overflow = detectOverflow(state, {\n    boundary: boundary,\n    rootBoundary: rootBoundary,\n    padding: padding,\n    altBoundary: altBoundary\n  });\n  var basePlacement = getBasePlacement(state.placement);\n  var variation = getVariation(state.placement);\n  var isBasePlacement = !variation;\n  var mainAxis = getMainAxisFromPlacement(basePlacement);\n  var altAxis = getAltAxis(mainAxis);\n  var popperOffsets = state.modifiersData.popperOffsets;\n  var referenceRect = state.rects.reference;\n  var popperRect = state.rects.popper;\n  var tetherOffsetValue = typeof tetherOffset === 'function' ? tetherOffset(Object.assign({}, state.rects, {\n    placement: state.placement\n  })) : tetherOffset;\n  var normalizedTetherOffsetValue = typeof tetherOffsetValue === 'number' ? {\n    mainAxis: tetherOffsetValue,\n    altAxis: tetherOffsetValue\n  } : Object.assign({\n    mainAxis: 0,\n    altAxis: 0\n  }, tetherOffsetValue);\n  var offsetModifierState = state.modifiersData.offset ? state.modifiersData.offset[state.placement] : null;\n  var data = {\n    x: 0,\n    y: 0\n  };\n\n  if (!popperOffsets) {\n    return;\n  }\n\n  if (checkMainAxis) {\n    var _offsetModifierState$;\n\n    var mainSide = mainAxis === 'y' ? top : left;\n    var altSide = mainAxis === 'y' ? bottom : right;\n    var len = mainAxis === 'y' ? 'height' : 'width';\n    var offset = popperOffsets[mainAxis];\n    var min = offset + overflow[mainSide];\n    var max = offset - overflow[altSide];\n    var additive = tether ? -popperRect[len] / 2 : 0;\n    var minLen = variation === start ? referenceRect[len] : popperRect[len];\n    var maxLen = variation === start ? -popperRect[len] : -referenceRect[len]; // We need to include the arrow in the calculation so the arrow doesn't go\n    // outside the reference bounds\n\n    var arrowElement = state.elements.arrow;\n    var arrowRect = tether && arrowElement ? getLayoutRect(arrowElement) : {\n      width: 0,\n      height: 0\n    };\n    var arrowPaddingObject = state.modifiersData['arrow#persistent'] ? state.modifiersData['arrow#persistent'].padding : getFreshSideObject();\n    var arrowPaddingMin = arrowPaddingObject[mainSide];\n    var arrowPaddingMax = arrowPaddingObject[altSide]; // If the reference length is smaller than the arrow length, we don't want\n    // to include its full size in the calculation. If the reference is small\n    // and near the edge of a boundary, the popper can overflow even if the\n    // reference is not overflowing as well (e.g. virtual elements with no\n    // width or height)\n\n    var arrowLen = within(0, referenceRect[len], arrowRect[len]);\n    var minOffset = isBasePlacement ? referenceRect[len] / 2 - additive - arrowLen - arrowPaddingMin - normalizedTetherOffsetValue.mainAxis : minLen - arrowLen - arrowPaddingMin - normalizedTetherOffsetValue.mainAxis;\n    var maxOffset = isBasePlacement ? -referenceRect[len] / 2 + additive + arrowLen + arrowPaddingMax + normalizedTetherOffsetValue.mainAxis : maxLen + arrowLen + arrowPaddingMax + normalizedTetherOffsetValue.mainAxis;\n    var arrowOffsetParent = state.elements.arrow && getOffsetParent(state.elements.arrow);\n    var clientOffset = arrowOffsetParent ? mainAxis === 'y' ? arrowOffsetParent.clientTop || 0 : arrowOffsetParent.clientLeft || 0 : 0;\n    var offsetModifierValue = (_offsetModifierState$ = offsetModifierState == null ? void 0 : offsetModifierState[mainAxis]) != null ? _offsetModifierState$ : 0;\n    var tetherMin = offset + minOffset - offsetModifierValue - clientOffset;\n    var tetherMax = offset + maxOffset - offsetModifierValue;\n    var preventedOffset = within(tether ? mathMin(min, tetherMin) : min, offset, tether ? mathMax(max, tetherMax) : max);\n    popperOffsets[mainAxis] = preventedOffset;\n    data[mainAxis] = preventedOffset - offset;\n  }\n\n  if (checkAltAxis) {\n    var _offsetModifierState$2;\n\n    var _mainSide = mainAxis === 'x' ? top : left;\n\n    var _altSide = mainAxis === 'x' ? bottom : right;\n\n    var _offset = popperOffsets[altAxis];\n\n    var _len = altAxis === 'y' ? 'height' : 'width';\n\n    var _min = _offset + overflow[_mainSide];\n\n    var _max = _offset - overflow[_altSide];\n\n    var isOriginSide = [top, left].indexOf(basePlacement) !== -1;\n\n    var _offsetModifierValue = (_offsetModifierState$2 = offsetModifierState == null ? void 0 : offsetModifierState[altAxis]) != null ? _offsetModifierState$2 : 0;\n\n    var _tetherMin = isOriginSide ? _min : _offset - referenceRect[_len] - popperRect[_len] - _offsetModifierValue + normalizedTetherOffsetValue.altAxis;\n\n    var _tetherMax = isOriginSide ? _offset + referenceRect[_len] + popperRect[_len] - _offsetModifierValue - normalizedTetherOffsetValue.altAxis : _max;\n\n    var _preventedOffset = tether && isOriginSide ? withinMaxClamp(_tetherMin, _offset, _tetherMax) : within(tether ? _tetherMin : _min, _offset, tether ? _tetherMax : _max);\n\n    popperOffsets[altAxis] = _preventedOffset;\n    data[altAxis] = _preventedOffset - _offset;\n  }\n\n  state.modifiersData[name] = data;\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'preventOverflow',\n  enabled: true,\n  phase: 'main',\n  fn: preventOverflow,\n  requiresIfExists: ['offset']\n};","export default function getAltAxis(axis) {\n  return axis === 'x' ? 'y' : 'x';\n}","import getBoundingClientRect from \"./getBoundingClientRect.js\";\nimport getNodeScroll from \"./getNodeScroll.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport { isHTMLElement } from \"./instanceOf.js\";\nimport getWindowScrollBarX from \"./getWindowScrollBarX.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport isScrollParent from \"./isScrollParent.js\";\nimport { round } from \"../utils/math.js\";\n\nfunction isElementScaled(element) {\n  var rect = element.getBoundingClientRect();\n  var scaleX = round(rect.width) / element.offsetWidth || 1;\n  var scaleY = round(rect.height) / element.offsetHeight || 1;\n  return scaleX !== 1 || scaleY !== 1;\n} // Returns the composite rect of an element relative to its offsetParent.\n// Composite means it takes into account transforms as well as layout.\n\n\nexport default function getCompositeRect(elementOrVirtualElement, offsetParent, isFixed) {\n  if (isFixed === void 0) {\n    isFixed = false;\n  }\n\n  var isOffsetParentAnElement = isHTMLElement(offsetParent);\n  var offsetParentIsScaled = isHTMLElement(offsetParent) && isElementScaled(offsetParent);\n  var documentElement = getDocumentElement(offsetParent);\n  var rect = getBoundingClientRect(elementOrVirtualElement, offsetParentIsScaled, isFixed);\n  var scroll = {\n    scrollLeft: 0,\n    scrollTop: 0\n  };\n  var offsets = {\n    x: 0,\n    y: 0\n  };\n\n  if (isOffsetParentAnElement || !isOffsetParentAnElement && !isFixed) {\n    if (getNodeName(offsetParent) !== 'body' || // https://github.com/popperjs/popper-core/issues/1078\n    isScrollParent(documentElement)) {\n      scroll = getNodeScroll(offsetParent);\n    }\n\n    if (isHTMLElement(offsetParent)) {\n      offsets = getBoundingClientRect(offsetParent, true);\n      offsets.x += offsetParent.clientLeft;\n      offsets.y += offsetParent.clientTop;\n    } else if (documentElement) {\n      offsets.x = getWindowScrollBarX(documentElement);\n    }\n  }\n\n  return {\n    x: rect.left + scroll.scrollLeft - offsets.x,\n    y: rect.top + scroll.scrollTop - offsets.y,\n    width: rect.width,\n    height: rect.height\n  };\n}","import getWindowScroll from \"./getWindowScroll.js\";\nimport getWindow from \"./getWindow.js\";\nimport { isHTMLElement } from \"./instanceOf.js\";\nimport getHTMLElementScroll from \"./getHTMLElementScroll.js\";\nexport default function getNodeScroll(node) {\n  if (node === getWindow(node) || !isHTMLElement(node)) {\n    return getWindowScroll(node);\n  } else {\n    return getHTMLElementScroll(node);\n  }\n}","export default function getHTMLElementScroll(element) {\n  return {\n    scrollLeft: element.scrollLeft,\n    scrollTop: element.scrollTop\n  };\n}","import { modifierPhases } from \"../enums.js\"; // source: https://stackoverflow.com/questions/49875255\n\nfunction order(modifiers) {\n  var map = new Map();\n  var visited = new Set();\n  var result = [];\n  modifiers.forEach(function (modifier) {\n    map.set(modifier.name, modifier);\n  }); // On visiting object, check for its dependencies and visit them recursively\n\n  function sort(modifier) {\n    visited.add(modifier.name);\n    var requires = [].concat(modifier.requires || [], modifier.requiresIfExists || []);\n    requires.forEach(function (dep) {\n      if (!visited.has(dep)) {\n        var depModifier = map.get(dep);\n\n        if (depModifier) {\n          sort(depModifier);\n        }\n      }\n    });\n    result.push(modifier);\n  }\n\n  modifiers.forEach(function (modifier) {\n    if (!visited.has(modifier.name)) {\n      // check for visited object\n      sort(modifier);\n    }\n  });\n  return result;\n}\n\nexport default function orderModifiers(modifiers) {\n  // order based on dependencies\n  var orderedModifiers = order(modifiers); // order based on phase\n\n  return modifierPhases.reduce(function (acc, phase) {\n    return acc.concat(orderedModifiers.filter(function (modifier) {\n      return modifier.phase === phase;\n    }));\n  }, []);\n}","import getCompositeRect from \"./dom-utils/getCompositeRect.js\";\nimport getLayoutRect from \"./dom-utils/getLayoutRect.js\";\nimport listScrollParents from \"./dom-utils/listScrollParents.js\";\nimport getOffsetParent from \"./dom-utils/getOffsetParent.js\";\nimport orderModifiers from \"./utils/orderModifiers.js\";\nimport debounce from \"./utils/debounce.js\";\nimport mergeByName from \"./utils/mergeByName.js\";\nimport detectOverflow from \"./utils/detectOverflow.js\";\nimport { isElement } from \"./dom-utils/instanceOf.js\";\nvar DEFAULT_OPTIONS = {\n  placement: 'bottom',\n  modifiers: [],\n  strategy: 'absolute'\n};\n\nfunction areValidElements() {\n  for (var _len = arguments.length, args = new Array(_len), _key = 0; _key < _len; _key++) {\n    args[_key] = arguments[_key];\n  }\n\n  return !args.some(function (element) {\n    return !(element && typeof element.getBoundingClientRect === 'function');\n  });\n}\n\nexport function popperGenerator(generatorOptions) {\n  if (generatorOptions === void 0) {\n    generatorOptions = {};\n  }\n\n  var _generatorOptions = generatorOptions,\n      _generatorOptions$def = _generatorOptions.defaultModifiers,\n      defaultModifiers = _generatorOptions$def === void 0 ? [] : _generatorOptions$def,\n      _generatorOptions$def2 = _generatorOptions.defaultOptions,\n      defaultOptions = _generatorOptions$def2 === void 0 ? DEFAULT_OPTIONS : _generatorOptions$def2;\n  return function createPopper(reference, popper, options) {\n    if (options === void 0) {\n      options = defaultOptions;\n    }\n\n    var state = {\n      placement: 'bottom',\n      orderedModifiers: [],\n      options: Object.assign({}, DEFAULT_OPTIONS, defaultOptions),\n      modifiersData: {},\n      elements: {\n        reference: reference,\n        popper: popper\n      },\n      attributes: {},\n      styles: {}\n    };\n    var effectCleanupFns = [];\n    var isDestroyed = false;\n    var instance = {\n      state: state,\n      setOptions: function setOptions(setOptionsAction) {\n        var options = typeof setOptionsAction === 'function' ? setOptionsAction(state.options) : setOptionsAction;\n        cleanupModifierEffects();\n        state.options = Object.assign({}, defaultOptions, state.options, options);\n        state.scrollParents = {\n          reference: isElement(reference) ? listScrollParents(reference) : reference.contextElement ? listScrollParents(reference.contextElement) : [],\n          popper: listScrollParents(popper)\n        }; // Orders the modifiers based on their dependencies and `phase`\n        // properties\n\n        var orderedModifiers = orderModifiers(mergeByName([].concat(defaultModifiers, state.options.modifiers))); // Strip out disabled modifiers\n\n        state.orderedModifiers = orderedModifiers.filter(function (m) {\n          return m.enabled;\n        });\n        runModifierEffects();\n        return instance.update();\n      },\n      // Sync update – it will always be executed, even if not necessary. This\n      // is useful for low frequency updates where sync behavior simplifies the\n      // logic.\n      // For high frequency updates (e.g. `resize` and `scroll` events), always\n      // prefer the async Popper#update method\n      forceUpdate: function forceUpdate() {\n        if (isDestroyed) {\n          return;\n        }\n\n        var _state$elements = state.elements,\n            reference = _state$elements.reference,\n            popper = _state$elements.popper; // Don't proceed if `reference` or `popper` are not valid elements\n        // anymore\n\n        if (!areValidElements(reference, popper)) {\n          return;\n        } // Store the reference and popper rects to be read by modifiers\n\n\n        state.rects = {\n          reference: getCompositeRect(reference, getOffsetParent(popper), state.options.strategy === 'fixed'),\n          popper: getLayoutRect(popper)\n        }; // Modifiers have the ability to reset the current update cycle. The\n        // most common use case for this is the `flip` modifier changing the\n        // placement, which then needs to re-run all the modifiers, because the\n        // logic was previously ran for the previous placement and is therefore\n        // stale/incorrect\n\n        state.reset = false;\n        state.placement = state.options.placement; // On each update cycle, the `modifiersData` property for each modifier\n        // is filled with the initial data specified by the modifier. This means\n        // it doesn't persist and is fresh on each update.\n        // To ensure persistent data, use `${name}#persistent`\n\n        state.orderedModifiers.forEach(function (modifier) {\n          return state.modifiersData[modifier.name] = Object.assign({}, modifier.data);\n        });\n\n        for (var index = 0; index < state.orderedModifiers.length; index++) {\n          if (state.reset === true) {\n            state.reset = false;\n            index = -1;\n            continue;\n          }\n\n          var _state$orderedModifie = state.orderedModifiers[index],\n              fn = _state$orderedModifie.fn,\n              _state$orderedModifie2 = _state$orderedModifie.options,\n              _options = _state$orderedModifie2 === void 0 ? {} : _state$orderedModifie2,\n              name = _state$orderedModifie.name;\n\n          if (typeof fn === 'function') {\n            state = fn({\n              state: state,\n              options: _options,\n              name: name,\n              instance: instance\n            }) || state;\n          }\n        }\n      },\n      // Async and optimistically optimized update – it will not be executed if\n      // not necessary (debounced to run at most once-per-tick)\n      update: debounce(function () {\n        return new Promise(function (resolve) {\n          instance.forceUpdate();\n          resolve(state);\n        });\n      }),\n      destroy: function destroy() {\n        cleanupModifierEffects();\n        isDestroyed = true;\n      }\n    };\n\n    if (!areValidElements(reference, popper)) {\n      return instance;\n    }\n\n    instance.setOptions(options).then(function (state) {\n      if (!isDestroyed && options.onFirstUpdate) {\n        options.onFirstUpdate(state);\n      }\n    }); // Modifiers have the ability to execute arbitrary code before the first\n    // update cycle runs. They will be executed in the same order as the update\n    // cycle. This is useful when a modifier adds some persistent data that\n    // other modifiers need to use, but the modifier is run after the dependent\n    // one.\n\n    function runModifierEffects() {\n      state.orderedModifiers.forEach(function (_ref) {\n        var name = _ref.name,\n            _ref$options = _ref.options,\n            options = _ref$options === void 0 ? {} : _ref$options,\n            effect = _ref.effect;\n\n        if (typeof effect === 'function') {\n          var cleanupFn = effect({\n            state: state,\n            name: name,\n            instance: instance,\n            options: options\n          });\n\n          var noopFn = function noopFn() {};\n\n          effectCleanupFns.push(cleanupFn || noopFn);\n        }\n      });\n    }\n\n    function cleanupModifierEffects() {\n      effectCleanupFns.forEach(function (fn) {\n        return fn();\n      });\n      effectCleanupFns = [];\n    }\n\n    return instance;\n  };\n}\nexport var createPopper = /*#__PURE__*/popperGenerator(); // eslint-disable-next-line import/no-unused-modules\n\nexport { detectOverflow };","export default function debounce(fn) {\n  var pending;\n  return function () {\n    if (!pending) {\n      pending = new Promise(function (resolve) {\n        Promise.resolve().then(function () {\n          pending = undefined;\n          resolve(fn());\n        });\n      });\n    }\n\n    return pending;\n  };\n}","export default function mergeByName(modifiers) {\n  var merged = modifiers.reduce(function (merged, current) {\n    var existing = merged[current.name];\n    merged[current.name] = existing ? Object.assign({}, existing, current, {\n      options: Object.assign({}, existing.options, current.options),\n      data: Object.assign({}, existing.data, current.data)\n    }) : current;\n    return merged;\n  }, {}); // IE11 does not support Object.values\n\n  return Object.keys(merged).map(function (key) {\n    return merged[key];\n  });\n}","import { popperGenerator, detectOverflow } from \"./createPopper.js\";\nimport eventListeners from \"./modifiers/eventListeners.js\";\nimport popperOffsets from \"./modifiers/popperOffsets.js\";\nimport computeStyles from \"./modifiers/computeStyles.js\";\nimport applyStyles from \"./modifiers/applyStyles.js\";\nimport offset from \"./modifiers/offset.js\";\nimport flip from \"./modifiers/flip.js\";\nimport preventOverflow from \"./modifiers/preventOverflow.js\";\nimport arrow from \"./modifiers/arrow.js\";\nimport hide from \"./modifiers/hide.js\";\nvar defaultModifiers = [eventListeners, popperOffsets, computeStyles, applyStyles, offset, flip, preventOverflow, arrow, hide];\nvar createPopper = /*#__PURE__*/popperGenerator({\n  defaultModifiers: defaultModifiers\n}); // eslint-disable-next-line import/no-unused-modules\n\nexport { createPopper, popperGenerator, defaultModifiers, detectOverflow }; // eslint-disable-next-line import/no-unused-modules\n\nexport { createPopper as createPopperLite } from \"./popper-lite.js\"; // eslint-disable-next-line import/no-unused-modules\n\nexport * from \"./modifiers/index.js\";","import { popperGenerator, detectOverflow } from \"./createPopper.js\";\nimport eventListeners from \"./modifiers/eventListeners.js\";\nimport popperOffsets from \"./modifiers/popperOffsets.js\";\nimport computeStyles from \"./modifiers/computeStyles.js\";\nimport applyStyles from \"./modifiers/applyStyles.js\";\nvar defaultModifiers = [eventListeners, popperOffsets, computeStyles, applyStyles];\nvar createPopper = /*#__PURE__*/popperGenerator({\n  defaultModifiers: defaultModifiers\n}); // eslint-disable-next-line import/no-unused-modules\n\nexport { createPopper, popperGenerator, defaultModifiers, detectOverflow };","/*!\n  * Bootstrap v5.3.3 (https://getbootstrap.com/)\n  * Copyright 2011-2024 The Bootstrap Authors (https://github.com/twbs/bootstrap/graphs/contributors)\n  * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n  */\nimport * as Popper from '@popperjs/core';\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dom/data.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n/**\n * Constants\n */\n\nconst elementMap = new Map();\nconst Data = {\n  set(element, key, instance) {\n    if (!elementMap.has(element)) {\n      elementMap.set(element, new Map());\n    }\n    const instanceMap = elementMap.get(element);\n\n    // make it clear we only want one instance per element\n    // can be removed later when multiple key/instances are fine to be used\n    if (!instanceMap.has(key) && instanceMap.size !== 0) {\n      // eslint-disable-next-line no-console\n      console.error(`Bootstrap doesn't allow more than one instance per element. Bound instance: ${Array.from(instanceMap.keys())[0]}.`);\n      return;\n    }\n    instanceMap.set(key, instance);\n  },\n  get(element, key) {\n    if (elementMap.has(element)) {\n      return elementMap.get(element).get(key) || null;\n    }\n    return null;\n  },\n  remove(element, key) {\n    if (!elementMap.has(element)) {\n      return;\n    }\n    const instanceMap = elementMap.get(element);\n    instanceMap.delete(key);\n\n    // free up element references if there are no instances left for an element\n    if (instanceMap.size === 0) {\n      elementMap.delete(element);\n    }\n  }\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/index.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nconst MAX_UID = 1000000;\nconst MILLISECONDS_MULTIPLIER = 1000;\nconst TRANSITION_END = 'transitionend';\n\n/**\n * Properly escape IDs selectors to handle weird IDs\n * @param {string} selector\n * @returns {string}\n */\nconst parseSelector = selector => {\n  if (selector && window.CSS && window.CSS.escape) {\n    // document.querySelector needs escaping to handle IDs (html5+) containing for instance /\n    selector = selector.replace(/#([^\\s\"#']+)/g, (match, id) => `#${CSS.escape(id)}`);\n  }\n  return selector;\n};\n\n// Shout-out Angus Croll (https://goo.gl/pxwQGp)\nconst toType = object => {\n  if (object === null || object === undefined) {\n    return `${object}`;\n  }\n  return Object.prototype.toString.call(object).match(/\\s([a-z]+)/i)[1].toLowerCase();\n};\n\n/**\n * Public Util API\n */\n\nconst getUID = prefix => {\n  do {\n    prefix += Math.floor(Math.random() * MAX_UID);\n  } while (document.getElementById(prefix));\n  return prefix;\n};\nconst getTransitionDurationFromElement = element => {\n  if (!element) {\n    return 0;\n  }\n\n  // Get transition-duration of the element\n  let {\n    transitionDuration,\n    transitionDelay\n  } = window.getComputedStyle(element);\n  const floatTransitionDuration = Number.parseFloat(transitionDuration);\n  const floatTransitionDelay = Number.parseFloat(transitionDelay);\n\n  // Return 0 if element or transition duration is not found\n  if (!floatTransitionDuration && !floatTransitionDelay) {\n    return 0;\n  }\n\n  // If multiple durations are defined, take the first\n  transitionDuration = transitionDuration.split(',')[0];\n  transitionDelay = transitionDelay.split(',')[0];\n  return (Number.parseFloat(transitionDuration) + Number.parseFloat(transitionDelay)) * MILLISECONDS_MULTIPLIER;\n};\nconst triggerTransitionEnd = element => {\n  element.dispatchEvent(new Event(TRANSITION_END));\n};\nconst isElement = object => {\n  if (!object || typeof object !== 'object') {\n    return false;\n  }\n  if (typeof object.jquery !== 'undefined') {\n    object = object[0];\n  }\n  return typeof object.nodeType !== 'undefined';\n};\nconst getElement = object => {\n  // it's a jQuery object or a node element\n  if (isElement(object)) {\n    return object.jquery ? object[0] : object;\n  }\n  if (typeof object === 'string' && object.length > 0) {\n    return document.querySelector(parseSelector(object));\n  }\n  return null;\n};\nconst isVisible = element => {\n  if (!isElement(element) || element.getClientRects().length === 0) {\n    return false;\n  }\n  const elementIsVisible = getComputedStyle(element).getPropertyValue('visibility') === 'visible';\n  // Handle `details` element as its content may falsie appear visible when it is closed\n  const closedDetails = element.closest('details:not([open])');\n  if (!closedDetails) {\n    return elementIsVisible;\n  }\n  if (closedDetails !== element) {\n    const summary = element.closest('summary');\n    if (summary && summary.parentNode !== closedDetails) {\n      return false;\n    }\n    if (summary === null) {\n      return false;\n    }\n  }\n  return elementIsVisible;\n};\nconst isDisabled = element => {\n  if (!element || element.nodeType !== Node.ELEMENT_NODE) {\n    return true;\n  }\n  if (element.classList.contains('disabled')) {\n    return true;\n  }\n  if (typeof element.disabled !== 'undefined') {\n    return element.disabled;\n  }\n  return element.hasAttribute('disabled') && element.getAttribute('disabled') !== 'false';\n};\nconst findShadowRoot = element => {\n  if (!document.documentElement.attachShadow) {\n    return null;\n  }\n\n  // Can find the shadow root otherwise it'll return the document\n  if (typeof element.getRootNode === 'function') {\n    const root = element.getRootNode();\n    return root instanceof ShadowRoot ? root : null;\n  }\n  if (element instanceof ShadowRoot) {\n    return element;\n  }\n\n  // when we don't find a shadow root\n  if (!element.parentNode) {\n    return null;\n  }\n  return findShadowRoot(element.parentNode);\n};\nconst noop = () => {};\n\n/**\n * Trick to restart an element's animation\n *\n * @param {HTMLElement} element\n * @return void\n *\n * @see https://www.charistheo.io/blog/2021/02/restart-a-css-animation-with-javascript/#restarting-a-css-animation\n */\nconst reflow = element => {\n  element.offsetHeight; // eslint-disable-line no-unused-expressions\n};\nconst getjQuery = () => {\n  if (window.jQuery && !document.body.hasAttribute('data-bs-no-jquery')) {\n    return window.jQuery;\n  }\n  return null;\n};\nconst DOMContentLoadedCallbacks = [];\nconst onDOMContentLoaded = callback => {\n  if (document.readyState === 'loading') {\n    // add listener on the first call when the document is in loading state\n    if (!DOMContentLoadedCallbacks.length) {\n      document.addEventListener('DOMContentLoaded', () => {\n        for (const callback of DOMContentLoadedCallbacks) {\n          callback();\n        }\n      });\n    }\n    DOMContentLoadedCallbacks.push(callback);\n  } else {\n    callback();\n  }\n};\nconst isRTL = () => document.documentElement.dir === 'rtl';\nconst defineJQueryPlugin = plugin => {\n  onDOMContentLoaded(() => {\n    const $ = getjQuery();\n    /* istanbul ignore if */\n    if ($) {\n      const name = plugin.NAME;\n      const JQUERY_NO_CONFLICT = $.fn[name];\n      $.fn[name] = plugin.jQueryInterface;\n      $.fn[name].Constructor = plugin;\n      $.fn[name].noConflict = () => {\n        $.fn[name] = JQUERY_NO_CONFLICT;\n        return plugin.jQueryInterface;\n      };\n    }\n  });\n};\nconst execute = (possibleCallback, args = [], defaultValue = possibleCallback) => {\n  return typeof possibleCallback === 'function' ? possibleCallback(...args) : defaultValue;\n};\nconst executeAfterTransition = (callback, transitionElement, waitForTransition = true) => {\n  if (!waitForTransition) {\n    execute(callback);\n    return;\n  }\n  const durationPadding = 5;\n  const emulatedDuration = getTransitionDurationFromElement(transitionElement) + durationPadding;\n  let called = false;\n  const handler = ({\n    target\n  }) => {\n    if (target !== transitionElement) {\n      return;\n    }\n    called = true;\n    transitionElement.removeEventListener(TRANSITION_END, handler);\n    execute(callback);\n  };\n  transitionElement.addEventListener(TRANSITION_END, handler);\n  setTimeout(() => {\n    if (!called) {\n      triggerTransitionEnd(transitionElement);\n    }\n  }, emulatedDuration);\n};\n\n/**\n * Return the previous/next element of a list.\n *\n * @param {array} list    The list of elements\n * @param activeElement   The active element\n * @param shouldGetNext   Choose to get next or previous element\n * @param isCycleAllowed\n * @return {Element|elem} The proper element\n */\nconst getNextActiveElement = (list, activeElement, shouldGetNext, isCycleAllowed) => {\n  const listLength = list.length;\n  let index = list.indexOf(activeElement);\n\n  // if the element does not exist in the list return an element\n  // depending on the direction and if cycle is allowed\n  if (index === -1) {\n    return !shouldGetNext && isCycleAllowed ? list[listLength - 1] : list[0];\n  }\n  index += shouldGetNext ? 1 : -1;\n  if (isCycleAllowed) {\n    index = (index + listLength) % listLength;\n  }\n  return list[Math.max(0, Math.min(index, listLength - 1))];\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dom/event-handler.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst namespaceRegex = /[^.]*(?=\\..*)\\.|.*/;\nconst stripNameRegex = /\\..*/;\nconst stripUidRegex = /::\\d+$/;\nconst eventRegistry = {}; // Events storage\nlet uidEvent = 1;\nconst customEvents = {\n  mouseenter: 'mouseover',\n  mouseleave: 'mouseout'\n};\nconst nativeEvents = new Set(['click', 'dblclick', 'mouseup', 'mousedown', 'contextmenu', 'mousewheel', 'DOMMouseScroll', 'mouseover', 'mouseout', 'mousemove', 'selectstart', 'selectend', 'keydown', 'keypress', 'keyup', 'orientationchange', 'touchstart', 'touchmove', 'touchend', 'touchcancel', 'pointerdown', 'pointermove', 'pointerup', 'pointerleave', 'pointercancel', 'gesturestart', 'gesturechange', 'gestureend', 'focus', 'blur', 'change', 'reset', 'select', 'submit', 'focusin', 'focusout', 'load', 'unload', 'beforeunload', 'resize', 'move', 'DOMContentLoaded', 'readystatechange', 'error', 'abort', 'scroll']);\n\n/**\n * Private methods\n */\n\nfunction makeEventUid(element, uid) {\n  return uid && `${uid}::${uidEvent++}` || element.uidEvent || uidEvent++;\n}\nfunction getElementEvents(element) {\n  const uid = makeEventUid(element);\n  element.uidEvent = uid;\n  eventRegistry[uid] = eventRegistry[uid] || {};\n  return eventRegistry[uid];\n}\nfunction bootstrapHandler(element, fn) {\n  return function handler(event) {\n    hydrateObj(event, {\n      delegateTarget: element\n    });\n    if (handler.oneOff) {\n      EventHandler.off(element, event.type, fn);\n    }\n    return fn.apply(element, [event]);\n  };\n}\nfunction bootstrapDelegationHandler(element, selector, fn) {\n  return function handler(event) {\n    const domElements = element.querySelectorAll(selector);\n    for (let {\n      target\n    } = event; target && target !== this; target = target.parentNode) {\n      for (const domElement of domElements) {\n        if (domElement !== target) {\n          continue;\n        }\n        hydrateObj(event, {\n          delegateTarget: target\n        });\n        if (handler.oneOff) {\n          EventHandler.off(element, event.type, selector, fn);\n        }\n        return fn.apply(target, [event]);\n      }\n    }\n  };\n}\nfunction findHandler(events, callable, delegationSelector = null) {\n  return Object.values(events).find(event => event.callable === callable && event.delegationSelector === delegationSelector);\n}\nfunction normalizeParameters(originalTypeEvent, handler, delegationFunction) {\n  const isDelegated = typeof handler === 'string';\n  // TODO: tooltip passes `false` instead of selector, so we need to check\n  const callable = isDelegated ? delegationFunction : handler || delegationFunction;\n  let typeEvent = getTypeEvent(originalTypeEvent);\n  if (!nativeEvents.has(typeEvent)) {\n    typeEvent = originalTypeEvent;\n  }\n  return [isDelegated, callable, typeEvent];\n}\nfunction addHandler(element, originalTypeEvent, handler, delegationFunction, oneOff) {\n  if (typeof originalTypeEvent !== 'string' || !element) {\n    return;\n  }\n  let [isDelegated, callable, typeEvent] = normalizeParameters(originalTypeEvent, handler, delegationFunction);\n\n  // in case of mouseenter or mouseleave wrap the handler within a function that checks for its DOM position\n  // this prevents the handler from being dispatched the same way as mouseover or mouseout does\n  if (originalTypeEvent in customEvents) {\n    const wrapFunction = fn => {\n      return function (event) {\n        if (!event.relatedTarget || event.relatedTarget !== event.delegateTarget && !event.delegateTarget.contains(event.relatedTarget)) {\n          return fn.call(this, event);\n        }\n      };\n    };\n    callable = wrapFunction(callable);\n  }\n  const events = getElementEvents(element);\n  const handlers = events[typeEvent] || (events[typeEvent] = {});\n  const previousFunction = findHandler(handlers, callable, isDelegated ? handler : null);\n  if (previousFunction) {\n    previousFunction.oneOff = previousFunction.oneOff && oneOff;\n    return;\n  }\n  const uid = makeEventUid(callable, originalTypeEvent.replace(namespaceRegex, ''));\n  const fn = isDelegated ? bootstrapDelegationHandler(element, handler, callable) : bootstrapHandler(element, callable);\n  fn.delegationSelector = isDelegated ? handler : null;\n  fn.callable = callable;\n  fn.oneOff = oneOff;\n  fn.uidEvent = uid;\n  handlers[uid] = fn;\n  element.addEventListener(typeEvent, fn, isDelegated);\n}\nfunction removeHandler(element, events, typeEvent, handler, delegationSelector) {\n  const fn = findHandler(events[typeEvent], handler, delegationSelector);\n  if (!fn) {\n    return;\n  }\n  element.removeEventListener(typeEvent, fn, Boolean(delegationSelector));\n  delete events[typeEvent][fn.uidEvent];\n}\nfunction removeNamespacedHandlers(element, events, typeEvent, namespace) {\n  const storeElementEvent = events[typeEvent] || {};\n  for (const [handlerKey, event] of Object.entries(storeElementEvent)) {\n    if (handlerKey.includes(namespace)) {\n      removeHandler(element, events, typeEvent, event.callable, event.delegationSelector);\n    }\n  }\n}\nfunction getTypeEvent(event) {\n  // allow to get the native events from namespaced events ('click.bs.button' --> 'click')\n  event = event.replace(stripNameRegex, '');\n  return customEvents[event] || event;\n}\nconst EventHandler = {\n  on(element, event, handler, delegationFunction) {\n    addHandler(element, event, handler, delegationFunction, false);\n  },\n  one(element, event, handler, delegationFunction) {\n    addHandler(element, event, handler, delegationFunction, true);\n  },\n  off(element, originalTypeEvent, handler, delegationFunction) {\n    if (typeof originalTypeEvent !== 'string' || !element) {\n      return;\n    }\n    const [isDelegated, callable, typeEvent] = normalizeParameters(originalTypeEvent, handler, delegationFunction);\n    const inNamespace = typeEvent !== originalTypeEvent;\n    const events = getElementEvents(element);\n    const storeElementEvent = events[typeEvent] || {};\n    const isNamespace = originalTypeEvent.startsWith('.');\n    if (typeof callable !== 'undefined') {\n      // Simplest case: handler is passed, remove that listener ONLY.\n      if (!Object.keys(storeElementEvent).length) {\n        return;\n      }\n      removeHandler(element, events, typeEvent, callable, isDelegated ? handler : null);\n      return;\n    }\n    if (isNamespace) {\n      for (const elementEvent of Object.keys(events)) {\n        removeNamespacedHandlers(element, events, elementEvent, originalTypeEvent.slice(1));\n      }\n    }\n    for (const [keyHandlers, event] of Object.entries(storeElementEvent)) {\n      const handlerKey = keyHandlers.replace(stripUidRegex, '');\n      if (!inNamespace || originalTypeEvent.includes(handlerKey)) {\n        removeHandler(element, events, typeEvent, event.callable, event.delegationSelector);\n      }\n    }\n  },\n  trigger(element, event, args) {\n    if (typeof event !== 'string' || !element) {\n      return null;\n    }\n    const $ = getjQuery();\n    const typeEvent = getTypeEvent(event);\n    const inNamespace = event !== typeEvent;\n    let jQueryEvent = null;\n    let bubbles = true;\n    let nativeDispatch = true;\n    let defaultPrevented = false;\n    if (inNamespace && $) {\n      jQueryEvent = $.Event(event, args);\n      $(element).trigger(jQueryEvent);\n      bubbles = !jQueryEvent.isPropagationStopped();\n      nativeDispatch = !jQueryEvent.isImmediatePropagationStopped();\n      defaultPrevented = jQueryEvent.isDefaultPrevented();\n    }\n    const evt = hydrateObj(new Event(event, {\n      bubbles,\n      cancelable: true\n    }), args);\n    if (defaultPrevented) {\n      evt.preventDefault();\n    }\n    if (nativeDispatch) {\n      element.dispatchEvent(evt);\n    }\n    if (evt.defaultPrevented && jQueryEvent) {\n      jQueryEvent.preventDefault();\n    }\n    return evt;\n  }\n};\nfunction hydrateObj(obj, meta = {}) {\n  for (const [key, value] of Object.entries(meta)) {\n    try {\n      obj[key] = value;\n    } catch (_unused) {\n      Object.defineProperty(obj, key, {\n        configurable: true,\n        get() {\n          return value;\n        }\n      });\n    }\n  }\n  return obj;\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dom/manipulator.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nfunction normalizeData(value) {\n  if (value === 'true') {\n    return true;\n  }\n  if (value === 'false') {\n    return false;\n  }\n  if (value === Number(value).toString()) {\n    return Number(value);\n  }\n  if (value === '' || value === 'null') {\n    return null;\n  }\n  if (typeof value !== 'string') {\n    return value;\n  }\n  try {\n    return JSON.parse(decodeURIComponent(value));\n  } catch (_unused) {\n    return value;\n  }\n}\nfunction normalizeDataKey(key) {\n  return key.replace(/[A-Z]/g, chr => `-${chr.toLowerCase()}`);\n}\nconst Manipulator = {\n  setDataAttribute(element, key, value) {\n    element.setAttribute(`data-bs-${normalizeDataKey(key)}`, value);\n  },\n  removeDataAttribute(element, key) {\n    element.removeAttribute(`data-bs-${normalizeDataKey(key)}`);\n  },\n  getDataAttributes(element) {\n    if (!element) {\n      return {};\n    }\n    const attributes = {};\n    const bsKeys = Object.keys(element.dataset).filter(key => key.startsWith('bs') && !key.startsWith('bsConfig'));\n    for (const key of bsKeys) {\n      let pureKey = key.replace(/^bs/, '');\n      pureKey = pureKey.charAt(0).toLowerCase() + pureKey.slice(1, pureKey.length);\n      attributes[pureKey] = normalizeData(element.dataset[key]);\n    }\n    return attributes;\n  },\n  getDataAttribute(element, key) {\n    return normalizeData(element.getAttribute(`data-bs-${normalizeDataKey(key)}`));\n  }\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/config.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Class definition\n */\n\nclass Config {\n  // Getters\n  static get Default() {\n    return {};\n  }\n  static get DefaultType() {\n    return {};\n  }\n  static get NAME() {\n    throw new Error('You have to implement the static method \"NAME\", for each component!');\n  }\n  _getConfig(config) {\n    config = this._mergeConfigObj(config);\n    config = this._configAfterMerge(config);\n    this._typeCheckConfig(config);\n    return config;\n  }\n  _configAfterMerge(config) {\n    return config;\n  }\n  _mergeConfigObj(config, element) {\n    const jsonConfig = isElement(element) ? Manipulator.getDataAttribute(element, 'config') : {}; // try to parse\n\n    return {\n      ...this.constructor.Default,\n      ...(typeof jsonConfig === 'object' ? jsonConfig : {}),\n      ...(isElement(element) ? Manipulator.getDataAttributes(element) : {}),\n      ...(typeof config === 'object' ? config : {})\n    };\n  }\n  _typeCheckConfig(config, configTypes = this.constructor.DefaultType) {\n    for (const [property, expectedTypes] of Object.entries(configTypes)) {\n      const value = config[property];\n      const valueType = isElement(value) ? 'element' : toType(value);\n      if (!new RegExp(expectedTypes).test(valueType)) {\n        throw new TypeError(`${this.constructor.NAME.toUpperCase()}: Option \"${property}\" provided type \"${valueType}\" but expected type \"${expectedTypes}\".`);\n      }\n    }\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap base-component.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst VERSION = '5.3.3';\n\n/**\n * Class definition\n */\n\nclass BaseComponent extends Config {\n  constructor(element, config) {\n    super();\n    element = getElement(element);\n    if (!element) {\n      return;\n    }\n    this._element = element;\n    this._config = this._getConfig(config);\n    Data.set(this._element, this.constructor.DATA_KEY, this);\n  }\n\n  // Public\n  dispose() {\n    Data.remove(this._element, this.constructor.DATA_KEY);\n    EventHandler.off(this._element, this.constructor.EVENT_KEY);\n    for (const propertyName of Object.getOwnPropertyNames(this)) {\n      this[propertyName] = null;\n    }\n  }\n  _queueCallback(callback, element, isAnimated = true) {\n    executeAfterTransition(callback, element, isAnimated);\n  }\n  _getConfig(config) {\n    config = this._mergeConfigObj(config, this._element);\n    config = this._configAfterMerge(config);\n    this._typeCheckConfig(config);\n    return config;\n  }\n\n  // Static\n  static getInstance(element) {\n    return Data.get(getElement(element), this.DATA_KEY);\n  }\n  static getOrCreateInstance(element, config = {}) {\n    return this.getInstance(element) || new this(element, typeof config === 'object' ? config : null);\n  }\n  static get VERSION() {\n    return VERSION;\n  }\n  static get DATA_KEY() {\n    return `bs.${this.NAME}`;\n  }\n  static get EVENT_KEY() {\n    return `.${this.DATA_KEY}`;\n  }\n  static eventName(name) {\n    return `${name}${this.EVENT_KEY}`;\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dom/selector-engine.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nconst getSelector = element => {\n  let selector = element.getAttribute('data-bs-target');\n  if (!selector || selector === '#') {\n    let hrefAttribute = element.getAttribute('href');\n\n    // The only valid content that could double as a selector are IDs or classes,\n    // so everything starting with `#` or `.`. If a \"real\" URL is used as the selector,\n    // `document.querySelector` will rightfully complain it is invalid.\n    // See https://github.com/twbs/bootstrap/issues/32273\n    if (!hrefAttribute || !hrefAttribute.includes('#') && !hrefAttribute.startsWith('.')) {\n      return null;\n    }\n\n    // Just in case some CMS puts out a full URL with the anchor appended\n    if (hrefAttribute.includes('#') && !hrefAttribute.startsWith('#')) {\n      hrefAttribute = `#${hrefAttribute.split('#')[1]}`;\n    }\n    selector = hrefAttribute && hrefAttribute !== '#' ? hrefAttribute.trim() : null;\n  }\n  return selector ? selector.split(',').map(sel => parseSelector(sel)).join(',') : null;\n};\nconst SelectorEngine = {\n  find(selector, element = document.documentElement) {\n    return [].concat(...Element.prototype.querySelectorAll.call(element, selector));\n  },\n  findOne(selector, element = document.documentElement) {\n    return Element.prototype.querySelector.call(element, selector);\n  },\n  children(element, selector) {\n    return [].concat(...element.children).filter(child => child.matches(selector));\n  },\n  parents(element, selector) {\n    const parents = [];\n    let ancestor = element.parentNode.closest(selector);\n    while (ancestor) {\n      parents.push(ancestor);\n      ancestor = ancestor.parentNode.closest(selector);\n    }\n    return parents;\n  },\n  prev(element, selector) {\n    let previous = element.previousElementSibling;\n    while (previous) {\n      if (previous.matches(selector)) {\n        return [previous];\n      }\n      previous = previous.previousElementSibling;\n    }\n    return [];\n  },\n  // TODO: this is now unused; remove later along with prev()\n  next(element, selector) {\n    let next = element.nextElementSibling;\n    while (next) {\n      if (next.matches(selector)) {\n        return [next];\n      }\n      next = next.nextElementSibling;\n    }\n    return [];\n  },\n  focusableChildren(element) {\n    const focusables = ['a', 'button', 'input', 'textarea', 'select', 'details', '[tabindex]', '[contenteditable=\"true\"]'].map(selector => `${selector}:not([tabindex^=\"-\"])`).join(',');\n    return this.find(focusables, element).filter(el => !isDisabled(el) && isVisible(el));\n  },\n  getSelectorFromElement(element) {\n    const selector = getSelector(element);\n    if (selector) {\n      return SelectorEngine.findOne(selector) ? selector : null;\n    }\n    return null;\n  },\n  getElementFromSelector(element) {\n    const selector = getSelector(element);\n    return selector ? SelectorEngine.findOne(selector) : null;\n  },\n  getMultipleElementsFromSelector(element) {\n    const selector = getSelector(element);\n    return selector ? SelectorEngine.find(selector) : [];\n  }\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/component-functions.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nconst enableDismissTrigger = (component, method = 'hide') => {\n  const clickEvent = `click.dismiss${component.EVENT_KEY}`;\n  const name = component.NAME;\n  EventHandler.on(document, clickEvent, `[data-bs-dismiss=\"${name}\"]`, function (event) {\n    if (['A', 'AREA'].includes(this.tagName)) {\n      event.preventDefault();\n    }\n    if (isDisabled(this)) {\n      return;\n    }\n    const target = SelectorEngine.getElementFromSelector(this) || this.closest(`.${name}`);\n    const instance = component.getOrCreateInstance(target);\n\n    // Method argument is left, for Alert and only, as it doesn't implement the 'hide' method\n    instance[method]();\n  });\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap alert.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$f = 'alert';\nconst DATA_KEY$a = 'bs.alert';\nconst EVENT_KEY$b = `.${DATA_KEY$a}`;\nconst EVENT_CLOSE = `close${EVENT_KEY$b}`;\nconst EVENT_CLOSED = `closed${EVENT_KEY$b}`;\nconst CLASS_NAME_FADE$5 = 'fade';\nconst CLASS_NAME_SHOW$8 = 'show';\n\n/**\n * Class definition\n */\n\nclass Alert extends BaseComponent {\n  // Getters\n  static get NAME() {\n    return NAME$f;\n  }\n\n  // Public\n  close() {\n    const closeEvent = EventHandler.trigger(this._element, EVENT_CLOSE);\n    if (closeEvent.defaultPrevented) {\n      return;\n    }\n    this._element.classList.remove(CLASS_NAME_SHOW$8);\n    const isAnimated = this._element.classList.contains(CLASS_NAME_FADE$5);\n    this._queueCallback(() => this._destroyElement(), this._element, isAnimated);\n  }\n\n  // Private\n  _destroyElement() {\n    this._element.remove();\n    EventHandler.trigger(this._element, EVENT_CLOSED);\n    this.dispose();\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Alert.getOrCreateInstance(this);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config](this);\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nenableDismissTrigger(Alert, 'close');\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Alert);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap button.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$e = 'button';\nconst DATA_KEY$9 = 'bs.button';\nconst EVENT_KEY$a = `.${DATA_KEY$9}`;\nconst DATA_API_KEY$6 = '.data-api';\nconst CLASS_NAME_ACTIVE$3 = 'active';\nconst SELECTOR_DATA_TOGGLE$5 = '[data-bs-toggle=\"button\"]';\nconst EVENT_CLICK_DATA_API$6 = `click${EVENT_KEY$a}${DATA_API_KEY$6}`;\n\n/**\n * Class definition\n */\n\nclass Button extends BaseComponent {\n  // Getters\n  static get NAME() {\n    return NAME$e;\n  }\n\n  // Public\n  toggle() {\n    // Toggle class and sync the `aria-pressed` attribute with the return value of the `.toggle()` method\n    this._element.setAttribute('aria-pressed', this._element.classList.toggle(CLASS_NAME_ACTIVE$3));\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Button.getOrCreateInstance(this);\n      if (config === 'toggle') {\n        data[config]();\n      }\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$6, SELECTOR_DATA_TOGGLE$5, event => {\n  event.preventDefault();\n  const button = event.target.closest(SELECTOR_DATA_TOGGLE$5);\n  const data = Button.getOrCreateInstance(button);\n  data.toggle();\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Button);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/swipe.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$d = 'swipe';\nconst EVENT_KEY$9 = '.bs.swipe';\nconst EVENT_TOUCHSTART = `touchstart${EVENT_KEY$9}`;\nconst EVENT_TOUCHMOVE = `touchmove${EVENT_KEY$9}`;\nconst EVENT_TOUCHEND = `touchend${EVENT_KEY$9}`;\nconst EVENT_POINTERDOWN = `pointerdown${EVENT_KEY$9}`;\nconst EVENT_POINTERUP = `pointerup${EVENT_KEY$9}`;\nconst POINTER_TYPE_TOUCH = 'touch';\nconst POINTER_TYPE_PEN = 'pen';\nconst CLASS_NAME_POINTER_EVENT = 'pointer-event';\nconst SWIPE_THRESHOLD = 40;\nconst Default$c = {\n  endCallback: null,\n  leftCallback: null,\n  rightCallback: null\n};\nconst DefaultType$c = {\n  endCallback: '(function|null)',\n  leftCallback: '(function|null)',\n  rightCallback: '(function|null)'\n};\n\n/**\n * Class definition\n */\n\nclass Swipe extends Config {\n  constructor(element, config) {\n    super();\n    this._element = element;\n    if (!element || !Swipe.isSupported()) {\n      return;\n    }\n    this._config = this._getConfig(config);\n    this._deltaX = 0;\n    this._supportPointerEvents = Boolean(window.PointerEvent);\n    this._initEvents();\n  }\n\n  // Getters\n  static get Default() {\n    return Default$c;\n  }\n  static get DefaultType() {\n    return DefaultType$c;\n  }\n  static get NAME() {\n    return NAME$d;\n  }\n\n  // Public\n  dispose() {\n    EventHandler.off(this._element, EVENT_KEY$9);\n  }\n\n  // Private\n  _start(event) {\n    if (!this._supportPointerEvents) {\n      this._deltaX = event.touches[0].clientX;\n      return;\n    }\n    if (this._eventIsPointerPenTouch(event)) {\n      this._deltaX = event.clientX;\n    }\n  }\n  _end(event) {\n    if (this._eventIsPointerPenTouch(event)) {\n      this._deltaX = event.clientX - this._deltaX;\n    }\n    this._handleSwipe();\n    execute(this._config.endCallback);\n  }\n  _move(event) {\n    this._deltaX = event.touches && event.touches.length > 1 ? 0 : event.touches[0].clientX - this._deltaX;\n  }\n  _handleSwipe() {\n    const absDeltaX = Math.abs(this._deltaX);\n    if (absDeltaX <= SWIPE_THRESHOLD) {\n      return;\n    }\n    const direction = absDeltaX / this._deltaX;\n    this._deltaX = 0;\n    if (!direction) {\n      return;\n    }\n    execute(direction > 0 ? this._config.rightCallback : this._config.leftCallback);\n  }\n  _initEvents() {\n    if (this._supportPointerEvents) {\n      EventHandler.on(this._element, EVENT_POINTERDOWN, event => this._start(event));\n      EventHandler.on(this._element, EVENT_POINTERUP, event => this._end(event));\n      this._element.classList.add(CLASS_NAME_POINTER_EVENT);\n    } else {\n      EventHandler.on(this._element, EVENT_TOUCHSTART, event => this._start(event));\n      EventHandler.on(this._element, EVENT_TOUCHMOVE, event => this._move(event));\n      EventHandler.on(this._element, EVENT_TOUCHEND, event => this._end(event));\n    }\n  }\n  _eventIsPointerPenTouch(event) {\n    return this._supportPointerEvents && (event.pointerType === POINTER_TYPE_PEN || event.pointerType === POINTER_TYPE_TOUCH);\n  }\n\n  // Static\n  static isSupported() {\n    return 'ontouchstart' in document.documentElement || navigator.maxTouchPoints > 0;\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap carousel.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$c = 'carousel';\nconst DATA_KEY$8 = 'bs.carousel';\nconst EVENT_KEY$8 = `.${DATA_KEY$8}`;\nconst DATA_API_KEY$5 = '.data-api';\nconst ARROW_LEFT_KEY$1 = 'ArrowLeft';\nconst ARROW_RIGHT_KEY$1 = 'ArrowRight';\nconst TOUCHEVENT_COMPAT_WAIT = 500; // Time for mouse compat events to fire after touch\n\nconst ORDER_NEXT = 'next';\nconst ORDER_PREV = 'prev';\nconst DIRECTION_LEFT = 'left';\nconst DIRECTION_RIGHT = 'right';\nconst EVENT_SLIDE = `slide${EVENT_KEY$8}`;\nconst EVENT_SLID = `slid${EVENT_KEY$8}`;\nconst EVENT_KEYDOWN$1 = `keydown${EVENT_KEY$8}`;\nconst EVENT_MOUSEENTER$1 = `mouseenter${EVENT_KEY$8}`;\nconst EVENT_MOUSELEAVE$1 = `mouseleave${EVENT_KEY$8}`;\nconst EVENT_DRAG_START = `dragstart${EVENT_KEY$8}`;\nconst EVENT_LOAD_DATA_API$3 = `load${EVENT_KEY$8}${DATA_API_KEY$5}`;\nconst EVENT_CLICK_DATA_API$5 = `click${EVENT_KEY$8}${DATA_API_KEY$5}`;\nconst CLASS_NAME_CAROUSEL = 'carousel';\nconst CLASS_NAME_ACTIVE$2 = 'active';\nconst CLASS_NAME_SLIDE = 'slide';\nconst CLASS_NAME_END = 'carousel-item-end';\nconst CLASS_NAME_START = 'carousel-item-start';\nconst CLASS_NAME_NEXT = 'carousel-item-next';\nconst CLASS_NAME_PREV = 'carousel-item-prev';\nconst SELECTOR_ACTIVE = '.active';\nconst SELECTOR_ITEM = '.carousel-item';\nconst SELECTOR_ACTIVE_ITEM = SELECTOR_ACTIVE + SELECTOR_ITEM;\nconst SELECTOR_ITEM_IMG = '.carousel-item img';\nconst SELECTOR_INDICATORS = '.carousel-indicators';\nconst SELECTOR_DATA_SLIDE = '[data-bs-slide], [data-bs-slide-to]';\nconst SELECTOR_DATA_RIDE = '[data-bs-ride=\"carousel\"]';\nconst KEY_TO_DIRECTION = {\n  [ARROW_LEFT_KEY$1]: DIRECTION_RIGHT,\n  [ARROW_RIGHT_KEY$1]: DIRECTION_LEFT\n};\nconst Default$b = {\n  interval: 5000,\n  keyboard: true,\n  pause: 'hover',\n  ride: false,\n  touch: true,\n  wrap: true\n};\nconst DefaultType$b = {\n  interval: '(number|boolean)',\n  // TODO:v6 remove boolean support\n  keyboard: 'boolean',\n  pause: '(string|boolean)',\n  ride: '(boolean|string)',\n  touch: 'boolean',\n  wrap: 'boolean'\n};\n\n/**\n * Class definition\n */\n\nclass Carousel extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._interval = null;\n    this._activeElement = null;\n    this._isSliding = false;\n    this.touchTimeout = null;\n    this._swipeHelper = null;\n    this._indicatorsElement = SelectorEngine.findOne(SELECTOR_INDICATORS, this._element);\n    this._addEventListeners();\n    if (this._config.ride === CLASS_NAME_CAROUSEL) {\n      this.cycle();\n    }\n  }\n\n  // Getters\n  static get Default() {\n    return Default$b;\n  }\n  static get DefaultType() {\n    return DefaultType$b;\n  }\n  static get NAME() {\n    return NAME$c;\n  }\n\n  // Public\n  next() {\n    this._slide(ORDER_NEXT);\n  }\n  nextWhenVisible() {\n    // FIXME TODO use `document.visibilityState`\n    // Don't call next when the page isn't visible\n    // or the carousel or its parent isn't visible\n    if (!document.hidden && isVisible(this._element)) {\n      this.next();\n    }\n  }\n  prev() {\n    this._slide(ORDER_PREV);\n  }\n  pause() {\n    if (this._isSliding) {\n      triggerTransitionEnd(this._element);\n    }\n    this._clearInterval();\n  }\n  cycle() {\n    this._clearInterval();\n    this._updateInterval();\n    this._interval = setInterval(() => this.nextWhenVisible(), this._config.interval);\n  }\n  _maybeEnableCycle() {\n    if (!this._config.ride) {\n      return;\n    }\n    if (this._isSliding) {\n      EventHandler.one(this._element, EVENT_SLID, () => this.cycle());\n      return;\n    }\n    this.cycle();\n  }\n  to(index) {\n    const items = this._getItems();\n    if (index > items.length - 1 || index < 0) {\n      return;\n    }\n    if (this._isSliding) {\n      EventHandler.one(this._element, EVENT_SLID, () => this.to(index));\n      return;\n    }\n    const activeIndex = this._getItemIndex(this._getActive());\n    if (activeIndex === index) {\n      return;\n    }\n    const order = index > activeIndex ? ORDER_NEXT : ORDER_PREV;\n    this._slide(order, items[index]);\n  }\n  dispose() {\n    if (this._swipeHelper) {\n      this._swipeHelper.dispose();\n    }\n    super.dispose();\n  }\n\n  // Private\n  _configAfterMerge(config) {\n    config.defaultInterval = config.interval;\n    return config;\n  }\n  _addEventListeners() {\n    if (this._config.keyboard) {\n      EventHandler.on(this._element, EVENT_KEYDOWN$1, event => this._keydown(event));\n    }\n    if (this._config.pause === 'hover') {\n      EventHandler.on(this._element, EVENT_MOUSEENTER$1, () => this.pause());\n      EventHandler.on(this._element, EVENT_MOUSELEAVE$1, () => this._maybeEnableCycle());\n    }\n    if (this._config.touch && Swipe.isSupported()) {\n      this._addTouchEventListeners();\n    }\n  }\n  _addTouchEventListeners() {\n    for (const img of SelectorEngine.find(SELECTOR_ITEM_IMG, this._element)) {\n      EventHandler.on(img, EVENT_DRAG_START, event => event.preventDefault());\n    }\n    const endCallBack = () => {\n      if (this._config.pause !== 'hover') {\n        return;\n      }\n\n      // If it's a touch-enabled device, mouseenter/leave are fired as\n      // part of the mouse compatibility events on first tap - the carousel\n      // would stop cycling until user tapped out of it;\n      // here, we listen for touchend, explicitly pause the carousel\n      // (as if it's the second time we tap on it, mouseenter compat event\n      // is NOT fired) and after a timeout (to allow for mouse compatibility\n      // events to fire) we explicitly restart cycling\n\n      this.pause();\n      if (this.touchTimeout) {\n        clearTimeout(this.touchTimeout);\n      }\n      this.touchTimeout = setTimeout(() => this._maybeEnableCycle(), TOUCHEVENT_COMPAT_WAIT + this._config.interval);\n    };\n    const swipeConfig = {\n      leftCallback: () => this._slide(this._directionToOrder(DIRECTION_LEFT)),\n      rightCallback: () => this._slide(this._directionToOrder(DIRECTION_RIGHT)),\n      endCallback: endCallBack\n    };\n    this._swipeHelper = new Swipe(this._element, swipeConfig);\n  }\n  _keydown(event) {\n    if (/input|textarea/i.test(event.target.tagName)) {\n      return;\n    }\n    const direction = KEY_TO_DIRECTION[event.key];\n    if (direction) {\n      event.preventDefault();\n      this._slide(this._directionToOrder(direction));\n    }\n  }\n  _getItemIndex(element) {\n    return this._getItems().indexOf(element);\n  }\n  _setActiveIndicatorElement(index) {\n    if (!this._indicatorsElement) {\n      return;\n    }\n    const activeIndicator = SelectorEngine.findOne(SELECTOR_ACTIVE, this._indicatorsElement);\n    activeIndicator.classList.remove(CLASS_NAME_ACTIVE$2);\n    activeIndicator.removeAttribute('aria-current');\n    const newActiveIndicator = SelectorEngine.findOne(`[data-bs-slide-to=\"${index}\"]`, this._indicatorsElement);\n    if (newActiveIndicator) {\n      newActiveIndicator.classList.add(CLASS_NAME_ACTIVE$2);\n      newActiveIndicator.setAttribute('aria-current', 'true');\n    }\n  }\n  _updateInterval() {\n    const element = this._activeElement || this._getActive();\n    if (!element) {\n      return;\n    }\n    const elementInterval = Number.parseInt(element.getAttribute('data-bs-interval'), 10);\n    this._config.interval = elementInterval || this._config.defaultInterval;\n  }\n  _slide(order, element = null) {\n    if (this._isSliding) {\n      return;\n    }\n    const activeElement = this._getActive();\n    const isNext = order === ORDER_NEXT;\n    const nextElement = element || getNextActiveElement(this._getItems(), activeElement, isNext, this._config.wrap);\n    if (nextElement === activeElement) {\n      return;\n    }\n    const nextElementIndex = this._getItemIndex(nextElement);\n    const triggerEvent = eventName => {\n      return EventHandler.trigger(this._element, eventName, {\n        relatedTarget: nextElement,\n        direction: this._orderToDirection(order),\n        from: this._getItemIndex(activeElement),\n        to: nextElementIndex\n      });\n    };\n    const slideEvent = triggerEvent(EVENT_SLIDE);\n    if (slideEvent.defaultPrevented) {\n      return;\n    }\n    if (!activeElement || !nextElement) {\n      // Some weirdness is happening, so we bail\n      // TODO: change tests that use empty divs to avoid this check\n      return;\n    }\n    const isCycling = Boolean(this._interval);\n    this.pause();\n    this._isSliding = true;\n    this._setActiveIndicatorElement(nextElementIndex);\n    this._activeElement = nextElement;\n    const directionalClassName = isNext ? CLASS_NAME_START : CLASS_NAME_END;\n    const orderClassName = isNext ? CLASS_NAME_NEXT : CLASS_NAME_PREV;\n    nextElement.classList.add(orderClassName);\n    reflow(nextElement);\n    activeElement.classList.add(directionalClassName);\n    nextElement.classList.add(directionalClassName);\n    const completeCallBack = () => {\n      nextElement.classList.remove(directionalClassName, orderClassName);\n      nextElement.classList.add(CLASS_NAME_ACTIVE$2);\n      activeElement.classList.remove(CLASS_NAME_ACTIVE$2, orderClassName, directionalClassName);\n      this._isSliding = false;\n      triggerEvent(EVENT_SLID);\n    };\n    this._queueCallback(completeCallBack, activeElement, this._isAnimated());\n    if (isCycling) {\n      this.cycle();\n    }\n  }\n  _isAnimated() {\n    return this._element.classList.contains(CLASS_NAME_SLIDE);\n  }\n  _getActive() {\n    return SelectorEngine.findOne(SELECTOR_ACTIVE_ITEM, this._element);\n  }\n  _getItems() {\n    return SelectorEngine.find(SELECTOR_ITEM, this._element);\n  }\n  _clearInterval() {\n    if (this._interval) {\n      clearInterval(this._interval);\n      this._interval = null;\n    }\n  }\n  _directionToOrder(direction) {\n    if (isRTL()) {\n      return direction === DIRECTION_LEFT ? ORDER_PREV : ORDER_NEXT;\n    }\n    return direction === DIRECTION_LEFT ? ORDER_NEXT : ORDER_PREV;\n  }\n  _orderToDirection(order) {\n    if (isRTL()) {\n      return order === ORDER_PREV ? DIRECTION_LEFT : DIRECTION_RIGHT;\n    }\n    return order === ORDER_PREV ? DIRECTION_RIGHT : DIRECTION_LEFT;\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Carousel.getOrCreateInstance(this, config);\n      if (typeof config === 'number') {\n        data.to(config);\n        return;\n      }\n      if (typeof config === 'string') {\n        if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n          throw new TypeError(`No method named \"${config}\"`);\n        }\n        data[config]();\n      }\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$5, SELECTOR_DATA_SLIDE, function (event) {\n  const target = SelectorEngine.getElementFromSelector(this);\n  if (!target || !target.classList.contains(CLASS_NAME_CAROUSEL)) {\n    return;\n  }\n  event.preventDefault();\n  const carousel = Carousel.getOrCreateInstance(target);\n  const slideIndex = this.getAttribute('data-bs-slide-to');\n  if (slideIndex) {\n    carousel.to(slideIndex);\n    carousel._maybeEnableCycle();\n    return;\n  }\n  if (Manipulator.getDataAttribute(this, 'slide') === 'next') {\n    carousel.next();\n    carousel._maybeEnableCycle();\n    return;\n  }\n  carousel.prev();\n  carousel._maybeEnableCycle();\n});\nEventHandler.on(window, EVENT_LOAD_DATA_API$3, () => {\n  const carousels = SelectorEngine.find(SELECTOR_DATA_RIDE);\n  for (const carousel of carousels) {\n    Carousel.getOrCreateInstance(carousel);\n  }\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Carousel);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap collapse.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$b = 'collapse';\nconst DATA_KEY$7 = 'bs.collapse';\nconst EVENT_KEY$7 = `.${DATA_KEY$7}`;\nconst DATA_API_KEY$4 = '.data-api';\nconst EVENT_SHOW$6 = `show${EVENT_KEY$7}`;\nconst EVENT_SHOWN$6 = `shown${EVENT_KEY$7}`;\nconst EVENT_HIDE$6 = `hide${EVENT_KEY$7}`;\nconst EVENT_HIDDEN$6 = `hidden${EVENT_KEY$7}`;\nconst EVENT_CLICK_DATA_API$4 = `click${EVENT_KEY$7}${DATA_API_KEY$4}`;\nconst CLASS_NAME_SHOW$7 = 'show';\nconst CLASS_NAME_COLLAPSE = 'collapse';\nconst CLASS_NAME_COLLAPSING = 'collapsing';\nconst CLASS_NAME_COLLAPSED = 'collapsed';\nconst CLASS_NAME_DEEPER_CHILDREN = `:scope .${CLASS_NAME_COLLAPSE} .${CLASS_NAME_COLLAPSE}`;\nconst CLASS_NAME_HORIZONTAL = 'collapse-horizontal';\nconst WIDTH = 'width';\nconst HEIGHT = 'height';\nconst SELECTOR_ACTIVES = '.collapse.show, .collapse.collapsing';\nconst SELECTOR_DATA_TOGGLE$4 = '[data-bs-toggle=\"collapse\"]';\nconst Default$a = {\n  parent: null,\n  toggle: true\n};\nconst DefaultType$a = {\n  parent: '(null|element)',\n  toggle: 'boolean'\n};\n\n/**\n * Class definition\n */\n\nclass Collapse extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._isTransitioning = false;\n    this._triggerArray = [];\n    const toggleList = SelectorEngine.find(SELECTOR_DATA_TOGGLE$4);\n    for (const elem of toggleList) {\n      const selector = SelectorEngine.getSelectorFromElement(elem);\n      const filterElement = SelectorEngine.find(selector).filter(foundElement => foundElement === this._element);\n      if (selector !== null && filterElement.length) {\n        this._triggerArray.push(elem);\n      }\n    }\n    this._initializeChildren();\n    if (!this._config.parent) {\n      this._addAriaAndCollapsedClass(this._triggerArray, this._isShown());\n    }\n    if (this._config.toggle) {\n      this.toggle();\n    }\n  }\n\n  // Getters\n  static get Default() {\n    return Default$a;\n  }\n  static get DefaultType() {\n    return DefaultType$a;\n  }\n  static get NAME() {\n    return NAME$b;\n  }\n\n  // Public\n  toggle() {\n    if (this._isShown()) {\n      this.hide();\n    } else {\n      this.show();\n    }\n  }\n  show() {\n    if (this._isTransitioning || this._isShown()) {\n      return;\n    }\n    let activeChildren = [];\n\n    // find active children\n    if (this._config.parent) {\n      activeChildren = this._getFirstLevelChildren(SELECTOR_ACTIVES).filter(element => element !== this._element).map(element => Collapse.getOrCreateInstance(element, {\n        toggle: false\n      }));\n    }\n    if (activeChildren.length && activeChildren[0]._isTransitioning) {\n      return;\n    }\n    const startEvent = EventHandler.trigger(this._element, EVENT_SHOW$6);\n    if (startEvent.defaultPrevented) {\n      return;\n    }\n    for (const activeInstance of activeChildren) {\n      activeInstance.hide();\n    }\n    const dimension = this._getDimension();\n    this._element.classList.remove(CLASS_NAME_COLLAPSE);\n    this._element.classList.add(CLASS_NAME_COLLAPSING);\n    this._element.style[dimension] = 0;\n    this._addAriaAndCollapsedClass(this._triggerArray, true);\n    this._isTransitioning = true;\n    const complete = () => {\n      this._isTransitioning = false;\n      this._element.classList.remove(CLASS_NAME_COLLAPSING);\n      this._element.classList.add(CLASS_NAME_COLLAPSE, CLASS_NAME_SHOW$7);\n      this._element.style[dimension] = '';\n      EventHandler.trigger(this._element, EVENT_SHOWN$6);\n    };\n    const capitalizedDimension = dimension[0].toUpperCase() + dimension.slice(1);\n    const scrollSize = `scroll${capitalizedDimension}`;\n    this._queueCallback(complete, this._element, true);\n    this._element.style[dimension] = `${this._element[scrollSize]}px`;\n  }\n  hide() {\n    if (this._isTransitioning || !this._isShown()) {\n      return;\n    }\n    const startEvent = EventHandler.trigger(this._element, EVENT_HIDE$6);\n    if (startEvent.defaultPrevented) {\n      return;\n    }\n    const dimension = this._getDimension();\n    this._element.style[dimension] = `${this._element.getBoundingClientRect()[dimension]}px`;\n    reflow(this._element);\n    this._element.classList.add(CLASS_NAME_COLLAPSING);\n    this._element.classList.remove(CLASS_NAME_COLLAPSE, CLASS_NAME_SHOW$7);\n    for (const trigger of this._triggerArray) {\n      const element = SelectorEngine.getElementFromSelector(trigger);\n      if (element && !this._isShown(element)) {\n        this._addAriaAndCollapsedClass([trigger], false);\n      }\n    }\n    this._isTransitioning = true;\n    const complete = () => {\n      this._isTransitioning = false;\n      this._element.classList.remove(CLASS_NAME_COLLAPSING);\n      this._element.classList.add(CLASS_NAME_COLLAPSE);\n      EventHandler.trigger(this._element, EVENT_HIDDEN$6);\n    };\n    this._element.style[dimension] = '';\n    this._queueCallback(complete, this._element, true);\n  }\n  _isShown(element = this._element) {\n    return element.classList.contains(CLASS_NAME_SHOW$7);\n  }\n\n  // Private\n  _configAfterMerge(config) {\n    config.toggle = Boolean(config.toggle); // Coerce string values\n    config.parent = getElement(config.parent);\n    return config;\n  }\n  _getDimension() {\n    return this._element.classList.contains(CLASS_NAME_HORIZONTAL) ? WIDTH : HEIGHT;\n  }\n  _initializeChildren() {\n    if (!this._config.parent) {\n      return;\n    }\n    const children = this._getFirstLevelChildren(SELECTOR_DATA_TOGGLE$4);\n    for (const element of children) {\n      const selected = SelectorEngine.getElementFromSelector(element);\n      if (selected) {\n        this._addAriaAndCollapsedClass([element], this._isShown(selected));\n      }\n    }\n  }\n  _getFirstLevelChildren(selector) {\n    const children = SelectorEngine.find(CLASS_NAME_DEEPER_CHILDREN, this._config.parent);\n    // remove children if greater depth\n    return SelectorEngine.find(selector, this._config.parent).filter(element => !children.includes(element));\n  }\n  _addAriaAndCollapsedClass(triggerArray, isOpen) {\n    if (!triggerArray.length) {\n      return;\n    }\n    for (const element of triggerArray) {\n      element.classList.toggle(CLASS_NAME_COLLAPSED, !isOpen);\n      element.setAttribute('aria-expanded', isOpen);\n    }\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    const _config = {};\n    if (typeof config === 'string' && /show|hide/.test(config)) {\n      _config.toggle = false;\n    }\n    return this.each(function () {\n      const data = Collapse.getOrCreateInstance(this, _config);\n      if (typeof config === 'string') {\n        if (typeof data[config] === 'undefined') {\n          throw new TypeError(`No method named \"${config}\"`);\n        }\n        data[config]();\n      }\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$4, SELECTOR_DATA_TOGGLE$4, function (event) {\n  // preventDefault only for <a> elements (which change the URL) not inside the collapsible element\n  if (event.target.tagName === 'A' || event.delegateTarget && event.delegateTarget.tagName === 'A') {\n    event.preventDefault();\n  }\n  for (const element of SelectorEngine.getMultipleElementsFromSelector(this)) {\n    Collapse.getOrCreateInstance(element, {\n      toggle: false\n    }).toggle();\n  }\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Collapse);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dropdown.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$a = 'dropdown';\nconst DATA_KEY$6 = 'bs.dropdown';\nconst EVENT_KEY$6 = `.${DATA_KEY$6}`;\nconst DATA_API_KEY$3 = '.data-api';\nconst ESCAPE_KEY$2 = 'Escape';\nconst TAB_KEY$1 = 'Tab';\nconst ARROW_UP_KEY$1 = 'ArrowUp';\nconst ARROW_DOWN_KEY$1 = 'ArrowDown';\nconst RIGHT_MOUSE_BUTTON = 2; // MouseEvent.button value for the secondary button, usually the right button\n\nconst EVENT_HIDE$5 = `hide${EVENT_KEY$6}`;\nconst EVENT_HIDDEN$5 = `hidden${EVENT_KEY$6}`;\nconst EVENT_SHOW$5 = `show${EVENT_KEY$6}`;\nconst EVENT_SHOWN$5 = `shown${EVENT_KEY$6}`;\nconst EVENT_CLICK_DATA_API$3 = `click${EVENT_KEY$6}${DATA_API_KEY$3}`;\nconst EVENT_KEYDOWN_DATA_API = `keydown${EVENT_KEY$6}${DATA_API_KEY$3}`;\nconst EVENT_KEYUP_DATA_API = `keyup${EVENT_KEY$6}${DATA_API_KEY$3}`;\nconst CLASS_NAME_SHOW$6 = 'show';\nconst CLASS_NAME_DROPUP = 'dropup';\nconst CLASS_NAME_DROPEND = 'dropend';\nconst CLASS_NAME_DROPSTART = 'dropstart';\nconst CLASS_NAME_DROPUP_CENTER = 'dropup-center';\nconst CLASS_NAME_DROPDOWN_CENTER = 'dropdown-center';\nconst SELECTOR_DATA_TOGGLE$3 = '[data-bs-toggle=\"dropdown\"]:not(.disabled):not(:disabled)';\nconst SELECTOR_DATA_TOGGLE_SHOWN = `${SELECTOR_DATA_TOGGLE$3}.${CLASS_NAME_SHOW$6}`;\nconst SELECTOR_MENU = '.dropdown-menu';\nconst SELECTOR_NAVBAR = '.navbar';\nconst SELECTOR_NAVBAR_NAV = '.navbar-nav';\nconst SELECTOR_VISIBLE_ITEMS = '.dropdown-menu .dropdown-item:not(.disabled):not(:disabled)';\nconst PLACEMENT_TOP = isRTL() ? 'top-end' : 'top-start';\nconst PLACEMENT_TOPEND = isRTL() ? 'top-start' : 'top-end';\nconst PLACEMENT_BOTTOM = isRTL() ? 'bottom-end' : 'bottom-start';\nconst PLACEMENT_BOTTOMEND = isRTL() ? 'bottom-start' : 'bottom-end';\nconst PLACEMENT_RIGHT = isRTL() ? 'left-start' : 'right-start';\nconst PLACEMENT_LEFT = isRTL() ? 'right-start' : 'left-start';\nconst PLACEMENT_TOPCENTER = 'top';\nconst PLACEMENT_BOTTOMCENTER = 'bottom';\nconst Default$9 = {\n  autoClose: true,\n  boundary: 'clippingParents',\n  display: 'dynamic',\n  offset: [0, 2],\n  popperConfig: null,\n  reference: 'toggle'\n};\nconst DefaultType$9 = {\n  autoClose: '(boolean|string)',\n  boundary: '(string|element)',\n  display: 'string',\n  offset: '(array|string|function)',\n  popperConfig: '(null|object|function)',\n  reference: '(string|element|object)'\n};\n\n/**\n * Class definition\n */\n\nclass Dropdown extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._popper = null;\n    this._parent = this._element.parentNode; // dropdown wrapper\n    // TODO: v6 revert #37011 & change markup https://getbootstrap.com/docs/5.3/forms/input-group/\n    this._menu = SelectorEngine.next(this._element, SELECTOR_MENU)[0] || SelectorEngine.prev(this._element, SELECTOR_MENU)[0] || SelectorEngine.findOne(SELECTOR_MENU, this._parent);\n    this._inNavbar = this._detectNavbar();\n  }\n\n  // Getters\n  static get Default() {\n    return Default$9;\n  }\n  static get DefaultType() {\n    return DefaultType$9;\n  }\n  static get NAME() {\n    return NAME$a;\n  }\n\n  // Public\n  toggle() {\n    return this._isShown() ? this.hide() : this.show();\n  }\n  show() {\n    if (isDisabled(this._element) || this._isShown()) {\n      return;\n    }\n    const relatedTarget = {\n      relatedTarget: this._element\n    };\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW$5, relatedTarget);\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n    this._createPopper();\n\n    // If this is a touch-enabled device we add extra\n    // empty mouseover listeners to the body's immediate children;\n    // only needed because of broken event delegation on iOS\n    // https://www.quirksmode.org/blog/archives/2014/02/mouse_event_bub.html\n    if ('ontouchstart' in document.documentElement && !this._parent.closest(SELECTOR_NAVBAR_NAV)) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.on(element, 'mouseover', noop);\n      }\n    }\n    this._element.focus();\n    this._element.setAttribute('aria-expanded', true);\n    this._menu.classList.add(CLASS_NAME_SHOW$6);\n    this._element.classList.add(CLASS_NAME_SHOW$6);\n    EventHandler.trigger(this._element, EVENT_SHOWN$5, relatedTarget);\n  }\n  hide() {\n    if (isDisabled(this._element) || !this._isShown()) {\n      return;\n    }\n    const relatedTarget = {\n      relatedTarget: this._element\n    };\n    this._completeHide(relatedTarget);\n  }\n  dispose() {\n    if (this._popper) {\n      this._popper.destroy();\n    }\n    super.dispose();\n  }\n  update() {\n    this._inNavbar = this._detectNavbar();\n    if (this._popper) {\n      this._popper.update();\n    }\n  }\n\n  // Private\n  _completeHide(relatedTarget) {\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE$5, relatedTarget);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n\n    // If this is a touch-enabled device we remove the extra\n    // empty mouseover listeners we added for iOS support\n    if ('ontouchstart' in document.documentElement) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.off(element, 'mouseover', noop);\n      }\n    }\n    if (this._popper) {\n      this._popper.destroy();\n    }\n    this._menu.classList.remove(CLASS_NAME_SHOW$6);\n    this._element.classList.remove(CLASS_NAME_SHOW$6);\n    this._element.setAttribute('aria-expanded', 'false');\n    Manipulator.removeDataAttribute(this._menu, 'popper');\n    EventHandler.trigger(this._element, EVENT_HIDDEN$5, relatedTarget);\n  }\n  _getConfig(config) {\n    config = super._getConfig(config);\n    if (typeof config.reference === 'object' && !isElement(config.reference) && typeof config.reference.getBoundingClientRect !== 'function') {\n      // Popper virtual elements require a getBoundingClientRect method\n      throw new TypeError(`${NAME$a.toUpperCase()}: Option \"reference\" provided type \"object\" without a required \"getBoundingClientRect\" method.`);\n    }\n    return config;\n  }\n  _createPopper() {\n    if (typeof Popper === 'undefined') {\n      throw new TypeError('Bootstrap\\'s dropdowns require Popper (https://popper.js.org)');\n    }\n    let referenceElement = this._element;\n    if (this._config.reference === 'parent') {\n      referenceElement = this._parent;\n    } else if (isElement(this._config.reference)) {\n      referenceElement = getElement(this._config.reference);\n    } else if (typeof this._config.reference === 'object') {\n      referenceElement = this._config.reference;\n    }\n    const popperConfig = this._getPopperConfig();\n    this._popper = Popper.createPopper(referenceElement, this._menu, popperConfig);\n  }\n  _isShown() {\n    return this._menu.classList.contains(CLASS_NAME_SHOW$6);\n  }\n  _getPlacement() {\n    const parentDropdown = this._parent;\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPEND)) {\n      return PLACEMENT_RIGHT;\n    }\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPSTART)) {\n      return PLACEMENT_LEFT;\n    }\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPUP_CENTER)) {\n      return PLACEMENT_TOPCENTER;\n    }\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPDOWN_CENTER)) {\n      return PLACEMENT_BOTTOMCENTER;\n    }\n\n    // We need to trim the value because custom properties can also include spaces\n    const isEnd = getComputedStyle(this._menu).getPropertyValue('--bs-position').trim() === 'end';\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPUP)) {\n      return isEnd ? PLACEMENT_TOPEND : PLACEMENT_TOP;\n    }\n    return isEnd ? PLACEMENT_BOTTOMEND : PLACEMENT_BOTTOM;\n  }\n  _detectNavbar() {\n    return this._element.closest(SELECTOR_NAVBAR) !== null;\n  }\n  _getOffset() {\n    const {\n      offset\n    } = this._config;\n    if (typeof offset === 'string') {\n      return offset.split(',').map(value => Number.parseInt(value, 10));\n    }\n    if (typeof offset === 'function') {\n      return popperData => offset(popperData, this._element);\n    }\n    return offset;\n  }\n  _getPopperConfig() {\n    const defaultBsPopperConfig = {\n      placement: this._getPlacement(),\n      modifiers: [{\n        name: 'preventOverflow',\n        options: {\n          boundary: this._config.boundary\n        }\n      }, {\n        name: 'offset',\n        options: {\n          offset: this._getOffset()\n        }\n      }]\n    };\n\n    // Disable Popper if we have a static display or Dropdown is in Navbar\n    if (this._inNavbar || this._config.display === 'static') {\n      Manipulator.setDataAttribute(this._menu, 'popper', 'static'); // TODO: v6 remove\n      defaultBsPopperConfig.modifiers = [{\n        name: 'applyStyles',\n        enabled: false\n      }];\n    }\n    return {\n      ...defaultBsPopperConfig,\n      ...execute(this._config.popperConfig, [defaultBsPopperConfig])\n    };\n  }\n  _selectMenuItem({\n    key,\n    target\n  }) {\n    const items = SelectorEngine.find(SELECTOR_VISIBLE_ITEMS, this._menu).filter(element => isVisible(element));\n    if (!items.length) {\n      return;\n    }\n\n    // if target isn't included in items (e.g. when expanding the dropdown)\n    // allow cycling to get the last item in case key equals ARROW_UP_KEY\n    getNextActiveElement(items, target, key === ARROW_DOWN_KEY$1, !items.includes(target)).focus();\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Dropdown.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n  static clearMenus(event) {\n    if (event.button === RIGHT_MOUSE_BUTTON || event.type === 'keyup' && event.key !== TAB_KEY$1) {\n      return;\n    }\n    const openToggles = SelectorEngine.find(SELECTOR_DATA_TOGGLE_SHOWN);\n    for (const toggle of openToggles) {\n      const context = Dropdown.getInstance(toggle);\n      if (!context || context._config.autoClose === false) {\n        continue;\n      }\n      const composedPath = event.composedPath();\n      const isMenuTarget = composedPath.includes(context._menu);\n      if (composedPath.includes(context._element) || context._config.autoClose === 'inside' && !isMenuTarget || context._config.autoClose === 'outside' && isMenuTarget) {\n        continue;\n      }\n\n      // Tab navigation through the dropdown menu or events from contained inputs shouldn't close the menu\n      if (context._menu.contains(event.target) && (event.type === 'keyup' && event.key === TAB_KEY$1 || /input|select|option|textarea|form/i.test(event.target.tagName))) {\n        continue;\n      }\n      const relatedTarget = {\n        relatedTarget: context._element\n      };\n      if (event.type === 'click') {\n        relatedTarget.clickEvent = event;\n      }\n      context._completeHide(relatedTarget);\n    }\n  }\n  static dataApiKeydownHandler(event) {\n    // If not an UP | DOWN | ESCAPE key => not a dropdown command\n    // If input/textarea && if key is other than ESCAPE => not a dropdown command\n\n    const isInput = /input|textarea/i.test(event.target.tagName);\n    const isEscapeEvent = event.key === ESCAPE_KEY$2;\n    const isUpOrDownEvent = [ARROW_UP_KEY$1, ARROW_DOWN_KEY$1].includes(event.key);\n    if (!isUpOrDownEvent && !isEscapeEvent) {\n      return;\n    }\n    if (isInput && !isEscapeEvent) {\n      return;\n    }\n    event.preventDefault();\n\n    // TODO: v6 revert #37011 & change markup https://getbootstrap.com/docs/5.3/forms/input-group/\n    const getToggleButton = this.matches(SELECTOR_DATA_TOGGLE$3) ? this : SelectorEngine.prev(this, SELECTOR_DATA_TOGGLE$3)[0] || SelectorEngine.next(this, SELECTOR_DATA_TOGGLE$3)[0] || SelectorEngine.findOne(SELECTOR_DATA_TOGGLE$3, event.delegateTarget.parentNode);\n    const instance = Dropdown.getOrCreateInstance(getToggleButton);\n    if (isUpOrDownEvent) {\n      event.stopPropagation();\n      instance.show();\n      instance._selectMenuItem(event);\n      return;\n    }\n    if (instance._isShown()) {\n      // else is escape and we check if it is shown\n      event.stopPropagation();\n      instance.hide();\n      getToggleButton.focus();\n    }\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_KEYDOWN_DATA_API, SELECTOR_DATA_TOGGLE$3, Dropdown.dataApiKeydownHandler);\nEventHandler.on(document, EVENT_KEYDOWN_DATA_API, SELECTOR_MENU, Dropdown.dataApiKeydownHandler);\nEventHandler.on(document, EVENT_CLICK_DATA_API$3, Dropdown.clearMenus);\nEventHandler.on(document, EVENT_KEYUP_DATA_API, Dropdown.clearMenus);\nEventHandler.on(document, EVENT_CLICK_DATA_API$3, SELECTOR_DATA_TOGGLE$3, function (event) {\n  event.preventDefault();\n  Dropdown.getOrCreateInstance(this).toggle();\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Dropdown);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/backdrop.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$9 = 'backdrop';\nconst CLASS_NAME_FADE$4 = 'fade';\nconst CLASS_NAME_SHOW$5 = 'show';\nconst EVENT_MOUSEDOWN = `mousedown.bs.${NAME$9}`;\nconst Default$8 = {\n  className: 'modal-backdrop',\n  clickCallback: null,\n  isAnimated: false,\n  isVisible: true,\n  // if false, we use the backdrop helper without adding any element to the dom\n  rootElement: 'body' // give the choice to place backdrop under different elements\n};\nconst DefaultType$8 = {\n  className: 'string',\n  clickCallback: '(function|null)',\n  isAnimated: 'boolean',\n  isVisible: 'boolean',\n  rootElement: '(element|string)'\n};\n\n/**\n * Class definition\n */\n\nclass Backdrop extends Config {\n  constructor(config) {\n    super();\n    this._config = this._getConfig(config);\n    this._isAppended = false;\n    this._element = null;\n  }\n\n  // Getters\n  static get Default() {\n    return Default$8;\n  }\n  static get DefaultType() {\n    return DefaultType$8;\n  }\n  static get NAME() {\n    return NAME$9;\n  }\n\n  // Public\n  show(callback) {\n    if (!this._config.isVisible) {\n      execute(callback);\n      return;\n    }\n    this._append();\n    const element = this._getElement();\n    if (this._config.isAnimated) {\n      reflow(element);\n    }\n    element.classList.add(CLASS_NAME_SHOW$5);\n    this._emulateAnimation(() => {\n      execute(callback);\n    });\n  }\n  hide(callback) {\n    if (!this._config.isVisible) {\n      execute(callback);\n      return;\n    }\n    this._getElement().classList.remove(CLASS_NAME_SHOW$5);\n    this._emulateAnimation(() => {\n      this.dispose();\n      execute(callback);\n    });\n  }\n  dispose() {\n    if (!this._isAppended) {\n      return;\n    }\n    EventHandler.off(this._element, EVENT_MOUSEDOWN);\n    this._element.remove();\n    this._isAppended = false;\n  }\n\n  // Private\n  _getElement() {\n    if (!this._element) {\n      const backdrop = document.createElement('div');\n      backdrop.className = this._config.className;\n      if (this._config.isAnimated) {\n        backdrop.classList.add(CLASS_NAME_FADE$4);\n      }\n      this._element = backdrop;\n    }\n    return this._element;\n  }\n  _configAfterMerge(config) {\n    // use getElement() with the default \"body\" to get a fresh Element on each instantiation\n    config.rootElement = getElement(config.rootElement);\n    return config;\n  }\n  _append() {\n    if (this._isAppended) {\n      return;\n    }\n    const element = this._getElement();\n    this._config.rootElement.append(element);\n    EventHandler.on(element, EVENT_MOUSEDOWN, () => {\n      execute(this._config.clickCallback);\n    });\n    this._isAppended = true;\n  }\n  _emulateAnimation(callback) {\n    executeAfterTransition(callback, this._getElement(), this._config.isAnimated);\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/focustrap.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$8 = 'focustrap';\nconst DATA_KEY$5 = 'bs.focustrap';\nconst EVENT_KEY$5 = `.${DATA_KEY$5}`;\nconst EVENT_FOCUSIN$2 = `focusin${EVENT_KEY$5}`;\nconst EVENT_KEYDOWN_TAB = `keydown.tab${EVENT_KEY$5}`;\nconst TAB_KEY = 'Tab';\nconst TAB_NAV_FORWARD = 'forward';\nconst TAB_NAV_BACKWARD = 'backward';\nconst Default$7 = {\n  autofocus: true,\n  trapElement: null // The element to trap focus inside of\n};\nconst DefaultType$7 = {\n  autofocus: 'boolean',\n  trapElement: 'element'\n};\n\n/**\n * Class definition\n */\n\nclass FocusTrap extends Config {\n  constructor(config) {\n    super();\n    this._config = this._getConfig(config);\n    this._isActive = false;\n    this._lastTabNavDirection = null;\n  }\n\n  // Getters\n  static get Default() {\n    return Default$7;\n  }\n  static get DefaultType() {\n    return DefaultType$7;\n  }\n  static get NAME() {\n    return NAME$8;\n  }\n\n  // Public\n  activate() {\n    if (this._isActive) {\n      return;\n    }\n    if (this._config.autofocus) {\n      this._config.trapElement.focus();\n    }\n    EventHandler.off(document, EVENT_KEY$5); // guard against infinite focus loop\n    EventHandler.on(document, EVENT_FOCUSIN$2, event => this._handleFocusin(event));\n    EventHandler.on(document, EVENT_KEYDOWN_TAB, event => this._handleKeydown(event));\n    this._isActive = true;\n  }\n  deactivate() {\n    if (!this._isActive) {\n      return;\n    }\n    this._isActive = false;\n    EventHandler.off(document, EVENT_KEY$5);\n  }\n\n  // Private\n  _handleFocusin(event) {\n    const {\n      trapElement\n    } = this._config;\n    if (event.target === document || event.target === trapElement || trapElement.contains(event.target)) {\n      return;\n    }\n    const elements = SelectorEngine.focusableChildren(trapElement);\n    if (elements.length === 0) {\n      trapElement.focus();\n    } else if (this._lastTabNavDirection === TAB_NAV_BACKWARD) {\n      elements[elements.length - 1].focus();\n    } else {\n      elements[0].focus();\n    }\n  }\n  _handleKeydown(event) {\n    if (event.key !== TAB_KEY) {\n      return;\n    }\n    this._lastTabNavDirection = event.shiftKey ? TAB_NAV_BACKWARD : TAB_NAV_FORWARD;\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/scrollBar.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst SELECTOR_FIXED_CONTENT = '.fixed-top, .fixed-bottom, .is-fixed, .sticky-top';\nconst SELECTOR_STICKY_CONTENT = '.sticky-top';\nconst PROPERTY_PADDING = 'padding-right';\nconst PROPERTY_MARGIN = 'margin-right';\n\n/**\n * Class definition\n */\n\nclass ScrollBarHelper {\n  constructor() {\n    this._element = document.body;\n  }\n\n  // Public\n  getWidth() {\n    // https://developer.mozilla.org/en-US/docs/Web/API/Window/innerWidth#usage_notes\n    const documentWidth = document.documentElement.clientWidth;\n    return Math.abs(window.innerWidth - documentWidth);\n  }\n  hide() {\n    const width = this.getWidth();\n    this._disableOverFlow();\n    // give padding to element to balance the hidden scrollbar width\n    this._setElementAttributes(this._element, PROPERTY_PADDING, calculatedValue => calculatedValue + width);\n    // trick: We adjust positive paddingRight and negative marginRight to sticky-top elements to keep showing fullwidth\n    this._setElementAttributes(SELECTOR_FIXED_CONTENT, PROPERTY_PADDING, calculatedValue => calculatedValue + width);\n    this._setElementAttributes(SELECTOR_STICKY_CONTENT, PROPERTY_MARGIN, calculatedValue => calculatedValue - width);\n  }\n  reset() {\n    this._resetElementAttributes(this._element, 'overflow');\n    this._resetElementAttributes(this._element, PROPERTY_PADDING);\n    this._resetElementAttributes(SELECTOR_FIXED_CONTENT, PROPERTY_PADDING);\n    this._resetElementAttributes(SELECTOR_STICKY_CONTENT, PROPERTY_MARGIN);\n  }\n  isOverflowing() {\n    return this.getWidth() > 0;\n  }\n\n  // Private\n  _disableOverFlow() {\n    this._saveInitialAttribute(this._element, 'overflow');\n    this._element.style.overflow = 'hidden';\n  }\n  _setElementAttributes(selector, styleProperty, callback) {\n    const scrollbarWidth = this.getWidth();\n    const manipulationCallBack = element => {\n      if (element !== this._element && window.innerWidth > element.clientWidth + scrollbarWidth) {\n        return;\n      }\n      this._saveInitialAttribute(element, styleProperty);\n      const calculatedValue = window.getComputedStyle(element).getPropertyValue(styleProperty);\n      element.style.setProperty(styleProperty, `${callback(Number.parseFloat(calculatedValue))}px`);\n    };\n    this._applyManipulationCallback(selector, manipulationCallBack);\n  }\n  _saveInitialAttribute(element, styleProperty) {\n    const actualValue = element.style.getPropertyValue(styleProperty);\n    if (actualValue) {\n      Manipulator.setDataAttribute(element, styleProperty, actualValue);\n    }\n  }\n  _resetElementAttributes(selector, styleProperty) {\n    const manipulationCallBack = element => {\n      const value = Manipulator.getDataAttribute(element, styleProperty);\n      // We only want to remove the property if the value is `null`; the value can also be zero\n      if (value === null) {\n        element.style.removeProperty(styleProperty);\n        return;\n      }\n      Manipulator.removeDataAttribute(element, styleProperty);\n      element.style.setProperty(styleProperty, value);\n    };\n    this._applyManipulationCallback(selector, manipulationCallBack);\n  }\n  _applyManipulationCallback(selector, callBack) {\n    if (isElement(selector)) {\n      callBack(selector);\n      return;\n    }\n    for (const sel of SelectorEngine.find(selector, this._element)) {\n      callBack(sel);\n    }\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap modal.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$7 = 'modal';\nconst DATA_KEY$4 = 'bs.modal';\nconst EVENT_KEY$4 = `.${DATA_KEY$4}`;\nconst DATA_API_KEY$2 = '.data-api';\nconst ESCAPE_KEY$1 = 'Escape';\nconst EVENT_HIDE$4 = `hide${EVENT_KEY$4}`;\nconst EVENT_HIDE_PREVENTED$1 = `hidePrevented${EVENT_KEY$4}`;\nconst EVENT_HIDDEN$4 = `hidden${EVENT_KEY$4}`;\nconst EVENT_SHOW$4 = `show${EVENT_KEY$4}`;\nconst EVENT_SHOWN$4 = `shown${EVENT_KEY$4}`;\nconst EVENT_RESIZE$1 = `resize${EVENT_KEY$4}`;\nconst EVENT_CLICK_DISMISS = `click.dismiss${EVENT_KEY$4}`;\nconst EVENT_MOUSEDOWN_DISMISS = `mousedown.dismiss${EVENT_KEY$4}`;\nconst EVENT_KEYDOWN_DISMISS$1 = `keydown.dismiss${EVENT_KEY$4}`;\nconst EVENT_CLICK_DATA_API$2 = `click${EVENT_KEY$4}${DATA_API_KEY$2}`;\nconst CLASS_NAME_OPEN = 'modal-open';\nconst CLASS_NAME_FADE$3 = 'fade';\nconst CLASS_NAME_SHOW$4 = 'show';\nconst CLASS_NAME_STATIC = 'modal-static';\nconst OPEN_SELECTOR$1 = '.modal.show';\nconst SELECTOR_DIALOG = '.modal-dialog';\nconst SELECTOR_MODAL_BODY = '.modal-body';\nconst SELECTOR_DATA_TOGGLE$2 = '[data-bs-toggle=\"modal\"]';\nconst Default$6 = {\n  backdrop: true,\n  focus: true,\n  keyboard: true\n};\nconst DefaultType$6 = {\n  backdrop: '(boolean|string)',\n  focus: 'boolean',\n  keyboard: 'boolean'\n};\n\n/**\n * Class definition\n */\n\nclass Modal extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._dialog = SelectorEngine.findOne(SELECTOR_DIALOG, this._element);\n    this._backdrop = this._initializeBackDrop();\n    this._focustrap = this._initializeFocusTrap();\n    this._isShown = false;\n    this._isTransitioning = false;\n    this._scrollBar = new ScrollBarHelper();\n    this._addEventListeners();\n  }\n\n  // Getters\n  static get Default() {\n    return Default$6;\n  }\n  static get DefaultType() {\n    return DefaultType$6;\n  }\n  static get NAME() {\n    return NAME$7;\n  }\n\n  // Public\n  toggle(relatedTarget) {\n    return this._isShown ? this.hide() : this.show(relatedTarget);\n  }\n  show(relatedTarget) {\n    if (this._isShown || this._isTransitioning) {\n      return;\n    }\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW$4, {\n      relatedTarget\n    });\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n    this._isShown = true;\n    this._isTransitioning = true;\n    this._scrollBar.hide();\n    document.body.classList.add(CLASS_NAME_OPEN);\n    this._adjustDialog();\n    this._backdrop.show(() => this._showElement(relatedTarget));\n  }\n  hide() {\n    if (!this._isShown || this._isTransitioning) {\n      return;\n    }\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE$4);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    this._isShown = false;\n    this._isTransitioning = true;\n    this._focustrap.deactivate();\n    this._element.classList.remove(CLASS_NAME_SHOW$4);\n    this._queueCallback(() => this._hideModal(), this._element, this._isAnimated());\n  }\n  dispose() {\n    EventHandler.off(window, EVENT_KEY$4);\n    EventHandler.off(this._dialog, EVENT_KEY$4);\n    this._backdrop.dispose();\n    this._focustrap.deactivate();\n    super.dispose();\n  }\n  handleUpdate() {\n    this._adjustDialog();\n  }\n\n  // Private\n  _initializeBackDrop() {\n    return new Backdrop({\n      isVisible: Boolean(this._config.backdrop),\n      // 'static' option will be translated to true, and booleans will keep their value,\n      isAnimated: this._isAnimated()\n    });\n  }\n  _initializeFocusTrap() {\n    return new FocusTrap({\n      trapElement: this._element\n    });\n  }\n  _showElement(relatedTarget) {\n    // try to append dynamic modal\n    if (!document.body.contains(this._element)) {\n      document.body.append(this._element);\n    }\n    this._element.style.display = 'block';\n    this._element.removeAttribute('aria-hidden');\n    this._element.setAttribute('aria-modal', true);\n    this._element.setAttribute('role', 'dialog');\n    this._element.scrollTop = 0;\n    const modalBody = SelectorEngine.findOne(SELECTOR_MODAL_BODY, this._dialog);\n    if (modalBody) {\n      modalBody.scrollTop = 0;\n    }\n    reflow(this._element);\n    this._element.classList.add(CLASS_NAME_SHOW$4);\n    const transitionComplete = () => {\n      if (this._config.focus) {\n        this._focustrap.activate();\n      }\n      this._isTransitioning = false;\n      EventHandler.trigger(this._element, EVENT_SHOWN$4, {\n        relatedTarget\n      });\n    };\n    this._queueCallback(transitionComplete, this._dialog, this._isAnimated());\n  }\n  _addEventListeners() {\n    EventHandler.on(this._element, EVENT_KEYDOWN_DISMISS$1, event => {\n      if (event.key !== ESCAPE_KEY$1) {\n        return;\n      }\n      if (this._config.keyboard) {\n        this.hide();\n        return;\n      }\n      this._triggerBackdropTransition();\n    });\n    EventHandler.on(window, EVENT_RESIZE$1, () => {\n      if (this._isShown && !this._isTransitioning) {\n        this._adjustDialog();\n      }\n    });\n    EventHandler.on(this._element, EVENT_MOUSEDOWN_DISMISS, event => {\n      // a bad trick to segregate clicks that may start inside dialog but end outside, and avoid listen to scrollbar clicks\n      EventHandler.one(this._element, EVENT_CLICK_DISMISS, event2 => {\n        if (this._element !== event.target || this._element !== event2.target) {\n          return;\n        }\n        if (this._config.backdrop === 'static') {\n          this._triggerBackdropTransition();\n          return;\n        }\n        if (this._config.backdrop) {\n          this.hide();\n        }\n      });\n    });\n  }\n  _hideModal() {\n    this._element.style.display = 'none';\n    this._element.setAttribute('aria-hidden', true);\n    this._element.removeAttribute('aria-modal');\n    this._element.removeAttribute('role');\n    this._isTransitioning = false;\n    this._backdrop.hide(() => {\n      document.body.classList.remove(CLASS_NAME_OPEN);\n      this._resetAdjustments();\n      this._scrollBar.reset();\n      EventHandler.trigger(this._element, EVENT_HIDDEN$4);\n    });\n  }\n  _isAnimated() {\n    return this._element.classList.contains(CLASS_NAME_FADE$3);\n  }\n  _triggerBackdropTransition() {\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE_PREVENTED$1);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    const isModalOverflowing = this._element.scrollHeight > document.documentElement.clientHeight;\n    const initialOverflowY = this._element.style.overflowY;\n    // return if the following background transition hasn't yet completed\n    if (initialOverflowY === 'hidden' || this._element.classList.contains(CLASS_NAME_STATIC)) {\n      return;\n    }\n    if (!isModalOverflowing) {\n      this._element.style.overflowY = 'hidden';\n    }\n    this._element.classList.add(CLASS_NAME_STATIC);\n    this._queueCallback(() => {\n      this._element.classList.remove(CLASS_NAME_STATIC);\n      this._queueCallback(() => {\n        this._element.style.overflowY = initialOverflowY;\n      }, this._dialog);\n    }, this._dialog);\n    this._element.focus();\n  }\n\n  /**\n   * The following methods are used to handle overflowing modals\n   */\n\n  _adjustDialog() {\n    const isModalOverflowing = this._element.scrollHeight > document.documentElement.clientHeight;\n    const scrollbarWidth = this._scrollBar.getWidth();\n    const isBodyOverflowing = scrollbarWidth > 0;\n    if (isBodyOverflowing && !isModalOverflowing) {\n      const property = isRTL() ? 'paddingLeft' : 'paddingRight';\n      this._element.style[property] = `${scrollbarWidth}px`;\n    }\n    if (!isBodyOverflowing && isModalOverflowing) {\n      const property = isRTL() ? 'paddingRight' : 'paddingLeft';\n      this._element.style[property] = `${scrollbarWidth}px`;\n    }\n  }\n  _resetAdjustments() {\n    this._element.style.paddingLeft = '';\n    this._element.style.paddingRight = '';\n  }\n\n  // Static\n  static jQueryInterface(config, relatedTarget) {\n    return this.each(function () {\n      const data = Modal.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config](relatedTarget);\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$2, SELECTOR_DATA_TOGGLE$2, function (event) {\n  const target = SelectorEngine.getElementFromSelector(this);\n  if (['A', 'AREA'].includes(this.tagName)) {\n    event.preventDefault();\n  }\n  EventHandler.one(target, EVENT_SHOW$4, showEvent => {\n    if (showEvent.defaultPrevented) {\n      // only register focus restorer if modal will actually get shown\n      return;\n    }\n    EventHandler.one(target, EVENT_HIDDEN$4, () => {\n      if (isVisible(this)) {\n        this.focus();\n      }\n    });\n  });\n\n  // avoid conflict when clicking modal toggler while another one is open\n  const alreadyOpen = SelectorEngine.findOne(OPEN_SELECTOR$1);\n  if (alreadyOpen) {\n    Modal.getInstance(alreadyOpen).hide();\n  }\n  const data = Modal.getOrCreateInstance(target);\n  data.toggle(this);\n});\nenableDismissTrigger(Modal);\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Modal);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap offcanvas.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$6 = 'offcanvas';\nconst DATA_KEY$3 = 'bs.offcanvas';\nconst EVENT_KEY$3 = `.${DATA_KEY$3}`;\nconst DATA_API_KEY$1 = '.data-api';\nconst EVENT_LOAD_DATA_API$2 = `load${EVENT_KEY$3}${DATA_API_KEY$1}`;\nconst ESCAPE_KEY = 'Escape';\nconst CLASS_NAME_SHOW$3 = 'show';\nconst CLASS_NAME_SHOWING$1 = 'showing';\nconst CLASS_NAME_HIDING = 'hiding';\nconst CLASS_NAME_BACKDROP = 'offcanvas-backdrop';\nconst OPEN_SELECTOR = '.offcanvas.show';\nconst EVENT_SHOW$3 = `show${EVENT_KEY$3}`;\nconst EVENT_SHOWN$3 = `shown${EVENT_KEY$3}`;\nconst EVENT_HIDE$3 = `hide${EVENT_KEY$3}`;\nconst EVENT_HIDE_PREVENTED = `hidePrevented${EVENT_KEY$3}`;\nconst EVENT_HIDDEN$3 = `hidden${EVENT_KEY$3}`;\nconst EVENT_RESIZE = `resize${EVENT_KEY$3}`;\nconst EVENT_CLICK_DATA_API$1 = `click${EVENT_KEY$3}${DATA_API_KEY$1}`;\nconst EVENT_KEYDOWN_DISMISS = `keydown.dismiss${EVENT_KEY$3}`;\nconst SELECTOR_DATA_TOGGLE$1 = '[data-bs-toggle=\"offcanvas\"]';\nconst Default$5 = {\n  backdrop: true,\n  keyboard: true,\n  scroll: false\n};\nconst DefaultType$5 = {\n  backdrop: '(boolean|string)',\n  keyboard: 'boolean',\n  scroll: 'boolean'\n};\n\n/**\n * Class definition\n */\n\nclass Offcanvas extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._isShown = false;\n    this._backdrop = this._initializeBackDrop();\n    this._focustrap = this._initializeFocusTrap();\n    this._addEventListeners();\n  }\n\n  // Getters\n  static get Default() {\n    return Default$5;\n  }\n  static get DefaultType() {\n    return DefaultType$5;\n  }\n  static get NAME() {\n    return NAME$6;\n  }\n\n  // Public\n  toggle(relatedTarget) {\n    return this._isShown ? this.hide() : this.show(relatedTarget);\n  }\n  show(relatedTarget) {\n    if (this._isShown) {\n      return;\n    }\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW$3, {\n      relatedTarget\n    });\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n    this._isShown = true;\n    this._backdrop.show();\n    if (!this._config.scroll) {\n      new ScrollBarHelper().hide();\n    }\n    this._element.setAttribute('aria-modal', true);\n    this._element.setAttribute('role', 'dialog');\n    this._element.classList.add(CLASS_NAME_SHOWING$1);\n    const completeCallBack = () => {\n      if (!this._config.scroll || this._config.backdrop) {\n        this._focustrap.activate();\n      }\n      this._element.classList.add(CLASS_NAME_SHOW$3);\n      this._element.classList.remove(CLASS_NAME_SHOWING$1);\n      EventHandler.trigger(this._element, EVENT_SHOWN$3, {\n        relatedTarget\n      });\n    };\n    this._queueCallback(completeCallBack, this._element, true);\n  }\n  hide() {\n    if (!this._isShown) {\n      return;\n    }\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE$3);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    this._focustrap.deactivate();\n    this._element.blur();\n    this._isShown = false;\n    this._element.classList.add(CLASS_NAME_HIDING);\n    this._backdrop.hide();\n    const completeCallback = () => {\n      this._element.classList.remove(CLASS_NAME_SHOW$3, CLASS_NAME_HIDING);\n      this._element.removeAttribute('aria-modal');\n      this._element.removeAttribute('role');\n      if (!this._config.scroll) {\n        new ScrollBarHelper().reset();\n      }\n      EventHandler.trigger(this._element, EVENT_HIDDEN$3);\n    };\n    this._queueCallback(completeCallback, this._element, true);\n  }\n  dispose() {\n    this._backdrop.dispose();\n    this._focustrap.deactivate();\n    super.dispose();\n  }\n\n  // Private\n  _initializeBackDrop() {\n    const clickCallback = () => {\n      if (this._config.backdrop === 'static') {\n        EventHandler.trigger(this._element, EVENT_HIDE_PREVENTED);\n        return;\n      }\n      this.hide();\n    };\n\n    // 'static' option will be translated to true, and booleans will keep their value\n    const isVisible = Boolean(this._config.backdrop);\n    return new Backdrop({\n      className: CLASS_NAME_BACKDROP,\n      isVisible,\n      isAnimated: true,\n      rootElement: this._element.parentNode,\n      clickCallback: isVisible ? clickCallback : null\n    });\n  }\n  _initializeFocusTrap() {\n    return new FocusTrap({\n      trapElement: this._element\n    });\n  }\n  _addEventListeners() {\n    EventHandler.on(this._element, EVENT_KEYDOWN_DISMISS, event => {\n      if (event.key !== ESCAPE_KEY) {\n        return;\n      }\n      if (this._config.keyboard) {\n        this.hide();\n        return;\n      }\n      EventHandler.trigger(this._element, EVENT_HIDE_PREVENTED);\n    });\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Offcanvas.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config](this);\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$1, SELECTOR_DATA_TOGGLE$1, function (event) {\n  const target = SelectorEngine.getElementFromSelector(this);\n  if (['A', 'AREA'].includes(this.tagName)) {\n    event.preventDefault();\n  }\n  if (isDisabled(this)) {\n    return;\n  }\n  EventHandler.one(target, EVENT_HIDDEN$3, () => {\n    // focus on trigger when it is closed\n    if (isVisible(this)) {\n      this.focus();\n    }\n  });\n\n  // avoid conflict when clicking a toggler of an offcanvas, while another is open\n  const alreadyOpen = SelectorEngine.findOne(OPEN_SELECTOR);\n  if (alreadyOpen && alreadyOpen !== target) {\n    Offcanvas.getInstance(alreadyOpen).hide();\n  }\n  const data = Offcanvas.getOrCreateInstance(target);\n  data.toggle(this);\n});\nEventHandler.on(window, EVENT_LOAD_DATA_API$2, () => {\n  for (const selector of SelectorEngine.find(OPEN_SELECTOR)) {\n    Offcanvas.getOrCreateInstance(selector).show();\n  }\n});\nEventHandler.on(window, EVENT_RESIZE, () => {\n  for (const element of SelectorEngine.find('[aria-modal][class*=show][class*=offcanvas-]')) {\n    if (getComputedStyle(element).position !== 'fixed') {\n      Offcanvas.getOrCreateInstance(element).hide();\n    }\n  }\n});\nenableDismissTrigger(Offcanvas);\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Offcanvas);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/sanitizer.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n// js-docs-start allow-list\nconst ARIA_ATTRIBUTE_PATTERN = /^aria-[\\w-]*$/i;\nconst DefaultAllowlist = {\n  // Global attributes allowed on any supplied element below.\n  '*': ['class', 'dir', 'id', 'lang', 'role', ARIA_ATTRIBUTE_PATTERN],\n  a: ['target', 'href', 'title', 'rel'],\n  area: [],\n  b: [],\n  br: [],\n  col: [],\n  code: [],\n  dd: [],\n  div: [],\n  dl: [],\n  dt: [],\n  em: [],\n  hr: [],\n  h1: [],\n  h2: [],\n  h3: [],\n  h4: [],\n  h5: [],\n  h6: [],\n  i: [],\n  img: ['src', 'srcset', 'alt', 'title', 'width', 'height'],\n  li: [],\n  ol: [],\n  p: [],\n  pre: [],\n  s: [],\n  small: [],\n  span: [],\n  sub: [],\n  sup: [],\n  strong: [],\n  u: [],\n  ul: []\n};\n// js-docs-end allow-list\n\nconst uriAttributes = new Set(['background', 'cite', 'href', 'itemtype', 'longdesc', 'poster', 'src', 'xlink:href']);\n\n/**\n * A pattern that recognizes URLs that are safe wrt. XSS in URL navigation\n * contexts.\n *\n * Shout-out to Angular https://github.com/angular/angular/blob/15.2.8/packages/core/src/sanitization/url_sanitizer.ts#L38\n */\n// eslint-disable-next-line unicorn/better-regex\nconst SAFE_URL_PATTERN = /^(?!javascript:)(?:[a-z0-9+.-]+:|[^&:/?#]*(?:[/?#]|$))/i;\nconst allowedAttribute = (attribute, allowedAttributeList) => {\n  const attributeName = attribute.nodeName.toLowerCase();\n  if (allowedAttributeList.includes(attributeName)) {\n    if (uriAttributes.has(attributeName)) {\n      return Boolean(SAFE_URL_PATTERN.test(attribute.nodeValue));\n    }\n    return true;\n  }\n\n  // Check if a regular expression validates the attribute.\n  return allowedAttributeList.filter(attributeRegex => attributeRegex instanceof RegExp).some(regex => regex.test(attributeName));\n};\nfunction sanitizeHtml(unsafeHtml, allowList, sanitizeFunction) {\n  if (!unsafeHtml.length) {\n    return unsafeHtml;\n  }\n  if (sanitizeFunction && typeof sanitizeFunction === 'function') {\n    return sanitizeFunction(unsafeHtml);\n  }\n  const domParser = new window.DOMParser();\n  const createdDocument = domParser.parseFromString(unsafeHtml, 'text/html');\n  const elements = [].concat(...createdDocument.body.querySelectorAll('*'));\n  for (const element of elements) {\n    const elementName = element.nodeName.toLowerCase();\n    if (!Object.keys(allowList).includes(elementName)) {\n      element.remove();\n      continue;\n    }\n    const attributeList = [].concat(...element.attributes);\n    const allowedAttributes = [].concat(allowList['*'] || [], allowList[elementName] || []);\n    for (const attribute of attributeList) {\n      if (!allowedAttribute(attribute, allowedAttributes)) {\n        element.removeAttribute(attribute.nodeName);\n      }\n    }\n  }\n  return createdDocument.body.innerHTML;\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/template-factory.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$5 = 'TemplateFactory';\nconst Default$4 = {\n  allowList: DefaultAllowlist,\n  content: {},\n  // { selector : text ,  selector2 : text2 , }\n  extraClass: '',\n  html: false,\n  sanitize: true,\n  sanitizeFn: null,\n  template: '<div></div>'\n};\nconst DefaultType$4 = {\n  allowList: 'object',\n  content: 'object',\n  extraClass: '(string|function)',\n  html: 'boolean',\n  sanitize: 'boolean',\n  sanitizeFn: '(null|function)',\n  template: 'string'\n};\nconst DefaultContentType = {\n  entry: '(string|element|function|null)',\n  selector: '(string|element)'\n};\n\n/**\n * Class definition\n */\n\nclass TemplateFactory extends Config {\n  constructor(config) {\n    super();\n    this._config = this._getConfig(config);\n  }\n\n  // Getters\n  static get Default() {\n    return Default$4;\n  }\n  static get DefaultType() {\n    return DefaultType$4;\n  }\n  static get NAME() {\n    return NAME$5;\n  }\n\n  // Public\n  getContent() {\n    return Object.values(this._config.content).map(config => this._resolvePossibleFunction(config)).filter(Boolean);\n  }\n  hasContent() {\n    return this.getContent().length > 0;\n  }\n  changeContent(content) {\n    this._checkContent(content);\n    this._config.content = {\n      ...this._config.content,\n      ...content\n    };\n    return this;\n  }\n  toHtml() {\n    const templateWrapper = document.createElement('div');\n    templateWrapper.innerHTML = this._maybeSanitize(this._config.template);\n    for (const [selector, text] of Object.entries(this._config.content)) {\n      this._setContent(templateWrapper, text, selector);\n    }\n    const template = templateWrapper.children[0];\n    const extraClass = this._resolvePossibleFunction(this._config.extraClass);\n    if (extraClass) {\n      template.classList.add(...extraClass.split(' '));\n    }\n    return template;\n  }\n\n  // Private\n  _typeCheckConfig(config) {\n    super._typeCheckConfig(config);\n    this._checkContent(config.content);\n  }\n  _checkContent(arg) {\n    for (const [selector, content] of Object.entries(arg)) {\n      super._typeCheckConfig({\n        selector,\n        entry: content\n      }, DefaultContentType);\n    }\n  }\n  _setContent(template, content, selector) {\n    const templateElement = SelectorEngine.findOne(selector, template);\n    if (!templateElement) {\n      return;\n    }\n    content = this._resolvePossibleFunction(content);\n    if (!content) {\n      templateElement.remove();\n      return;\n    }\n    if (isElement(content)) {\n      this._putElementInTemplate(getElement(content), templateElement);\n      return;\n    }\n    if (this._config.html) {\n      templateElement.innerHTML = this._maybeSanitize(content);\n      return;\n    }\n    templateElement.textContent = content;\n  }\n  _maybeSanitize(arg) {\n    return this._config.sanitize ? sanitizeHtml(arg, this._config.allowList, this._config.sanitizeFn) : arg;\n  }\n  _resolvePossibleFunction(arg) {\n    return execute(arg, [this]);\n  }\n  _putElementInTemplate(element, templateElement) {\n    if (this._config.html) {\n      templateElement.innerHTML = '';\n      templateElement.append(element);\n      return;\n    }\n    templateElement.textContent = element.textContent;\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap tooltip.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$4 = 'tooltip';\nconst DISALLOWED_ATTRIBUTES = new Set(['sanitize', 'allowList', 'sanitizeFn']);\nconst CLASS_NAME_FADE$2 = 'fade';\nconst CLASS_NAME_MODAL = 'modal';\nconst CLASS_NAME_SHOW$2 = 'show';\nconst SELECTOR_TOOLTIP_INNER = '.tooltip-inner';\nconst SELECTOR_MODAL = `.${CLASS_NAME_MODAL}`;\nconst EVENT_MODAL_HIDE = 'hide.bs.modal';\nconst TRIGGER_HOVER = 'hover';\nconst TRIGGER_FOCUS = 'focus';\nconst TRIGGER_CLICK = 'click';\nconst TRIGGER_MANUAL = 'manual';\nconst EVENT_HIDE$2 = 'hide';\nconst EVENT_HIDDEN$2 = 'hidden';\nconst EVENT_SHOW$2 = 'show';\nconst EVENT_SHOWN$2 = 'shown';\nconst EVENT_INSERTED = 'inserted';\nconst EVENT_CLICK$1 = 'click';\nconst EVENT_FOCUSIN$1 = 'focusin';\nconst EVENT_FOCUSOUT$1 = 'focusout';\nconst EVENT_MOUSEENTER = 'mouseenter';\nconst EVENT_MOUSELEAVE = 'mouseleave';\nconst AttachmentMap = {\n  AUTO: 'auto',\n  TOP: 'top',\n  RIGHT: isRTL() ? 'left' : 'right',\n  BOTTOM: 'bottom',\n  LEFT: isRTL() ? 'right' : 'left'\n};\nconst Default$3 = {\n  allowList: DefaultAllowlist,\n  animation: true,\n  boundary: 'clippingParents',\n  container: false,\n  customClass: '',\n  delay: 0,\n  fallbackPlacements: ['top', 'right', 'bottom', 'left'],\n  html: false,\n  offset: [0, 6],\n  placement: 'top',\n  popperConfig: null,\n  sanitize: true,\n  sanitizeFn: null,\n  selector: false,\n  template: '<div class=\"tooltip\" role=\"tooltip\">' + '<div class=\"tooltip-arrow\"></div>' + '<div class=\"tooltip-inner\"></div>' + '</div>',\n  title: '',\n  trigger: 'hover focus'\n};\nconst DefaultType$3 = {\n  allowList: 'object',\n  animation: 'boolean',\n  boundary: '(string|element)',\n  container: '(string|element|boolean)',\n  customClass: '(string|function)',\n  delay: '(number|object)',\n  fallbackPlacements: 'array',\n  html: 'boolean',\n  offset: '(array|string|function)',\n  placement: '(string|function)',\n  popperConfig: '(null|object|function)',\n  sanitize: 'boolean',\n  sanitizeFn: '(null|function)',\n  selector: '(string|boolean)',\n  template: 'string',\n  title: '(string|element|function)',\n  trigger: 'string'\n};\n\n/**\n * Class definition\n */\n\nclass Tooltip extends BaseComponent {\n  constructor(element, config) {\n    if (typeof Popper === 'undefined') {\n      throw new TypeError('Bootstrap\\'s tooltips require Popper (https://popper.js.org)');\n    }\n    super(element, config);\n\n    // Private\n    this._isEnabled = true;\n    this._timeout = 0;\n    this._isHovered = null;\n    this._activeTrigger = {};\n    this._popper = null;\n    this._templateFactory = null;\n    this._newContent = null;\n\n    // Protected\n    this.tip = null;\n    this._setListeners();\n    if (!this._config.selector) {\n      this._fixTitle();\n    }\n  }\n\n  // Getters\n  static get Default() {\n    return Default$3;\n  }\n  static get DefaultType() {\n    return DefaultType$3;\n  }\n  static get NAME() {\n    return NAME$4;\n  }\n\n  // Public\n  enable() {\n    this._isEnabled = true;\n  }\n  disable() {\n    this._isEnabled = false;\n  }\n  toggleEnabled() {\n    this._isEnabled = !this._isEnabled;\n  }\n  toggle() {\n    if (!this._isEnabled) {\n      return;\n    }\n    this._activeTrigger.click = !this._activeTrigger.click;\n    if (this._isShown()) {\n      this._leave();\n      return;\n    }\n    this._enter();\n  }\n  dispose() {\n    clearTimeout(this._timeout);\n    EventHandler.off(this._element.closest(SELECTOR_MODAL), EVENT_MODAL_HIDE, this._hideModalHandler);\n    if (this._element.getAttribute('data-bs-original-title')) {\n      this._element.setAttribute('title', this._element.getAttribute('data-bs-original-title'));\n    }\n    this._disposePopper();\n    super.dispose();\n  }\n  show() {\n    if (this._element.style.display === 'none') {\n      throw new Error('Please use show on visible elements');\n    }\n    if (!(this._isWithContent() && this._isEnabled)) {\n      return;\n    }\n    const showEvent = EventHandler.trigger(this._element, this.constructor.eventName(EVENT_SHOW$2));\n    const shadowRoot = findShadowRoot(this._element);\n    const isInTheDom = (shadowRoot || this._element.ownerDocument.documentElement).contains(this._element);\n    if (showEvent.defaultPrevented || !isInTheDom) {\n      return;\n    }\n\n    // TODO: v6 remove this or make it optional\n    this._disposePopper();\n    const tip = this._getTipElement();\n    this._element.setAttribute('aria-describedby', tip.getAttribute('id'));\n    const {\n      container\n    } = this._config;\n    if (!this._element.ownerDocument.documentElement.contains(this.tip)) {\n      container.append(tip);\n      EventHandler.trigger(this._element, this.constructor.eventName(EVENT_INSERTED));\n    }\n    this._popper = this._createPopper(tip);\n    tip.classList.add(CLASS_NAME_SHOW$2);\n\n    // If this is a touch-enabled device we add extra\n    // empty mouseover listeners to the body's immediate children;\n    // only needed because of broken event delegation on iOS\n    // https://www.quirksmode.org/blog/archives/2014/02/mouse_event_bub.html\n    if ('ontouchstart' in document.documentElement) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.on(element, 'mouseover', noop);\n      }\n    }\n    const complete = () => {\n      EventHandler.trigger(this._element, this.constructor.eventName(EVENT_SHOWN$2));\n      if (this._isHovered === false) {\n        this._leave();\n      }\n      this._isHovered = false;\n    };\n    this._queueCallback(complete, this.tip, this._isAnimated());\n  }\n  hide() {\n    if (!this._isShown()) {\n      return;\n    }\n    const hideEvent = EventHandler.trigger(this._element, this.constructor.eventName(EVENT_HIDE$2));\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    const tip = this._getTipElement();\n    tip.classList.remove(CLASS_NAME_SHOW$2);\n\n    // If this is a touch-enabled device we remove the extra\n    // empty mouseover listeners we added for iOS support\n    if ('ontouchstart' in document.documentElement) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.off(element, 'mouseover', noop);\n      }\n    }\n    this._activeTrigger[TRIGGER_CLICK] = false;\n    this._activeTrigger[TRIGGER_FOCUS] = false;\n    this._activeTrigger[TRIGGER_HOVER] = false;\n    this._isHovered = null; // it is a trick to support manual triggering\n\n    const complete = () => {\n      if (this._isWithActiveTrigger()) {\n        return;\n      }\n      if (!this._isHovered) {\n        this._disposePopper();\n      }\n      this._element.removeAttribute('aria-describedby');\n      EventHandler.trigger(this._element, this.constructor.eventName(EVENT_HIDDEN$2));\n    };\n    this._queueCallback(complete, this.tip, this._isAnimated());\n  }\n  update() {\n    if (this._popper) {\n      this._popper.update();\n    }\n  }\n\n  // Protected\n  _isWithContent() {\n    return Boolean(this._getTitle());\n  }\n  _getTipElement() {\n    if (!this.tip) {\n      this.tip = this._createTipElement(this._newContent || this._getContentForTemplate());\n    }\n    return this.tip;\n  }\n  _createTipElement(content) {\n    const tip = this._getTemplateFactory(content).toHtml();\n\n    // TODO: remove this check in v6\n    if (!tip) {\n      return null;\n    }\n    tip.classList.remove(CLASS_NAME_FADE$2, CLASS_NAME_SHOW$2);\n    // TODO: v6 the following can be achieved with CSS only\n    tip.classList.add(`bs-${this.constructor.NAME}-auto`);\n    const tipId = getUID(this.constructor.NAME).toString();\n    tip.setAttribute('id', tipId);\n    if (this._isAnimated()) {\n      tip.classList.add(CLASS_NAME_FADE$2);\n    }\n    return tip;\n  }\n  setContent(content) {\n    this._newContent = content;\n    if (this._isShown()) {\n      this._disposePopper();\n      this.show();\n    }\n  }\n  _getTemplateFactory(content) {\n    if (this._templateFactory) {\n      this._templateFactory.changeContent(content);\n    } else {\n      this._templateFactory = new TemplateFactory({\n        ...this._config,\n        // the `content` var has to be after `this._config`\n        // to override config.content in case of popover\n        content,\n        extraClass: this._resolvePossibleFunction(this._config.customClass)\n      });\n    }\n    return this._templateFactory;\n  }\n  _getContentForTemplate() {\n    return {\n      [SELECTOR_TOOLTIP_INNER]: this._getTitle()\n    };\n  }\n  _getTitle() {\n    return this._resolvePossibleFunction(this._config.title) || this._element.getAttribute('data-bs-original-title');\n  }\n\n  // Private\n  _initializeOnDelegatedTarget(event) {\n    return this.constructor.getOrCreateInstance(event.delegateTarget, this._getDelegateConfig());\n  }\n  _isAnimated() {\n    return this._config.animation || this.tip && this.tip.classList.contains(CLASS_NAME_FADE$2);\n  }\n  _isShown() {\n    return this.tip && this.tip.classList.contains(CLASS_NAME_SHOW$2);\n  }\n  _createPopper(tip) {\n    const placement = execute(this._config.placement, [this, tip, this._element]);\n    const attachment = AttachmentMap[placement.toUpperCase()];\n    return Popper.createPopper(this._element, tip, this._getPopperConfig(attachment));\n  }\n  _getOffset() {\n    const {\n      offset\n    } = this._config;\n    if (typeof offset === 'string') {\n      return offset.split(',').map(value => Number.parseInt(value, 10));\n    }\n    if (typeof offset === 'function') {\n      return popperData => offset(popperData, this._element);\n    }\n    return offset;\n  }\n  _resolvePossibleFunction(arg) {\n    return execute(arg, [this._element]);\n  }\n  _getPopperConfig(attachment) {\n    const defaultBsPopperConfig = {\n      placement: attachment,\n      modifiers: [{\n        name: 'flip',\n        options: {\n          fallbackPlacements: this._config.fallbackPlacements\n        }\n      }, {\n        name: 'offset',\n        options: {\n          offset: this._getOffset()\n        }\n      }, {\n        name: 'preventOverflow',\n        options: {\n          boundary: this._config.boundary\n        }\n      }, {\n        name: 'arrow',\n        options: {\n          element: `.${this.constructor.NAME}-arrow`\n        }\n      }, {\n        name: 'preSetPlacement',\n        enabled: true,\n        phase: 'beforeMain',\n        fn: data => {\n          // Pre-set Popper's placement attribute in order to read the arrow sizes properly.\n          // Otherwise, Popper mixes up the width and height dimensions since the initial arrow style is for top placement\n          this._getTipElement().setAttribute('data-popper-placement', data.state.placement);\n        }\n      }]\n    };\n    return {\n      ...defaultBsPopperConfig,\n      ...execute(this._config.popperConfig, [defaultBsPopperConfig])\n    };\n  }\n  _setListeners() {\n    const triggers = this._config.trigger.split(' ');\n    for (const trigger of triggers) {\n      if (trigger === 'click') {\n        EventHandler.on(this._element, this.constructor.eventName(EVENT_CLICK$1), this._config.selector, event => {\n          const context = this._initializeOnDelegatedTarget(event);\n          context.toggle();\n        });\n      } else if (trigger !== TRIGGER_MANUAL) {\n        const eventIn = trigger === TRIGGER_HOVER ? this.constructor.eventName(EVENT_MOUSEENTER) : this.constructor.eventName(EVENT_FOCUSIN$1);\n        const eventOut = trigger === TRIGGER_HOVER ? this.constructor.eventName(EVENT_MOUSELEAVE) : this.constructor.eventName(EVENT_FOCUSOUT$1);\n        EventHandler.on(this._element, eventIn, this._config.selector, event => {\n          const context = this._initializeOnDelegatedTarget(event);\n          context._activeTrigger[event.type === 'focusin' ? TRIGGER_FOCUS : TRIGGER_HOVER] = true;\n          context._enter();\n        });\n        EventHandler.on(this._element, eventOut, this._config.selector, event => {\n          const context = this._initializeOnDelegatedTarget(event);\n          context._activeTrigger[event.type === 'focusout' ? TRIGGER_FOCUS : TRIGGER_HOVER] = context._element.contains(event.relatedTarget);\n          context._leave();\n        });\n      }\n    }\n    this._hideModalHandler = () => {\n      if (this._element) {\n        this.hide();\n      }\n    };\n    EventHandler.on(this._element.closest(SELECTOR_MODAL), EVENT_MODAL_HIDE, this._hideModalHandler);\n  }\n  _fixTitle() {\n    const title = this._element.getAttribute('title');\n    if (!title) {\n      return;\n    }\n    if (!this._element.getAttribute('aria-label') && !this._element.textContent.trim()) {\n      this._element.setAttribute('aria-label', title);\n    }\n    this._element.setAttribute('data-bs-original-title', title); // DO NOT USE IT. Is only for backwards compatibility\n    this._element.removeAttribute('title');\n  }\n  _enter() {\n    if (this._isShown() || this._isHovered) {\n      this._isHovered = true;\n      return;\n    }\n    this._isHovered = true;\n    this._setTimeout(() => {\n      if (this._isHovered) {\n        this.show();\n      }\n    }, this._config.delay.show);\n  }\n  _leave() {\n    if (this._isWithActiveTrigger()) {\n      return;\n    }\n    this._isHovered = false;\n    this._setTimeout(() => {\n      if (!this._isHovered) {\n        this.hide();\n      }\n    }, this._config.delay.hide);\n  }\n  _setTimeout(handler, timeout) {\n    clearTimeout(this._timeout);\n    this._timeout = setTimeout(handler, timeout);\n  }\n  _isWithActiveTrigger() {\n    return Object.values(this._activeTrigger).includes(true);\n  }\n  _getConfig(config) {\n    const dataAttributes = Manipulator.getDataAttributes(this._element);\n    for (const dataAttribute of Object.keys(dataAttributes)) {\n      if (DISALLOWED_ATTRIBUTES.has(dataAttribute)) {\n        delete dataAttributes[dataAttribute];\n      }\n    }\n    config = {\n      ...dataAttributes,\n      ...(typeof config === 'object' && config ? config : {})\n    };\n    config = this._mergeConfigObj(config);\n    config = this._configAfterMerge(config);\n    this._typeCheckConfig(config);\n    return config;\n  }\n  _configAfterMerge(config) {\n    config.container = config.container === false ? document.body : getElement(config.container);\n    if (typeof config.delay === 'number') {\n      config.delay = {\n        show: config.delay,\n        hide: config.delay\n      };\n    }\n    if (typeof config.title === 'number') {\n      config.title = config.title.toString();\n    }\n    if (typeof config.content === 'number') {\n      config.content = config.content.toString();\n    }\n    return config;\n  }\n  _getDelegateConfig() {\n    const config = {};\n    for (const [key, value] of Object.entries(this._config)) {\n      if (this.constructor.Default[key] !== value) {\n        config[key] = value;\n      }\n    }\n    config.selector = false;\n    config.trigger = 'manual';\n\n    // In the future can be replaced with:\n    // const keysWithDifferentValues = Object.entries(this._config).filter(entry => this.constructor.Default[entry[0]] !== this._config[entry[0]])\n    // `Object.fromEntries(keysWithDifferentValues)`\n    return config;\n  }\n  _disposePopper() {\n    if (this._popper) {\n      this._popper.destroy();\n      this._popper = null;\n    }\n    if (this.tip) {\n      this.tip.remove();\n      this.tip = null;\n    }\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Tooltip.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n}\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Tooltip);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap popover.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$3 = 'popover';\nconst SELECTOR_TITLE = '.popover-header';\nconst SELECTOR_CONTENT = '.popover-body';\nconst Default$2 = {\n  ...Tooltip.Default,\n  content: '',\n  offset: [0, 8],\n  placement: 'right',\n  template: '<div class=\"popover\" role=\"tooltip\">' + '<div class=\"popover-arrow\"></div>' + '<h3 class=\"popover-header\"></h3>' + '<div class=\"popover-body\"></div>' + '</div>',\n  trigger: 'click'\n};\nconst DefaultType$2 = {\n  ...Tooltip.DefaultType,\n  content: '(null|string|element|function)'\n};\n\n/**\n * Class definition\n */\n\nclass Popover extends Tooltip {\n  // Getters\n  static get Default() {\n    return Default$2;\n  }\n  static get DefaultType() {\n    return DefaultType$2;\n  }\n  static get NAME() {\n    return NAME$3;\n  }\n\n  // Overrides\n  _isWithContent() {\n    return this._getTitle() || this._getContent();\n  }\n\n  // Private\n  _getContentForTemplate() {\n    return {\n      [SELECTOR_TITLE]: this._getTitle(),\n      [SELECTOR_CONTENT]: this._getContent()\n    };\n  }\n  _getContent() {\n    return this._resolvePossibleFunction(this._config.content);\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Popover.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n}\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Popover);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap scrollspy.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$2 = 'scrollspy';\nconst DATA_KEY$2 = 'bs.scrollspy';\nconst EVENT_KEY$2 = `.${DATA_KEY$2}`;\nconst DATA_API_KEY = '.data-api';\nconst EVENT_ACTIVATE = `activate${EVENT_KEY$2}`;\nconst EVENT_CLICK = `click${EVENT_KEY$2}`;\nconst EVENT_LOAD_DATA_API$1 = `load${EVENT_KEY$2}${DATA_API_KEY}`;\nconst CLASS_NAME_DROPDOWN_ITEM = 'dropdown-item';\nconst CLASS_NAME_ACTIVE$1 = 'active';\nconst SELECTOR_DATA_SPY = '[data-bs-spy=\"scroll\"]';\nconst SELECTOR_TARGET_LINKS = '[href]';\nconst SELECTOR_NAV_LIST_GROUP = '.nav, .list-group';\nconst SELECTOR_NAV_LINKS = '.nav-link';\nconst SELECTOR_NAV_ITEMS = '.nav-item';\nconst SELECTOR_LIST_ITEMS = '.list-group-item';\nconst SELECTOR_LINK_ITEMS = `${SELECTOR_NAV_LINKS}, ${SELECTOR_NAV_ITEMS} > ${SELECTOR_NAV_LINKS}, ${SELECTOR_LIST_ITEMS}`;\nconst SELECTOR_DROPDOWN = '.dropdown';\nconst SELECTOR_DROPDOWN_TOGGLE$1 = '.dropdown-toggle';\nconst Default$1 = {\n  offset: null,\n  // TODO: v6 @deprecated, keep it for backwards compatibility reasons\n  rootMargin: '0px 0px -25%',\n  smoothScroll: false,\n  target: null,\n  threshold: [0.1, 0.5, 1]\n};\nconst DefaultType$1 = {\n  offset: '(number|null)',\n  // TODO v6 @deprecated, keep it for backwards compatibility reasons\n  rootMargin: 'string',\n  smoothScroll: 'boolean',\n  target: 'element',\n  threshold: 'array'\n};\n\n/**\n * Class definition\n */\n\nclass ScrollSpy extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n\n    // this._element is the observablesContainer and config.target the menu links wrapper\n    this._targetLinks = new Map();\n    this._observableSections = new Map();\n    this._rootElement = getComputedStyle(this._element).overflowY === 'visible' ? null : this._element;\n    this._activeTarget = null;\n    this._observer = null;\n    this._previousScrollData = {\n      visibleEntryTop: 0,\n      parentScrollTop: 0\n    };\n    this.refresh(); // initialize\n  }\n\n  // Getters\n  static get Default() {\n    return Default$1;\n  }\n  static get DefaultType() {\n    return DefaultType$1;\n  }\n  static get NAME() {\n    return NAME$2;\n  }\n\n  // Public\n  refresh() {\n    this._initializeTargetsAndObservables();\n    this._maybeEnableSmoothScroll();\n    if (this._observer) {\n      this._observer.disconnect();\n    } else {\n      this._observer = this._getNewObserver();\n    }\n    for (const section of this._observableSections.values()) {\n      this._observer.observe(section);\n    }\n  }\n  dispose() {\n    this._observer.disconnect();\n    super.dispose();\n  }\n\n  // Private\n  _configAfterMerge(config) {\n    // TODO: on v6 target should be given explicitly & remove the {target: 'ss-target'} case\n    config.target = getElement(config.target) || document.body;\n\n    // TODO: v6 Only for backwards compatibility reasons. Use rootMargin only\n    config.rootMargin = config.offset ? `${config.offset}px 0px -30%` : config.rootMargin;\n    if (typeof config.threshold === 'string') {\n      config.threshold = config.threshold.split(',').map(value => Number.parseFloat(value));\n    }\n    return config;\n  }\n  _maybeEnableSmoothScroll() {\n    if (!this._config.smoothScroll) {\n      return;\n    }\n\n    // unregister any previous listeners\n    EventHandler.off(this._config.target, EVENT_CLICK);\n    EventHandler.on(this._config.target, EVENT_CLICK, SELECTOR_TARGET_LINKS, event => {\n      const observableSection = this._observableSections.get(event.target.hash);\n      if (observableSection) {\n        event.preventDefault();\n        const root = this._rootElement || window;\n        const height = observableSection.offsetTop - this._element.offsetTop;\n        if (root.scrollTo) {\n          root.scrollTo({\n            top: height,\n            behavior: 'smooth'\n          });\n          return;\n        }\n\n        // Chrome 60 doesn't support `scrollTo`\n        root.scrollTop = height;\n      }\n    });\n  }\n  _getNewObserver() {\n    const options = {\n      root: this._rootElement,\n      threshold: this._config.threshold,\n      rootMargin: this._config.rootMargin\n    };\n    return new IntersectionObserver(entries => this._observerCallback(entries), options);\n  }\n\n  // The logic of selection\n  _observerCallback(entries) {\n    const targetElement = entry => this._targetLinks.get(`#${entry.target.id}`);\n    const activate = entry => {\n      this._previousScrollData.visibleEntryTop = entry.target.offsetTop;\n      this._process(targetElement(entry));\n    };\n    const parentScrollTop = (this._rootElement || document.documentElement).scrollTop;\n    const userScrollsDown = parentScrollTop >= this._previousScrollData.parentScrollTop;\n    this._previousScrollData.parentScrollTop = parentScrollTop;\n    for (const entry of entries) {\n      if (!entry.isIntersecting) {\n        this._activeTarget = null;\n        this._clearActiveClass(targetElement(entry));\n        continue;\n      }\n      const entryIsLowerThanPrevious = entry.target.offsetTop >= this._previousScrollData.visibleEntryTop;\n      // if we are scrolling down, pick the bigger offsetTop\n      if (userScrollsDown && entryIsLowerThanPrevious) {\n        activate(entry);\n        // if parent isn't scrolled, let's keep the first visible item, breaking the iteration\n        if (!parentScrollTop) {\n          return;\n        }\n        continue;\n      }\n\n      // if we are scrolling up, pick the smallest offsetTop\n      if (!userScrollsDown && !entryIsLowerThanPrevious) {\n        activate(entry);\n      }\n    }\n  }\n  _initializeTargetsAndObservables() {\n    this._targetLinks = new Map();\n    this._observableSections = new Map();\n    const targetLinks = SelectorEngine.find(SELECTOR_TARGET_LINKS, this._config.target);\n    for (const anchor of targetLinks) {\n      // ensure that the anchor has an id and is not disabled\n      if (!anchor.hash || isDisabled(anchor)) {\n        continue;\n      }\n      const observableSection = SelectorEngine.findOne(decodeURI(anchor.hash), this._element);\n\n      // ensure that the observableSection exists & is visible\n      if (isVisible(observableSection)) {\n        this._targetLinks.set(decodeURI(anchor.hash), anchor);\n        this._observableSections.set(anchor.hash, observableSection);\n      }\n    }\n  }\n  _process(target) {\n    if (this._activeTarget === target) {\n      return;\n    }\n    this._clearActiveClass(this._config.target);\n    this._activeTarget = target;\n    target.classList.add(CLASS_NAME_ACTIVE$1);\n    this._activateParents(target);\n    EventHandler.trigger(this._element, EVENT_ACTIVATE, {\n      relatedTarget: target\n    });\n  }\n  _activateParents(target) {\n    // Activate dropdown parents\n    if (target.classList.contains(CLASS_NAME_DROPDOWN_ITEM)) {\n      SelectorEngine.findOne(SELECTOR_DROPDOWN_TOGGLE$1, target.closest(SELECTOR_DROPDOWN)).classList.add(CLASS_NAME_ACTIVE$1);\n      return;\n    }\n    for (const listGroup of SelectorEngine.parents(target, SELECTOR_NAV_LIST_GROUP)) {\n      // Set triggered links parents as active\n      // With both <ul> and <nav> markup a parent is the previous sibling of any nav ancestor\n      for (const item of SelectorEngine.prev(listGroup, SELECTOR_LINK_ITEMS)) {\n        item.classList.add(CLASS_NAME_ACTIVE$1);\n      }\n    }\n  }\n  _clearActiveClass(parent) {\n    parent.classList.remove(CLASS_NAME_ACTIVE$1);\n    const activeNodes = SelectorEngine.find(`${SELECTOR_TARGET_LINKS}.${CLASS_NAME_ACTIVE$1}`, parent);\n    for (const node of activeNodes) {\n      node.classList.remove(CLASS_NAME_ACTIVE$1);\n    }\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = ScrollSpy.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(window, EVENT_LOAD_DATA_API$1, () => {\n  for (const spy of SelectorEngine.find(SELECTOR_DATA_SPY)) {\n    ScrollSpy.getOrCreateInstance(spy);\n  }\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(ScrollSpy);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap tab.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$1 = 'tab';\nconst DATA_KEY$1 = 'bs.tab';\nconst EVENT_KEY$1 = `.${DATA_KEY$1}`;\nconst EVENT_HIDE$1 = `hide${EVENT_KEY$1}`;\nconst EVENT_HIDDEN$1 = `hidden${EVENT_KEY$1}`;\nconst EVENT_SHOW$1 = `show${EVENT_KEY$1}`;\nconst EVENT_SHOWN$1 = `shown${EVENT_KEY$1}`;\nconst EVENT_CLICK_DATA_API = `click${EVENT_KEY$1}`;\nconst EVENT_KEYDOWN = `keydown${EVENT_KEY$1}`;\nconst EVENT_LOAD_DATA_API = `load${EVENT_KEY$1}`;\nconst ARROW_LEFT_KEY = 'ArrowLeft';\nconst ARROW_RIGHT_KEY = 'ArrowRight';\nconst ARROW_UP_KEY = 'ArrowUp';\nconst ARROW_DOWN_KEY = 'ArrowDown';\nconst HOME_KEY = 'Home';\nconst END_KEY = 'End';\nconst CLASS_NAME_ACTIVE = 'active';\nconst CLASS_NAME_FADE$1 = 'fade';\nconst CLASS_NAME_SHOW$1 = 'show';\nconst CLASS_DROPDOWN = 'dropdown';\nconst SELECTOR_DROPDOWN_TOGGLE = '.dropdown-toggle';\nconst SELECTOR_DROPDOWN_MENU = '.dropdown-menu';\nconst NOT_SELECTOR_DROPDOWN_TOGGLE = `:not(${SELECTOR_DROPDOWN_TOGGLE})`;\nconst SELECTOR_TAB_PANEL = '.list-group, .nav, [role=\"tablist\"]';\nconst SELECTOR_OUTER = '.nav-item, .list-group-item';\nconst SELECTOR_INNER = `.nav-link${NOT_SELECTOR_DROPDOWN_TOGGLE}, .list-group-item${NOT_SELECTOR_DROPDOWN_TOGGLE}, [role=\"tab\"]${NOT_SELECTOR_DROPDOWN_TOGGLE}`;\nconst SELECTOR_DATA_TOGGLE = '[data-bs-toggle=\"tab\"], [data-bs-toggle=\"pill\"], [data-bs-toggle=\"list\"]'; // TODO: could only be `tab` in v6\nconst SELECTOR_INNER_ELEM = `${SELECTOR_INNER}, ${SELECTOR_DATA_TOGGLE}`;\nconst SELECTOR_DATA_TOGGLE_ACTIVE = `.${CLASS_NAME_ACTIVE}[data-bs-toggle=\"tab\"], .${CLASS_NAME_ACTIVE}[data-bs-toggle=\"pill\"], .${CLASS_NAME_ACTIVE}[data-bs-toggle=\"list\"]`;\n\n/**\n * Class definition\n */\n\nclass Tab extends BaseComponent {\n  constructor(element) {\n    super(element);\n    this._parent = this._element.closest(SELECTOR_TAB_PANEL);\n    if (!this._parent) {\n      return;\n      // TODO: should throw exception in v6\n      // throw new TypeError(`${element.outerHTML} has not a valid parent ${SELECTOR_INNER_ELEM}`)\n    }\n\n    // Set up initial aria attributes\n    this._setInitialAttributes(this._parent, this._getChildren());\n    EventHandler.on(this._element, EVENT_KEYDOWN, event => this._keydown(event));\n  }\n\n  // Getters\n  static get NAME() {\n    return NAME$1;\n  }\n\n  // Public\n  show() {\n    // Shows this elem and deactivate the active sibling if exists\n    const innerElem = this._element;\n    if (this._elemIsActive(innerElem)) {\n      return;\n    }\n\n    // Search for active tab on same parent to deactivate it\n    const active = this._getActiveElem();\n    const hideEvent = active ? EventHandler.trigger(active, EVENT_HIDE$1, {\n      relatedTarget: innerElem\n    }) : null;\n    const showEvent = EventHandler.trigger(innerElem, EVENT_SHOW$1, {\n      relatedTarget: active\n    });\n    if (showEvent.defaultPrevented || hideEvent && hideEvent.defaultPrevented) {\n      return;\n    }\n    this._deactivate(active, innerElem);\n    this._activate(innerElem, active);\n  }\n\n  // Private\n  _activate(element, relatedElem) {\n    if (!element) {\n      return;\n    }\n    element.classList.add(CLASS_NAME_ACTIVE);\n    this._activate(SelectorEngine.getElementFromSelector(element)); // Search and activate/show the proper section\n\n    const complete = () => {\n      if (element.getAttribute('role') !== 'tab') {\n        element.classList.add(CLASS_NAME_SHOW$1);\n        return;\n      }\n      element.removeAttribute('tabindex');\n      element.setAttribute('aria-selected', true);\n      this._toggleDropDown(element, true);\n      EventHandler.trigger(element, EVENT_SHOWN$1, {\n        relatedTarget: relatedElem\n      });\n    };\n    this._queueCallback(complete, element, element.classList.contains(CLASS_NAME_FADE$1));\n  }\n  _deactivate(element, relatedElem) {\n    if (!element) {\n      return;\n    }\n    element.classList.remove(CLASS_NAME_ACTIVE);\n    element.blur();\n    this._deactivate(SelectorEngine.getElementFromSelector(element)); // Search and deactivate the shown section too\n\n    const complete = () => {\n      if (element.getAttribute('role') !== 'tab') {\n        element.classList.remove(CLASS_NAME_SHOW$1);\n        return;\n      }\n      element.setAttribute('aria-selected', false);\n      element.setAttribute('tabindex', '-1');\n      this._toggleDropDown(element, false);\n      EventHandler.trigger(element, EVENT_HIDDEN$1, {\n        relatedTarget: relatedElem\n      });\n    };\n    this._queueCallback(complete, element, element.classList.contains(CLASS_NAME_FADE$1));\n  }\n  _keydown(event) {\n    if (![ARROW_LEFT_KEY, ARROW_RIGHT_KEY, ARROW_UP_KEY, ARROW_DOWN_KEY, HOME_KEY, END_KEY].includes(event.key)) {\n      return;\n    }\n    event.stopPropagation(); // stopPropagation/preventDefault both added to support up/down keys without scrolling the page\n    event.preventDefault();\n    const children = this._getChildren().filter(element => !isDisabled(element));\n    let nextActiveElement;\n    if ([HOME_KEY, END_KEY].includes(event.key)) {\n      nextActiveElement = children[event.key === HOME_KEY ? 0 : children.length - 1];\n    } else {\n      const isNext = [ARROW_RIGHT_KEY, ARROW_DOWN_KEY].includes(event.key);\n      nextActiveElement = getNextActiveElement(children, event.target, isNext, true);\n    }\n    if (nextActiveElement) {\n      nextActiveElement.focus({\n        preventScroll: true\n      });\n      Tab.getOrCreateInstance(nextActiveElement).show();\n    }\n  }\n  _getChildren() {\n    // collection of inner elements\n    return SelectorEngine.find(SELECTOR_INNER_ELEM, this._parent);\n  }\n  _getActiveElem() {\n    return this._getChildren().find(child => this._elemIsActive(child)) || null;\n  }\n  _setInitialAttributes(parent, children) {\n    this._setAttributeIfNotExists(parent, 'role', 'tablist');\n    for (const child of children) {\n      this._setInitialAttributesOnChild(child);\n    }\n  }\n  _setInitialAttributesOnChild(child) {\n    child = this._getInnerElement(child);\n    const isActive = this._elemIsActive(child);\n    const outerElem = this._getOuterElement(child);\n    child.setAttribute('aria-selected', isActive);\n    if (outerElem !== child) {\n      this._setAttributeIfNotExists(outerElem, 'role', 'presentation');\n    }\n    if (!isActive) {\n      child.setAttribute('tabindex', '-1');\n    }\n    this._setAttributeIfNotExists(child, 'role', 'tab');\n\n    // set attributes to the related panel too\n    this._setInitialAttributesOnTargetPanel(child);\n  }\n  _setInitialAttributesOnTargetPanel(child) {\n    const target = SelectorEngine.getElementFromSelector(child);\n    if (!target) {\n      return;\n    }\n    this._setAttributeIfNotExists(target, 'role', 'tabpanel');\n    if (child.id) {\n      this._setAttributeIfNotExists(target, 'aria-labelledby', `${child.id}`);\n    }\n  }\n  _toggleDropDown(element, open) {\n    const outerElem = this._getOuterElement(element);\n    if (!outerElem.classList.contains(CLASS_DROPDOWN)) {\n      return;\n    }\n    const toggle = (selector, className) => {\n      const element = SelectorEngine.findOne(selector, outerElem);\n      if (element) {\n        element.classList.toggle(className, open);\n      }\n    };\n    toggle(SELECTOR_DROPDOWN_TOGGLE, CLASS_NAME_ACTIVE);\n    toggle(SELECTOR_DROPDOWN_MENU, CLASS_NAME_SHOW$1);\n    outerElem.setAttribute('aria-expanded', open);\n  }\n  _setAttributeIfNotExists(element, attribute, value) {\n    if (!element.hasAttribute(attribute)) {\n      element.setAttribute(attribute, value);\n    }\n  }\n  _elemIsActive(elem) {\n    return elem.classList.contains(CLASS_NAME_ACTIVE);\n  }\n\n  // Try to get the inner element (usually the .nav-link)\n  _getInnerElement(elem) {\n    return elem.matches(SELECTOR_INNER_ELEM) ? elem : SelectorEngine.findOne(SELECTOR_INNER_ELEM, elem);\n  }\n\n  // Try to get the outer element (usually the .nav-item)\n  _getOuterElement(elem) {\n    return elem.closest(SELECTOR_OUTER) || elem;\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Tab.getOrCreateInstance(this);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API, SELECTOR_DATA_TOGGLE, function (event) {\n  if (['A', 'AREA'].includes(this.tagName)) {\n    event.preventDefault();\n  }\n  if (isDisabled(this)) {\n    return;\n  }\n  Tab.getOrCreateInstance(this).show();\n});\n\n/**\n * Initialize on focus\n */\nEventHandler.on(window, EVENT_LOAD_DATA_API, () => {\n  for (const element of SelectorEngine.find(SELECTOR_DATA_TOGGLE_ACTIVE)) {\n    Tab.getOrCreateInstance(element);\n  }\n});\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Tab);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap toast.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME = 'toast';\nconst DATA_KEY = 'bs.toast';\nconst EVENT_KEY = `.${DATA_KEY}`;\nconst EVENT_MOUSEOVER = `mouseover${EVENT_KEY}`;\nconst EVENT_MOUSEOUT = `mouseout${EVENT_KEY}`;\nconst EVENT_FOCUSIN = `focusin${EVENT_KEY}`;\nconst EVENT_FOCUSOUT = `focusout${EVENT_KEY}`;\nconst EVENT_HIDE = `hide${EVENT_KEY}`;\nconst EVENT_HIDDEN = `hidden${EVENT_KEY}`;\nconst EVENT_SHOW = `show${EVENT_KEY}`;\nconst EVENT_SHOWN = `shown${EVENT_KEY}`;\nconst CLASS_NAME_FADE = 'fade';\nconst CLASS_NAME_HIDE = 'hide'; // @deprecated - kept here only for backwards compatibility\nconst CLASS_NAME_SHOW = 'show';\nconst CLASS_NAME_SHOWING = 'showing';\nconst DefaultType = {\n  animation: 'boolean',\n  autohide: 'boolean',\n  delay: 'number'\n};\nconst Default = {\n  animation: true,\n  autohide: true,\n  delay: 5000\n};\n\n/**\n * Class definition\n */\n\nclass Toast extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._timeout = null;\n    this._hasMouseInteraction = false;\n    this._hasKeyboardInteraction = false;\n    this._setListeners();\n  }\n\n  // Getters\n  static get Default() {\n    return Default;\n  }\n  static get DefaultType() {\n    return DefaultType;\n  }\n  static get NAME() {\n    return NAME;\n  }\n\n  // Public\n  show() {\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW);\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n    this._clearTimeout();\n    if (this._config.animation) {\n      this._element.classList.add(CLASS_NAME_FADE);\n    }\n    const complete = () => {\n      this._element.classList.remove(CLASS_NAME_SHOWING);\n      EventHandler.trigger(this._element, EVENT_SHOWN);\n      this._maybeScheduleHide();\n    };\n    this._element.classList.remove(CLASS_NAME_HIDE); // @deprecated\n    reflow(this._element);\n    this._element.classList.add(CLASS_NAME_SHOW, CLASS_NAME_SHOWING);\n    this._queueCallback(complete, this._element, this._config.animation);\n  }\n  hide() {\n    if (!this.isShown()) {\n      return;\n    }\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    const complete = () => {\n      this._element.classList.add(CLASS_NAME_HIDE); // @deprecated\n      this._element.classList.remove(CLASS_NAME_SHOWING, CLASS_NAME_SHOW);\n      EventHandler.trigger(this._element, EVENT_HIDDEN);\n    };\n    this._element.classList.add(CLASS_NAME_SHOWING);\n    this._queueCallback(complete, this._element, this._config.animation);\n  }\n  dispose() {\n    this._clearTimeout();\n    if (this.isShown()) {\n      this._element.classList.remove(CLASS_NAME_SHOW);\n    }\n    super.dispose();\n  }\n  isShown() {\n    return this._element.classList.contains(CLASS_NAME_SHOW);\n  }\n\n  // Private\n\n  _maybeScheduleHide() {\n    if (!this._config.autohide) {\n      return;\n    }\n    if (this._hasMouseInteraction || this._hasKeyboardInteraction) {\n      return;\n    }\n    this._timeout = setTimeout(() => {\n      this.hide();\n    }, this._config.delay);\n  }\n  _onInteraction(event, isInteracting) {\n    switch (event.type) {\n      case 'mouseover':\n      case 'mouseout':\n        {\n          this._hasMouseInteraction = isInteracting;\n          break;\n        }\n      case 'focusin':\n      case 'focusout':\n        {\n          this._hasKeyboardInteraction = isInteracting;\n          break;\n        }\n    }\n    if (isInteracting) {\n      this._clearTimeout();\n      return;\n    }\n    const nextElement = event.relatedTarget;\n    if (this._element === nextElement || this._element.contains(nextElement)) {\n      return;\n    }\n    this._maybeScheduleHide();\n  }\n  _setListeners() {\n    EventHandler.on(this._element, EVENT_MOUSEOVER, event => this._onInteraction(event, true));\n    EventHandler.on(this._element, EVENT_MOUSEOUT, event => this._onInteraction(event, false));\n    EventHandler.on(this._element, EVENT_FOCUSIN, event => this._onInteraction(event, true));\n    EventHandler.on(this._element, EVENT_FOCUSOUT, event => this._onInteraction(event, false));\n  }\n  _clearTimeout() {\n    clearTimeout(this._timeout);\n    this._timeout = null;\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Toast.getOrCreateInstance(this, config);\n      if (typeof config === 'string') {\n        if (typeof data[config] === 'undefined') {\n          throw new TypeError(`No method named \"${config}\"`);\n        }\n        data[config](this);\n      }\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nenableDismissTrigger(Toast);\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Toast);\n\nexport { Alert, Button, Carousel, Collapse, Dropdown, Modal, Offcanvas, Popover, ScrollSpy, Tab, Toast, Tooltip };\n//# sourceMappingURL=bootstrap.esm.js.map\n","/* define several functions to replace jQuery methods\n * inspired by https://tobiasahlin.com/blog/move-from-jquery-to-vanilla-javascript/\n */\n\n/**\n * Execute a method if DOM has finished loading\n *\n * @param {function} callback the method to execute\n */\nexport function documentReady(callback) {\n  if 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define several functions to replace jQuery methods\n * inspired by https://tobiasahlin.com/blog/move-from-jquery-to-vanilla-javascript/\n */\n\n/**\n * Execute a method if DOM has finished loading\n *\n * @param {function} callback the method to execute\n */\nexport function documentReady(callback) {\n  if (document.readyState != \"loading\") callback();\n  else document.addEventListener(\"DOMContentLoaded\", callback);\n}\n","/**\n * Compare [semver](https://semver.org/) version strings to find greater, equal or lesser.\n * This library supports the full semver specification, including comparing versions with different number of digits like `1.0.0`, `1.0`, `1`, and pre-release versions like `1.0.0-alpha`.\n * @param v1 - First version to compare\n * @param v2 - Second version to compare\n * @returns Numeric value compatible with the [Array.sort(fn) interface](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/sort#Parameters).\n */\nexport const compareVersions = (v1, v2) => {\n    // validate input and split into segments\n    const n1 = validateAndParse(v1);\n    const n2 = validateAndParse(v2);\n    // pop off the patch\n    const p1 = n1.pop();\n    const p2 = n2.pop();\n    // validate numbers\n    const r = compareSegments(n1, n2);\n    if (r !== 0)\n        return r;\n    // validate pre-release\n    if (p1 && p2) {\n        return compareSegments(p1.split('.'), p2.split('.'));\n    }\n    else if (p1 || p2) {\n        return p1 ? -1 : 1;\n    }\n    return 0;\n};\n/**\n * Validate [semver](https://semver.org/) version strings.\n *\n * @param version Version number to validate\n * @returns `true` if the version number is a valid semver version number, `false` otherwise.\n *\n * @example\n * ```\n * validate('1.0.0-rc.1'); // return true\n * validate('1.0-rc.1'); // return false\n * validate('foo'); // return false\n * ```\n */\nexport const validate = (version) => typeof version === 'string' && /^[v\\d]/.test(version) && semver.test(version);\n/**\n * Compare [semver](https://semver.org/) version strings using the specified operator.\n *\n * @param v1 First version to compare\n * @param v2 Second version to compare\n * @param operator Allowed arithmetic operator to use\n * @returns `true` if the comparison between the firstVersion and the secondVersion satisfies the operator, `false` otherwise.\n *\n * @example\n * ```\n * compare('10.1.8', '10.0.4', '>'); // return true\n * compare('10.0.1', '10.0.1', '='); // return true\n * compare('10.1.1', '10.2.2', '<'); // return true\n * compare('10.1.1', '10.2.2', '<='); // return true\n * compare('10.1.1', '10.2.2', '>='); // return false\n * ```\n */\nexport const compare = (v1, v2, operator) => {\n    // validate input operator\n    assertValidOperator(operator);\n    // since result of compareVersions can only be -1 or 0 or 1\n    // a simple map can be used to replace switch\n    const res = compareVersions(v1, v2);\n    return operatorResMap[operator].includes(res);\n};\n/**\n * Match [npm semver](https://docs.npmjs.com/cli/v6/using-npm/semver) version range.\n *\n * @param version Version number to match\n * @param range Range pattern for version\n * @returns `true` if the version number is within the range, `false` otherwise.\n *\n * @example\n * ```\n * satisfies('1.1.0', '^1.0.0'); // return true\n * satisfies('1.1.0', '~1.0.0'); // return false\n * ```\n */\nexport const satisfies = (version, range) => {\n    // if no range operator then \"=\"\n    const m = range.match(/^([<>=~^]+)/);\n    const op = m ? m[1] : '=';\n    // if gt/lt/eq then operator compare\n    if (op !== '^' && op !== '~')\n        return compare(version, range, op);\n    // else range of either \"~\" or \"^\" is assumed\n    const [v1, v2, v3, , vp] = validateAndParse(version);\n    const [r1, r2, r3, , rp] = validateAndParse(range);\n    const v = [v1, v2, v3];\n    const r = [r1, r2 !== null && r2 !== void 0 ? r2 : 'x', r3 !== null && r3 !== void 0 ? r3 : 'x'];\n    // validate pre-release\n    if (rp) {\n        if (!vp)\n            return false;\n        if (compareSegments(v, r) !== 0)\n            return false;\n        if (compareSegments(vp.split('.'), rp.split('.')) === -1)\n            return false;\n    }\n    // first non-zero number\n    const nonZero = r.findIndex((v) => v !== '0') + 1;\n    // pointer to where segments can be >=\n    const i = op === '~' ? 2 : nonZero > 1 ? nonZero : 1;\n    // before pointer must be equal\n    if (compareSegments(v.slice(0, i), r.slice(0, i)) !== 0)\n        return false;\n    // after pointer must be >=\n    if (compareSegments(v.slice(i), r.slice(i)) === -1)\n        return false;\n    return true;\n};\nconst semver = /^[v^~<>=]*?(\\d+)(?:\\.([x*]|\\d+)(?:\\.([x*]|\\d+)(?:\\.([x*]|\\d+))?(?:-([\\da-z\\-]+(?:\\.[\\da-z\\-]+)*))?(?:\\+[\\da-z\\-]+(?:\\.[\\da-z\\-]+)*)?)?)?$/i;\nconst validateAndParse = (version) => {\n    if (typeof version !== 'string') {\n        throw new TypeError('Invalid argument expected string');\n    }\n    const match = version.match(semver);\n    if (!match) {\n        throw new Error(`Invalid argument not valid semver ('${version}' received)`);\n    }\n    match.shift();\n    return match;\n};\nconst isWildcard = (s) => s === '*' || s === 'x' || s === 'X';\nconst tryParse = (v) => {\n    const n = parseInt(v, 10);\n    return isNaN(n) ? v : n;\n};\nconst forceType = (a, b) => typeof a !== typeof b ? [String(a), String(b)] : [a, b];\nconst compareStrings = (a, b) => {\n    if (isWildcard(a) || isWildcard(b))\n        return 0;\n    const [ap, bp] = forceType(tryParse(a), tryParse(b));\n    if (ap > bp)\n        return 1;\n    if (ap < bp)\n        return -1;\n    return 0;\n};\nconst compareSegments = (a, b) => {\n    for (let i = 0; i < Math.max(a.length, b.length); i++) {\n        const r = compareStrings(a[i] || '0', b[i] || '0');\n        if (r !== 0)\n            return r;\n    }\n    return 0;\n};\nconst operatorResMap = {\n    '>': [1],\n    '>=': [0, 1],\n    '=': [0],\n    '<=': [-1, 0],\n    '<': [-1],\n};\nconst allowedOperators = Object.keys(operatorResMap);\nconst assertValidOperator = (op) => {\n    if (typeof op !== 'string') {\n        throw new TypeError(`Invalid operator type, expected string but got ${typeof op}`);\n    }\n    if (allowedOperators.indexOf(op) === -1) {\n        throw new Error(`Invalid operator, expected one of ${allowedOperators.join('|')}`);\n    }\n};\n//# sourceMappingURL=index.js.map","// Define the custom behavior of the page\nimport { documentReady } from \"./mixin\";\nimport { compare, validate } from \"compare-versions\";\n\nimport \"../styles/pydata-sphinx-theme.scss\";\n\n/*******************************************************************************\n * Theme interaction\n */\n\nvar prefersDark = window.matchMedia(\"(prefers-color-scheme: dark)\");\n\n/**\n * set the the body theme to the one specified by the user browser\n *\n * @param {event} e\n */\nfunction autoTheme(e) {\n  document.documentElement.dataset.theme = prefersDark.matches\n    ? \"dark\"\n    : \"light\";\n}\n\n/**\n * Set the theme using the specified mode.\n * It can be one of [\"auto\", \"dark\", \"light\"]\n *\n * @param {str} mode\n */\nfunction setTheme(mode) {\n  if (mode !== \"light\" && mode !== \"dark\" && mode !== \"auto\") {\n    console.error(`Got invalid theme mode: ${mode}. Resetting to auto.`);\n    mode = \"auto\";\n  }\n\n  // get the theme\n  var colorScheme = prefersDark.matches ? \"dark\" : \"light\";\n  document.documentElement.dataset.mode = mode;\n  var theme = mode == \"auto\" ? colorScheme : mode;\n  document.documentElement.dataset.theme = theme;\n  // TODO: remove this line after Bootstrap upgrade\n  // v5.3 has a colors mode: https://getbootstrap.com/docs/5.3/customize/color-modes/\n  document.querySelectorAll(\".dropdown-menu\").forEach((el) => {\n    if (theme === \"dark\") {\n      el.classList.add(\"dropdown-menu-dark\");\n    } else {\n      el.classList.remove(\"dropdown-menu-dark\");\n    }\n  });\n\n  // save mode and theme\n  localStorage.setItem(\"mode\", mode);\n  localStorage.setItem(\"theme\", theme);\n  console.log(`[PST]: Changed to ${mode} mode using the ${theme} theme.`);\n\n  // add a listener if set on auto\n  prefersDark.onchange = mode == \"auto\" ? autoTheme : \"\";\n}\n\n/**\n * Change the theme option order so that clicking on the btn is always a change\n * from \"auto\"\n */\nfunction cycleMode() {\n  const defaultMode = document.documentElement.dataset.defaultMode || \"auto\";\n  const currentMode = localStorage.getItem(\"mode\") || defaultMode;\n\n  var loopArray = (arr, current) => {\n    var nextPosition = arr.indexOf(current) + 1;\n    if (nextPosition === arr.length) {\n      nextPosition = 0;\n    }\n    return arr[nextPosition];\n  };\n\n  // make sure the next theme after auto is always a change\n  var modeList = prefersDark.matches\n    ? [\"auto\", \"light\", \"dark\"]\n    : [\"auto\", \"dark\", \"light\"];\n  var newMode = loopArray(modeList, currentMode);\n  setTheme(newMode);\n}\n\n/**\n * add the theme listener on the btns of the navbar\n */\nfunction addModeListener() {\n  // the theme was set a first time using the initial mini-script\n  // running setMode will ensure the use of the dark mode if auto is selected\n  setTheme(document.documentElement.dataset.mode);\n\n  // Attach event handlers for toggling themes colors\n  document.querySelectorAll(\".theme-switch-button\").forEach((el) => {\n    el.addEventListener(\"click\", cycleMode);\n  });\n}\n\n/*******************************************************************************\n * TOC interactivity\n */\n\n/**\n * TOC sidebar - add \"active\" class to parent list\n *\n * Bootstrap's scrollspy adds the active class to the <a> link,\n * but for the automatic collapsing we need this on the parent list item.\n *\n * The event is triggered on \"window\" (and not the nav item as documented),\n * see https://github.com/twbs/bootstrap/issues/20086\n */\nfunction addTOCInteractivity() {\n  window.addEventListener(\"activate.bs.scrollspy\", function () {\n    const navLinks = document.querySelectorAll(\".bd-toc-nav a\");\n\n    navLinks.forEach((navLink) => {\n      navLink.parentElement.classList.remove(\"active\");\n    });\n\n    const activeNavLinks = document.querySelectorAll(\".bd-toc-nav a.active\");\n    activeNavLinks.forEach((navLink) => {\n      navLink.parentElement.classList.add(\"active\");\n    });\n  });\n}\n\n/*******************************************************************************\n * Scroll\n */\n\n/**\n * Navigation sidebar scrolling to active page\n */\nfunction scrollToActive() {\n  // If the docs nav doesn't exist, do nothing (e.g., on search page)\n  if (!document.querySelector(\".bd-docs-nav\")) {\n    return;\n  }\n\n  var sidebar = document.querySelector(\"div.bd-sidebar\");\n\n  // Remember the sidebar scroll position between page loads\n  // Inspired on source of revealjs.com\n  let storedScrollTop = parseInt(\n    sessionStorage.getItem(\"sidebar-scroll-top\"),\n    10,\n  );\n\n  if (!isNaN(storedScrollTop)) {\n    // If we've got a saved scroll position, just use that\n    sidebar.scrollTop = storedScrollTop;\n    console.log(\"[PST]: Scrolled sidebar using stored browser position...\");\n  } else {\n    // Otherwise, calculate a position to scroll to based on the lowest `active` link\n    var sidebarNav = document.querySelector(\".bd-docs-nav\");\n    var active_pages = sidebarNav.querySelectorAll(\".active\");\n    if (active_pages.length > 0) {\n      // Use the last active page as the offset since it's the page we're on\n      var latest_active = active_pages[active_pages.length - 1];\n      var offset =\n        latest_active.getBoundingClientRect().y -\n        sidebar.getBoundingClientRect().y;\n      // Only scroll the navbar if the active link is lower than 50% of the page\n      if (latest_active.getBoundingClientRect().y > window.innerHeight * 0.5) {\n        let buffer = 0.25; // Buffer so we have some space above the scrolled item\n        sidebar.scrollTop = offset - sidebar.clientHeight * buffer;\n        console.log(\"[PST]: Scrolled sidebar using last active link...\");\n      }\n    }\n  }\n\n  // Store the sidebar scroll position\n  window.addEventListener(\"beforeunload\", () => {\n    sessionStorage.setItem(\"sidebar-scroll-top\", sidebar.scrollTop);\n  });\n}\n\n/*******************************************************************************\n * Search\n */\n\n/**\n * Find any search forms on the page and return their input element\n */\nvar findSearchInput = () => {\n  let forms = document.querySelectorAll(\"form.bd-search\");\n  if (!forms.length) {\n    // no search form found\n    return;\n  } else {\n    var form;\n    if (forms.length == 1) {\n      // there is exactly one search form (persistent or hidden)\n      form = forms[0];\n    } else {\n      // must be at least one persistent form, use the first persistent one\n      form = document.querySelector(\n        \"div:not(.search-button__search-container) > form.bd-search\",\n      );\n    }\n    return form.querySelector(\"input\");\n  }\n};\n\n/**\n * Activate the search field on the page.\n * - If there is a search field already visible it will be activated.\n * - If not, then a search field will pop up.\n */\nvar toggleSearchField = () => {\n  // Find the search input to highlight\n  let input = findSearchInput();\n\n  // if the input field is the hidden one (the one associated with the\n  // search button) then toggle the button state (to show/hide the field)\n  let searchPopupWrapper = document.querySelector(\".search-button__wrapper\");\n  let hiddenInput = searchPopupWrapper.querySelector(\"input\");\n  if (input === hiddenInput) {\n    searchPopupWrapper.classList.toggle(\"show\");\n  }\n  // when toggling off the search field, remove its focus\n  if (document.activeElement === input) {\n    input.blur();\n  } else {\n    input.focus();\n    input.select();\n    input.scrollIntoView({ block: \"center\" });\n  }\n};\n\n/**\n * Add an event listener for toggleSearchField() for Ctrl/Cmd + K\n */\nvar addEventListenerForSearchKeyboard = () => {\n  window.addEventListener(\n    \"keydown\",\n    (event) => {\n      let input = findSearchInput();\n      // toggle on Ctrl+k or ⌘+k\n      if (\n        // Ignore if shift or alt are pressed\n        !event.shiftKey &&\n        !event.altKey &&\n        // On Mac use ⌘, all other OS use Ctrl\n        (useCommandKey\n          ? event.metaKey && !event.ctrlKey\n          : !event.metaKey && event.ctrlKey) &&\n        // Case-insensitive so the shortcut still works with caps lock\n        /^k$/i.test(event.key)\n      ) {\n        event.preventDefault();\n        toggleSearchField();\n      }\n      // also allow Escape key to hide (but not show) the dynamic search field\n      else if (document.activeElement === input && /Escape/i.test(event.key)) {\n        toggleSearchField();\n      }\n    },\n    true,\n  );\n};\n\n/**\n * If the user is on a Mac, use command (⌘) instead of control (ctrl) key\n *\n * Note: `navigator.platform` is deprecated; however MDN still recommends using\n * it for the one specific use case of detecting whether a keyboard shortcut\n * should use control or command:\n * https://developer.mozilla.org/en-US/docs/Web/API/Navigator/platform#examples\n */\nvar useCommandKey =\n  navigator.platform.indexOf(\"Mac\") === 0 || navigator.platform === \"iPhone\";\n\n/**\n * Change the search hint to `meta key` if we are a Mac\n */\n\nvar changeSearchShortcutKey = () => {\n  let shortcuts = document.querySelectorAll(\".search-button__kbd-shortcut\");\n  if (useCommandKey) {\n    shortcuts.forEach(\n      (f) => (f.querySelector(\"kbd.kbd-shortcut__modifier\").innerText = \"⌘\"),\n    );\n  }\n};\n\n/**\n * Activate callbacks for search button popup\n */\nvar setupSearchButtons = () => {\n  changeSearchShortcutKey();\n  addEventListenerForSearchKeyboard();\n\n  // Add the search button trigger event callback\n  document.querySelectorAll(\".search-button__button\").forEach((btn) => {\n    btn.onclick = toggleSearchField;\n  });\n\n  // Add the search button overlay event callback\n  let overlay = document.querySelector(\".search-button__overlay\");\n  if (overlay) {\n    overlay.onclick = toggleSearchField;\n  }\n};\n\n/*******************************************************************************\n * Version Switcher\n * Note that this depends on two variables existing that are defined in\n * and `html-page-context` hook:\n *\n * - DOCUMENTATION_OPTIONS.pagename\n * - DOCUMENTATION_OPTIONS.theme_switcher_url\n */\n\n/**\n * path component of URL\n */\nvar getCurrentUrlPath = () => {\n  if (DOCUMENTATION_OPTIONS.BUILDER == \"dirhtml\") {\n    return DOCUMENTATION_OPTIONS.pagename == \"index\"\n      ? `/`\n      : `${DOCUMENTATION_OPTIONS.pagename}/`;\n  }\n  return `${DOCUMENTATION_OPTIONS.pagename}.html`;\n};\n\n/**\n * Allow user to dismiss the warning banner about the docs version being dev / old.\n * We store the dismissal date and version, to give us flexibility about making the\n * dismissal last for longer than one browser session, if we decide to do that.\n *\n * @param {event} event the event that trigger the check\n */\nasync function DismissBannerAndStorePref(event) {\n  const banner = document.querySelector(\"#bd-header-version-warning\");\n  banner.remove();\n  const version = DOCUMENTATION_OPTIONS.VERSION;\n  const now = new Date();\n  const banner_pref = JSON.parse(\n    localStorage.getItem(\"pst_banner_pref\") || \"{}\",\n  );\n  console.debug(\n    `[PST] Dismissing the version warning banner on ${version} starting ${now}.`,\n  );\n  banner_pref[version] = now;\n  localStorage.setItem(\"pst_banner_pref\", JSON.stringify(banner_pref));\n}\n\n/**\n * Check if corresponding page path exists in other version of docs\n * and, if so, go there instead of the homepage of the other docs version\n *\n * @param {event} event the event that trigger the check\n */\nasync function checkPageExistsAndRedirect(event) {\n  // ensure we don't follow the initial link\n  event.preventDefault();\n  const currentFilePath = getCurrentUrlPath();\n  let tryUrl = event.currentTarget.getAttribute(\"href\");\n  let otherDocsHomepage = tryUrl.replace(currentFilePath, \"\");\n  try {\n    let head = await fetch(tryUrl, { method: \"HEAD\" });\n    if (head.ok) {\n      location.href = tryUrl; // the page exists, go there\n    } else {\n      location.href = otherDocsHomepage;\n    }\n  } catch (err) {\n    // something went wrong, probably CORS restriction, fallback to other docs homepage\n    location.href = otherDocsHomepage;\n  }\n}\n\n/**\n * Load and parse the version switcher JSON file from an absolute or relative URL.\n *\n * @param {string} url The URL to load version switcher entries from.\n */\nasync function fetchVersionSwitcherJSON(url) {\n  // first check if it's a valid URL\n  try {\n    var result = new URL(url);\n  } catch (err) {\n    if (err instanceof TypeError) {\n      if (!window.location.origin) {\n        // window.location.origin is null for local static sites\n        // (ie. window.location.protocol == 'file:')\n        //\n        // TODO: Fix this to return the static version switcher by working out\n        // how to get the correct path to the switcher JSON file on local static builds\n        return null;\n      }\n      // assume we got a relative path, and fix accordingly. But first, we need to\n      // use `fetch()` to follow redirects so we get the correct final base URL\n      const origin = await fetch(window.location.origin, { method: \"HEAD\" });\n      result = new URL(url, origin.url);\n    } else {\n      // something unexpected happened\n      throw err;\n    }\n  }\n  // load and return the JSON\n  const response = await fetch(result);\n  const data = await response.json();\n  return data;\n}\n\n// Populate the version switcher from the JSON data\nfunction populateVersionSwitcher(data, versionSwitcherBtns) {\n  const currentFilePath = getCurrentUrlPath();\n  versionSwitcherBtns.forEach((btn) => {\n    // Set empty strings by default so that these attributes exist and can be used in CSS selectors\n    btn.dataset[\"activeVersionName\"] = \"\";\n    btn.dataset[\"activeVersion\"] = \"\";\n  });\n  // in case there are multiple entries with the same version string, this helps us\n  // decide which entry's `name` to put on the button itself. Without this, it would\n  // always be the *last* version-matching entry; now it will be either the\n  // version-matching entry that is also marked as `\"preferred\": true`, or if that\n  // doesn't exist: the *first* version-matching entry.\n  data = data.map((entry) => {\n    // does this entry match the version that we're currently building/viewing?\n    entry.match =\n      entry.version == DOCUMENTATION_OPTIONS.theme_switcher_version_match;\n    entry.preferred = entry.preferred || false;\n    // if no custom name specified (e.g., \"latest\"), use version string\n    if (!(\"name\" in entry)) {\n      entry.name = entry.version;\n    }\n    return entry;\n  });\n  const hasMatchingPreferredEntry = data\n    .map((entry) => entry.preferred && entry.match)\n    .some(Boolean);\n  var foundMatch = false;\n  // create links to the corresponding page in the other docs versions\n  data.forEach((entry) => {\n    // create the node\n    const anchor = document.createElement(\"a\");\n    anchor.setAttribute(\n      \"class\",\n      \"dropdown-item list-group-item list-group-item-action py-1\",\n    );\n    anchor.setAttribute(\"href\", `${entry.url}${currentFilePath}`);\n    anchor.setAttribute(\"role\", \"option\");\n    const span = document.createElement(\"span\");\n    span.textContent = `${entry.name}`;\n    anchor.appendChild(span);\n    // Add dataset values for the version and name in case people want\n    // to apply CSS styling based on this information.\n    anchor.dataset[\"versionName\"] = entry.name;\n    anchor.dataset[\"version\"] = entry.version;\n    // replace dropdown button text with the preferred display name of the\n    // currently-viewed version, rather than using sphinx's {{ version }} variable.\n    // also highlight the dropdown entry for the currently-viewed version's entry\n    let matchesAndIsPreferred = hasMatchingPreferredEntry && entry.preferred;\n    let matchesAndIsFirst =\n      !hasMatchingPreferredEntry && !foundMatch && entry.match;\n    if (matchesAndIsPreferred || matchesAndIsFirst) {\n      anchor.classList.add(\"active\");\n      versionSwitcherBtns.forEach((btn) => {\n        btn.innerText = entry.name;\n        btn.dataset[\"activeVersionName\"] = entry.name;\n        btn.dataset[\"activeVersion\"] = entry.version;\n      });\n      foundMatch = true;\n    }\n    // There may be multiple version-switcher elements, e.g. one\n    // in a slide-over panel displayed on smaller screens.\n    document.querySelectorAll(\".version-switcher__menu\").forEach((menu) => {\n      // we need to clone the node for each menu, but onclick attributes are not\n      // preserved by `.cloneNode()` so we add onclick here after cloning.\n      let node = anchor.cloneNode(true);\n      node.onclick = checkPageExistsAndRedirect;\n      // on click, AJAX calls will check if the linked page exists before\n      // trying to redirect, and if not, will redirect to the homepage\n      // for that version of the docs.\n      menu.append(node);\n    });\n  });\n}\n\n/*******************************************************************************\n * Warning banner when viewing non-stable version of the docs.\n */\n\n/**\n * Show a warning banner when viewing a non-stable version of the docs.\n *\n * adapted 2023-06 from https://mne.tools/versionwarning.js, which was\n * originally adapted 2020-05 from https://scikit-learn.org/versionwarning.js\n *\n * @param {Array} data The version data used to populate the switcher menu.\n */\nfunction showVersionWarningBanner(data) {\n  var version = DOCUMENTATION_OPTIONS.VERSION;\n  // figure out what latest stable version is\n  var preferredEntries = data.filter((entry) => entry.preferred);\n  if (preferredEntries.length !== 1) {\n    const howMany = preferredEntries.length == 0 ? \"No\" : \"Multiple\";\n    console.log(\n      `[PST] ${howMany} versions marked \"preferred\" found in versions JSON, ignoring.`,\n    );\n    return;\n  }\n  const preferredVersion = preferredEntries[0].version;\n  const preferredURL = preferredEntries[0].url;\n  // if already on preferred version, nothing to do\n  const versionsAreComparable = validate(version) && validate(preferredVersion);\n  if (versionsAreComparable && compare(version, preferredVersion, \"=\")) {\n    console.log(\n      \"This is the prefered version of the docs, not showing the warning banner.\",\n    );\n    return;\n  }\n  // check if banner has been dismissed recently\n  const dismiss_date_str = JSON.parse(\n    localStorage.getItem(\"pst_banner_pref\") || \"{}\",\n  )[version];\n  if (dismiss_date_str != null) {\n    const dismiss_date = new Date(dismiss_date_str);\n    const now = new Date();\n    const milliseconds_in_a_day = 24 * 60 * 60 * 1000;\n    const days_passed = (now - dismiss_date) / milliseconds_in_a_day;\n    const timeout_in_days = 14;\n    if (days_passed < timeout_in_days) {\n      console.info(\n        `[PST] Suppressing version warning banner; was dismissed ${Math.floor(days_passed)} day(s) ago`,\n      );\n      return;\n    }\n  }\n\n  // now construct the warning banner\n  const banner = document.querySelector(\"#bd-header-version-warning\");\n  const middle = document.createElement(\"div\");\n  const inner = document.createElement(\"div\");\n  const bold = document.createElement(\"strong\");\n  const button = document.createElement(\"a\");\n  const close_btn = document.createElement(\"a\");\n  // these classes exist since pydata-sphinx-theme v0.10.0\n  // the init class is used for animation\n  middle.classList = \"bd-header-announcement__content  ms-auto me-auto\";\n  inner.classList = \"sidebar-message\";\n  button.classList =\n    \"btn text-wrap font-weight-bold ms-3 my-1 align-baseline pst-button-link-to-stable-version\";\n  button.href = `${preferredURL}${getCurrentUrlPath()}`;\n  button.innerText = \"Switch to stable version\";\n  button.onclick = checkPageExistsAndRedirect;\n  close_btn.classList = \"ms-3 my-1 align-baseline\";\n  const close_x = document.createElement(\"i\");\n  close_btn.append(close_x);\n  close_x.classList = \"fa-solid fa-xmark\";\n  close_btn.onclick = DismissBannerAndStorePref;\n  // add the version-dependent text\n  inner.innerText = \"This is documentation for \";\n  const isDev =\n    version.includes(\"dev\") ||\n    version.includes(\"rc\") ||\n    version.includes(\"pre\");\n  const newerThanPreferred =\n    versionsAreComparable && compare(version, preferredVersion, \">\");\n  if (isDev || newerThanPreferred) {\n    bold.innerText = \"an unstable development version\";\n  } else if (versionsAreComparable && compare(version, preferredVersion, \"<\")) {\n    bold.innerText = `an old version (${version})`;\n  } else if (!version) {\n    bold.innerText = \"an unknown version\"; // e.g., an empty string\n  } else {\n    bold.innerText = `version ${version}`;\n  }\n  banner.appendChild(middle);\n  banner.append(close_btn);\n  middle.appendChild(inner);\n  inner.appendChild(bold);\n  inner.appendChild(document.createTextNode(\".\"));\n  inner.appendChild(button);\n  banner.classList.remove(\"d-none\");\n}\n\n/*******************************************************************************\n * MutationObserver to move the ReadTheDocs button\n */\n\n/**\n * intercept the RTD flyout and place it in the rtd-footer-container if existing\n * if not it stays where on top of the page\n */\nfunction initRTDObserver() {\n  const mutatedCallback = (mutationList, observer) => {\n    mutationList.forEach((mutation) => {\n      // Check whether the mutation is for RTD, which will have a specific structure\n      if (mutation.addedNodes.length === 0) {\n        return;\n      }\n      if (mutation.addedNodes[0].data === undefined) {\n        return;\n      }\n      if (mutation.addedNodes[0].data.search(\"Inserted RTD Footer\") != -1) {\n        mutation.addedNodes.forEach((node) => {\n          document.getElementById(\"rtd-footer-container\").append(node);\n        });\n      }\n    });\n  };\n\n  const observer = new MutationObserver(mutatedCallback);\n  const config = { childList: true };\n  observer.observe(document.body, config);\n}\n\nasync function fetchAndUseVersions() {\n  // fetch the JSON version data (only once), then use it to populate the version\n  // switcher and maybe show the version warning bar\n  var versionSwitcherBtns = document.querySelectorAll(\n    \".version-switcher__button\",\n  );\n  const hasSwitcherMenu = versionSwitcherBtns.length > 0;\n  const hasVersionsJSON = DOCUMENTATION_OPTIONS.hasOwnProperty(\n    \"theme_switcher_json_url\",\n  );\n  const wantsWarningBanner = DOCUMENTATION_OPTIONS.show_version_warning_banner;\n\n  if (hasVersionsJSON && (hasSwitcherMenu || wantsWarningBanner)) {\n    const data = await fetchVersionSwitcherJSON(\n      DOCUMENTATION_OPTIONS.theme_switcher_json_url,\n    );\n    // TODO: remove the `if(data)` once the `return null` is fixed within fetchVersionSwitcherJSON.\n    // We don't really want the switcher and warning bar to silently not work.\n    if (data) {\n      populateVersionSwitcher(data, versionSwitcherBtns);\n      if (wantsWarningBanner) {\n        showVersionWarningBanner(data);\n      }\n    }\n  }\n}\n\n/*******************************************************************************\n * Add keyboard functionality to mobile sidebars.\n *\n * Wire up the hamburger-style buttons using the click event which (on buttons)\n * handles both mouse clicks and the space and enter keys.\n */\nfunction setupMobileSidebarKeyboardHandlers() {\n  // These are hidden checkboxes at the top of the page whose :checked property\n  // allows the mobile sidebars to be hidden or revealed via CSS.\n  const primaryToggle = document.getElementById(\"pst-primary-sidebar-checkbox\");\n  const secondaryToggle = document.getElementById(\n    \"pst-secondary-sidebar-checkbox\",\n  );\n  const primarySidebar = document.querySelector(\".bd-sidebar-primary\");\n  const secondarySidebar = document.querySelector(\".bd-sidebar-secondary\");\n\n  // Toggle buttons -\n  //\n  // These are the hamburger-style buttons in the header nav bar. When the user\n  // clicks, the button transmits the click to the hidden checkboxes used by the\n  // CSS to control whether the sidebar is open or closed.\n  const primaryClickTransmitter = document.querySelector(\".primary-toggle\");\n  const secondaryClickTransmitter = document.querySelector(\".secondary-toggle\");\n  [\n    [primaryClickTransmitter, primaryToggle, primarySidebar],\n    [secondaryClickTransmitter, secondaryToggle, secondarySidebar],\n  ].forEach(([clickTransmitter, toggle, sidebar]) => {\n    if (!clickTransmitter) {\n      return;\n    }\n    clickTransmitter.addEventListener(\"click\", (event) => {\n      event.preventDefault();\n      event.stopPropagation();\n      toggle.checked = !toggle.checked;\n\n      // If we are opening the sidebar, move focus to the first focusable item\n      // in the sidebar\n      if (toggle.checked) {\n        // Note: this selector is not exhaustive, and we may need to update it\n        // in the future\n        const tabStop = sidebar.querySelector(\"a, button\");\n        // use setTimeout because you cannot move focus synchronously during a\n        // click in the handler for the click event\n        setTimeout(() => tabStop.focus(), 100);\n      }\n    });\n  });\n\n  // Escape key -\n  //\n  // When sidebar is open, user should be able to press escape key to close the\n  // sidebar.\n  [\n    [primarySidebar, primaryToggle, primaryClickTransmitter],\n    [secondarySidebar, secondaryToggle, secondaryClickTransmitter],\n  ].forEach(([sidebar, toggle, transmitter]) => {\n    if (!sidebar) {\n      return;\n    }\n    sidebar.addEventListener(\"keydown\", (event) => {\n      if (event.key === \"Escape\") {\n        event.preventDefault();\n        event.stopPropagation();\n        toggle.checked = false;\n        transmitter.focus();\n      }\n    });\n  });\n\n  // When the <label> overlay is clicked to close the sidebar, return focus to\n  // the opener button in the nav bar.\n  [\n    [primaryToggle, primaryClickTransmitter],\n    [secondaryToggle, secondaryClickTransmitter],\n  ].forEach(([toggle, transmitter]) => {\n    toggle.addEventListener(\"change\", (event) => {\n      if (!event.currentTarget.checked) {\n        transmitter.focus();\n      }\n    });\n  });\n}\n\n/**\n * When the page loads, or the window resizes, or descendant nodes are added or\n * removed from the main element, check all code blocks and Jupyter notebook\n * outputs, and for each one that has scrollable overflow, set tabIndex = 0.\n */\nfunction addTabStopsToScrollableElements() {\n  const updateTabStops = () => {\n    document\n      .querySelectorAll(\n        \"pre, \" + // code blocks\n          \".nboutput > .output_area, \" + // NBSphinx notebook output\n          \".cell_output > .output, \" + // Myst-NB\n          \".jp-RenderedHTMLCommon\", // ipywidgets\n      )\n      .forEach((el) => {\n        el.tabIndex =\n          el.scrollWidth > el.clientWidth || el.scrollHeight > el.clientHeight\n            ? 0\n            : -1;\n      });\n  };\n  const debouncedUpdateTabStops = debounce(updateTabStops, 300);\n\n  // On window resize\n  window.addEventListener(\"resize\", debouncedUpdateTabStops);\n\n  // The following MutationObserver is for ipywidgets, which take some time to\n  // finish loading and rendering on the page (so even after the \"load\" event is\n  // fired, they still have not finished rendering). Would be nice to replace\n  // the MutationObserver if there is a way to hook into the ipywidgets code to\n  // know when it is done.\n  const mainObserver = new MutationObserver(debouncedUpdateTabStops);\n\n  // On descendant nodes added/removed from main element\n  mainObserver.observe(document.getElementById(\"main-content\"), {\n    subtree: true,\n    childList: true,\n  });\n\n  // On page load (when this function gets called)\n  updateTabStops();\n}\nfunction debounce(callback, wait) {\n  let timeoutId = null;\n  return (...args) => {\n    clearTimeout(timeoutId);\n    timeoutId = setTimeout(() => {\n      callback(...args);\n    }, wait);\n  };\n}\n\n/*******************************************************************************\n * Announcement banner - fetch and load remote HTML\n */\nasync function setupAnnouncementBanner() {\n  const banner = document.querySelector(\".bd-header-announcement\");\n  const { pstAnnouncementUrl } = banner ? banner.dataset : null;\n\n  if (!pstAnnouncementUrl) {\n    return;\n  }\n\n  try {\n    const response = await fetch(pstAnnouncementUrl);\n    if (!response.ok) {\n      throw new Error(\n        `[PST]: HTTP response status not ok: ${response.status} ${response.statusText}`,\n      );\n    }\n    const data = await response.text();\n    if (data.length === 0) {\n      console.log(`[PST]: Empty announcement at: ${pstAnnouncementUrl}`);\n      return;\n    }\n    banner.innerHTML = `<div class=\"bd-header-announcement__content\">${data}</div>`;\n    banner.classList.remove(\"d-none\");\n  } catch (_error) {\n    console.log(`[PST]: Failed to load announcement at: ${pstAnnouncementUrl}`);\n    console.error(_error);\n  }\n}\n\n/*******************************************************************************\n * Reveal (and animate) the banners (version warning, announcement) together\n */\nasync function fetchRevealBannersTogether() {\n  // Wait until finished fetching and loading banners\n  await 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diff --git a/dev/docs/html/_static/scripts/sphinx-book-theme.js b/dev/docs/html/_static/scripts/sphinx-book-theme.js
new file mode 100644
index 0000000..481ba0c
--- /dev/null
+++ b/dev/docs/html/_static/scripts/sphinx-book-theme.js
@@ -0,0 +1,2 @@
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diff --git a/dev/docs/html/_static/scripts/sphinx-book-theme.js.map b/dev/docs/html/_static/scripts/sphinx-book-theme.js.map
new file mode 100644
index 0000000..2cc145a
--- /dev/null
+++ b/dev/docs/html/_static/scripts/sphinx-book-theme.js.map
@@ -0,0 +1 @@
+{"version":3,"file":"scripts/sphinx-book-theme.js","mappings":"mBASA,IAAIA,EAAsBC,IACG,WAAvBC,SAASC,WACXF,IACSC,SAASE,iBAClBF,SAASE,iBAAiB,mBAAoBH,GAE9CC,SAASG,YAAY,sBAAsB,WACd,YAAvBH,SAASC,YAA0BF,GACzC,GACF,EAkKFK,OAAOC,aAtCY,KACjB,IAAIC,EAAQN,SAASO,cAAc,cACrBD,EAAME,mBAGPC,UAAUC,SAAS,wBAC9BJ,EAAMK,mBACJ,WACA,iDAIJC,WAAW,EA2BbR,OAAOS,iBAzJgB,KACrB,IAAIC,EACDd,SAASe,mBAAoD,OAA/Bf,SAASe,mBACvCf,SAASgB,yBAC6B,OAArChB,SAASgB,wBACb,IAAIC,EAASjB,SAASkB,gBACjBJ,GAQHK,QAAQC,IAAI,8BACRpB,SAASqB,eACXrB,SAASqB,iBACArB,SAASsB,sBAClBtB,SAASsB,yBAXXH,QAAQC,IAAI,+BACRH,EAAOM,kBACTN,EAAOM,oBACEN,EAAOO,yBAChBP,EAAOO,0BASX,EA0IF1B,GAzHkB,KAChB,IAAI2B,EAAgB,GACpB,IAwCIC,EAAc,IAAIC,sBAxCA,CAACC,EAASC,KAE9BD,EAAQE,SAASC,IACf,GAAIA,EAAMC,eAERP,EAAcQ,KAAKF,EAAMG,aAGzB,IAAK,IAAIC,EAAK,EAAGA,EAAKV,EAAcW,OAAQD,IAC1C,GAAIV,EAAcU,KAAQJ,EAAMG,OAAQ,CACtCT,EAAcY,OAAOF,EAAI,GACzB,KACF,CAEJ,IAIEV,EAAcW,OAAS,EACzBpC,SAASO,cAAc,4BAA4BE,UAAU6B,IAAI,QAEjEtC,SACGO,cAAc,4BACdE,UAAU8B,OAAO,OACtB,GAYY,CAEZC,WAAY,qBAaVC,EAAiB,GATG,CACtB,aACA,WACA,SACA,iBACA,aACA,UACA,UAGcX,SAASK,IAEvBM,EAAeR,KAEX,IAAIE,IACJ,QAAQA,IACR,IAAIA,EAAGO,QAAQ,IAAK,OACpB,QAAQP,EAAGO,QAAQ,IAAK,OAE3B,IAEH1C,SAAS2C,iBAAiBF,EAAeG,KAAK,OAAOd,SAASK,IAC5DT,EAAYmB,QAAQV,EAAG,IAIJ,IAAIR,sBA1CO,CAACC,EAASC,KAEpCD,EAAQ,GAAGkB,mBAAmBC,EAAI,EACpC/C,SAASgD,KAAKvC,UAAU6B,IAAI,YAE5BtC,SAASgD,KAAKvC,UAAU8B,OAAO,WACjC,IAqCaM,QAAQ7C,SAASO,cAAc,4BAA4B,IAmD5ET,GA1BA,WACE,IAAImD,EAAkB,CACpB,0BACA,aACA,qBACA,sBACA,wBACA,qBACA,qBACA,cACAL,KAAK,KACP5C,SAAS2C,iBAAiBM,GAAiBnB,SAASK,IAClDA,EAAG1B,UAAU6B,IAAI,UAAU,GAE/B,G","sources":["webpack://sphinx_book_theme/./src/sphinx_book_theme/assets/scripts/index.js"],"sourcesContent":["// Import CSS variables\n// ref: https://css-tricks.com/getting-javascript-to-talk-to-css-and-sass/\nimport \"../styles/index.scss\";\n\n/**\n * A helper function to load scripts when the DOM is loaded.\n * This waits for everything to be on the page first before running, since\n * some functionality doesn't behave properly until everything is ready.\n */\nvar sbRunWhenDOMLoaded = (cb) => {\n  if (document.readyState != \"loading\") {\n    cb();\n  } else if (document.addEventListener) {\n    document.addEventListener(\"DOMContentLoaded\", cb);\n  } else {\n    document.attachEvent(\"onreadystatechange\", function () {\n      if (document.readyState == \"complete\") cb();\n    });\n  }\n};\n\n/**\n * Toggle full-screen with button\n *\n * There are some browser-specific hacks in here:\n * - Safari requires a `webkit` prefix, so this uses conditionals to check for that\n *   ref: https://developer.mozilla.org/en-US/docs/Web/API/Fullscreen_API\n */\nvar toggleFullScreen = () => {\n  var isInFullScreen =\n    (document.fullscreenElement && document.fullscreenElement !== null) ||\n    (document.webkitFullscreenElement &&\n      document.webkitFullscreenElement !== null);\n  let docElm = document.documentElement;\n  if (!isInFullScreen) {\n    console.log(\"[SBT]: Entering full screen\");\n    if (docElm.requestFullscreen) {\n      docElm.requestFullscreen();\n    } else if (docElm.webkitRequestFullscreen) {\n      docElm.webkitRequestFullscreen();\n    }\n  } else {\n    console.log(\"[SBT]: Exiting full screen\");\n    if (document.exitFullscreen) {\n      document.exitFullscreen();\n    } else if (document.webkitExitFullscreen) {\n      document.webkitExitFullscreen();\n    }\n  }\n};\n\n/**\n * Manage scrolling behavior. This is primarily two things:\n *\n * 1. Hide the Table of Contents any time sidebar content is on the screen.\n *\n * This will be triggered any time a sidebar item enters or exits the screen.\n * It adds/removes items from an array if they have entered the screen, and\n * removes them when they exit the screen. It hides the TOC if anything is\n * on-screen.\n *\n * ref: https://developer.mozilla.org/en-US/docs/Web/API/Intersection_Observer_API\n *\n * 2. Add a `scrolled` class to <body> to trigger CSS changes.\n */\nvar initTocHide = () => {\n  var onScreenItems = [];\n  let hideTocCallback = (entries, observer) => {\n    // Check whether any sidebar item is displayed\n    entries.forEach((entry) => {\n      if (entry.isIntersecting) {\n        // If an element just came on screen, add it our list\n        onScreenItems.push(entry.target);\n      } else {\n        // Otherwise, if it's in our list then remove it\n        for (let ii = 0; ii < onScreenItems.length; ii++) {\n          if (onScreenItems[ii] === entry.target) {\n            onScreenItems.splice(ii, 1);\n            break;\n          }\n        }\n      }\n    });\n\n    // Hide the TOC if any margin content is displayed on the screen\n    if (onScreenItems.length > 0) {\n      document.querySelector(\"div.bd-sidebar-secondary\").classList.add(\"hide\");\n    } else {\n      document\n        .querySelector(\"div.bd-sidebar-secondary\")\n        .classList.remove(\"hide\");\n    }\n  };\n  let manageScrolledClassOnBody = (entries, observer) => {\n    // The pixel is at the top, so if we're < 0 that it means we've scrolled\n    if (entries[0].boundingClientRect.y < 0) {\n      document.body.classList.add(\"scrolled\");\n    } else {\n      document.body.classList.remove(\"scrolled\");\n    }\n  };\n\n  // Set up the intersection observer to watch all margin content\n  let options = {\n    // Trigger callback when the top of a margin item is 1/3 up the screen\n    rootMargin: \"0px 0px -33% 0px\",\n  };\n  let tocObserver = new IntersectionObserver(hideTocCallback, options);\n  // TODO: deprecate popout after v0.5.0\n  const selectorClasses = [\n    \"marginnote\",\n    \"sidenote\",\n    \"margin\",\n    \"margin-caption\",\n    \"full-width\",\n    \"sidebar\",\n    \"popout\",\n  ];\n  let marginSelector = [];\n  selectorClasses.forEach((ii) => {\n    // Use three permutations of each class name because `tag_` and `_` used to be supported\n    marginSelector.push(\n      ...[\n        `.${ii}`,\n        `.tag_${ii}`,\n        `.${ii.replace(\"-\", \"_\")}`,\n        `.tag_${ii.replace(\"-\", \"_\")}`,\n      ],\n    );\n  });\n  document.querySelectorAll(marginSelector.join(\", \")).forEach((ii) => {\n    tocObserver.observe(ii);\n  });\n\n  // Set up the observer to check if we've scrolled from top of page\n  let scrollObserver = new IntersectionObserver(manageScrolledClassOnBody);\n  scrollObserver.observe(document.querySelector(\".sbt-scroll-pixel-helper\"));\n};\n\n/**\n * Activate Thebe with a custom button click.\n */\nvar initThebeSBT = () => {\n  var title = document.querySelector(\"section h1\");\n  var sibling = title.nextElementSibling;\n  // If the next element after the title isn't a thebe button, add one now.\n  // That way it is initiatlized when thebe is first-clicked and isn't re-added after.\n  if (!sibling.classList.contains(\"thebe-launch-button\")) {\n    title.insertAdjacentHTML(\n      \"afterend\",\n      \"<button class='thebe-launch-button'></button>\",\n    );\n  }\n  // This function is provided by sphinx-thebe\n  initThebe();\n};\n\n/**\n * Add no print class to certain DOM elements\n */\n\nfunction addNoPrint() {\n  var noPrintSelector = [\n    \".bd-header-announcement\",\n    \".bd-header\",\n    \".bd-header-article\",\n    \".bd-sidebar-primary\",\n    \".bd-sidebar-secondary\",\n    \".bd-footer-article\",\n    \".bd-footer-content\",\n    \".bd-footer\",\n  ].join(\",\");\n  document.querySelectorAll(noPrintSelector).forEach((ii) => {\n    ii.classList.add(\"noprint\");\n  });\n}\n\n/**\n * Set up callback functions for UI click actions\n */\nwindow.initThebeSBT = initThebeSBT;\nwindow.toggleFullScreen = toggleFullScreen;\n\n/**\n * Set up functions to load when the DOM is ready\n */\nsbRunWhenDOMLoaded(initTocHide);\nsbRunWhenDOMLoaded(addNoPrint);\n"],"names":["sbRunWhenDOMLoaded","cb","document","readyState","addEventListener","attachEvent","window","initThebeSBT","title","querySelector","nextElementSibling","classList","contains","insertAdjacentHTML","initThebe","toggleFullScreen","isInFullScreen","fullscreenElement","webkitFullscreenElement","docElm","documentElement","console","log","exitFullscreen","webkitExitFullscreen","requestFullscreen","webkitRequestFullscreen","onScreenItems","tocObserver","IntersectionObserver","entries","observer","forEach","entry","isIntersecting","push","target","ii","length","splice","add","remove","rootMargin","marginSelector","replace","querySelectorAll","join","observe","boundingClientRect","y","body","noPrintSelector"],"sourceRoot":""}
\ No newline at end of file
diff --git a/dev/docs/html/_static/searchtools.js b/dev/docs/html/_static/searchtools.js
new file mode 100644
index 0000000..92da3f8
--- /dev/null
+++ b/dev/docs/html/_static/searchtools.js
@@ -0,0 +1,619 @@
+/*
+ * searchtools.js
+ * ~~~~~~~~~~~~~~~~
+ *
+ * Sphinx JavaScript utilities for the full-text search.
+ *
+ * :copyright: Copyright 2007-2024 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+"use strict";
+
+/**
+ * Simple result scoring code.
+ */
+if (typeof Scorer === "undefined") {
+  var Scorer = {
+    // Implement the following function to further tweak the score for each result
+    // The function takes a result array [docname, title, anchor, descr, score, filename]
+    // and returns the new score.
+    /*
+    score: result => {
+      const [docname, title, anchor, descr, score, filename] = result
+      return score
+    },
+    */
+
+    // query matches the full name of an object
+    objNameMatch: 11,
+    // or matches in the last dotted part of the object name
+    objPartialMatch: 6,
+    // Additive scores depending on the priority of the object
+    objPrio: {
+      0: 15, // used to be importantResults
+      1: 5, // used to be objectResults
+      2: -5, // used to be unimportantResults
+    },
+    //  Used when the priority is not in the mapping.
+    objPrioDefault: 0,
+
+    // query found in title
+    title: 15,
+    partialTitle: 7,
+    // query found in terms
+    term: 5,
+    partialTerm: 2,
+  };
+}
+
+const _removeChildren = (element) => {
+  while (element && element.lastChild) element.removeChild(element.lastChild);
+};
+
+/**
+ * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping
+ */
+const _escapeRegExp = (string) =>
+  string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string
+
+const _displayItem = (item, searchTerms, highlightTerms) => {
+  const docBuilder = DOCUMENTATION_OPTIONS.BUILDER;
+  const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX;
+  const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX;
+  const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY;
+  const contentRoot = document.documentElement.dataset.content_root;
+
+  const [docName, title, anchor, descr, score, _filename] = item;
+
+  let listItem = document.createElement("li");
+  let requestUrl;
+  let linkUrl;
+  if (docBuilder === "dirhtml") {
+    // dirhtml builder
+    let dirname = docName + "/";
+    if (dirname.match(/\/index\/$/))
+      dirname = dirname.substring(0, dirname.length - 6);
+    else if (dirname === "index/") dirname = "";
+    requestUrl = contentRoot + dirname;
+    linkUrl = requestUrl;
+  } else {
+    // normal html builders
+    requestUrl = contentRoot + docName + docFileSuffix;
+    linkUrl = docName + docLinkSuffix;
+  }
+  let linkEl = listItem.appendChild(document.createElement("a"));
+  linkEl.href = linkUrl + anchor;
+  linkEl.dataset.score = score;
+  linkEl.innerHTML = title;
+  if (descr) {
+    listItem.appendChild(document.createElement("span")).innerHTML =
+      " (" + descr + ")";
+    // highlight search terms in the description
+    if (SPHINX_HIGHLIGHT_ENABLED)  // set in sphinx_highlight.js
+      highlightTerms.forEach((term) => _highlightText(listItem, term, "highlighted"));
+  }
+  else if (showSearchSummary)
+    fetch(requestUrl)
+      .then((responseData) => responseData.text())
+      .then((data) => {
+        if (data)
+          listItem.appendChild(
+            Search.makeSearchSummary(data, searchTerms, anchor)
+          );
+        // highlight search terms in the summary
+        if (SPHINX_HIGHLIGHT_ENABLED)  // set in sphinx_highlight.js
+          highlightTerms.forEach((term) => _highlightText(listItem, term, "highlighted"));
+      });
+  Search.output.appendChild(listItem);
+};
+const _finishSearch = (resultCount) => {
+  Search.stopPulse();
+  Search.title.innerText = _("Search Results");
+  if (!resultCount)
+    Search.status.innerText = Documentation.gettext(
+      "Your search did not match any documents. Please make sure that all words are spelled correctly and that you've selected enough categories."
+    );
+  else
+    Search.status.innerText = _(
+      "Search finished, found ${resultCount} page(s) matching the search query."
+    ).replace('${resultCount}', resultCount);
+};
+const _displayNextItem = (
+  results,
+  resultCount,
+  searchTerms,
+  highlightTerms,
+) => {
+  // results left, load the summary and display it
+  // this is intended to be dynamic (don't sub resultsCount)
+  if (results.length) {
+    _displayItem(results.pop(), searchTerms, highlightTerms);
+    setTimeout(
+      () => _displayNextItem(results, resultCount, searchTerms, highlightTerms),
+      5
+    );
+  }
+  // search finished, update title and status message
+  else _finishSearch(resultCount);
+};
+// Helper function used by query() to order search results.
+// Each input is an array of [docname, title, anchor, descr, score, filename].
+// Order the results by score (in opposite order of appearance, since the
+// `_displayNextItem` function uses pop() to retrieve items) and then alphabetically.
+const _orderResultsByScoreThenName = (a, b) => {
+  const leftScore = a[4];
+  const rightScore = b[4];
+  if (leftScore === rightScore) {
+    // same score: sort alphabetically
+    const leftTitle = a[1].toLowerCase();
+    const rightTitle = b[1].toLowerCase();
+    if (leftTitle === rightTitle) return 0;
+    return leftTitle > rightTitle ? -1 : 1; // inverted is intentional
+  }
+  return leftScore > rightScore ? 1 : -1;
+};
+
+/**
+ * Default splitQuery function. Can be overridden in ``sphinx.search`` with a
+ * custom function per language.
+ *
+ * The regular expression works by splitting the string on consecutive characters
+ * that are not Unicode letters, numbers, underscores, or emoji characters.
+ * This is the same as ``\W+`` in Python, preserving the surrogate pair area.
+ */
+if (typeof splitQuery === "undefined") {
+  var splitQuery = (query) => query
+      .split(/[^\p{Letter}\p{Number}_\p{Emoji_Presentation}]+/gu)
+      .filter(term => term)  // remove remaining empty strings
+}
+
+/**
+ * Search Module
+ */
+const Search = {
+  _index: null,
+  _queued_query: null,
+  _pulse_status: -1,
+
+  htmlToText: (htmlString, anchor) => {
+    const htmlElement = new DOMParser().parseFromString(htmlString, 'text/html');
+    for (const removalQuery of [".headerlinks", "script", "style"]) {
+      htmlElement.querySelectorAll(removalQuery).forEach((el) => { el.remove() });
+    }
+    if (anchor) {
+      const anchorContent = htmlElement.querySelector(`[role="main"] ${anchor}`);
+      if (anchorContent) return anchorContent.textContent;
+
+      console.warn(
+        `Anchored content block not found. Sphinx search tries to obtain it via DOM query '[role=main] ${anchor}'. Check your theme or template.`
+      );
+    }
+
+    // if anchor not specified or not found, fall back to main content
+    const docContent = htmlElement.querySelector('[role="main"]');
+    if (docContent) return docContent.textContent;
+
+    console.warn(
+      "Content block not found. Sphinx search tries to obtain it via DOM query '[role=main]'. Check your theme or template."
+    );
+    return "";
+  },
+
+  init: () => {
+    const query = new URLSearchParams(window.location.search).get("q");
+    document
+      .querySelectorAll('input[name="q"]')
+      .forEach((el) => (el.value = query));
+    if (query) Search.performSearch(query);
+  },
+
+  loadIndex: (url) =>
+    (document.body.appendChild(document.createElement("script")).src = url),
+
+  setIndex: (index) => {
+    Search._index = index;
+    if (Search._queued_query !== null) {
+      const query = Search._queued_query;
+      Search._queued_query = null;
+      Search.query(query);
+    }
+  },
+
+  hasIndex: () => Search._index !== null,
+
+  deferQuery: (query) => (Search._queued_query = query),
+
+  stopPulse: () => (Search._pulse_status = -1),
+
+  startPulse: () => {
+    if (Search._pulse_status >= 0) return;
+
+    const pulse = () => {
+      Search._pulse_status = (Search._pulse_status + 1) % 4;
+      Search.dots.innerText = ".".repeat(Search._pulse_status);
+      if (Search._pulse_status >= 0) window.setTimeout(pulse, 500);
+    };
+    pulse();
+  },
+
+  /**
+   * perform a search for something (or wait until index is loaded)
+   */
+  performSearch: (query) => {
+    // create the required interface elements
+    const searchText = document.createElement("h2");
+    searchText.textContent = _("Searching");
+    const searchSummary = document.createElement("p");
+    searchSummary.classList.add("search-summary");
+    searchSummary.innerText = "";
+    const searchList = document.createElement("ul");
+    searchList.classList.add("search");
+
+    const out = document.getElementById("search-results");
+    Search.title = out.appendChild(searchText);
+    Search.dots = Search.title.appendChild(document.createElement("span"));
+    Search.status = out.appendChild(searchSummary);
+    Search.output = out.appendChild(searchList);
+
+    const searchProgress = document.getElementById("search-progress");
+    // Some themes don't use the search progress node
+    if (searchProgress) {
+      searchProgress.innerText = _("Preparing search...");
+    }
+    Search.startPulse();
+
+    // index already loaded, the browser was quick!
+    if (Search.hasIndex()) Search.query(query);
+    else Search.deferQuery(query);
+  },
+
+  _parseQuery: (query) => {
+    // stem the search terms and add them to the correct list
+    const stemmer = new Stemmer();
+    const searchTerms = new Set();
+    const excludedTerms = new Set();
+    const highlightTerms = new Set();
+    const objectTerms = new Set(splitQuery(query.toLowerCase().trim()));
+    splitQuery(query.trim()).forEach((queryTerm) => {
+      const queryTermLower = queryTerm.toLowerCase();
+
+      // maybe skip this "word"
+      // stopwords array is from language_data.js
+      if (
+        stopwords.indexOf(queryTermLower) !== -1 ||
+        queryTerm.match(/^\d+$/)
+      )
+        return;
+
+      // stem the word
+      let word = stemmer.stemWord(queryTermLower);
+      // select the correct list
+      if (word[0] === "-") excludedTerms.add(word.substr(1));
+      else {
+        searchTerms.add(word);
+        highlightTerms.add(queryTermLower);
+      }
+    });
+
+    if (SPHINX_HIGHLIGHT_ENABLED) {  // set in sphinx_highlight.js
+      localStorage.setItem("sphinx_highlight_terms", [...highlightTerms].join(" "))
+    }
+
+    // console.debug("SEARCH: searching for:");
+    // console.info("required: ", [...searchTerms]);
+    // console.info("excluded: ", [...excludedTerms]);
+
+    return [query, searchTerms, excludedTerms, highlightTerms, objectTerms];
+  },
+
+  /**
+   * execute search (requires search index to be loaded)
+   */
+  _performSearch: (query, searchTerms, excludedTerms, highlightTerms, objectTerms) => {
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const titles = Search._index.titles;
+    const allTitles = Search._index.alltitles;
+    const indexEntries = Search._index.indexentries;
+
+    // Collect multiple result groups to be sorted separately and then ordered.
+    // Each is an array of [docname, title, anchor, descr, score, filename].
+    const normalResults = [];
+    const nonMainIndexResults = [];
+
+    _removeChildren(document.getElementById("search-progress"));
+
+    const queryLower = query.toLowerCase().trim();
+    for (const [title, foundTitles] of Object.entries(allTitles)) {
+      if (title.toLowerCase().trim().includes(queryLower) && (queryLower.length >= title.length/2)) {
+        for (const [file, id] of foundTitles) {
+          let score = Math.round(100 * queryLower.length / title.length)
+          normalResults.push([
+            docNames[file],
+            titles[file] !== title ? `${titles[file]} > ${title}` : title,
+            id !== null ? "#" + id : "",
+            null,
+            score,
+            filenames[file],
+          ]);
+        }
+      }
+    }
+
+    // search for explicit entries in index directives
+    for (const [entry, foundEntries] of Object.entries(indexEntries)) {
+      if (entry.includes(queryLower) && (queryLower.length >= entry.length/2)) {
+        for (const [file, id, isMain] of foundEntries) {
+          const score = Math.round(100 * queryLower.length / entry.length);
+          const result = [
+            docNames[file],
+            titles[file],
+            id ? "#" + id : "",
+            null,
+            score,
+            filenames[file],
+          ];
+          if (isMain) {
+            normalResults.push(result);
+          } else {
+            nonMainIndexResults.push(result);
+          }
+        }
+      }
+    }
+
+    // lookup as object
+    objectTerms.forEach((term) =>
+      normalResults.push(...Search.performObjectSearch(term, objectTerms))
+    );
+
+    // lookup as search terms in fulltext
+    normalResults.push(...Search.performTermsSearch(searchTerms, excludedTerms));
+
+    // let the scorer override scores with a custom scoring function
+    if (Scorer.score) {
+      normalResults.forEach((item) => (item[4] = Scorer.score(item)));
+      nonMainIndexResults.forEach((item) => (item[4] = Scorer.score(item)));
+    }
+
+    // Sort each group of results by score and then alphabetically by name.
+    normalResults.sort(_orderResultsByScoreThenName);
+    nonMainIndexResults.sort(_orderResultsByScoreThenName);
+
+    // Combine the result groups in (reverse) order.
+    // Non-main index entries are typically arbitrary cross-references,
+    // so display them after other results.
+    let results = [...nonMainIndexResults, ...normalResults];
+
+    // remove duplicate search results
+    // note the reversing of results, so that in the case of duplicates, the highest-scoring entry is kept
+    let seen = new Set();
+    results = results.reverse().reduce((acc, result) => {
+      let resultStr = result.slice(0, 4).concat([result[5]]).map(v => String(v)).join(',');
+      if (!seen.has(resultStr)) {
+        acc.push(result);
+        seen.add(resultStr);
+      }
+      return acc;
+    }, []);
+
+    return results.reverse();
+  },
+
+  query: (query) => {
+    const [searchQuery, searchTerms, excludedTerms, highlightTerms, objectTerms] = Search._parseQuery(query);
+    const results = Search._performSearch(searchQuery, searchTerms, excludedTerms, highlightTerms, objectTerms);
+
+    // for debugging
+    //Search.lastresults = results.slice();  // a copy
+    // console.info("search results:", Search.lastresults);
+
+    // print the results
+    _displayNextItem(results, results.length, searchTerms, highlightTerms);
+  },
+
+  /**
+   * search for object names
+   */
+  performObjectSearch: (object, objectTerms) => {
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const objects = Search._index.objects;
+    const objNames = Search._index.objnames;
+    const titles = Search._index.titles;
+
+    const results = [];
+
+    const objectSearchCallback = (prefix, match) => {
+      const name = match[4]
+      const fullname = (prefix ? prefix + "." : "") + name;
+      const fullnameLower = fullname.toLowerCase();
+      if (fullnameLower.indexOf(object) < 0) return;
+
+      let score = 0;
+      const parts = fullnameLower.split(".");
+
+      // check for different match types: exact matches of full name or
+      // "last name" (i.e. last dotted part)
+      if (fullnameLower === object || parts.slice(-1)[0] === object)
+        score += Scorer.objNameMatch;
+      else if (parts.slice(-1)[0].indexOf(object) > -1)
+        score += Scorer.objPartialMatch; // matches in last name
+
+      const objName = objNames[match[1]][2];
+      const title = titles[match[0]];
+
+      // If more than one term searched for, we require other words to be
+      // found in the name/title/description
+      const otherTerms = new Set(objectTerms);
+      otherTerms.delete(object);
+      if (otherTerms.size > 0) {
+        const haystack = `${prefix} ${name} ${objName} ${title}`.toLowerCase();
+        if (
+          [...otherTerms].some((otherTerm) => haystack.indexOf(otherTerm) < 0)
+        )
+          return;
+      }
+
+      let anchor = match[3];
+      if (anchor === "") anchor = fullname;
+      else if (anchor === "-") anchor = objNames[match[1]][1] + "-" + fullname;
+
+      const descr = objName + _(", in ") + title;
+
+      // add custom score for some objects according to scorer
+      if (Scorer.objPrio.hasOwnProperty(match[2]))
+        score += Scorer.objPrio[match[2]];
+      else score += Scorer.objPrioDefault;
+
+      results.push([
+        docNames[match[0]],
+        fullname,
+        "#" + anchor,
+        descr,
+        score,
+        filenames[match[0]],
+      ]);
+    };
+    Object.keys(objects).forEach((prefix) =>
+      objects[prefix].forEach((array) =>
+        objectSearchCallback(prefix, array)
+      )
+    );
+    return results;
+  },
+
+  /**
+   * search for full-text terms in the index
+   */
+  performTermsSearch: (searchTerms, excludedTerms) => {
+    // prepare search
+    const terms = Search._index.terms;
+    const titleTerms = Search._index.titleterms;
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const titles = Search._index.titles;
+
+    const scoreMap = new Map();
+    const fileMap = new Map();
+
+    // perform the search on the required terms
+    searchTerms.forEach((word) => {
+      const files = [];
+      const arr = [
+        { files: terms[word], score: Scorer.term },
+        { files: titleTerms[word], score: Scorer.title },
+      ];
+      // add support for partial matches
+      if (word.length > 2) {
+        const escapedWord = _escapeRegExp(word);
+        if (!terms.hasOwnProperty(word)) {
+          Object.keys(terms).forEach((term) => {
+            if (term.match(escapedWord))
+              arr.push({ files: terms[term], score: Scorer.partialTerm });
+          });
+        }
+        if (!titleTerms.hasOwnProperty(word)) {
+          Object.keys(titleTerms).forEach((term) => {
+            if (term.match(escapedWord))
+              arr.push({ files: titleTerms[term], score: Scorer.partialTitle });
+          });
+        }
+      }
+
+      // no match but word was a required one
+      if (arr.every((record) => record.files === undefined)) return;
+
+      // found search word in contents
+      arr.forEach((record) => {
+        if (record.files === undefined) return;
+
+        let recordFiles = record.files;
+        if (recordFiles.length === undefined) recordFiles = [recordFiles];
+        files.push(...recordFiles);
+
+        // set score for the word in each file
+        recordFiles.forEach((file) => {
+          if (!scoreMap.has(file)) scoreMap.set(file, {});
+          scoreMap.get(file)[word] = record.score;
+        });
+      });
+
+      // create the mapping
+      files.forEach((file) => {
+        if (!fileMap.has(file)) fileMap.set(file, [word]);
+        else if (fileMap.get(file).indexOf(word) === -1) fileMap.get(file).push(word);
+      });
+    });
+
+    // now check if the files don't contain excluded terms
+    const results = [];
+    for (const [file, wordList] of fileMap) {
+      // check if all requirements are matched
+
+      // as search terms with length < 3 are discarded
+      const filteredTermCount = [...searchTerms].filter(
+        (term) => term.length > 2
+      ).length;
+      if (
+        wordList.length !== searchTerms.size &&
+        wordList.length !== filteredTermCount
+      )
+        continue;
+
+      // ensure that none of the excluded terms is in the search result
+      if (
+        [...excludedTerms].some(
+          (term) =>
+            terms[term] === file ||
+            titleTerms[term] === file ||
+            (terms[term] || []).includes(file) ||
+            (titleTerms[term] || []).includes(file)
+        )
+      )
+        break;
+
+      // select one (max) score for the file.
+      const score = Math.max(...wordList.map((w) => scoreMap.get(file)[w]));
+      // add result to the result list
+      results.push([
+        docNames[file],
+        titles[file],
+        "",
+        null,
+        score,
+        filenames[file],
+      ]);
+    }
+    return results;
+  },
+
+  /**
+   * helper function to return a node containing the
+   * search summary for a given text. keywords is a list
+   * of stemmed words.
+   */
+  makeSearchSummary: (htmlText, keywords, anchor) => {
+    const text = Search.htmlToText(htmlText, anchor);
+    if (text === "") return null;
+
+    const textLower = text.toLowerCase();
+    const actualStartPosition = [...keywords]
+      .map((k) => textLower.indexOf(k.toLowerCase()))
+      .filter((i) => i > -1)
+      .slice(-1)[0];
+    const startWithContext = Math.max(actualStartPosition - 120, 0);
+
+    const top = startWithContext === 0 ? "" : "...";
+    const tail = startWithContext + 240 < text.length ? "..." : "";
+
+    let summary = document.createElement("p");
+    summary.classList.add("context");
+    summary.textContent = top + text.substr(startWithContext, 240).trim() + tail;
+
+    return summary;
+  },
+};
+
+_ready(Search.init);
diff --git a/dev/docs/html/_static/sphinx_highlight.js b/dev/docs/html/_static/sphinx_highlight.js
new file mode 100644
index 0000000..8a96c69
--- /dev/null
+++ b/dev/docs/html/_static/sphinx_highlight.js
@@ -0,0 +1,154 @@
+/* Highlighting utilities for Sphinx HTML documentation. */
+"use strict";
+
+const SPHINX_HIGHLIGHT_ENABLED = true
+
+/**
+ * highlight a given string on a node by wrapping it in
+ * span elements with the given class name.
+ */
+const _highlight = (node, addItems, text, className) => {
+  if (node.nodeType === Node.TEXT_NODE) {
+    const val = node.nodeValue;
+    const parent = node.parentNode;
+    const pos = val.toLowerCase().indexOf(text);
+    if (
+      pos >= 0 &&
+      !parent.classList.contains(className) &&
+      !parent.classList.contains("nohighlight")
+    ) {
+      let span;
+
+      const closestNode = parent.closest("body, svg, foreignObject");
+      const isInSVG = closestNode && closestNode.matches("svg");
+      if (isInSVG) {
+        span = document.createElementNS("http://www.w3.org/2000/svg", "tspan");
+      } else {
+        span = document.createElement("span");
+        span.classList.add(className);
+      }
+
+      span.appendChild(document.createTextNode(val.substr(pos, text.length)));
+      const rest = document.createTextNode(val.substr(pos + text.length));
+      parent.insertBefore(
+        span,
+        parent.insertBefore(
+          rest,
+          node.nextSibling
+        )
+      );
+      node.nodeValue = val.substr(0, pos);
+      /* There may be more occurrences of search term in this node. So call this
+       * function recursively on the remaining fragment.
+       */
+      _highlight(rest, addItems, text, className);
+
+      if (isInSVG) {
+        const rect = document.createElementNS(
+          "http://www.w3.org/2000/svg",
+          "rect"
+        );
+        const bbox = parent.getBBox();
+        rect.x.baseVal.value = bbox.x;
+        rect.y.baseVal.value = bbox.y;
+        rect.width.baseVal.value = bbox.width;
+        rect.height.baseVal.value = bbox.height;
+        rect.setAttribute("class", className);
+        addItems.push({ parent: parent, target: rect });
+      }
+    }
+  } else if (node.matches && !node.matches("button, select, textarea")) {
+    node.childNodes.forEach((el) => _highlight(el, addItems, text, className));
+  }
+};
+const _highlightText = (thisNode, text, className) => {
+  let addItems = [];
+  _highlight(thisNode, addItems, text, className);
+  addItems.forEach((obj) =>
+    obj.parent.insertAdjacentElement("beforebegin", obj.target)
+  );
+};
+
+/**
+ * Small JavaScript module for the documentation.
+ */
+const SphinxHighlight = {
+
+  /**
+   * highlight the search words provided in localstorage in the text
+   */
+  highlightSearchWords: () => {
+    if (!SPHINX_HIGHLIGHT_ENABLED) return;  // bail if no highlight
+
+    // get and clear terms from localstorage
+    const url = new URL(window.location);
+    const highlight =
+        localStorage.getItem("sphinx_highlight_terms")
+        || url.searchParams.get("highlight")
+        || "";
+    localStorage.removeItem("sphinx_highlight_terms")
+    url.searchParams.delete("highlight");
+    window.history.replaceState({}, "", url);
+
+    // get individual terms from highlight string
+    const terms = highlight.toLowerCase().split(/\s+/).filter(x => x);
+    if (terms.length === 0) return; // nothing to do
+
+    // There should never be more than one element matching "div.body"
+    const divBody = document.querySelectorAll("div.body");
+    const body = divBody.length ? divBody[0] : document.querySelector("body");
+    window.setTimeout(() => {
+      terms.forEach((term) => _highlightText(body, term, "highlighted"));
+    }, 10);
+
+    const searchBox = document.getElementById("searchbox");
+    if (searchBox === null) return;
+    searchBox.appendChild(
+      document
+        .createRange()
+        .createContextualFragment(
+          '<p class="highlight-link">' +
+            '<a href="javascript:SphinxHighlight.hideSearchWords()">' +
+            _("Hide Search Matches") +
+            "</a></p>"
+        )
+    );
+  },
+
+  /**
+   * helper function to hide the search marks again
+   */
+  hideSearchWords: () => {
+    document
+      .querySelectorAll("#searchbox .highlight-link")
+      .forEach((el) => el.remove());
+    document
+      .querySelectorAll("span.highlighted")
+      .forEach((el) => el.classList.remove("highlighted"));
+    localStorage.removeItem("sphinx_highlight_terms")
+  },
+
+  initEscapeListener: () => {
+    // only install a listener if it is really needed
+    if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) return;
+
+    document.addEventListener("keydown", (event) => {
+      // bail for input elements
+      if (BLACKLISTED_KEY_CONTROL_ELEMENTS.has(document.activeElement.tagName)) return;
+      // bail with special keys
+      if (event.shiftKey || event.altKey || event.ctrlKey || event.metaKey) return;
+      if (DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS && (event.key === "Escape")) {
+        SphinxHighlight.hideSearchWords();
+        event.preventDefault();
+      }
+    });
+  },
+};
+
+_ready(() => {
+  /* Do not call highlightSearchWords() when we are on the search page.
+   * It will highlight words from the *previous* search query.
+   */
+  if (typeof Search === "undefined") SphinxHighlight.highlightSearchWords();
+  SphinxHighlight.initEscapeListener();
+});
diff --git a/dev/docs/html/_static/stationary_synthetic_FDC.png b/dev/docs/html/_static/stationary_synthetic_FDC.png
new file mode 100644
index 0000000..822cfef
Binary files /dev/null and b/dev/docs/html/_static/stationary_synthetic_FDC.png differ
diff --git a/dev/docs/html/_static/styles/bootstrap.css b/dev/docs/html/_static/styles/bootstrap.css
new file mode 100644
index 0000000..5ac709f
--- /dev/null
+++ b/dev/docs/html/_static/styles/bootstrap.css
@@ -0,0 +1,6 @@
+/*!
+ * Bootstrap  v5.3.3 (https://getbootstrap.com/)
+ * Copyright 2011-2024 The Bootstrap Authors
+ * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)
+ */:root,[data-bs-theme=light]{--bs-blue:#0d6efd;--bs-indigo:#6610f2;--bs-purple:#6f42c1;--bs-pink:#d63384;--bs-red:#dc3545;--bs-orange:#fd7e14;--bs-yellow:#ffc107;--bs-green:#198754;--bs-teal:#20c997;--bs-cyan:#0dcaf0;--bs-black:#000;--bs-white:#fff;--bs-gray:#6c757d;--bs-gray-dark:#343a40;--bs-gray-100:#f8f9fa;--bs-gray-200:#e9ecef;--bs-gray-300:#dee2e6;--bs-gray-400:#ced4da;--bs-gray-500:#adb5bd;--bs-gray-600:#6c757d;--bs-gray-700:#495057;--bs-gray-800:#343a40;--bs-gray-900:#212529;--bs-primary:#0d6efd;--bs-secondary:#6c757d;--bs-success:#198754;--bs-info:#0dcaf0;--bs-warning:#ffc107;--bs-danger:#dc3545;--bs-light:#f8f9fa;--bs-dark:#212529;--bs-primary-rgb:13,110,253;--bs-secondary-rgb:108,117,125;--bs-success-rgb:25,135,84;--bs-info-rgb:13,202,240;--bs-warning-rgb:255,193,7;--bs-danger-rgb:220,53,69;--bs-light-rgb:248,249,250;--bs-dark-rgb:33,37,41;--bs-primary-text-emphasis:#052c65;--bs-secondary-text-emphasis:#2b2f32;--bs-success-text-emphasis:#0a3622;--bs-info-text-emphasis:#055160;--bs-warning-text-emphasis:#664d03;--bs-danger-text-emphasis:#58151c;--bs-light-text-emphasis:#495057;--bs-dark-text-emphasis:#495057;--bs-primary-bg-subtle:#cfe2ff;--bs-secondary-bg-subtle:#e2e3e5;--bs-success-bg-subtle:#d1e7dd;--bs-info-bg-subtle:#cff4fc;--bs-warning-bg-subtle:#fff3cd;--bs-danger-bg-subtle:#f8d7da;--bs-light-bg-subtle:#fcfcfd;--bs-dark-bg-subtle:#ced4da;--bs-primary-border-subtle:#9ec5fe;--bs-secondary-border-subtle:#c4c8cb;--bs-success-border-subtle:#a3cfbb;--bs-info-border-subtle:#9eeaf9;--bs-warning-border-subtle:#ffe69c;--bs-danger-border-subtle:#f1aeb5;--bs-light-border-subtle:#e9ecef;--bs-dark-border-subtle:#adb5bd;--bs-white-rgb:255,255,255;--bs-black-rgb:0,0,0;--bs-font-sans-serif:system-ui,-apple-system,"Segoe UI",Roboto,"Helvetica Neue","Noto Sans","Liberation Sans",Arial,sans-serif,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol","Noto Color Emoji";--bs-font-monospace:SFMono-Regular,Menlo,Monaco,Consolas,"Liberation Mono","Courier 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.offcanvas{background-color:transparent!important;border:0!important;flex-grow:1;height:auto!important;position:static;transform:none!important;transition:none;visibility:visible!important;width:auto!important;z-index:auto}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;flex-grow:0;overflow-y:visible;padding:0}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color:hsla(0,0%,100%,.55);--bs-navbar-hover-color:hsla(0,0%,100%,.75);--bs-navbar-disabled-color:hsla(0,0%,100%,.25);--bs-navbar-active-color:#fff;--bs-navbar-brand-color:#fff;--bs-navbar-brand-hover-color:#fff;--bs-navbar-toggler-border-color:hsla(0,0%,100%,.1)}.navbar-dark,.navbar[data-bs-theme=dark],[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg:url("data:image/svg+xml;charset=utf-8,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3E%3Cpath stroke='rgba(255, 255, 255, 0.55)' stroke-linecap='round' stroke-miterlimit='10' 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.nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-left:calc(var(--bs-card-cap-padding-x)*-.5);margin-right:calc(var(--bs-card-cap-padding-x)*-.5)}.card-img-overlay{border-radius:var(--bs-card-inner-border-radius);bottom:0;left:0;padding:var(--bs-card-img-overlay-padding);position:absolute;right:0;top:0}.card-img,.card-img-bottom,.card-img-top{width:100%}.card-img,.card-img-top{border-top-left-radius:var(--bs-card-inner-border-radius);border-top-right-radius:var(--bs-card-inner-border-radius)}.card-img,.card-img-bottom{border-bottom-left-radius:var(--bs-card-inner-border-radius);border-bottom-right-radius:var(--bs-card-inner-border-radius)}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width:540px){.card-group{display:flex;flex-flow:row wrap}.card-group>.card{flex:1 0 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var(--bs-accordion-border-color);color:var(--bs-accordion-active-color)}.accordion-button:not(.collapsed):after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button:after{background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);content:"";flex-shrink:0;height:var(--bs-accordion-btn-icon-width);margin-left:auto;transition:var(--bs-accordion-btn-icon-transition);width:var(--bs-accordion-btn-icon-width)}@media(prefers-reduced-motion:reduce){.accordion-button:after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{box-shadow:var(--bs-accordion-btn-focus-box-shadow);outline:0;z-index:3}.accordion-header{margin-bottom:0}.accordion-item{background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color);color:var(--bs-accordion-color)}.accordion-item:first-of-type{border-top-left-radius:var(--bs-accordion-border-radius);border-top-right-radius:var(--bs-accordion-border-radius)}.accordion-item:first-of-type>.accordion-header .accordion-button{border-top-left-radius:var(--bs-accordion-inner-border-radius);border-top-right-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:not(:first-of-type){border-top:0}.accordion-item:last-of-type{border-bottom-left-radius:var(--bs-accordion-border-radius);border-bottom-right-radius:var(--bs-accordion-border-radius)}.accordion-item:last-of-type>.accordion-header .accordion-button.collapsed{border-bottom-left-radius:var(--bs-accordion-inner-border-radius);border-bottom-right-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:last-of-type>.accordion-collapse{border-bottom-left-radius:var(--bs-accordion-border-radius);border-bottom-right-radius:var(--bs-accordion-border-radius)}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush>.accordion-item{border-left:0;border-radius:0;border-right:0}.accordion-flush>.accordion-item:first-child{border-top:0}.accordion-flush>.accordion-item:last-child{border-bottom:0}.accordion-flush>.accordion-item>.accordion-collapse,.accordion-flush>.accordion-item>.accordion-header .accordion-button,.accordion-flush>.accordion-item>.accordion-header .accordion-button.collapsed{border-radius:0}[data-bs-theme=dark] .accordion-button:after{--bs-accordion-btn-icon:url("data:image/svg+xml;charset=utf-8,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3E%3Cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3E%3C/svg%3E");--bs-accordion-btn-active-icon:url("data:image/svg+xml;charset=utf-8,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3E%3Cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3E%3C/svg%3E")}.breadcrumb{--bs-breadcrumb-padding-x:0;--bs-breadcrumb-padding-y:0;--bs-breadcrumb-margin-bottom:1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color:var(--bs-secondary-color);--bs-breadcrumb-item-padding-x:0.5rem;--bs-breadcrumb-item-active-color:var(--bs-secondary-color);background-color:var(--bs-breadcrumb-bg);border-radius:var(--bs-breadcrumb-border-radius);display:flex;flex-wrap:wrap;font-size:var(--bs-breadcrumb-font-size);list-style:none;margin-bottom:var(--bs-breadcrumb-margin-bottom);padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item:before{color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider,"/");float:left;padding-right:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x:0.75rem;--bs-pagination-padding-y:0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color:var(--bs-link-color);--bs-pagination-bg:var(--bs-body-bg);--bs-pagination-border-width:var(--bs-border-width);--bs-pagination-border-color:var(--bs-border-color);--bs-pagination-border-radius:var(--bs-border-radius);--bs-pagination-hover-color:var(--bs-link-hover-color);--bs-pagination-hover-bg:var(--bs-tertiary-bg);--bs-pagination-hover-border-color:var(--bs-border-color);--bs-pagination-focus-color:var(--bs-link-hover-color);--bs-pagination-focus-bg:var(--bs-secondary-bg);--bs-pagination-focus-box-shadow:0 0 0 0.1875rem var(--pst-color-accent);--bs-pagination-active-color:#fff;--bs-pagination-active-bg:#0d6efd;--bs-pagination-active-border-color:#0d6efd;--bs-pagination-disabled-color:var(--bs-secondary-color);--bs-pagination-disabled-bg:var(--bs-secondary-bg);--bs-pagination-disabled-border-color:var(--bs-border-color);display:flex;list-style:none;padding-left:0}.page-link{background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);color:var(--bs-pagination-color);display:block;font-size:var(--bs-pagination-font-size);padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);position:relative;text-decoration:none;transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion:reduce){.page-link{transition:none}}.page-link:hover{background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color);color:var(--bs-pagination-hover-color);z-index:2}.page-link:focus{background-color:var(--bs-pagination-focus-bg);box-shadow:var(--bs-pagination-focus-box-shadow);color:var(--bs-pagination-focus-color);outline:0;z-index:3}.active>.page-link,.page-link.active{background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color);color:var(--bs-pagination-active-color);z-index:3}.disabled>.page-link,.page-link.disabled{background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color);color:var(--bs-pagination-disabled-color);pointer-events:none}.page-item:not(:first-child) .page-link{margin-left:calc(var(--bs-border-width)*-1)}.page-item:first-child .page-link{border-bottom-left-radius:var(--bs-pagination-border-radius);border-top-left-radius:var(--bs-pagination-border-radius)}.page-item:last-child .page-link{border-bottom-right-radius:var(--bs-pagination-border-radius);border-top-right-radius:var(--bs-pagination-border-radius)}.pagination-lg{--bs-pagination-padding-x:1.5rem;--bs-pagination-padding-y:0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius:var(--bs-border-radius-lg)}.pagination-sm{--bs-pagination-padding-x:0.5rem;--bs-pagination-padding-y:0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius:var(--bs-border-radius-sm)}.badge{--bs-badge-padding-x:0.65em;--bs-badge-padding-y:0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight:700;--bs-badge-color:#fff;--bs-badge-border-radius:var(--bs-border-radius);border-radius:var(--bs-badge-border-radius);color:var(--bs-badge-color);display:inline-block;font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);text-align:center;vertical-align:baseline;white-space:nowrap}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg:transparent;--bs-alert-padding-x:1rem;--bs-alert-padding-y:1rem;--bs-alert-margin-bottom:1rem;--bs-alert-color:inherit;--bs-alert-border-color:transparent;--bs-alert-border:var(--bs-border-width) solid var(--bs-alert-border-color);--bs-alert-border-radius:var(--bs-border-radius);--bs-alert-link-color:inherit;background-color:var(--bs-alert-bg);border:var(--bs-alert-border);border-radius:var(--bs-alert-border-radius);color:var(--bs-alert-color);margin-bottom:var(--bs-alert-margin-bottom);padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);position:relative}.alert-heading{color:inherit}.alert-link{color:var(--bs-alert-link-color);font-weight:700}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{padding:1.25rem 1rem;position:absolute;right:0;top:0;z-index:2}.alert-primary{--bs-alert-color:var(--bs-primary-text-emphasis);--bs-alert-bg:var(--bs-primary-bg-subtle);--bs-alert-border-color:var(--bs-primary-border-subtle);--bs-alert-link-color:var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color:var(--bs-secondary-text-emphasis);--bs-alert-bg:var(--bs-secondary-bg-subtle);--bs-alert-border-color:var(--bs-secondary-border-subtle);--bs-alert-link-color:var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color:var(--bs-success-text-emphasis);--bs-alert-bg:var(--bs-success-bg-subtle);--bs-alert-border-color:var(--bs-success-border-subtle);--bs-alert-link-color:var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color:var(--bs-info-text-emphasis);--bs-alert-bg:var(--bs-info-bg-subtle);--bs-alert-border-color:var(--bs-info-border-subtle);--bs-alert-link-color:var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color:var(--bs-warning-text-emphasis);--bs-alert-bg:var(--bs-warning-bg-subtle);--bs-alert-border-color:var(--bs-warning-border-subtle);--bs-alert-link-color:var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color:var(--bs-danger-text-emphasis);--bs-alert-bg:var(--bs-danger-bg-subtle);--bs-alert-border-color:var(--bs-danger-border-subtle);--bs-alert-link-color:var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color:var(--bs-light-text-emphasis);--bs-alert-bg:var(--bs-light-bg-subtle);--bs-alert-border-color:var(--bs-light-border-subtle);--bs-alert-link-color:var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color:var(--bs-dark-text-emphasis);--bs-alert-bg:var(--bs-dark-bg-subtle);--bs-alert-border-color:var(--bs-dark-border-subtle);--bs-alert-link-color:var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:1rem}}.progress,.progress-stacked{--bs-progress-height:1rem;--bs-progress-font-size:0.75rem;--bs-progress-bg:var(--bs-secondary-bg);--bs-progress-border-radius:var(--bs-border-radius);--bs-progress-box-shadow:var(--bs-box-shadow-inset);--bs-progress-bar-color:#fff;--bs-progress-bar-bg:#0d6efd;--bs-progress-bar-transition:width 0.6s ease;background-color:var(--bs-progress-bg);border-radius:var(--bs-progress-border-radius);display:flex;font-size:var(--bs-progress-font-size);height:var(--bs-progress-height);overflow:hidden}.progress-bar{background-color:var(--bs-progress-bar-bg);color:var(--bs-progress-bar-color);display:flex;flex-direction:column;justify-content:center;overflow:hidden;text-align:center;transition:var(--bs-progress-bar-transition);white-space:nowrap}@media(prefers-reduced-motion:reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg,hsla(0,0%,100%,.15) 25%,transparent 0,transparent 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";counter-increment:section}.list-group-item-action{color:var(--bs-list-group-action-color);text-align:inherit;width:100%}.list-group-item-action:focus,.list-group-item-action:hover{background-color:var(--bs-list-group-action-hover-bg);color:var(--bs-list-group-action-hover-color);text-decoration:none;z-index:1}.list-group-item-action:active{background-color:var(--bs-list-group-action-active-bg);color:var(--bs-list-group-action-active-color)}.list-group-item{background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color);color:var(--bs-list-group-color);display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);position:relative;text-decoration:none}.list-group-item:first-child{border-top-left-radius:inherit;border-top-right-radius:inherit}.list-group-item:last-child{border-bottom-left-radius:inherit;border-bottom-right-radius:inherit}.list-group-item.disabled,.list-group-item:disabled{background-color:var(--bs-list-group-disabled-bg);color:var(--bs-list-group-disabled-color);pointer-events:none}.list-group-item.active{background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color);color:var(--bs-list-group-active-color);z-index:2}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{border-top-width:var(--bs-list-group-border-width);margin-top:calc(var(--bs-list-group-border-width)*-1)}.list-group-horizontal{flex-direction:row}.list-group-horizontal>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}@media(min-width:540px){.list-group-horizontal-sm{flex-direction:row}.list-group-horizontal-sm>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-sm>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}}@media(min-width:720px){.list-group-horizontal-md{flex-direction:row}.list-group-horizontal-md>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-md>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal-md>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}}@media(min-width:960px){.list-group-horizontal-lg{flex-direction:row}.list-group-horizontal-lg>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-lg>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}}@media(min-width:1200px){.list-group-horizontal-xl{flex-direction:row}.list-group-horizontal-xl>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-xl>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}}.list-group-flush{border-radius:0}.list-group-flush>.list-group-item{border-width:0 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var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-primary{--bs-list-group-color:var(--bs-primary-text-emphasis);--bs-list-group-bg:var(--bs-primary-bg-subtle);--bs-list-group-border-color:var(--bs-primary-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-primary-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-primary-border-subtle);--bs-list-group-active-color:var(--bs-primary-bg-subtle);--bs-list-group-active-bg:var(--bs-primary-text-emphasis);--bs-list-group-active-border-color:var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color:var(--bs-secondary-text-emphasis);--bs-list-group-bg:var(--bs-secondary-bg-subtle);--bs-list-group-border-color:var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-secondary-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-secondary-border-subtle);--bs-list-group-active-color:var(--bs-secondary-bg-subtle);--bs-list-group-active-bg:var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color:var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color:var(--bs-success-text-emphasis);--bs-list-group-bg:var(--bs-success-bg-subtle);--bs-list-group-border-color:var(--bs-success-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-success-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-success-border-subtle);--bs-list-group-active-color:var(--bs-success-bg-subtle);--bs-list-group-active-bg:var(--bs-success-text-emphasis);--bs-list-group-active-border-color:var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color:var(--bs-info-text-emphasis);--bs-list-group-bg:var(--bs-info-bg-subtle);--bs-list-group-border-color:var(--bs-info-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-info-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-info-border-subtle);--bs-list-group-active-color:var(--bs-info-bg-subtle);--bs-list-group-active-bg:var(--bs-info-text-emphasis);--bs-list-group-active-border-color:var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color:var(--bs-warning-text-emphasis);--bs-list-group-bg:var(--bs-warning-bg-subtle);--bs-list-group-border-color:var(--bs-warning-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-warning-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-warning-border-subtle);--bs-list-group-active-color:var(--bs-warning-bg-subtle);--bs-list-group-active-bg:var(--bs-warning-text-emphasis);--bs-list-group-active-border-color:var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color:var(--bs-danger-text-emphasis);--bs-list-group-bg:var(--bs-danger-bg-subtle);--bs-list-group-border-color:var(--bs-danger-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-danger-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-danger-border-subtle);--bs-list-group-active-color:var(--bs-danger-bg-subtle);--bs-list-group-active-bg:var(--bs-danger-text-emphasis);--bs-list-group-active-border-color:var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color:var(--bs-light-text-emphasis);--bs-list-group-bg:var(--bs-light-bg-subtle);--bs-list-group-border-color:var(--bs-light-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-light-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-light-border-subtle);--bs-list-group-active-color:var(--bs-light-bg-subtle);--bs-list-group-active-bg:var(--bs-light-text-emphasis);--bs-list-group-active-border-color:var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color:var(--bs-dark-text-emphasis);--bs-list-group-bg:var(--bs-dark-bg-subtle);--bs-list-group-border-color:var(--bs-dark-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-dark-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-dark-border-subtle);--bs-list-group-active-color:var(--bs-dark-bg-subtle);--bs-list-group-active-bg:var(--bs-dark-text-emphasis);--bs-list-group-active-border-color:var(--bs-dark-text-emphasis)}.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0)}.dropdown-item:hover,.dropdown-item:focus{color:var(--bs-dropdown-link-hover-color);background-color:var(--bs-dropdown-link-hover-bg)}.dropdown-item.active,.dropdown-item:active{color:var(--bs-dropdown-link-active-color);text-decoration:none;background-color:var(--bs-dropdown-link-active-bg)}.dropdown-item.disabled,.dropdown-item:disabled{color:var(--bs-dropdown-link-disabled-color);pointer-events:none;background-color:rgba(0,0,0,0)}.dropdown-menu.show{display:block}.dropdown-header{display:block;padding:var(--bs-dropdown-header-padding-y) var(--bs-dropdown-header-padding-x);margin-bottom:0;font-size:0.875rem;color:var(--bs-dropdown-header-color);white-space:nowrap}.dropdown-item-text{display:block;padding:var(--bs-dropdown-item-padding-y) var(--bs-dropdown-item-padding-x);color:var(--bs-dropdown-link-color)}.dropdown-menu-dark{--bs-dropdown-color: #dee2e6;--bs-dropdown-bg: #343a40;--bs-dropdown-border-color: var(--bs-border-color-translucent);--bs-dropdown-box-shadow: ;--bs-dropdown-link-color: #dee2e6;--bs-dropdown-link-hover-color: #fff;--bs-dropdown-divider-bg: var(--bs-border-color-translucent);--bs-dropdown-link-hover-bg: var(--pst-color-surface);--bs-dropdown-link-active-color: #fff;--bs-dropdown-link-active-bg: var(--pst-color-surface);--bs-dropdown-link-disabled-color: #adb5bd;--bs-dropdown-header-color: #adb5bd}.btn-group,.btn-group-vertical{position:relative;display:inline-flex;vertical-align:middle}.btn-group>.btn,.btn-group-vertical>.btn{position:relative;flex:1 1 auto}.btn-group>.btn-check:checked+.btn,.btn-group>.btn-check:focus+.btn,.btn-group>.btn:hover,.btn-group>.btn:focus,.btn-group>.btn:active,.btn-group>.btn.active,.btn-group-vertical>.btn-check:checked+.btn,.btn-group-vertical>.btn-check:focus+.btn,.btn-group-vertical>.btn:hover,.btn-group-vertical>.btn:focus,.btn-group-vertical>.btn:active,.btn-group-vertical>.btn.active{z-index:1}.btn-toolbar{display:flex;flex-wrap:wrap;justify-content:flex-start}.btn-toolbar .input-group{width:auto}.btn-group{border-radius:var(--bs-border-radius)}.btn-group>:not(.btn-check:first-child)+.btn,.btn-group>.btn-group:not(:first-child){margin-left:calc(var(--bs-border-width)*-1)}.btn-group>.btn:not(:last-child):not(.dropdown-toggle),.btn-group>.btn.dropdown-toggle-split:first-child,.btn-group>.btn-group:not(:last-child)>.btn{border-top-right-radius:0;border-bottom-right-radius:0}.btn-group>.btn:nth-child(n+3),.btn-group>:not(.btn-check)+.btn,.btn-group>.btn-group:not(:first-child)>.btn{border-top-left-radius:0;border-bottom-left-radius:0}.dropdown-toggle-split{padding-right:.5625rem;padding-left:.5625rem}.dropdown-toggle-split::after,.dropup .dropdown-toggle-split::after,.dropend .dropdown-toggle-split::after{margin-left:0}.dropstart .dropdown-toggle-split::before{margin-right:0}.btn-sm+.dropdown-toggle-split,.btn-group-sm>.btn+.dropdown-toggle-split{padding-right:.375rem;padding-left:.375rem}.btn-lg+.dropdown-toggle-split,.btn-group-lg>.btn+.dropdown-toggle-split{padding-right:.75rem;padding-left:.75rem}.btn-group-vertical{flex-direction:column;align-items:flex-start;justify-content:center}.btn-group-vertical>.btn,.btn-group-vertical>.btn-group{width:100%}.btn-group-vertical>.btn:not(:first-child),.btn-group-vertical>.btn-group:not(:first-child){margin-top:calc(var(--bs-border-width)*-1)}.btn-group-vertical>.btn:not(:last-child):not(.dropdown-toggle),.btn-group-vertical>.btn-group:not(:last-child)>.btn{border-bottom-right-radius:0;border-bottom-left-radius:0}.btn-group-vertical>.btn~.btn,.btn-group-vertical>.btn-group:not(:first-child)>.btn{border-top-left-radius:0;border-top-right-radius:0}.nav{--bs-nav-link-padding-x: 1rem;--bs-nav-link-padding-y: 0.5rem;--bs-nav-link-font-weight: ;--bs-nav-link-color: 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1.25rem;--bs-navbar-brand-color: rgba(var(--bs-emphasis-color-rgb), 1);--bs-navbar-brand-hover-color: rgba(var(--bs-emphasis-color-rgb), 1);--bs-navbar-nav-link-padding-x: 0.5rem;--bs-navbar-toggler-padding-y: 0.25rem;--bs-navbar-toggler-padding-x: 0.75rem;--bs-navbar-toggler-font-size: 1.25rem;--bs-navbar-toggler-icon-bg: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%2833, 37, 41, 0.75%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e\");--bs-navbar-toggler-border-color: rgba(var(--bs-emphasis-color-rgb), 0.15);--bs-navbar-toggler-border-radius: var(--bs-border-radius);--bs-navbar-toggler-focus-width: 0.1875rem;--bs-navbar-toggler-transition: box-shadow 0.15s ease-in-out;position:relative;display:flex;flex-wrap:wrap;align-items:center;justify-content:space-between;padding:var(--bs-navbar-padding-y) 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.offcanvas-body{display:flex;flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;flex-basis:auto}.navbar-expand .navbar-toggler{display:none}.navbar-expand .offcanvas{position:static;z-index:auto;flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;flex-grow:0;padding:0;overflow-y:visible}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color: rgba(255, 255, 255, 0.55);--bs-navbar-hover-color: rgba(255, 255, 255, 0.75);--bs-navbar-disabled-color: rgba(255, 255, 255, 0.25);--bs-navbar-active-color: #fff;--bs-navbar-brand-color: #fff;--bs-navbar-brand-hover-color: #fff;--bs-navbar-toggler-border-color: rgba(255, 255, 255, 0.1);--bs-navbar-toggler-icon-bg: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%28255, 255, 255, 0.55%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e\")}[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%28255, 255, 255, 0.55%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e\")}.card{--bs-card-spacer-y: 1rem;--bs-card-spacer-x: 1rem;--bs-card-title-spacer-y: 0.5rem;--bs-card-title-color: ;--bs-card-subtitle-color: ;--bs-card-border-width: var(--bs-border-width);--bs-card-border-color: var(--bs-border-color-translucent);--bs-card-border-radius: var(--bs-border-radius);--bs-card-box-shadow: ;--bs-card-inner-border-radius: calc(var(--bs-border-radius) - (var(--bs-border-width)));--bs-card-cap-padding-y: 0.5rem;--bs-card-cap-padding-x: 1rem;--bs-card-cap-bg: rgba(var(--bs-body-color-rgb), 0.03);--bs-card-cap-color: ;--bs-card-height: ;--bs-card-color: ;--bs-card-bg: var(--bs-body-bg);--bs-card-img-overlay-padding: 1rem;--bs-card-group-margin: 0.75rem;position:relative;display:flex;flex-direction:column;min-width:0;height:var(--bs-card-height);color:var(--bs-body-color);word-wrap:break-word;background-color:var(--bs-card-bg);background-clip:border-box;border:var(--bs-card-border-width) solid var(--bs-card-border-color);border-radius:var(--bs-card-border-radius)}.card>hr{margin-right:0;margin-left:0}.card>.list-group{border-top:inherit;border-bottom:inherit}.card>.list-group:first-child{border-top-width:0;border-top-left-radius:var(--bs-card-inner-border-radius);border-top-right-radius:var(--bs-card-inner-border-radius)}.card>.list-group:last-child{border-bottom-width:0;border-bottom-right-radius:var(--bs-card-inner-border-radius);border-bottom-left-radius:var(--bs-card-inner-border-radius)}.card>.card-header+.list-group,.card>.list-group+.card-footer{border-top:0}.card-body{flex:1 1 auto;padding:var(--bs-card-spacer-y) var(--bs-card-spacer-x);color:var(--bs-card-color)}.card-title{margin-bottom:var(--bs-card-title-spacer-y);color:var(--bs-card-title-color)}.card-subtitle{margin-top:calc(-0.5*var(--bs-card-title-spacer-y));margin-bottom:0;color:var(--bs-card-subtitle-color)}.card-text:last-child{margin-bottom:0}.card-link+.card-link{margin-left:var(--bs-card-spacer-x)}.card-header{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);margin-bottom:0;color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-bottom:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-header:first-child{border-radius:var(--bs-card-inner-border-radius) var(--bs-card-inner-border-radius) 0 0}.card-footer{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-top:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-footer:last-child{border-radius:0 0 var(--bs-card-inner-border-radius) var(--bs-card-inner-border-radius)}.card-header-tabs{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-bottom:calc(-1*var(--bs-card-cap-padding-y));margin-left:calc(-0.5*var(--bs-card-cap-padding-x));border-bottom:0}.card-header-tabs .nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-left:calc(-0.5*var(--bs-card-cap-padding-x))}.card-img-overlay{position:absolute;top:0;right:0;bottom:0;left:0;padding:var(--bs-card-img-overlay-padding);border-radius:var(--bs-card-inner-border-radius)}.card-img,.card-img-top,.card-img-bottom{width:100%}.card-img,.card-img-top{border-top-left-radius:var(--bs-card-inner-border-radius);border-top-right-radius:var(--bs-card-inner-border-radius)}.card-img,.card-img-bottom{border-bottom-right-radius:var(--bs-card-inner-border-radius);border-bottom-left-radius:var(--bs-card-inner-border-radius)}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width: 540px){.card-group{display:flex;flex-flow:row wrap}.card-group>.card{flex:1 0 0%;margin-bottom:0}.card-group>.card+.card{margin-left:0;border-left:0}.card-group>.card:not(:last-child){border-top-right-radius:0;border-bottom-right-radius:0}.card-group>.card:not(:last-child) .card-img-top,.card-group>.card:not(:last-child) .card-header{border-top-right-radius:0}.card-group>.card:not(:last-child) .card-img-bottom,.card-group>.card:not(:last-child) .card-footer{border-bottom-right-radius:0}.card-group>.card:not(:first-child){border-top-left-radius:0;border-bottom-left-radius:0}.card-group>.card:not(:first-child) .card-img-top,.card-group>.card:not(:first-child) .card-header{border-top-left-radius:0}.card-group>.card:not(:first-child) .card-img-bottom,.card-group>.card:not(:first-child) .card-footer{border-bottom-left-radius:0}}.accordion{--bs-accordion-color: var(--bs-body-color);--bs-accordion-bg: var(--bs-body-bg);--bs-accordion-transition: color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, border-radius 0.15s ease;--bs-accordion-border-color: var(--bs-border-color);--bs-accordion-border-width: var(--bs-border-width);--bs-accordion-border-radius: var(--bs-border-radius);--bs-accordion-inner-border-radius: calc(var(--bs-border-radius) - (var(--bs-border-width)));--bs-accordion-btn-padding-x: 1.25rem;--bs-accordion-btn-padding-y: 1rem;--bs-accordion-btn-color: var(--bs-body-color);--bs-accordion-btn-bg: var(--bs-accordion-bg);--bs-accordion-btn-icon: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='none' stroke='%23212529' stroke-linecap='round' stroke-linejoin='round'%3e%3cpath d='M2 5L8 11L14 5'/%3e%3c/svg%3e\");--bs-accordion-btn-icon-width: 1.25rem;--bs-accordion-btn-icon-transform: rotate(-180deg);--bs-accordion-btn-icon-transition: transform 0.2s ease-in-out;--bs-accordion-btn-active-icon: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='none' stroke='%23052c65' stroke-linecap='round' stroke-linejoin='round'%3e%3cpath d='M2 5L8 11L14 5'/%3e%3c/svg%3e\");--bs-accordion-btn-focus-box-shadow: 0 0 0 0.1875rem var(--pst-color-accent);--bs-accordion-body-padding-x: 1.25rem;--bs-accordion-body-padding-y: 1rem;--bs-accordion-active-color: var(--bs-primary-text-emphasis);--bs-accordion-active-bg: var(--bs-primary-bg-subtle)}.accordion-button{position:relative;display:flex;align-items:center;width:100%;padding:var(--bs-accordion-btn-padding-y) var(--bs-accordion-btn-padding-x);font-size:1rem;color:var(--bs-accordion-btn-color);text-align:left;background-color:var(--bs-accordion-btn-bg);border:0;border-radius:0;overflow-anchor:none;transition:var(--bs-accordion-transition)}@media(prefers-reduced-motion: reduce){.accordion-button{transition:none}}.accordion-button:not(.collapsed){color:var(--bs-accordion-active-color);background-color:var(--bs-accordion-active-bg);box-shadow:inset 0 calc(-1*var(--bs-accordion-border-width)) 0 var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:\"\";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:first-of-type{border-top-left-radius:var(--bs-accordion-border-radius);border-top-right-radius:var(--bs-accordion-border-radius)}.accordion-item:first-of-type>.accordion-header .accordion-button{border-top-left-radius:var(--bs-accordion-inner-border-radius);border-top-right-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:not(:first-of-type){border-top:0}.accordion-item:last-of-type{border-bottom-right-radius:var(--bs-accordion-border-radius);border-bottom-left-radius:var(--bs-accordion-border-radius)}.accordion-item:last-of-type>.accordion-header .accordion-button.collapsed{border-bottom-right-radius:var(--bs-accordion-inner-border-radius);border-bottom-left-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:last-of-type>.accordion-collapse{border-bottom-right-radius:var(--bs-accordion-border-radius);border-bottom-left-radius:var(--bs-accordion-border-radius)}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush>.accordion-item{border-right:0;border-left:0;border-radius:0}.accordion-flush>.accordion-item:first-child{border-top:0}.accordion-flush>.accordion-item:last-child{border-bottom:0}.accordion-flush>.accordion-item>.accordion-header .accordion-button,.accordion-flush>.accordion-item>.accordion-header .accordion-button.collapsed{border-radius:0}.accordion-flush>.accordion-item>.accordion-collapse{border-radius:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e\");--bs-accordion-btn-active-icon: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e\")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: var(--bs-secondary-color);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: var(--bs-secondary-color);display:flex;flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg);border-radius:var(--bs-breadcrumb-border-radius)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, \"/\") /* rtl: var(--bs-breadcrumb-divider, \"/\") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: var(--bs-link-color);--bs-pagination-bg: var(--bs-body-bg);--bs-pagination-border-width: var(--bs-border-width);--bs-pagination-border-color: var(--bs-border-color);--bs-pagination-border-radius: var(--bs-border-radius);--bs-pagination-hover-color: var(--bs-link-hover-color);--bs-pagination-hover-bg: var(--bs-tertiary-bg);--bs-pagination-hover-border-color: var(--bs-border-color);--bs-pagination-focus-color: var(--bs-link-hover-color);--bs-pagination-focus-bg: var(--bs-secondary-bg);--bs-pagination-focus-box-shadow: 0 0 0 0.1875rem var(--pst-color-accent);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #0d6efd;--bs-pagination-active-border-color: #0d6efd;--bs-pagination-disabled-color: var(--bs-secondary-color);--bs-pagination-disabled-bg: var(--bs-secondary-bg);--bs-pagination-disabled-border-color: var(--bs-border-color);display:flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(var(--bs-border-width)*-1)}.page-item:first-child .page-link{border-top-left-radius:var(--bs-pagination-border-radius);border-bottom-left-radius:var(--bs-pagination-border-radius)}.page-item:last-child .page-link{border-top-right-radius:var(--bs-pagination-border-radius);border-bottom-right-radius:var(--bs-pagination-border-radius)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: var(--bs-border-radius-lg)}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: var(--bs-border-radius-sm)}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: var(--bs-border-radius);display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline;border-radius:var(--bs-badge-border-radius)}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: var(--bs-border-width) solid var(--bs-alert-border-color);--bs-alert-border-radius: var(--bs-border-radius);--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border);border-radius:var(--bs-alert-border-radius)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:1rem}}.progress,.progress-stacked{--bs-progress-height: 1rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: var(--bs-secondary-bg);--bs-progress-border-radius: var(--bs-border-radius);--bs-progress-box-shadow: var(--bs-box-shadow-inset);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #0d6efd;--bs-progress-bar-transition: width 0.6s ease;display:flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg);border-radius:var(--bs-progress-border-radius)}.progress-bar{display:flex;flex-direction:column;justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: var(--bs-body-color);--bs-list-group-bg: var(--bs-body-bg);--bs-list-group-border-color: var(--bs-border-color);--bs-list-group-border-width: var(--bs-border-width);--bs-list-group-border-radius: var(--bs-border-radius);--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: var(--bs-secondary-color);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-tertiary-bg);--bs-list-group-action-active-color: var(--bs-body-color);--bs-list-group-action-active-bg: var(--bs-secondary-bg);--bs-list-group-disabled-color: var(--bs-secondary-color);--bs-list-group-disabled-bg: var(--bs-body-bg);--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #0d6efd;--bs-list-group-active-border-color: 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var(--bs-list-group-border-color)}.list-group-item:first-child{border-top-left-radius:inherit;border-top-right-radius:inherit}.list-group-item:last-child{border-bottom-right-radius:inherit;border-bottom-left-radius:inherit}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row}.list-group-horizontal>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 540px){.list-group-horizontal-sm{flex-direction:row}.list-group-horizontal-sm>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-sm>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 720px){.list-group-horizontal-md{flex-direction:row}.list-group-horizontal-md>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-md>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 960px){.list-group-horizontal-lg{flex-direction:row}.list-group-horizontal-lg>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-lg>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row}.list-group-horizontal-xl>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-xl>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush{border-radius:0}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e\");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.1875rem var(--pst-color-accent);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;border-radius:.375rem;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(var(--bs-body-bg-rgb), 0.85);--bs-toast-border-width: var(--bs-border-width);--bs-toast-border-color: var(--bs-border-color-translucent);--bs-toast-border-radius: var(--bs-border-radius);--bs-toast-box-shadow: var(--bs-box-shadow);--bs-toast-header-color: var(--bs-secondary-color);--bs-toast-header-bg: rgba(var(--bs-body-bg-rgb), 0.85);--bs-toast-header-border-color: var(--bs-border-color-translucent);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow);border-radius:var(--bs-toast-border-radius)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 1090;position:absolute;z-index:var(--bs-toast-zindex);width:max-content;max-width:100%;pointer-events:none}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{display:flex;align-items:center;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x);color:var(--bs-toast-header-color);background-color:var(--bs-toast-header-bg);background-clip:padding-box;border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color);border-top-left-radius:calc(var(--bs-toast-border-radius) - var(--bs-toast-border-width));border-top-right-radius:calc(var(--bs-toast-border-radius) - var(--bs-toast-border-width))}.toast-header .btn-close{margin-right:calc(-0.5*var(--bs-toast-padding-x));margin-left:var(--bs-toast-padding-x)}.toast-body{padding:var(--bs-toast-padding-x);word-wrap:break-word}.modal{--bs-modal-zindex: 1055;--bs-modal-width: 500px;--bs-modal-padding: 1rem;--bs-modal-margin: 0.5rem;--bs-modal-color: ;--bs-modal-bg: var(--bs-body-bg);--bs-modal-border-color: var(--bs-border-color-translucent);--bs-modal-border-width: var(--bs-border-width);--bs-modal-border-radius: var(--bs-border-radius-lg);--bs-modal-box-shadow: var(--bs-box-shadow-sm);--bs-modal-inner-border-radius: calc(var(--bs-border-radius-lg) - (var(--bs-border-width)));--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: var(--bs-border-color);--bs-modal-header-border-width: var(--bs-border-width);--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: var(--bs-border-color);--bs-modal-footer-border-width: var(--bs-border-width);position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static 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0.5;position:fixed;top:0;left:0;z-index:var(--bs-backdrop-zindex);width:100vw;height:100vh;background-color:var(--bs-backdrop-bg)}.modal-backdrop.fade{opacity:0}.modal-backdrop.show{opacity:var(--bs-backdrop-opacity)}.modal-header{display:flex;flex-shrink:0;align-items:center;padding:var(--bs-modal-header-padding);border-bottom:var(--bs-modal-header-border-width) solid var(--bs-modal-header-border-color);border-top-left-radius:var(--bs-modal-inner-border-radius);border-top-right-radius:var(--bs-modal-inner-border-radius)}.modal-header .btn-close{padding:calc(var(--bs-modal-header-padding-y)*.5) calc(var(--bs-modal-header-padding-x)*.5);margin:calc(-0.5*var(--bs-modal-header-padding-y)) calc(-0.5*var(--bs-modal-header-padding-x)) calc(-0.5*var(--bs-modal-header-padding-y)) auto}.modal-title{margin-bottom:0;line-height:var(--bs-modal-title-line-height)}.modal-body{position:relative;flex:1 1 auto;padding:var(--bs-modal-padding)}.modal-footer{display:flex;flex-shrink:0;flex-wrap:wrap;align-items:center;justify-content:flex-end;padding:calc(var(--bs-modal-padding) - var(--bs-modal-footer-gap)*.5);background-color:var(--bs-modal-footer-bg);border-top:var(--bs-modal-footer-border-width) solid var(--bs-modal-footer-border-color);border-bottom-right-radius:var(--bs-modal-inner-border-radius);border-bottom-left-radius:var(--bs-modal-inner-border-radius)}.modal-footer>*{margin:calc(var(--bs-modal-footer-gap)*.5)}@media(min-width: 540px){.modal{--bs-modal-margin: 1.75rem;--bs-modal-box-shadow: var(--bs-box-shadow)}.modal-dialog{max-width:var(--bs-modal-width);margin-right:auto;margin-left:auto}.modal-sm{--bs-modal-width: 300px}}@media(min-width: 960px){.modal-lg,.modal-xl{--bs-modal-width: 800px}}@media(min-width: 1200px){.modal-xl{--bs-modal-width: 1140px}}.modal-fullscreen{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen 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!important}.p-lg-5{padding:3rem !important}.px-lg-0{padding-right:0 !important;padding-left:0 !important}.px-lg-1{padding-right:.25rem !important;padding-left:.25rem !important}.px-lg-2{padding-right:.5rem !important;padding-left:.5rem !important}.px-lg-3{padding-right:1rem !important;padding-left:1rem !important}.px-lg-4{padding-right:1.5rem !important;padding-left:1.5rem !important}.px-lg-5{padding-right:3rem !important;padding-left:3rem !important}.py-lg-0{padding-top:0 !important;padding-bottom:0 !important}.py-lg-1{padding-top:.25rem !important;padding-bottom:.25rem !important}.py-lg-2{padding-top:.5rem !important;padding-bottom:.5rem !important}.py-lg-3{padding-top:1rem !important;padding-bottom:1rem !important}.py-lg-4{padding-top:1.5rem !important;padding-bottom:1.5rem !important}.py-lg-5{padding-top:3rem !important;padding-bottom:3rem !important}.pt-lg-0{padding-top:0 !important}.pt-lg-1{padding-top:.25rem !important}.pt-lg-2{padding-top:.5rem !important}.pt-lg-3{padding-top:1rem !important}.pt-lg-4{padding-top:1.5rem !important}.pt-lg-5{padding-top:3rem !important}.pe-lg-0{padding-right:0 !important}.pe-lg-1{padding-right:.25rem !important}.pe-lg-2{padding-right:.5rem !important}.pe-lg-3{padding-right:1rem !important}.pe-lg-4{padding-right:1.5rem !important}.pe-lg-5{padding-right:3rem !important}.pb-lg-0{padding-bottom:0 !important}.pb-lg-1{padding-bottom:.25rem !important}.pb-lg-2{padding-bottom:.5rem !important}.pb-lg-3{padding-bottom:1rem !important}.pb-lg-4{padding-bottom:1.5rem !important}.pb-lg-5{padding-bottom:3rem !important}.ps-lg-0{padding-left:0 !important}.ps-lg-1{padding-left:.25rem !important}.ps-lg-2{padding-left:.5rem !important}.ps-lg-3{padding-left:1rem !important}.ps-lg-4{padding-left:1.5rem !important}.ps-lg-5{padding-left:3rem !important}.gap-lg-0{gap:0 !important}.gap-lg-1{gap:.25rem !important}.gap-lg-2{gap:.5rem !important}.gap-lg-3{gap:1rem !important}.gap-lg-4{gap:1.5rem !important}.gap-lg-5{gap:3rem !important}.row-gap-lg-0{row-gap:0 !important}.row-gap-lg-1{row-gap:.25rem !important}.row-gap-lg-2{row-gap:.5rem !important}.row-gap-lg-3{row-gap:1rem !important}.row-gap-lg-4{row-gap:1.5rem !important}.row-gap-lg-5{row-gap:3rem !important}.column-gap-lg-0{column-gap:0 !important}.column-gap-lg-1{column-gap:.25rem !important}.column-gap-lg-2{column-gap:.5rem !important}.column-gap-lg-3{column-gap:1rem !important}.column-gap-lg-4{column-gap:1.5rem !important}.column-gap-lg-5{column-gap:3rem !important}.text-lg-start{text-align:left !important}.text-lg-end{text-align:right !important}.text-lg-center{text-align:center !important}}@media(min-width: 1200px){.float-xl-start{float:left !important}.float-xl-end{float:right !important}.float-xl-none{float:none !important}.object-fit-xl-contain{object-fit:contain !important}.object-fit-xl-cover{object-fit:cover !important}.object-fit-xl-fill{object-fit:fill !important}.object-fit-xl-scale{object-fit:scale-down !important}.object-fit-xl-none{object-fit:none !important}.d-xl-inline{display:inline !important}.d-xl-inline-block{display:inline-block !important}.d-xl-block{display:block !important}.d-xl-grid{display:grid !important}.d-xl-inline-grid{display:inline-grid !important}.d-xl-table{display:table !important}.d-xl-table-row{display:table-row !important}.d-xl-table-cell{display:table-cell !important}.d-xl-flex{display:flex !important}.d-xl-inline-flex{display:inline-flex !important}.d-xl-none{display:none !important}.flex-xl-fill{flex:1 1 auto !important}.flex-xl-row{flex-direction:row !important}.flex-xl-column{flex-direction:column !important}.flex-xl-row-reverse{flex-direction:row-reverse !important}.flex-xl-column-reverse{flex-direction:column-reverse !important}.flex-xl-grow-0{flex-grow:0 !important}.flex-xl-grow-1{flex-grow:1 !important}.flex-xl-shrink-0{flex-shrink:0 !important}.flex-xl-shrink-1{flex-shrink:1 !important}.flex-xl-wrap{flex-wrap:wrap !important}.flex-xl-nowrap{flex-wrap:nowrap !important}.flex-xl-wrap-reverse{flex-wrap:wrap-reverse !important}.justify-content-xl-start{justify-content:flex-start !important}.justify-content-xl-end{justify-content:flex-end !important}.justify-content-xl-center{justify-content:center !important}.justify-content-xl-between{justify-content:space-between !important}.justify-content-xl-around{justify-content:space-around !important}.justify-content-xl-evenly{justify-content:space-evenly !important}.align-items-xl-start{align-items:flex-start !important}.align-items-xl-end{align-items:flex-end !important}.align-items-xl-center{align-items:center !important}.align-items-xl-baseline{align-items:baseline !important}.align-items-xl-stretch{align-items:stretch !important}.align-content-xl-start{align-content:flex-start !important}.align-content-xl-end{align-content:flex-end !important}.align-content-xl-center{align-content:center !important}.align-content-xl-between{align-content:space-between !important}.align-content-xl-around{align-content:space-around !important}.align-content-xl-stretch{align-content:stretch !important}.align-self-xl-auto{align-self:auto !important}.align-self-xl-start{align-self:flex-start !important}.align-self-xl-end{align-self:flex-end !important}.align-self-xl-center{align-self:center !important}.align-self-xl-baseline{align-self:baseline !important}.align-self-xl-stretch{align-self:stretch !important}.order-xl-first{order:-1 !important}.order-xl-0{order:0 !important}.order-xl-1{order:1 !important}.order-xl-2{order:2 !important}.order-xl-3{order:3 !important}.order-xl-4{order:4 !important}.order-xl-5{order:5 !important}.order-xl-last{order:6 !important}.m-xl-0{margin:0 !important}.m-xl-1{margin:.25rem !important}.m-xl-2{margin:.5rem !important}.m-xl-3{margin:1rem !important}.m-xl-4{margin:1.5rem !important}.m-xl-5{margin:3rem !important}.m-xl-auto{margin:auto !important}.mx-xl-0{margin-right:0 !important;margin-left:0 !important}.mx-xl-1{margin-right:.25rem !important;margin-left:.25rem !important}.mx-xl-2{margin-right:.5rem !important;margin-left:.5rem !important}.mx-xl-3{margin-right:1rem !important;margin-left:1rem !important}.mx-xl-4{margin-right:1.5rem !important;margin-left:1.5rem !important}.mx-xl-5{margin-right:3rem !important;margin-left:3rem !important}.mx-xl-auto{margin-right:auto !important;margin-left:auto !important}.my-xl-0{margin-top:0 !important;margin-bottom:0 !important}.my-xl-1{margin-top:.25rem !important;margin-bottom:.25rem !important}.my-xl-2{margin-top:.5rem !important;margin-bottom:.5rem !important}.my-xl-3{margin-top:1rem !important;margin-bottom:1rem !important}.my-xl-4{margin-top:1.5rem !important;margin-bottom:1.5rem !important}.my-xl-5{margin-top:3rem !important;margin-bottom:3rem !important}.my-xl-auto{margin-top:auto !important;margin-bottom:auto !important}.mt-xl-0{margin-top:0 !important}.mt-xl-1{margin-top:.25rem !important}.mt-xl-2{margin-top:.5rem !important}.mt-xl-3{margin-top:1rem !important}.mt-xl-4{margin-top:1.5rem !important}.mt-xl-5{margin-top:3rem !important}.mt-xl-auto{margin-top:auto !important}.me-xl-0{margin-right:0 !important}.me-xl-1{margin-right:.25rem !important}.me-xl-2{margin-right:.5rem !important}.me-xl-3{margin-right:1rem !important}.me-xl-4{margin-right:1.5rem !important}.me-xl-5{margin-right:3rem !important}.me-xl-auto{margin-right:auto !important}.mb-xl-0{margin-bottom:0 !important}.mb-xl-1{margin-bottom:.25rem !important}.mb-xl-2{margin-bottom:.5rem !important}.mb-xl-3{margin-bottom:1rem !important}.mb-xl-4{margin-bottom:1.5rem !important}.mb-xl-5{margin-bottom:3rem !important}.mb-xl-auto{margin-bottom:auto !important}.ms-xl-0{margin-left:0 !important}.ms-xl-1{margin-left:.25rem !important}.ms-xl-2{margin-left:.5rem !important}.ms-xl-3{margin-left:1rem !important}.ms-xl-4{margin-left:1.5rem !important}.ms-xl-5{margin-left:3rem !important}.ms-xl-auto{margin-left:auto !important}.p-xl-0{padding:0 !important}.p-xl-1{padding:.25rem !important}.p-xl-2{padding:.5rem !important}.p-xl-3{padding:1rem !important}.p-xl-4{padding:1.5rem !important}.p-xl-5{padding:3rem !important}.px-xl-0{padding-right:0 !important;padding-left:0 !important}.px-xl-1{padding-right:.25rem !important;padding-left:.25rem !important}.px-xl-2{padding-right:.5rem !important;padding-left:.5rem !important}.px-xl-3{padding-right:1rem !important;padding-left:1rem !important}.px-xl-4{padding-right:1.5rem !important;padding-left:1.5rem !important}.px-xl-5{padding-right:3rem !important;padding-left:3rem !important}.py-xl-0{padding-top:0 !important;padding-bottom:0 !important}.py-xl-1{padding-top:.25rem !important;padding-bottom:.25rem !important}.py-xl-2{padding-top:.5rem !important;padding-bottom:.5rem !important}.py-xl-3{padding-top:1rem !important;padding-bottom:1rem !important}.py-xl-4{padding-top:1.5rem !important;padding-bottom:1.5rem !important}.py-xl-5{padding-top:3rem !important;padding-bottom:3rem !important}.pt-xl-0{padding-top:0 !important}.pt-xl-1{padding-top:.25rem !important}.pt-xl-2{padding-top:.5rem !important}.pt-xl-3{padding-top:1rem !important}.pt-xl-4{padding-top:1.5rem !important}.pt-xl-5{padding-top:3rem !important}.pe-xl-0{padding-right:0 !important}.pe-xl-1{padding-right:.25rem !important}.pe-xl-2{padding-right:.5rem !important}.pe-xl-3{padding-right:1rem !important}.pe-xl-4{padding-right:1.5rem !important}.pe-xl-5{padding-right:3rem !important}.pb-xl-0{padding-bottom:0 !important}.pb-xl-1{padding-bottom:.25rem !important}.pb-xl-2{padding-bottom:.5rem !important}.pb-xl-3{padding-bottom:1rem !important}.pb-xl-4{padding-bottom:1.5rem !important}.pb-xl-5{padding-bottom:3rem !important}.ps-xl-0{padding-left:0 !important}.ps-xl-1{padding-left:.25rem !important}.ps-xl-2{padding-left:.5rem !important}.ps-xl-3{padding-left:1rem !important}.ps-xl-4{padding-left:1.5rem !important}.ps-xl-5{padding-left:3rem !important}.gap-xl-0{gap:0 !important}.gap-xl-1{gap:.25rem !important}.gap-xl-2{gap:.5rem !important}.gap-xl-3{gap:1rem !important}.gap-xl-4{gap:1.5rem !important}.gap-xl-5{gap:3rem !important}.row-gap-xl-0{row-gap:0 !important}.row-gap-xl-1{row-gap:.25rem !important}.row-gap-xl-2{row-gap:.5rem !important}.row-gap-xl-3{row-gap:1rem !important}.row-gap-xl-4{row-gap:1.5rem !important}.row-gap-xl-5{row-gap:3rem !important}.column-gap-xl-0{column-gap:0 !important}.column-gap-xl-1{column-gap:.25rem !important}.column-gap-xl-2{column-gap:.5rem !important}.column-gap-xl-3{column-gap:1rem !important}.column-gap-xl-4{column-gap:1.5rem !important}.column-gap-xl-5{column-gap:3rem !important}.text-xl-start{text-align:left !important}.text-xl-end{text-align:right !important}.text-xl-center{text-align:center !important}}@media(min-width: 1200px){.fs-1{font-size:2.5rem !important}.fs-2{font-size:2rem !important}.fs-3{font-size:1.75rem !important}.fs-4{font-size:1.5rem !important}}@media print{.d-print-inline{display:inline !important}.d-print-inline-block{display:inline-block !important}.d-print-block{display:block !important}.d-print-grid{display:grid !important}.d-print-inline-grid{display:inline-grid !important}.d-print-table{display:table !important}.d-print-table-row{display:table-row !important}.d-print-table-cell{display:table-cell !important}.d-print-flex{display:flex !important}.d-print-inline-flex{display:inline-flex !important}.d-print-none{display:none !important}}","@mixin bsBanner($file) {\n  /*!\n   * Bootstrap #{$file} v5.3.3 (https://getbootstrap.com/)\n   * Copyright 2011-2024 The Bootstrap Authors\n   * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n   */\n}\n",":root,\n[data-bs-theme=\"light\"] {\n  // Note: Custom variable values only support SassScript inside `#{}`.\n\n  // Colors\n  //\n  // Generate palettes for full colors, grays, and theme colors.\n\n  @each $color, $value in $colors {\n    --#{$prefix}#{$color}: #{$value};\n  }\n\n  @each $color, $value in $grays {\n    --#{$prefix}gray-#{$color}: #{$value};\n  }\n\n  @each $color, $value in $theme-colors {\n    --#{$prefix}#{$color}: #{$value};\n  }\n\n  @each $color, $value in $theme-colors-rgb {\n    --#{$prefix}#{$color}-rgb: #{$value};\n  }\n\n  @each $color, $value in $theme-colors-text {\n    --#{$prefix}#{$color}-text-emphasis: #{$value};\n  }\n\n  @each $color, $value in $theme-colors-bg-subtle {\n    --#{$prefix}#{$color}-bg-subtle: #{$value};\n  }\n\n  @each $color, $value in $theme-colors-border-subtle {\n    --#{$prefix}#{$color}-border-subtle: #{$value};\n  }\n\n  --#{$prefix}white-rgb: #{to-rgb($white)};\n  --#{$prefix}black-rgb: #{to-rgb($black)};\n\n  // Fonts\n\n  // Note: Use `inspect` for lists so that quoted items keep the quotes.\n  // See https://github.com/sass/sass/issues/2383#issuecomment-336349172\n  --#{$prefix}font-sans-serif: #{inspect($font-family-sans-serif)};\n  --#{$prefix}font-monospace: #{inspect($font-family-monospace)};\n  --#{$prefix}gradient: #{$gradient};\n\n  // Root and body\n  // scss-docs-start root-body-variables\n  @if $font-size-root != null {\n    --#{$prefix}root-font-size: #{$font-size-root};\n  }\n  --#{$prefix}body-font-family: #{inspect($font-family-base)};\n  @include rfs($font-size-base, --#{$prefix}body-font-size);\n  --#{$prefix}body-font-weight: #{$font-weight-base};\n  --#{$prefix}body-line-height: #{$line-height-base};\n  @if $body-text-align != null {\n    --#{$prefix}body-text-align: #{$body-text-align};\n  }\n\n  --#{$prefix}body-color: #{$body-color};\n  --#{$prefix}body-color-rgb: #{to-rgb($body-color)};\n  --#{$prefix}body-bg: #{$body-bg};\n  --#{$prefix}body-bg-rgb: #{to-rgb($body-bg)};\n\n  --#{$prefix}emphasis-color: #{$body-emphasis-color};\n  --#{$prefix}emphasis-color-rgb: #{to-rgb($body-emphasis-color)};\n\n  --#{$prefix}secondary-color: #{$body-secondary-color};\n  --#{$prefix}secondary-color-rgb: #{to-rgb($body-secondary-color)};\n  --#{$prefix}secondary-bg: #{$body-secondary-bg};\n  --#{$prefix}secondary-bg-rgb: #{to-rgb($body-secondary-bg)};\n\n  --#{$prefix}tertiary-color: #{$body-tertiary-color};\n  --#{$prefix}tertiary-color-rgb: #{to-rgb($body-tertiary-color)};\n  --#{$prefix}tertiary-bg: #{$body-tertiary-bg};\n  --#{$prefix}tertiary-bg-rgb: #{to-rgb($body-tertiary-bg)};\n  // scss-docs-end root-body-variables\n\n  --#{$prefix}heading-color: #{$headings-color};\n\n  --#{$prefix}link-color: #{$link-color};\n  --#{$prefix}link-color-rgb: #{to-rgb($link-color)};\n  --#{$prefix}link-decoration: #{$link-decoration};\n\n  --#{$prefix}link-hover-color: #{$link-hover-color};\n  --#{$prefix}link-hover-color-rgb: #{to-rgb($link-hover-color)};\n\n  @if $link-hover-decoration != null {\n    --#{$prefix}link-hover-decoration: #{$link-hover-decoration};\n  }\n\n  --#{$prefix}code-color: #{$code-color};\n  --#{$prefix}highlight-color: #{$mark-color};\n  --#{$prefix}highlight-bg: #{$mark-bg};\n\n  // scss-docs-start root-border-var\n  --#{$prefix}border-width: #{$border-width};\n  --#{$prefix}border-style: #{$border-style};\n  --#{$prefix}border-color: #{$border-color};\n  --#{$prefix}border-color-translucent: #{$border-color-translucent};\n\n  --#{$prefix}border-radius: #{$border-radius};\n  --#{$prefix}border-radius-sm: #{$border-radius-sm};\n  --#{$prefix}border-radius-lg: #{$border-radius-lg};\n  --#{$prefix}border-radius-xl: #{$border-radius-xl};\n  --#{$prefix}border-radius-xxl: #{$border-radius-xxl};\n  --#{$prefix}border-radius-2xl: var(--#{$prefix}border-radius-xxl); // Deprecated in v5.3.0 for consistency\n  --#{$prefix}border-radius-pill: #{$border-radius-pill};\n  // scss-docs-end root-border-var\n\n  --#{$prefix}box-shadow: #{$box-shadow};\n  --#{$prefix}box-shadow-sm: #{$box-shadow-sm};\n  --#{$prefix}box-shadow-lg: #{$box-shadow-lg};\n  --#{$prefix}box-shadow-inset: #{$box-shadow-inset};\n\n  // Focus styles\n  // scss-docs-start root-focus-variables\n  --#{$prefix}focus-ring-width: #{$focus-ring-width};\n  --#{$prefix}focus-ring-opacity: #{$focus-ring-opacity};\n  --#{$prefix}focus-ring-color: #{$focus-ring-color};\n  // scss-docs-end root-focus-variables\n\n  // scss-docs-start root-form-validation-variables\n  --#{$prefix}form-valid-color: #{$form-valid-color};\n  --#{$prefix}form-valid-border-color: #{$form-valid-border-color};\n  --#{$prefix}form-invalid-color: #{$form-invalid-color};\n  --#{$prefix}form-invalid-border-color: #{$form-invalid-border-color};\n  // scss-docs-end root-form-validation-variables\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark, true) {\n    color-scheme: dark;\n\n    // scss-docs-start root-dark-mode-vars\n    --#{$prefix}body-color: #{$body-color-dark};\n    --#{$prefix}body-color-rgb: #{to-rgb($body-color-dark)};\n    --#{$prefix}body-bg: #{$body-bg-dark};\n    --#{$prefix}body-bg-rgb: #{to-rgb($body-bg-dark)};\n\n    --#{$prefix}emphasis-color: #{$body-emphasis-color-dark};\n    --#{$prefix}emphasis-color-rgb: #{to-rgb($body-emphasis-color-dark)};\n\n    --#{$prefix}secondary-color: #{$body-secondary-color-dark};\n    --#{$prefix}secondary-color-rgb: #{to-rgb($body-secondary-color-dark)};\n    --#{$prefix}secondary-bg: #{$body-secondary-bg-dark};\n    --#{$prefix}secondary-bg-rgb: #{to-rgb($body-secondary-bg-dark)};\n\n    --#{$prefix}tertiary-color: #{$body-tertiary-color-dark};\n    --#{$prefix}tertiary-color-rgb: #{to-rgb($body-tertiary-color-dark)};\n    --#{$prefix}tertiary-bg: #{$body-tertiary-bg-dark};\n    --#{$prefix}tertiary-bg-rgb: #{to-rgb($body-tertiary-bg-dark)};\n\n    @each $color, $value in $theme-colors-text-dark {\n      --#{$prefix}#{$color}-text-emphasis: #{$value};\n    }\n\n    @each $color, $value in $theme-colors-bg-subtle-dark {\n      --#{$prefix}#{$color}-bg-subtle: #{$value};\n    }\n\n    @each $color, $value in $theme-colors-border-subtle-dark {\n      --#{$prefix}#{$color}-border-subtle: #{$value};\n    }\n\n    --#{$prefix}heading-color: #{$headings-color-dark};\n\n    --#{$prefix}link-color: #{$link-color-dark};\n    --#{$prefix}link-hover-color: #{$link-hover-color-dark};\n    --#{$prefix}link-color-rgb: #{to-rgb($link-color-dark)};\n    --#{$prefix}link-hover-color-rgb: #{to-rgb($link-hover-color-dark)};\n\n    --#{$prefix}code-color: #{$code-color-dark};\n    --#{$prefix}highlight-color: #{$mark-color-dark};\n    --#{$prefix}highlight-bg: #{$mark-bg-dark};\n\n    --#{$prefix}border-color: #{$border-color-dark};\n    --#{$prefix}border-color-translucent: #{$border-color-translucent-dark};\n\n    --#{$prefix}form-valid-color: #{$form-valid-color-dark};\n    --#{$prefix}form-valid-border-color: #{$form-valid-border-color-dark};\n    --#{$prefix}form-invalid-color: #{$form-invalid-color-dark};\n    --#{$prefix}form-invalid-border-color: #{$form-invalid-border-color-dark};\n    // scss-docs-end root-dark-mode-vars\n  }\n}\n","// stylelint-disable scss/dimension-no-non-numeric-values\n\n// SCSS RFS mixin\n//\n// Automated responsive values for font sizes, paddings, margins and much more\n//\n// Licensed under MIT (https://github.com/twbs/rfs/blob/main/LICENSE)\n\n// Configuration\n\n// Base value\n$rfs-base-value: 1.25rem !default;\n$rfs-unit: rem !default;\n\n@if $rfs-unit != rem and $rfs-unit != px {\n  @error \"`#{$rfs-unit}` is not a valid unit for $rfs-unit. Use `px` or `rem`.\";\n}\n\n// Breakpoint at where values start decreasing if screen width is smaller\n$rfs-breakpoint: 1200px !default;\n$rfs-breakpoint-unit: px !default;\n\n@if $rfs-breakpoint-unit != px and $rfs-breakpoint-unit != em and $rfs-breakpoint-unit != rem {\n  @error \"`#{$rfs-breakpoint-unit}` is not a valid unit for $rfs-breakpoint-unit. Use `px`, `em` or `rem`.\";\n}\n\n// Resize values based on screen height and width\n$rfs-two-dimensional: false !default;\n\n// Factor of decrease\n$rfs-factor: 10 !default;\n\n@if type-of($rfs-factor) != number or $rfs-factor <= 1 {\n  @error \"`#{$rfs-factor}` is not a valid  $rfs-factor, it must be greater than 1.\";\n}\n\n// Mode. Possibilities: \"min-media-query\", \"max-media-query\"\n$rfs-mode: min-media-query !default;\n\n// Generate enable or disable classes. Possibilities: false, \"enable\" or \"disable\"\n$rfs-class: false !default;\n\n// 1 rem = $rfs-rem-value px\n$rfs-rem-value: 16 !default;\n\n// Safari iframe resize bug: https://github.com/twbs/rfs/issues/14\n$rfs-safari-iframe-resize-bug-fix: false !default;\n\n// Disable RFS by setting $enable-rfs to false\n$enable-rfs: true !default;\n\n// Cache $rfs-base-value unit\n$rfs-base-value-unit: unit($rfs-base-value);\n\n@function divide($dividend, $divisor, $precision: 10) {\n  $sign: if($dividend > 0 and $divisor > 0 or $dividend < 0 and $divisor < 0, 1, -1);\n  $dividend: abs($dividend);\n  $divisor: abs($divisor);\n  @if $dividend == 0 {\n    @return 0;\n  }\n  @if $divisor == 0 {\n    @error \"Cannot divide by 0\";\n  }\n  $remainder: $dividend;\n  $result: 0;\n  $factor: 10;\n  @while ($remainder > 0 and $precision >= 0) {\n    $quotient: 0;\n    @while ($remainder >= $divisor) {\n      $remainder: $remainder - $divisor;\n      $quotient: $quotient + 1;\n    }\n    $result: $result * 10 + $quotient;\n    $factor: $factor * .1;\n    $remainder: $remainder * 10;\n    $precision: $precision - 1;\n    @if ($precision < 0 and $remainder >= $divisor * 5) {\n      $result: $result + 1;\n    }\n  }\n  $result: $result * $factor * $sign;\n  $dividend-unit: unit($dividend);\n  $divisor-unit: unit($divisor);\n  $unit-map: (\n    \"px\": 1px,\n    \"rem\": 1rem,\n    \"em\": 1em,\n    \"%\": 1%\n  );\n  @if ($dividend-unit != $divisor-unit and map-has-key($unit-map, $dividend-unit)) {\n    $result: $result * map-get($unit-map, $dividend-unit);\n  }\n  @return $result;\n}\n\n// Remove px-unit from $rfs-base-value for calculations\n@if $rfs-base-value-unit == px {\n  $rfs-base-value: divide($rfs-base-value, $rfs-base-value * 0 + 1);\n}\n@else if $rfs-base-value-unit == rem {\n  $rfs-base-value: divide($rfs-base-value, divide($rfs-base-value * 0 + 1, $rfs-rem-value));\n}\n\n// Cache $rfs-breakpoint unit to prevent multiple calls\n$rfs-breakpoint-unit-cache: unit($rfs-breakpoint);\n\n// Remove unit from $rfs-breakpoint for calculations\n@if $rfs-breakpoint-unit-cache == px {\n  $rfs-breakpoint: divide($rfs-breakpoint, $rfs-breakpoint * 0 + 1);\n}\n@else if $rfs-breakpoint-unit-cache == rem or $rfs-breakpoint-unit-cache == \"em\" {\n  $rfs-breakpoint: divide($rfs-breakpoint, divide($rfs-breakpoint * 0 + 1, $rfs-rem-value));\n}\n\n// Calculate the media query value\n$rfs-mq-value: if($rfs-breakpoint-unit == px, #{$rfs-breakpoint}px, #{divide($rfs-breakpoint, $rfs-rem-value)}#{$rfs-breakpoint-unit});\n$rfs-mq-property-width: if($rfs-mode == max-media-query, max-width, min-width);\n$rfs-mq-property-height: if($rfs-mode == max-media-query, max-height, min-height);\n\n// Internal mixin used to determine which media query needs to be used\n@mixin _rfs-media-query {\n  @if $rfs-two-dimensional {\n    @if $rfs-mode == max-media-query {\n      @media (#{$rfs-mq-property-width}: #{$rfs-mq-value}), (#{$rfs-mq-property-height}: #{$rfs-mq-value}) {\n        @content;\n      }\n    }\n    @else {\n      @media (#{$rfs-mq-property-width}: #{$rfs-mq-value}) and (#{$rfs-mq-property-height}: #{$rfs-mq-value}) {\n        @content;\n      }\n    }\n  }\n  @else {\n    @media (#{$rfs-mq-property-width}: #{$rfs-mq-value}) {\n      @content;\n    }\n  }\n}\n\n// Internal mixin that adds disable classes to the selector if needed.\n@mixin _rfs-rule {\n  @if $rfs-class == disable and $rfs-mode == max-media-query {\n    // Adding an extra class increases specificity, which prevents the media query to override the property\n    &,\n    .disable-rfs &,\n    &.disable-rfs {\n      @content;\n    }\n  }\n  @else if $rfs-class == enable and $rfs-mode == min-media-query {\n    .enable-rfs &,\n    &.enable-rfs {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Internal mixin that adds enable classes to the selector if needed.\n@mixin _rfs-media-query-rule {\n\n  @if $rfs-class == enable {\n    @if $rfs-mode == min-media-query {\n      @content;\n    }\n\n    @include _rfs-media-query () {\n      .enable-rfs &,\n      &.enable-rfs {\n        @content;\n      }\n    }\n  }\n  @else {\n    @if $rfs-class == disable and $rfs-mode == min-media-query {\n      .disable-rfs &,\n      &.disable-rfs {\n        @content;\n      }\n    }\n    @include _rfs-media-query () {\n      @content;\n    }\n  }\n}\n\n// Helper function to get the formatted non-responsive value\n@function rfs-value($values) {\n  // Convert to list\n  $values: if(type-of($values) != list, ($values,), $values);\n\n  $val: \"\";\n\n  // Loop over each value and calculate value\n  @each $value in $values {\n    @if $value == 0 {\n      $val: $val + \" 0\";\n    }\n    @else {\n      // Cache $value unit\n      $unit: if(type-of($value) == \"number\", unit($value), false);\n\n      @if $unit == px {\n        // Convert to rem if needed\n        $val: $val + \" \" + if($rfs-unit == rem, #{divide($value, $value * 0 + $rfs-rem-value)}rem, $value);\n      }\n      @else if $unit == rem {\n        // Convert to px if needed\n        $val: $val + \" \" + if($rfs-unit == px, #{divide($value, $value * 0 + 1) * $rfs-rem-value}px, $value);\n      } @else {\n        // If $value isn't a number (like inherit) or $value has a unit (not px or rem, like 1.5em) or $ is 0, just print the value\n        $val: $val + \" \" + $value;\n      }\n    }\n  }\n\n  // Remove first space\n  @return unquote(str-slice($val, 2));\n}\n\n// Helper function to get the responsive value calculated by RFS\n@function rfs-fluid-value($values) {\n  // Convert to list\n  $values: if(type-of($values) != list, ($values,), $values);\n\n  $val: \"\";\n\n  // Loop over each value and calculate value\n  @each $value in $values {\n    @if $value == 0 {\n      $val: $val + \" 0\";\n    } @else {\n      // Cache $value unit\n      $unit: if(type-of($value) == \"number\", unit($value), false);\n\n      // If $value isn't a number (like inherit) or $value has a unit (not px or rem, like 1.5em) or $ is 0, just print the value\n      @if not $unit or $unit != px and $unit != rem {\n        $val: $val + \" \" + $value;\n      } @else {\n        // Remove unit from $value for calculations\n        $value: divide($value, $value * 0 + if($unit == px, 1, divide(1, $rfs-rem-value)));\n\n        // Only add the media query if the value is greater than the minimum value\n        @if abs($value) <= $rfs-base-value or not $enable-rfs {\n          $val: $val + \" \" + if($rfs-unit == rem, #{divide($value, $rfs-rem-value)}rem, #{$value}px);\n        }\n        @else {\n          // Calculate the minimum value\n          $value-min: $rfs-base-value + divide(abs($value) - $rfs-base-value, $rfs-factor);\n\n          // Calculate difference between $value and the minimum value\n          $value-diff: abs($value) - $value-min;\n\n          // Base value formatting\n          $min-width: if($rfs-unit == rem, #{divide($value-min, $rfs-rem-value)}rem, #{$value-min}px);\n\n          // Use negative value if needed\n          $min-width: if($value < 0, -$min-width, $min-width);\n\n          // Use `vmin` if two-dimensional is enabled\n          $variable-unit: if($rfs-two-dimensional, vmin, vw);\n\n          // Calculate the variable width between 0 and $rfs-breakpoint\n          $variable-width: #{divide($value-diff * 100, $rfs-breakpoint)}#{$variable-unit};\n\n          // Return the calculated value\n          $val: $val + \" calc(\" + $min-width + if($value < 0, \" - \", \" + \") + $variable-width + \")\";\n        }\n      }\n    }\n  }\n\n  // Remove first space\n  @return unquote(str-slice($val, 2));\n}\n\n// RFS mixin\n@mixin rfs($values, $property: font-size) {\n  @if $values != null {\n    $val: rfs-value($values);\n    $fluid-val: rfs-fluid-value($values);\n\n    // Do not print the media query if responsive & non-responsive values are the same\n    @if $val == $fluid-val {\n      #{$property}: $val;\n    }\n    @else {\n      @include _rfs-rule () {\n        #{$property}: if($rfs-mode == max-media-query, $val, $fluid-val);\n\n        // Include safari iframe resize fix if needed\n        min-width: if($rfs-safari-iframe-resize-bug-fix, (0 * 1vw), null);\n      }\n\n      @include _rfs-media-query-rule () {\n        #{$property}: if($rfs-mode == max-media-query, $fluid-val, $val);\n      }\n    }\n  }\n}\n\n// Shorthand helper mixins\n@mixin font-size($value) {\n  @include rfs($value);\n}\n\n@mixin padding($value) {\n  @include rfs($value, padding);\n}\n\n@mixin padding-top($value) {\n  @include rfs($value, padding-top);\n}\n\n@mixin padding-right($value) {\n  @include rfs($value, padding-right);\n}\n\n@mixin padding-bottom($value) {\n  @include rfs($value, padding-bottom);\n}\n\n@mixin padding-left($value) {\n  @include rfs($value, padding-left);\n}\n\n@mixin margin($value) {\n  @include rfs($value, margin);\n}\n\n@mixin margin-top($value) {\n  @include rfs($value, margin-top);\n}\n\n@mixin margin-right($value) {\n  @include rfs($value, margin-right);\n}\n\n@mixin margin-bottom($value) {\n  @include rfs($value, margin-bottom);\n}\n\n@mixin margin-left($value) {\n  @include rfs($value, margin-left);\n}\n","// scss-docs-start color-mode-mixin\n@mixin color-mode($mode: light, $root: false) {\n  @if $color-mode-type == \"media-query\" {\n    @if $root == true {\n      @media (prefers-color-scheme: $mode) {\n        :root {\n          @content;\n        }\n      }\n    } @else {\n      @media (prefers-color-scheme: $mode) {\n        @content;\n      }\n    }\n  } @else {\n    [data-bs-theme=\"#{$mode}\"] {\n      @content;\n    }\n  }\n}\n// scss-docs-end color-mode-mixin\n","// stylelint-disable declaration-no-important, selector-no-qualifying-type, property-no-vendor-prefix\n\n\n// Reboot\n//\n// Normalization of HTML elements, manually forked from Normalize.css to remove\n// styles targeting irrelevant browsers while applying new styles.\n//\n// Normalize is licensed MIT. https://github.com/necolas/normalize.css\n\n\n// Document\n//\n// Change from `box-sizing: content-box` so that `width` is not affected by `padding` or `border`.\n\n*,\n*::before,\n*::after {\n  box-sizing: border-box;\n}\n\n\n// Root\n//\n// Ability to the value of the root font sizes, affecting the value of `rem`.\n// null by default, thus nothing is generated.\n\n:root {\n  @if $font-size-root != null {\n    @include font-size(var(--#{$prefix}root-font-size));\n  }\n\n  @if $enable-smooth-scroll {\n    @media (prefers-reduced-motion: no-preference) {\n      scroll-behavior: smooth;\n    }\n  }\n}\n\n\n// Body\n//\n// 1. Remove the margin in all browsers.\n// 2. As a best practice, apply a default `background-color`.\n// 3. Prevent adjustments of font size after orientation changes in iOS.\n// 4. Change the default tap highlight to be completely transparent in iOS.\n\n// scss-docs-start reboot-body-rules\nbody {\n  margin: 0; // 1\n  font-family: var(--#{$prefix}body-font-family);\n  @include font-size(var(--#{$prefix}body-font-size));\n  font-weight: var(--#{$prefix}body-font-weight);\n  line-height: var(--#{$prefix}body-line-height);\n  color: var(--#{$prefix}body-color);\n  text-align: var(--#{$prefix}body-text-align);\n  background-color: var(--#{$prefix}body-bg); // 2\n  -webkit-text-size-adjust: 100%; // 3\n  -webkit-tap-highlight-color: rgba($black, 0); // 4\n}\n// scss-docs-end reboot-body-rules\n\n\n// Content grouping\n//\n// 1. Reset Firefox's gray color\n\nhr {\n  margin: $hr-margin-y 0;\n  color: $hr-color; // 1\n  border: 0;\n  border-top: $hr-border-width solid $hr-border-color;\n  opacity: $hr-opacity;\n}\n\n\n// Typography\n//\n// 1. Remove top margins from headings\n//    By default, `<h1>`-`<h6>` all receive top and bottom margins. We nuke the top\n//    margin for easier control within type scales as it avoids margin collapsing.\n\n%heading {\n  margin-top: 0; // 1\n  margin-bottom: $headings-margin-bottom;\n  font-family: $headings-font-family;\n  font-style: $headings-font-style;\n  font-weight: $headings-font-weight;\n  line-height: $headings-line-height;\n  color: var(--#{$prefix}heading-color);\n}\n\nh1 {\n  @extend %heading;\n  @include font-size($h1-font-size);\n}\n\nh2 {\n  @extend %heading;\n  @include font-size($h2-font-size);\n}\n\nh3 {\n  @extend %heading;\n  @include font-size($h3-font-size);\n}\n\nh4 {\n  @extend %heading;\n  @include font-size($h4-font-size);\n}\n\nh5 {\n  @extend %heading;\n  @include font-size($h5-font-size);\n}\n\nh6 {\n  @extend %heading;\n  @include font-size($h6-font-size);\n}\n\n\n// Reset margins on paragraphs\n//\n// Similarly, the top margin on `<p>`s get reset. However, we also reset the\n// bottom margin to use `rem` units instead of `em`.\n\np {\n  margin-top: 0;\n  margin-bottom: $paragraph-margin-bottom;\n}\n\n\n// Abbreviations\n//\n// 1. Add the correct text decoration in Chrome, Edge, Opera, and Safari.\n// 2. Add explicit cursor to indicate changed behavior.\n// 3. Prevent the text-decoration to be skipped.\n\nabbr[title] {\n  text-decoration: underline dotted; // 1\n  cursor: help; // 2\n  text-decoration-skip-ink: none; // 3\n}\n\n\n// Address\n\naddress {\n  margin-bottom: 1rem;\n  font-style: normal;\n  line-height: inherit;\n}\n\n\n// Lists\n\nol,\nul {\n  padding-left: 2rem;\n}\n\nol,\nul,\ndl {\n  margin-top: 0;\n  margin-bottom: 1rem;\n}\n\nol ol,\nul ul,\nol ul,\nul ol {\n  margin-bottom: 0;\n}\n\ndt {\n  font-weight: $dt-font-weight;\n}\n\n// 1. Undo browser default\n\ndd {\n  margin-bottom: .5rem;\n  margin-left: 0; // 1\n}\n\n\n// Blockquote\n\nblockquote {\n  margin: 0 0 1rem;\n}\n\n\n// Strong\n//\n// Add the correct font weight in Chrome, Edge, and Safari\n\nb,\nstrong {\n  font-weight: $font-weight-bolder;\n}\n\n\n// Small\n//\n// Add the correct font size in all browsers\n\nsmall {\n  @include font-size($small-font-size);\n}\n\n\n// Mark\n\nmark {\n  padding: $mark-padding;\n  color: var(--#{$prefix}highlight-color);\n  background-color: var(--#{$prefix}highlight-bg);\n}\n\n\n// Sub and Sup\n//\n// Prevent `sub` and `sup` elements from affecting the line height in\n// all browsers.\n\nsub,\nsup {\n  position: relative;\n  @include font-size($sub-sup-font-size);\n  line-height: 0;\n  vertical-align: baseline;\n}\n\nsub { bottom: -.25em; }\nsup { top: -.5em; }\n\n\n// Links\n\na {\n  color: rgba(var(--#{$prefix}link-color-rgb), var(--#{$prefix}link-opacity, 1));\n  text-decoration: $link-decoration;\n\n  &:hover {\n    --#{$prefix}link-color-rgb: var(--#{$prefix}link-hover-color-rgb);\n    text-decoration: $link-hover-decoration;\n  }\n}\n\n// And undo these styles for placeholder links/named anchors (without href).\n// It would be more straightforward to just use a[href] in previous block, but that\n// causes specificity issues in many other styles that are too complex to fix.\n// See https://github.com/twbs/bootstrap/issues/19402\n\na:not([href]):not([class]) {\n  &,\n  &:hover {\n    color: inherit;\n    text-decoration: none;\n  }\n}\n\n\n// Code\n\npre,\ncode,\nkbd,\nsamp {\n  font-family: $font-family-code;\n  @include font-size(1em); // Correct the odd `em` font sizing in all browsers.\n}\n\n// 1. Remove browser default top margin\n// 2. Reset browser default of `1em` to use `rem`s\n// 3. Don't allow content to break outside\n\npre {\n  display: block;\n  margin-top: 0; // 1\n  margin-bottom: 1rem; // 2\n  overflow: auto; // 3\n  @include font-size($code-font-size);\n  color: $pre-color;\n\n  // Account for some code outputs that place code tags in pre tags\n  code {\n    @include font-size(inherit);\n    color: inherit;\n    word-break: normal;\n  }\n}\n\ncode {\n  @include font-size($code-font-size);\n  color: var(--#{$prefix}code-color);\n  word-wrap: break-word;\n\n  // Streamline the style when inside anchors to avoid broken underline and more\n  a > & {\n    color: inherit;\n  }\n}\n\nkbd {\n  padding: $kbd-padding-y $kbd-padding-x;\n  @include font-size($kbd-font-size);\n  color: $kbd-color;\n  background-color: $kbd-bg;\n  @include border-radius($border-radius-sm);\n\n  kbd {\n    padding: 0;\n    @include font-size(1em);\n    font-weight: $nested-kbd-font-weight;\n  }\n}\n\n\n// Figures\n//\n// Apply a consistent margin strategy (matches our type styles).\n\nfigure {\n  margin: 0 0 1rem;\n}\n\n\n// Images and content\n\nimg,\nsvg {\n  vertical-align: middle;\n}\n\n\n// Tables\n//\n// Prevent double borders\n\ntable {\n  caption-side: bottom;\n  border-collapse: collapse;\n}\n\ncaption {\n  padding-top: $table-cell-padding-y;\n  padding-bottom: $table-cell-padding-y;\n  color: $table-caption-color;\n  text-align: left;\n}\n\n// 1. Removes font-weight bold by inheriting\n// 2. Matches default `<td>` alignment by inheriting `text-align`.\n// 3. Fix alignment for Safari\n\nth {\n  font-weight: $table-th-font-weight; // 1\n  text-align: inherit; // 2\n  text-align: -webkit-match-parent; // 3\n}\n\nthead,\ntbody,\ntfoot,\ntr,\ntd,\nth {\n  border-color: inherit;\n  border-style: solid;\n  border-width: 0;\n}\n\n\n// Forms\n//\n// 1. Allow labels to use `margin` for spacing.\n\nlabel {\n  display: inline-block; // 1\n}\n\n// Remove the default `border-radius` that macOS Chrome adds.\n// See https://github.com/twbs/bootstrap/issues/24093\n\nbutton {\n  // stylelint-disable-next-line property-disallowed-list\n  border-radius: 0;\n}\n\n// Explicitly remove focus outline in Chromium when it shouldn't be\n// visible (e.g. as result of mouse click or touch tap). It already\n// should be doing this automatically, but seems to currently be\n// confused and applies its very visible two-tone outline anyway.\n\nbutton:focus:not(:focus-visible) {\n  outline: 0;\n}\n\n// 1. Remove the margin in Firefox and Safari\n\ninput,\nbutton,\nselect,\noptgroup,\ntextarea {\n  margin: 0; // 1\n  font-family: inherit;\n  @include font-size(inherit);\n  line-height: inherit;\n}\n\n// Remove the inheritance of text transform in Firefox\nbutton,\nselect {\n  text-transform: none;\n}\n// Set the cursor for non-`<button>` buttons\n//\n// Details at https://github.com/twbs/bootstrap/pull/30562\n[role=\"button\"] {\n  cursor: pointer;\n}\n\nselect {\n  // Remove the inheritance of word-wrap in Safari.\n  // See https://github.com/twbs/bootstrap/issues/24990\n  word-wrap: normal;\n\n  // Undo the opacity change from Chrome\n  &:disabled {\n    opacity: 1;\n  }\n}\n\n// Remove the dropdown arrow only from text type inputs built with datalists in Chrome.\n// See https://stackoverflow.com/a/54997118\n\n[list]:not([type=\"date\"]):not([type=\"datetime-local\"]):not([type=\"month\"]):not([type=\"week\"]):not([type=\"time\"])::-webkit-calendar-picker-indicator {\n  display: none !important;\n}\n\n// 1. Prevent a WebKit bug where (2) destroys native `audio` and `video`\n//    controls in Android 4.\n// 2. Correct the inability to style clickable types in iOS and Safari.\n// 3. Opinionated: add \"hand\" cursor to non-disabled button elements.\n\nbutton,\n[type=\"button\"], // 1\n[type=\"reset\"],\n[type=\"submit\"] {\n  -webkit-appearance: button; // 2\n\n  @if $enable-button-pointers {\n    &:not(:disabled) {\n      cursor: pointer; // 3\n    }\n  }\n}\n\n// Remove inner border and padding from Firefox, but don't restore the outline like Normalize.\n\n::-moz-focus-inner {\n  padding: 0;\n  border-style: none;\n}\n\n// 1. Textareas should really only resize vertically so they don't break their (horizontal) containers.\n\ntextarea {\n  resize: vertical; // 1\n}\n\n// 1. Browsers set a default `min-width: min-content;` on fieldsets,\n//    unlike e.g. `<div>`s, which have `min-width: 0;` by default.\n//    So we reset that to ensure fieldsets behave more like a standard block element.\n//    See https://github.com/twbs/bootstrap/issues/12359\n//    and https://html.spec.whatwg.org/multipage/#the-fieldset-and-legend-elements\n// 2. Reset the default outline behavior of fieldsets so they don't affect page layout.\n\nfieldset {\n  min-width: 0; // 1\n  padding: 0; // 2\n  margin: 0; // 2\n  border: 0; // 2\n}\n\n// 1. By using `float: left`, the legend will behave like a block element.\n//    This way the border of a fieldset wraps around the legend if present.\n// 2. Fix wrapping bug.\n//    See https://github.com/twbs/bootstrap/issues/29712\n\nlegend {\n  float: left; // 1\n  width: 100%;\n  padding: 0;\n  margin-bottom: $legend-margin-bottom;\n  @include font-size($legend-font-size);\n  font-weight: $legend-font-weight;\n  line-height: inherit;\n\n  + * {\n    clear: left; // 2\n  }\n}\n\n// Fix height of inputs with a type of datetime-local, date, month, week, or time\n// See https://github.com/twbs/bootstrap/issues/18842\n\n::-webkit-datetime-edit-fields-wrapper,\n::-webkit-datetime-edit-text,\n::-webkit-datetime-edit-minute,\n::-webkit-datetime-edit-hour-field,\n::-webkit-datetime-edit-day-field,\n::-webkit-datetime-edit-month-field,\n::-webkit-datetime-edit-year-field {\n  padding: 0;\n}\n\n::-webkit-inner-spin-button {\n  height: auto;\n}\n\n// 1. This overrides the extra rounded corners on search inputs in iOS so that our\n//    `.form-control` class can properly style them. Note that this cannot simply\n//    be added to `.form-control` as it's not specific enough. For details, see\n//    https://github.com/twbs/bootstrap/issues/11586.\n// 2. Correct the outline style in Safari.\n\n[type=\"search\"] {\n  -webkit-appearance: textfield; // 1\n  outline-offset: -2px; // 2\n}\n\n// 1. A few input types should stay LTR\n// See https://rtlstyling.com/posts/rtl-styling#form-inputs\n// 2. RTL only output\n// See https://rtlcss.com/learn/usage-guide/control-directives/#raw\n\n/* rtl:raw:\n[type=\"tel\"],\n[type=\"url\"],\n[type=\"email\"],\n[type=\"number\"] {\n  direction: ltr;\n}\n*/\n\n// Remove the inner padding in Chrome and Safari on macOS.\n\n::-webkit-search-decoration {\n  -webkit-appearance: none;\n}\n\n// Remove padding around color pickers in webkit browsers\n\n::-webkit-color-swatch-wrapper {\n  padding: 0;\n}\n\n\n// 1. Inherit font family and line height for file input buttons\n// 2. Correct the inability to style clickable types in iOS and Safari.\n\n::file-selector-button {\n  font: inherit; // 1\n  -webkit-appearance: button; // 2\n}\n\n// Correct element displays\n\noutput {\n  display: inline-block;\n}\n\n// Remove border from iframe\n\niframe {\n  border: 0;\n}\n\n// Summary\n//\n// 1. Add the correct display in all browsers\n\nsummary {\n  display: list-item; // 1\n  cursor: pointer;\n}\n\n\n// Progress\n//\n// Add the correct vertical alignment in Chrome, Firefox, and Opera.\n\nprogress {\n  vertical-align: baseline;\n}\n\n\n// Hidden attribute\n//\n// Always hide an element with the `hidden` HTML attribute.\n\n[hidden] {\n  display: none !important;\n}\n","// Variables\n//\n// Variables should follow the `$component-state-property-size` formula for\n// consistent naming. Ex: $nav-link-disabled-color and $modal-content-box-shadow-xs.\n\n// Color system\n\n// scss-docs-start gray-color-variables\n$white:    #fff !default;\n$gray-100: #f8f9fa !default;\n$gray-200: #e9ecef !default;\n$gray-300: #dee2e6 !default;\n$gray-400: #ced4da !default;\n$gray-500: #adb5bd !default;\n$gray-600: #6c757d !default;\n$gray-700: #495057 !default;\n$gray-800: #343a40 !default;\n$gray-900: #212529 !default;\n$black:    #000 !default;\n// scss-docs-end gray-color-variables\n\n// fusv-disable\n// scss-docs-start gray-colors-map\n$grays: (\n  \"100\": $gray-100,\n  \"200\": $gray-200,\n  \"300\": $gray-300,\n  \"400\": $gray-400,\n  \"500\": $gray-500,\n  \"600\": $gray-600,\n  \"700\": $gray-700,\n  \"800\": $gray-800,\n  \"900\": $gray-900\n) !default;\n// scss-docs-end gray-colors-map\n// fusv-enable\n\n// scss-docs-start color-variables\n$blue:    #0d6efd !default;\n$indigo:  #6610f2 !default;\n$purple:  #6f42c1 !default;\n$pink:    #d63384 !default;\n$red:     #dc3545 !default;\n$orange:  #fd7e14 !default;\n$yellow:  #ffc107 !default;\n$green:   #198754 !default;\n$teal:    #20c997 !default;\n$cyan:    #0dcaf0 !default;\n// scss-docs-end color-variables\n\n// scss-docs-start colors-map\n$colors: (\n  \"blue\":       $blue,\n  \"indigo\":     $indigo,\n  \"purple\":     $purple,\n  \"pink\":       $pink,\n  \"red\":        $red,\n  \"orange\":     $orange,\n  \"yellow\":     $yellow,\n  \"green\":      $green,\n  \"teal\":       $teal,\n  \"cyan\":       $cyan,\n  \"black\":      $black,\n  \"white\":      $white,\n  \"gray\":       $gray-600,\n  \"gray-dark\":  $gray-800\n) !default;\n// scss-docs-end colors-map\n\n// The contrast ratio to reach against white, to determine if color changes from \"light\" to \"dark\". Acceptable values for WCAG 2.0 are 3, 4.5 and 7.\n// See https://www.w3.org/TR/WCAG20/#visual-audio-contrast-contrast\n$min-contrast-ratio:   4.5 !default;\n\n// Customize the light and dark text colors for use in our color contrast function.\n$color-contrast-dark:      $black !default;\n$color-contrast-light:     $white !default;\n\n// fusv-disable\n$blue-100: tint-color($blue, 80%) !default;\n$blue-200: tint-color($blue, 60%) !default;\n$blue-300: tint-color($blue, 40%) !default;\n$blue-400: tint-color($blue, 20%) !default;\n$blue-500: $blue !default;\n$blue-600: shade-color($blue, 20%) !default;\n$blue-700: shade-color($blue, 40%) !default;\n$blue-800: shade-color($blue, 60%) !default;\n$blue-900: shade-color($blue, 80%) !default;\n\n$indigo-100: tint-color($indigo, 80%) !default;\n$indigo-200: tint-color($indigo, 60%) !default;\n$indigo-300: tint-color($indigo, 40%) !default;\n$indigo-400: tint-color($indigo, 20%) !default;\n$indigo-500: $indigo !default;\n$indigo-600: shade-color($indigo, 20%) !default;\n$indigo-700: shade-color($indigo, 40%) !default;\n$indigo-800: shade-color($indigo, 60%) !default;\n$indigo-900: shade-color($indigo, 80%) !default;\n\n$purple-100: tint-color($purple, 80%) !default;\n$purple-200: tint-color($purple, 60%) !default;\n$purple-300: tint-color($purple, 40%) !default;\n$purple-400: tint-color($purple, 20%) !default;\n$purple-500: $purple !default;\n$purple-600: shade-color($purple, 20%) !default;\n$purple-700: shade-color($purple, 40%) !default;\n$purple-800: shade-color($purple, 60%) !default;\n$purple-900: shade-color($purple, 80%) !default;\n\n$pink-100: tint-color($pink, 80%) !default;\n$pink-200: tint-color($pink, 60%) !default;\n$pink-300: tint-color($pink, 40%) !default;\n$pink-400: tint-color($pink, 20%) !default;\n$pink-500: $pink !default;\n$pink-600: shade-color($pink, 20%) !default;\n$pink-700: shade-color($pink, 40%) !default;\n$pink-800: shade-color($pink, 60%) !default;\n$pink-900: shade-color($pink, 80%) !default;\n\n$red-100: tint-color($red, 80%) !default;\n$red-200: tint-color($red, 60%) !default;\n$red-300: tint-color($red, 40%) !default;\n$red-400: tint-color($red, 20%) !default;\n$red-500: $red !default;\n$red-600: shade-color($red, 20%) !default;\n$red-700: shade-color($red, 40%) !default;\n$red-800: shade-color($red, 60%) !default;\n$red-900: shade-color($red, 80%) !default;\n\n$orange-100: tint-color($orange, 80%) !default;\n$orange-200: tint-color($orange, 60%) !default;\n$orange-300: tint-color($orange, 40%) !default;\n$orange-400: tint-color($orange, 20%) !default;\n$orange-500: $orange !default;\n$orange-600: shade-color($orange, 20%) !default;\n$orange-700: shade-color($orange, 40%) !default;\n$orange-800: shade-color($orange, 60%) !default;\n$orange-900: shade-color($orange, 80%) !default;\n\n$yellow-100: tint-color($yellow, 80%) !default;\n$yellow-200: tint-color($yellow, 60%) !default;\n$yellow-300: tint-color($yellow, 40%) !default;\n$yellow-400: tint-color($yellow, 20%) !default;\n$yellow-500: $yellow !default;\n$yellow-600: shade-color($yellow, 20%) !default;\n$yellow-700: shade-color($yellow, 40%) !default;\n$yellow-800: shade-color($yellow, 60%) !default;\n$yellow-900: shade-color($yellow, 80%) !default;\n\n$green-100: tint-color($green, 80%) !default;\n$green-200: tint-color($green, 60%) !default;\n$green-300: tint-color($green, 40%) !default;\n$green-400: tint-color($green, 20%) !default;\n$green-500: $green !default;\n$green-600: shade-color($green, 20%) !default;\n$green-700: shade-color($green, 40%) !default;\n$green-800: shade-color($green, 60%) !default;\n$green-900: shade-color($green, 80%) !default;\n\n$teal-100: tint-color($teal, 80%) !default;\n$teal-200: tint-color($teal, 60%) !default;\n$teal-300: tint-color($teal, 40%) !default;\n$teal-400: tint-color($teal, 20%) !default;\n$teal-500: $teal !default;\n$teal-600: shade-color($teal, 20%) !default;\n$teal-700: shade-color($teal, 40%) !default;\n$teal-800: shade-color($teal, 60%) !default;\n$teal-900: shade-color($teal, 80%) !default;\n\n$cyan-100: tint-color($cyan, 80%) !default;\n$cyan-200: tint-color($cyan, 60%) !default;\n$cyan-300: tint-color($cyan, 40%) !default;\n$cyan-400: tint-color($cyan, 20%) !default;\n$cyan-500: $cyan !default;\n$cyan-600: shade-color($cyan, 20%) !default;\n$cyan-700: shade-color($cyan, 40%) !default;\n$cyan-800: shade-color($cyan, 60%) !default;\n$cyan-900: shade-color($cyan, 80%) !default;\n\n$blues: (\n  \"blue-100\": $blue-100,\n  \"blue-200\": $blue-200,\n  \"blue-300\": $blue-300,\n  \"blue-400\": $blue-400,\n  \"blue-500\": $blue-500,\n  \"blue-600\": $blue-600,\n  \"blue-700\": $blue-700,\n  \"blue-800\": $blue-800,\n  \"blue-900\": $blue-900\n) !default;\n\n$indigos: (\n  \"indigo-100\": $indigo-100,\n  \"indigo-200\": $indigo-200,\n  \"indigo-300\": $indigo-300,\n  \"indigo-400\": $indigo-400,\n  \"indigo-500\": $indigo-500,\n  \"indigo-600\": $indigo-600,\n  \"indigo-700\": $indigo-700,\n  \"indigo-800\": $indigo-800,\n  \"indigo-900\": $indigo-900\n) !default;\n\n$purples: (\n  \"purple-100\": $purple-100,\n  \"purple-200\": $purple-200,\n  \"purple-300\": $purple-300,\n  \"purple-400\": $purple-400,\n  \"purple-500\": $purple-500,\n  \"purple-600\": $purple-600,\n  \"purple-700\": $purple-700,\n  \"purple-800\": $purple-800,\n  \"purple-900\": $purple-900\n) !default;\n\n$pinks: (\n  \"pink-100\": $pink-100,\n  \"pink-200\": $pink-200,\n  \"pink-300\": $pink-300,\n  \"pink-400\": $pink-400,\n  \"pink-500\": $pink-500,\n  \"pink-600\": $pink-600,\n  \"pink-700\": $pink-700,\n  \"pink-800\": $pink-800,\n  \"pink-900\": $pink-900\n) !default;\n\n$reds: (\n  \"red-100\": $red-100,\n  \"red-200\": $red-200,\n  \"red-300\": $red-300,\n  \"red-400\": $red-400,\n  \"red-500\": $red-500,\n  \"red-600\": $red-600,\n  \"red-700\": $red-700,\n  \"red-800\": $red-800,\n  \"red-900\": $red-900\n) !default;\n\n$oranges: (\n  \"orange-100\": $orange-100,\n  \"orange-200\": $orange-200,\n  \"orange-300\": $orange-300,\n  \"orange-400\": $orange-400,\n  \"orange-500\": $orange-500,\n  \"orange-600\": $orange-600,\n  \"orange-700\": $orange-700,\n  \"orange-800\": $orange-800,\n  \"orange-900\": $orange-900\n) !default;\n\n$yellows: (\n  \"yellow-100\": $yellow-100,\n  \"yellow-200\": $yellow-200,\n  \"yellow-300\": $yellow-300,\n  \"yellow-400\": $yellow-400,\n  \"yellow-500\": $yellow-500,\n  \"yellow-600\": $yellow-600,\n  \"yellow-700\": $yellow-700,\n  \"yellow-800\": $yellow-800,\n  \"yellow-900\": $yellow-900\n) !default;\n\n$greens: (\n  \"green-100\": $green-100,\n  \"green-200\": $green-200,\n  \"green-300\": $green-300,\n  \"green-400\": $green-400,\n  \"green-500\": $green-500,\n  \"green-600\": $green-600,\n  \"green-700\": $green-700,\n  \"green-800\": $green-800,\n  \"green-900\": $green-900\n) !default;\n\n$teals: (\n  \"teal-100\": $teal-100,\n  \"teal-200\": $teal-200,\n  \"teal-300\": $teal-300,\n  \"teal-400\": $teal-400,\n  \"teal-500\": $teal-500,\n  \"teal-600\": $teal-600,\n  \"teal-700\": $teal-700,\n  \"teal-800\": $teal-800,\n  \"teal-900\": $teal-900\n) !default;\n\n$cyans: (\n  \"cyan-100\": $cyan-100,\n  \"cyan-200\": $cyan-200,\n  \"cyan-300\": $cyan-300,\n  \"cyan-400\": $cyan-400,\n  \"cyan-500\": $cyan-500,\n  \"cyan-600\": $cyan-600,\n  \"cyan-700\": $cyan-700,\n  \"cyan-800\": $cyan-800,\n  \"cyan-900\": $cyan-900\n) !default;\n// fusv-enable\n\n// scss-docs-start theme-color-variables\n$primary:       $blue !default;\n$secondary:     $gray-600 !default;\n$success:       $green !default;\n$info:          $cyan !default;\n$warning:       $yellow !default;\n$danger:        $red !default;\n$light:         $gray-100 !default;\n$dark:          $gray-900 !default;\n// scss-docs-end theme-color-variables\n\n// scss-docs-start theme-colors-map\n$theme-colors: (\n  \"primary\":    $primary,\n  \"secondary\":  $secondary,\n  \"success\":    $success,\n  \"info\":       $info,\n  \"warning\":    $warning,\n  \"danger\":     $danger,\n  \"light\":      $light,\n  \"dark\":       $dark\n) !default;\n// scss-docs-end theme-colors-map\n\n// scss-docs-start theme-text-variables\n$primary-text-emphasis:   shade-color($primary, 60%) !default;\n$secondary-text-emphasis: shade-color($secondary, 60%) !default;\n$success-text-emphasis:   shade-color($success, 60%) !default;\n$info-text-emphasis:      shade-color($info, 60%) !default;\n$warning-text-emphasis:   shade-color($warning, 60%) !default;\n$danger-text-emphasis:    shade-color($danger, 60%) !default;\n$light-text-emphasis:     $gray-700 !default;\n$dark-text-emphasis:      $gray-700 !default;\n// scss-docs-end theme-text-variables\n\n// scss-docs-start theme-bg-subtle-variables\n$primary-bg-subtle:       tint-color($primary, 80%) !default;\n$secondary-bg-subtle:     tint-color($secondary, 80%) !default;\n$success-bg-subtle:       tint-color($success, 80%) !default;\n$info-bg-subtle:          tint-color($info, 80%) !default;\n$warning-bg-subtle:       tint-color($warning, 80%) !default;\n$danger-bg-subtle:        tint-color($danger, 80%) !default;\n$light-bg-subtle:         mix($gray-100, $white) !default;\n$dark-bg-subtle:          $gray-400 !default;\n// scss-docs-end theme-bg-subtle-variables\n\n// scss-docs-start theme-border-subtle-variables\n$primary-border-subtle:   tint-color($primary, 60%) !default;\n$secondary-border-subtle: tint-color($secondary, 60%) !default;\n$success-border-subtle:   tint-color($success, 60%) !default;\n$info-border-subtle:      tint-color($info, 60%) !default;\n$warning-border-subtle:   tint-color($warning, 60%) !default;\n$danger-border-subtle:    tint-color($danger, 60%) !default;\n$light-border-subtle:     $gray-200 !default;\n$dark-border-subtle:      $gray-500 !default;\n// scss-docs-end theme-border-subtle-variables\n\n// Characters which are escaped by the escape-svg function\n$escaped-characters: (\n  (\"<\", \"%3c\"),\n  (\">\", \"%3e\"),\n  (\"#\", \"%23\"),\n  (\"(\", \"%28\"),\n  (\")\", \"%29\"),\n) !default;\n\n// Options\n//\n// Quickly modify global styling by enabling or disabling optional features.\n\n$enable-caret:                true !default;\n$enable-rounded:              true !default;\n$enable-shadows:              false !default;\n$enable-gradients:            false !default;\n$enable-transitions:          true !default;\n$enable-reduced-motion:       true !default;\n$enable-smooth-scroll:        true !default;\n$enable-grid-classes:         true !default;\n$enable-container-classes:    true !default;\n$enable-cssgrid:              false !default;\n$enable-button-pointers:      true !default;\n$enable-rfs:                  true !default;\n$enable-validation-icons:     true !default;\n$enable-negative-margins:     false !default;\n$enable-deprecation-messages: true !default;\n$enable-important-utilities:  true !default;\n\n$enable-dark-mode:            true !default;\n$color-mode-type:             data !default; // `data` or `media-query`\n\n// Prefix for :root CSS variables\n\n$variable-prefix:             bs- !default; // Deprecated in v5.2.0 for the shorter `$prefix`\n$prefix:                      $variable-prefix !default;\n\n// Gradient\n//\n// The gradient which is added to components if `$enable-gradients` is `true`\n// This gradient is also added to elements with `.bg-gradient`\n// scss-docs-start variable-gradient\n$gradient: linear-gradient(180deg, rgba($white, .15), rgba($white, 0)) !default;\n// scss-docs-end variable-gradient\n\n// Spacing\n//\n// Control the default styling of most Bootstrap elements by modifying these\n// variables. Mostly focused on spacing.\n// You can add more entries to the $spacers map, should you need more variation.\n\n// scss-docs-start spacer-variables-maps\n$spacer: 1rem !default;\n$spacers: (\n  0: 0,\n  1: $spacer * .25,\n  2: $spacer * .5,\n  3: $spacer,\n  4: $spacer * 1.5,\n  5: $spacer * 3,\n) !default;\n// scss-docs-end spacer-variables-maps\n\n// Position\n//\n// Define the edge positioning anchors of the position utilities.\n\n// scss-docs-start position-map\n$position-values: (\n  0: 0,\n  50: 50%,\n  100: 100%\n) !default;\n// scss-docs-end position-map\n\n// Body\n//\n// Settings for the `<body>` element.\n\n$body-text-align:           null !default;\n$body-color:                $gray-900 !default;\n$body-bg:                   $white !default;\n\n$body-secondary-color:      rgba($body-color, .75) !default;\n$body-secondary-bg:         $gray-200 !default;\n\n$body-tertiary-color:       rgba($body-color, .5) !default;\n$body-tertiary-bg:          $gray-100 !default;\n\n$body-emphasis-color:       $black !default;\n\n// Links\n//\n// Style anchor elements.\n\n$link-color:                              $primary !default;\n$link-decoration:                         underline !default;\n$link-shade-percentage:                   20% !default;\n$link-hover-color:                        shift-color($link-color, $link-shade-percentage) !default;\n$link-hover-decoration:                   null !default;\n\n$stretched-link-pseudo-element:           after !default;\n$stretched-link-z-index:                  1 !default;\n\n// Icon links\n// scss-docs-start icon-link-variables\n$icon-link-gap:               .375rem !default;\n$icon-link-underline-offset:  .25em !default;\n$icon-link-icon-size:         1em !default;\n$icon-link-icon-transition:   .2s ease-in-out transform !default;\n$icon-link-icon-transform:    translate3d(.25em, 0, 0) !default;\n// scss-docs-end icon-link-variables\n\n// Paragraphs\n//\n// Style p element.\n\n$paragraph-margin-bottom:   1rem !default;\n\n\n// Grid breakpoints\n//\n// Define the minimum dimensions at which your layout will change,\n// adapting to different screen sizes, for use in media queries.\n\n// scss-docs-start grid-breakpoints\n$grid-breakpoints: (\n  xs: 0,\n  sm: 576px,\n  md: 768px,\n  lg: 992px,\n  xl: 1200px,\n  xxl: 1400px\n) !default;\n// scss-docs-end grid-breakpoints\n\n@include _assert-ascending($grid-breakpoints, \"$grid-breakpoints\");\n@include _assert-starts-at-zero($grid-breakpoints, \"$grid-breakpoints\");\n\n\n// Grid containers\n//\n// Define the maximum width of `.container` for different screen sizes.\n\n// scss-docs-start container-max-widths\n$container-max-widths: (\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1140px,\n  xxl: 1320px\n) !default;\n// scss-docs-end container-max-widths\n\n@include _assert-ascending($container-max-widths, \"$container-max-widths\");\n\n\n// Grid columns\n//\n// Set the number of columns and specify the width of the gutters.\n\n$grid-columns:                12 !default;\n$grid-gutter-width:           1.5rem !default;\n$grid-row-columns:            6 !default;\n\n// Container padding\n\n$container-padding-x: $grid-gutter-width !default;\n\n\n// Components\n//\n// Define common padding and border radius sizes and more.\n\n// scss-docs-start border-variables\n$border-width:                1px !default;\n$border-widths: (\n  1: 1px,\n  2: 2px,\n  3: 3px,\n  4: 4px,\n  5: 5px\n) !default;\n$border-style:                solid !default;\n$border-color:                $gray-300 !default;\n$border-color-translucent:    rgba($black, .175) !default;\n// scss-docs-end border-variables\n\n// scss-docs-start border-radius-variables\n$border-radius:               .375rem !default;\n$border-radius-sm:            .25rem !default;\n$border-radius-lg:            .5rem !default;\n$border-radius-xl:            1rem !default;\n$border-radius-xxl:           2rem !default;\n$border-radius-pill:          50rem !default;\n// scss-docs-end border-radius-variables\n// fusv-disable\n$border-radius-2xl:           $border-radius-xxl !default; // Deprecated in v5.3.0\n// fusv-enable\n\n// scss-docs-start box-shadow-variables\n$box-shadow:                  0 .5rem 1rem rgba($black, .15) !default;\n$box-shadow-sm:               0 .125rem .25rem rgba($black, .075) !default;\n$box-shadow-lg:               0 1rem 3rem rgba($black, .175) !default;\n$box-shadow-inset:            inset 0 1px 2px rgba($black, .075) !default;\n// scss-docs-end box-shadow-variables\n\n$component-active-color:      $white !default;\n$component-active-bg:         $primary !default;\n\n// scss-docs-start focus-ring-variables\n$focus-ring-width:      .25rem !default;\n$focus-ring-opacity:    .25 !default;\n$focus-ring-color:      rgba($primary, $focus-ring-opacity) !default;\n$focus-ring-blur:       0 !default;\n$focus-ring-box-shadow: 0 0 $focus-ring-blur $focus-ring-width $focus-ring-color !default;\n// scss-docs-end focus-ring-variables\n\n// scss-docs-start caret-variables\n$caret-width:                 .3em !default;\n$caret-vertical-align:        $caret-width * .85 !default;\n$caret-spacing:               $caret-width * .85 !default;\n// scss-docs-end caret-variables\n\n$transition-base:             all .2s ease-in-out !default;\n$transition-fade:             opacity .15s linear !default;\n// scss-docs-start collapse-transition\n$transition-collapse:         height .35s ease !default;\n$transition-collapse-width:   width .35s ease !default;\n// scss-docs-end collapse-transition\n\n// stylelint-disable function-disallowed-list\n// scss-docs-start aspect-ratios\n$aspect-ratios: (\n  \"1x1\": 100%,\n  \"4x3\": calc(3 / 4 * 100%),\n  \"16x9\": calc(9 / 16 * 100%),\n  \"21x9\": calc(9 / 21 * 100%)\n) !default;\n// scss-docs-end aspect-ratios\n// stylelint-enable function-disallowed-list\n\n// Typography\n//\n// Font, line-height, and color for body text, headings, and more.\n\n// scss-docs-start font-variables\n// stylelint-disable value-keyword-case\n$font-family-sans-serif:      system-ui, -apple-system, \"Segoe UI\", Roboto, \"Helvetica Neue\", \"Noto Sans\", \"Liberation Sans\", Arial, sans-serif, \"Apple Color Emoji\", \"Segoe UI Emoji\", \"Segoe UI Symbol\", \"Noto Color Emoji\" !default;\n$font-family-monospace:       SFMono-Regular, Menlo, Monaco, Consolas, \"Liberation Mono\", \"Courier New\", monospace !default;\n// stylelint-enable value-keyword-case\n$font-family-base:            var(--#{$prefix}font-sans-serif) !default;\n$font-family-code:            var(--#{$prefix}font-monospace) !default;\n\n// $font-size-root affects the value of `rem`, which is used for as well font sizes, paddings, and margins\n// $font-size-base affects the font size of the body text\n$font-size-root:              null !default;\n$font-size-base:              1rem !default; // Assumes the browser default, typically `16px`\n$font-size-sm:                $font-size-base * .875 !default;\n$font-size-lg:                $font-size-base * 1.25 !default;\n\n$font-weight-lighter:         lighter !default;\n$font-weight-light:           300 !default;\n$font-weight-normal:          400 !default;\n$font-weight-medium:          500 !default;\n$font-weight-semibold:        600 !default;\n$font-weight-bold:            700 !default;\n$font-weight-bolder:          bolder !default;\n\n$font-weight-base:            $font-weight-normal !default;\n\n$line-height-base:            1.5 !default;\n$line-height-sm:              1.25 !default;\n$line-height-lg:              2 !default;\n\n$h1-font-size:                $font-size-base * 2.5 !default;\n$h2-font-size:                $font-size-base * 2 !default;\n$h3-font-size:                $font-size-base * 1.75 !default;\n$h4-font-size:                $font-size-base * 1.5 !default;\n$h5-font-size:                $font-size-base * 1.25 !default;\n$h6-font-size:                $font-size-base !default;\n// scss-docs-end font-variables\n\n// scss-docs-start font-sizes\n$font-sizes: (\n  1: $h1-font-size,\n  2: $h2-font-size,\n  3: $h3-font-size,\n  4: $h4-font-size,\n  5: $h5-font-size,\n  6: $h6-font-size\n) !default;\n// scss-docs-end font-sizes\n\n// scss-docs-start headings-variables\n$headings-margin-bottom:      $spacer * .5 !default;\n$headings-font-family:        null !default;\n$headings-font-style:         null !default;\n$headings-font-weight:        500 !default;\n$headings-line-height:        1.2 !default;\n$headings-color:              inherit !default;\n// scss-docs-end headings-variables\n\n// scss-docs-start display-headings\n$display-font-sizes: (\n  1: 5rem,\n  2: 4.5rem,\n  3: 4rem,\n  4: 3.5rem,\n  5: 3rem,\n  6: 2.5rem\n) !default;\n\n$display-font-family: null !default;\n$display-font-style:  null !default;\n$display-font-weight: 300 !default;\n$display-line-height: $headings-line-height !default;\n// scss-docs-end display-headings\n\n// scss-docs-start type-variables\n$lead-font-size:              $font-size-base * 1.25 !default;\n$lead-font-weight:            300 !default;\n\n$small-font-size:             .875em !default;\n\n$sub-sup-font-size:           .75em !default;\n\n// fusv-disable\n$text-muted:                  var(--#{$prefix}secondary-color) !default; // Deprecated in 5.3.0\n// fusv-enable\n\n$initialism-font-size:        $small-font-size !default;\n\n$blockquote-margin-y:         $spacer !default;\n$blockquote-font-size:        $font-size-base * 1.25 !default;\n$blockquote-footer-color:     $gray-600 !default;\n$blockquote-footer-font-size: $small-font-size !default;\n\n$hr-margin-y:                 $spacer !default;\n$hr-color:                    inherit !default;\n\n// fusv-disable\n$hr-bg-color:                 null !default; // Deprecated in v5.2.0\n$hr-height:                   null !default; // Deprecated in v5.2.0\n// fusv-enable\n\n$hr-border-color:             null !default; // Allows for inherited colors\n$hr-border-width:             var(--#{$prefix}border-width) !default;\n$hr-opacity:                  .25 !default;\n\n// scss-docs-start vr-variables\n$vr-border-width:             var(--#{$prefix}border-width) !default;\n// scss-docs-end vr-variables\n\n$legend-margin-bottom:        .5rem !default;\n$legend-font-size:            1.5rem !default;\n$legend-font-weight:          null !default;\n\n$dt-font-weight:              $font-weight-bold !default;\n\n$list-inline-padding:         .5rem !default;\n\n$mark-padding:                .1875em !default;\n$mark-color:                  $body-color !default;\n$mark-bg:                     $yellow-100 !default;\n// scss-docs-end type-variables\n\n\n// Tables\n//\n// Customizes the `.table` component with basic values, each used across all table variations.\n\n// scss-docs-start table-variables\n$table-cell-padding-y:        .5rem !default;\n$table-cell-padding-x:        .5rem !default;\n$table-cell-padding-y-sm:     .25rem !default;\n$table-cell-padding-x-sm:     .25rem !default;\n\n$table-cell-vertical-align:   top !default;\n\n$table-color:                 var(--#{$prefix}emphasis-color) !default;\n$table-bg:                    var(--#{$prefix}body-bg) !default;\n$table-accent-bg:             transparent !default;\n\n$table-th-font-weight:        null !default;\n\n$table-striped-color:         $table-color !default;\n$table-striped-bg-factor:     .05 !default;\n$table-striped-bg:            rgba(var(--#{$prefix}emphasis-color-rgb), $table-striped-bg-factor) !default;\n\n$table-active-color:          $table-color !default;\n$table-active-bg-factor:      .1 !default;\n$table-active-bg:             rgba(var(--#{$prefix}emphasis-color-rgb), $table-active-bg-factor) !default;\n\n$table-hover-color:           $table-color !default;\n$table-hover-bg-factor:       .075 !default;\n$table-hover-bg:              rgba(var(--#{$prefix}emphasis-color-rgb), $table-hover-bg-factor) !default;\n\n$table-border-factor:         .2 !default;\n$table-border-width:          var(--#{$prefix}border-width) !default;\n$table-border-color:          var(--#{$prefix}border-color) !default;\n\n$table-striped-order:         odd !default;\n$table-striped-columns-order: even !default;\n\n$table-group-separator-color: currentcolor !default;\n\n$table-caption-color:         var(--#{$prefix}secondary-color) !default;\n\n$table-bg-scale:              -80% !default;\n// scss-docs-end table-variables\n\n// scss-docs-start table-loop\n$table-variants: (\n  \"primary\":    shift-color($primary, $table-bg-scale),\n  \"secondary\":  shift-color($secondary, $table-bg-scale),\n  \"success\":    shift-color($success, $table-bg-scale),\n  \"info\":       shift-color($info, $table-bg-scale),\n  \"warning\":    shift-color($warning, $table-bg-scale),\n  \"danger\":     shift-color($danger, $table-bg-scale),\n  \"light\":      $light,\n  \"dark\":       $dark,\n) !default;\n// scss-docs-end table-loop\n\n\n// Buttons + Forms\n//\n// Shared variables that are reassigned to `$input-` and `$btn-` specific variables.\n\n// scss-docs-start input-btn-variables\n$input-btn-padding-y:         .375rem !default;\n$input-btn-padding-x:         .75rem !default;\n$input-btn-font-family:       null !default;\n$input-btn-font-size:         $font-size-base !default;\n$input-btn-line-height:       $line-height-base !default;\n\n$input-btn-focus-width:         $focus-ring-width !default;\n$input-btn-focus-color-opacity: $focus-ring-opacity !default;\n$input-btn-focus-color:         $focus-ring-color !default;\n$input-btn-focus-blur:          $focus-ring-blur !default;\n$input-btn-focus-box-shadow:    $focus-ring-box-shadow !default;\n\n$input-btn-padding-y-sm:      .25rem !default;\n$input-btn-padding-x-sm:      .5rem !default;\n$input-btn-font-size-sm:      $font-size-sm !default;\n\n$input-btn-padding-y-lg:      .5rem !default;\n$input-btn-padding-x-lg:      1rem !default;\n$input-btn-font-size-lg:      $font-size-lg !default;\n\n$input-btn-border-width:      var(--#{$prefix}border-width) !default;\n// scss-docs-end input-btn-variables\n\n\n// Buttons\n//\n// For each of Bootstrap's buttons, define text, background, and border color.\n\n// scss-docs-start btn-variables\n$btn-color:                   var(--#{$prefix}body-color) !default;\n$btn-padding-y:               $input-btn-padding-y !default;\n$btn-padding-x:               $input-btn-padding-x !default;\n$btn-font-family:             $input-btn-font-family !default;\n$btn-font-size:               $input-btn-font-size !default;\n$btn-line-height:             $input-btn-line-height !default;\n$btn-white-space:             null !default; // Set to `nowrap` to prevent text wrapping\n\n$btn-padding-y-sm:            $input-btn-padding-y-sm !default;\n$btn-padding-x-sm:            $input-btn-padding-x-sm !default;\n$btn-font-size-sm:            $input-btn-font-size-sm !default;\n\n$btn-padding-y-lg:            $input-btn-padding-y-lg !default;\n$btn-padding-x-lg:            $input-btn-padding-x-lg !default;\n$btn-font-size-lg:            $input-btn-font-size-lg !default;\n\n$btn-border-width:            $input-btn-border-width !default;\n\n$btn-font-weight:             $font-weight-normal !default;\n$btn-box-shadow:              inset 0 1px 0 rgba($white, .15), 0 1px 1px rgba($black, .075) !default;\n$btn-focus-width:             $input-btn-focus-width !default;\n$btn-focus-box-shadow:        $input-btn-focus-box-shadow !default;\n$btn-disabled-opacity:        .65 !default;\n$btn-active-box-shadow:       inset 0 3px 5px rgba($black, .125) !default;\n\n$btn-link-color:              var(--#{$prefix}link-color) !default;\n$btn-link-hover-color:        var(--#{$prefix}link-hover-color) !default;\n$btn-link-disabled-color:     $gray-600 !default;\n$btn-link-focus-shadow-rgb:   to-rgb(mix(color-contrast($link-color), $link-color, 15%)) !default;\n\n// Allows for customizing button radius independently from global border radius\n$btn-border-radius:           var(--#{$prefix}border-radius) !default;\n$btn-border-radius-sm:        var(--#{$prefix}border-radius-sm) !default;\n$btn-border-radius-lg:        var(--#{$prefix}border-radius-lg) !default;\n\n$btn-transition:              color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$btn-hover-bg-shade-amount:       15% !default;\n$btn-hover-bg-tint-amount:        15% !default;\n$btn-hover-border-shade-amount:   20% !default;\n$btn-hover-border-tint-amount:    10% !default;\n$btn-active-bg-shade-amount:      20% !default;\n$btn-active-bg-tint-amount:       20% !default;\n$btn-active-border-shade-amount:  25% !default;\n$btn-active-border-tint-amount:   10% !default;\n// scss-docs-end btn-variables\n\n\n// Forms\n\n// scss-docs-start form-text-variables\n$form-text-margin-top:                  .25rem !default;\n$form-text-font-size:                   $small-font-size !default;\n$form-text-font-style:                  null !default;\n$form-text-font-weight:                 null !default;\n$form-text-color:                       var(--#{$prefix}secondary-color) !default;\n// scss-docs-end form-text-variables\n\n// scss-docs-start form-label-variables\n$form-label-margin-bottom:              .5rem !default;\n$form-label-font-size:                  null !default;\n$form-label-font-style:                 null !default;\n$form-label-font-weight:                null !default;\n$form-label-color:                      null !default;\n// scss-docs-end form-label-variables\n\n// scss-docs-start form-input-variables\n$input-padding-y:                       $input-btn-padding-y !default;\n$input-padding-x:                       $input-btn-padding-x !default;\n$input-font-family:                     $input-btn-font-family !default;\n$input-font-size:                       $input-btn-font-size !default;\n$input-font-weight:                     $font-weight-base !default;\n$input-line-height:                     $input-btn-line-height !default;\n\n$input-padding-y-sm:                    $input-btn-padding-y-sm !default;\n$input-padding-x-sm:                    $input-btn-padding-x-sm !default;\n$input-font-size-sm:                    $input-btn-font-size-sm !default;\n\n$input-padding-y-lg:                    $input-btn-padding-y-lg !default;\n$input-padding-x-lg:                    $input-btn-padding-x-lg !default;\n$input-font-size-lg:                    $input-btn-font-size-lg !default;\n\n$input-bg:                              var(--#{$prefix}body-bg) !default;\n$input-disabled-color:                  null !default;\n$input-disabled-bg:                     var(--#{$prefix}secondary-bg) !default;\n$input-disabled-border-color:           null !default;\n\n$input-color:                           var(--#{$prefix}body-color) !default;\n$input-border-color:                    var(--#{$prefix}border-color) !default;\n$input-border-width:                    $input-btn-border-width !default;\n$input-box-shadow:                      var(--#{$prefix}box-shadow-inset) !default;\n\n$input-border-radius:                   var(--#{$prefix}border-radius) !default;\n$input-border-radius-sm:                var(--#{$prefix}border-radius-sm) !default;\n$input-border-radius-lg:                var(--#{$prefix}border-radius-lg) !default;\n\n$input-focus-bg:                        $input-bg !default;\n$input-focus-border-color:              tint-color($component-active-bg, 50%) !default;\n$input-focus-color:                     $input-color !default;\n$input-focus-width:                     $input-btn-focus-width !default;\n$input-focus-box-shadow:                $input-btn-focus-box-shadow !default;\n\n$input-placeholder-color:               var(--#{$prefix}secondary-color) !default;\n$input-plaintext-color:                 var(--#{$prefix}body-color) !default;\n\n$input-height-border:                   calc(#{$input-border-width} * 2) !default; // stylelint-disable-line function-disallowed-list\n\n$input-height-inner:                    add($input-line-height * 1em, $input-padding-y * 2) !default;\n$input-height-inner-half:               add($input-line-height * .5em, $input-padding-y) !default;\n$input-height-inner-quarter:            add($input-line-height * .25em, $input-padding-y * .5) !default;\n\n$input-height:                          add($input-line-height * 1em, add($input-padding-y * 2, $input-height-border, false)) !default;\n$input-height-sm:                       add($input-line-height * 1em, add($input-padding-y-sm * 2, $input-height-border, false)) !default;\n$input-height-lg:                       add($input-line-height * 1em, add($input-padding-y-lg * 2, $input-height-border, false)) !default;\n\n$input-transition:                      border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$form-color-width:                      3rem !default;\n// scss-docs-end form-input-variables\n\n// scss-docs-start form-check-variables\n$form-check-input-width:                  1em !default;\n$form-check-min-height:                   $font-size-base * $line-height-base !default;\n$form-check-padding-start:                $form-check-input-width + .5em !default;\n$form-check-margin-bottom:                .125rem !default;\n$form-check-label-color:                  null !default;\n$form-check-label-cursor:                 null !default;\n$form-check-transition:                   null !default;\n\n$form-check-input-active-filter:          brightness(90%) !default;\n\n$form-check-input-bg:                     $input-bg !default;\n$form-check-input-border:                 var(--#{$prefix}border-width) solid var(--#{$prefix}border-color) !default;\n$form-check-input-border-radius:          .25em !default;\n$form-check-radio-border-radius:          50% !default;\n$form-check-input-focus-border:           $input-focus-border-color !default;\n$form-check-input-focus-box-shadow:       $focus-ring-box-shadow !default;\n\n$form-check-input-checked-color:          $component-active-color !default;\n$form-check-input-checked-bg-color:       $component-active-bg !default;\n$form-check-input-checked-border-color:   $form-check-input-checked-bg-color !default;\n$form-check-input-checked-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 20 20'><path fill='none' stroke='#{$form-check-input-checked-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='3' d='m6 10 3 3 6-6'/></svg>\") !default;\n$form-check-radio-checked-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='2' fill='#{$form-check-input-checked-color}'/></svg>\") !default;\n\n$form-check-input-indeterminate-color:          $component-active-color !default;\n$form-check-input-indeterminate-bg-color:       $component-active-bg !default;\n$form-check-input-indeterminate-border-color:   $form-check-input-indeterminate-bg-color !default;\n$form-check-input-indeterminate-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 20 20'><path fill='none' stroke='#{$form-check-input-indeterminate-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='3' d='M6 10h8'/></svg>\") !default;\n\n$form-check-input-disabled-opacity:        .5 !default;\n$form-check-label-disabled-opacity:        $form-check-input-disabled-opacity !default;\n$form-check-btn-check-disabled-opacity:    $btn-disabled-opacity !default;\n\n$form-check-inline-margin-end:    1rem !default;\n// scss-docs-end form-check-variables\n\n// scss-docs-start form-switch-variables\n$form-switch-color:               rgba($black, .25) !default;\n$form-switch-width:               2em !default;\n$form-switch-padding-start:       $form-switch-width + .5em !default;\n$form-switch-bg-image:            url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-color}'/></svg>\") !default;\n$form-switch-border-radius:       $form-switch-width !default;\n$form-switch-transition:          background-position .15s ease-in-out !default;\n\n$form-switch-focus-color:         $input-focus-border-color !default;\n$form-switch-focus-bg-image:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-focus-color}'/></svg>\") !default;\n\n$form-switch-checked-color:       $component-active-color !default;\n$form-switch-checked-bg-image:    url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-checked-color}'/></svg>\") !default;\n$form-switch-checked-bg-position: right center !default;\n// scss-docs-end form-switch-variables\n\n// scss-docs-start input-group-variables\n$input-group-addon-padding-y:           $input-padding-y !default;\n$input-group-addon-padding-x:           $input-padding-x !default;\n$input-group-addon-font-weight:         $input-font-weight !default;\n$input-group-addon-color:               $input-color !default;\n$input-group-addon-bg:                  var(--#{$prefix}tertiary-bg) !default;\n$input-group-addon-border-color:        $input-border-color !default;\n// scss-docs-end input-group-variables\n\n// scss-docs-start form-select-variables\n$form-select-padding-y:             $input-padding-y !default;\n$form-select-padding-x:             $input-padding-x !default;\n$form-select-font-family:           $input-font-family !default;\n$form-select-font-size:             $input-font-size !default;\n$form-select-indicator-padding:     $form-select-padding-x * 3 !default; // Extra padding for background-image\n$form-select-font-weight:           $input-font-weight !default;\n$form-select-line-height:           $input-line-height !default;\n$form-select-color:                 $input-color !default;\n$form-select-bg:                    $input-bg !default;\n$form-select-disabled-color:        null !default;\n$form-select-disabled-bg:           $input-disabled-bg !default;\n$form-select-disabled-border-color: $input-disabled-border-color !default;\n$form-select-bg-position:           right $form-select-padding-x center !default;\n$form-select-bg-size:               16px 12px !default; // In pixels because image dimensions\n$form-select-indicator-color:       $gray-800 !default;\n$form-select-indicator:             url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16'><path fill='none' stroke='#{$form-select-indicator-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='2' d='m2 5 6 6 6-6'/></svg>\") !default;\n\n$form-select-feedback-icon-padding-end: $form-select-padding-x * 2.5 + $form-select-indicator-padding !default;\n$form-select-feedback-icon-position:    center right $form-select-indicator-padding !default;\n$form-select-feedback-icon-size:        $input-height-inner-half $input-height-inner-half !default;\n\n$form-select-border-width:        $input-border-width !default;\n$form-select-border-color:        $input-border-color !default;\n$form-select-border-radius:       $input-border-radius !default;\n$form-select-box-shadow:          var(--#{$prefix}box-shadow-inset) !default;\n\n$form-select-focus-border-color:  $input-focus-border-color !default;\n$form-select-focus-width:         $input-focus-width !default;\n$form-select-focus-box-shadow:    0 0 0 $form-select-focus-width $input-btn-focus-color !default;\n\n$form-select-padding-y-sm:        $input-padding-y-sm !default;\n$form-select-padding-x-sm:        $input-padding-x-sm !default;\n$form-select-font-size-sm:        $input-font-size-sm !default;\n$form-select-border-radius-sm:    $input-border-radius-sm !default;\n\n$form-select-padding-y-lg:        $input-padding-y-lg !default;\n$form-select-padding-x-lg:        $input-padding-x-lg !default;\n$form-select-font-size-lg:        $input-font-size-lg !default;\n$form-select-border-radius-lg:    $input-border-radius-lg !default;\n\n$form-select-transition:          $input-transition !default;\n// scss-docs-end form-select-variables\n\n// scss-docs-start form-range-variables\n$form-range-track-width:          100% !default;\n$form-range-track-height:         .5rem !default;\n$form-range-track-cursor:         pointer !default;\n$form-range-track-bg:             var(--#{$prefix}secondary-bg) !default;\n$form-range-track-border-radius:  1rem !default;\n$form-range-track-box-shadow:     var(--#{$prefix}box-shadow-inset) !default;\n\n$form-range-thumb-width:                   1rem !default;\n$form-range-thumb-height:                  $form-range-thumb-width !default;\n$form-range-thumb-bg:                      $component-active-bg !default;\n$form-range-thumb-border:                  0 !default;\n$form-range-thumb-border-radius:           1rem !default;\n$form-range-thumb-box-shadow:              0 .1rem .25rem rgba($black, .1) !default;\n$form-range-thumb-focus-box-shadow:        0 0 0 1px $body-bg, $input-focus-box-shadow !default;\n$form-range-thumb-focus-box-shadow-width:  $input-focus-width !default; // For focus box shadow issue in Edge\n$form-range-thumb-active-bg:               tint-color($component-active-bg, 70%) !default;\n$form-range-thumb-disabled-bg:             var(--#{$prefix}secondary-color) !default;\n$form-range-thumb-transition:              background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n// scss-docs-end form-range-variables\n\n// scss-docs-start form-file-variables\n$form-file-button-color:          $input-color !default;\n$form-file-button-bg:             var(--#{$prefix}tertiary-bg) !default;\n$form-file-button-hover-bg:       var(--#{$prefix}secondary-bg) !default;\n// scss-docs-end form-file-variables\n\n// scss-docs-start form-floating-variables\n$form-floating-height:                  add(3.5rem, $input-height-border) !default;\n$form-floating-line-height:             1.25 !default;\n$form-floating-padding-x:               $input-padding-x !default;\n$form-floating-padding-y:               1rem !default;\n$form-floating-input-padding-t:         1.625rem !default;\n$form-floating-input-padding-b:         .625rem !default;\n$form-floating-label-height:            1.5em !default;\n$form-floating-label-opacity:           .65 !default;\n$form-floating-label-transform:         scale(.85) translateY(-.5rem) translateX(.15rem) !default;\n$form-floating-label-disabled-color:    $gray-600 !default;\n$form-floating-transition:              opacity .1s ease-in-out, transform .1s ease-in-out !default;\n// scss-docs-end form-floating-variables\n\n// Form validation\n\n// scss-docs-start form-feedback-variables\n$form-feedback-margin-top:          $form-text-margin-top !default;\n$form-feedback-font-size:           $form-text-font-size !default;\n$form-feedback-font-style:          $form-text-font-style !default;\n$form-feedback-valid-color:         $success !default;\n$form-feedback-invalid-color:       $danger !default;\n\n$form-feedback-icon-valid-color:    $form-feedback-valid-color !default;\n$form-feedback-icon-valid:          url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 8 8'><path fill='#{$form-feedback-icon-valid-color}' d='M2.3 6.73.6 4.53c-.4-1.04.46-1.4 1.1-.8l1.1 1.4 3.4-3.8c.6-.63 1.6-.27 1.2.7l-4 4.6c-.43.5-.8.4-1.1.1z'/></svg>\") !default;\n$form-feedback-icon-invalid-color:  $form-feedback-invalid-color !default;\n$form-feedback-icon-invalid:        url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 12 12' width='12' height='12' fill='none' stroke='#{$form-feedback-icon-invalid-color}'><circle cx='6' cy='6' r='4.5'/><path stroke-linejoin='round' d='M5.8 3.6h.4L6 6.5z'/><circle cx='6' cy='8.2' r='.6' fill='#{$form-feedback-icon-invalid-color}' stroke='none'/></svg>\") !default;\n// scss-docs-end form-feedback-variables\n\n// scss-docs-start form-validation-colors\n$form-valid-color:                  $form-feedback-valid-color !default;\n$form-valid-border-color:           $form-feedback-valid-color !default;\n$form-invalid-color:                $form-feedback-invalid-color !default;\n$form-invalid-border-color:         $form-feedback-invalid-color !default;\n// scss-docs-end form-validation-colors\n\n// scss-docs-start form-validation-states\n$form-validation-states: (\n  \"valid\": (\n    \"color\": var(--#{$prefix}form-valid-color),\n    \"icon\": $form-feedback-icon-valid,\n    \"tooltip-color\": #fff,\n    \"tooltip-bg-color\": var(--#{$prefix}success),\n    \"focus-box-shadow\": 0 0 $input-btn-focus-blur $input-focus-width rgba(var(--#{$prefix}success-rgb), $input-btn-focus-color-opacity),\n    \"border-color\": var(--#{$prefix}form-valid-border-color),\n  ),\n  \"invalid\": (\n    \"color\": var(--#{$prefix}form-invalid-color),\n    \"icon\": $form-feedback-icon-invalid,\n    \"tooltip-color\": #fff,\n    \"tooltip-bg-color\": var(--#{$prefix}danger),\n    \"focus-box-shadow\": 0 0 $input-btn-focus-blur $input-focus-width rgba(var(--#{$prefix}danger-rgb), $input-btn-focus-color-opacity),\n    \"border-color\": var(--#{$prefix}form-invalid-border-color),\n  )\n) !default;\n// scss-docs-end form-validation-states\n\n// Z-index master list\n//\n// Warning: Avoid customizing these values. They're used for a bird's eye view\n// of components dependent on the z-axis and are designed to all work together.\n\n// scss-docs-start zindex-stack\n$zindex-dropdown:                   1000 !default;\n$zindex-sticky:                     1020 !default;\n$zindex-fixed:                      1030 !default;\n$zindex-offcanvas-backdrop:         1040 !default;\n$zindex-offcanvas:                  1045 !default;\n$zindex-modal-backdrop:             1050 !default;\n$zindex-modal:                      1055 !default;\n$zindex-popover:                    1070 !default;\n$zindex-tooltip:                    1080 !default;\n$zindex-toast:                      1090 !default;\n// scss-docs-end zindex-stack\n\n// scss-docs-start zindex-levels-map\n$zindex-levels: (\n  n1: -1,\n  0: 0,\n  1: 1,\n  2: 2,\n  3: 3\n) !default;\n// scss-docs-end zindex-levels-map\n\n\n// Navs\n\n// scss-docs-start nav-variables\n$nav-link-padding-y:                .5rem !default;\n$nav-link-padding-x:                1rem !default;\n$nav-link-font-size:                null !default;\n$nav-link-font-weight:              null !default;\n$nav-link-color:                    var(--#{$prefix}link-color) !default;\n$nav-link-hover-color:              var(--#{$prefix}link-hover-color) !default;\n$nav-link-transition:               color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out !default;\n$nav-link-disabled-color:           var(--#{$prefix}secondary-color) !default;\n$nav-link-focus-box-shadow:         $focus-ring-box-shadow !default;\n\n$nav-tabs-border-color:             var(--#{$prefix}border-color) !default;\n$nav-tabs-border-width:             var(--#{$prefix}border-width) !default;\n$nav-tabs-border-radius:            var(--#{$prefix}border-radius) !default;\n$nav-tabs-link-hover-border-color:  var(--#{$prefix}secondary-bg) var(--#{$prefix}secondary-bg) $nav-tabs-border-color !default;\n$nav-tabs-link-active-color:        var(--#{$prefix}emphasis-color) !default;\n$nav-tabs-link-active-bg:           var(--#{$prefix}body-bg) !default;\n$nav-tabs-link-active-border-color: var(--#{$prefix}border-color) var(--#{$prefix}border-color) $nav-tabs-link-active-bg !default;\n\n$nav-pills-border-radius:           var(--#{$prefix}border-radius) !default;\n$nav-pills-link-active-color:       $component-active-color !default;\n$nav-pills-link-active-bg:          $component-active-bg !default;\n\n$nav-underline-gap:                 1rem !default;\n$nav-underline-border-width:        .125rem !default;\n$nav-underline-link-active-color:   var(--#{$prefix}emphasis-color) !default;\n// scss-docs-end nav-variables\n\n\n// Navbar\n\n// scss-docs-start navbar-variables\n$navbar-padding-y:                  $spacer * .5 !default;\n$navbar-padding-x:                  null !default;\n\n$navbar-nav-link-padding-x:         .5rem !default;\n\n$navbar-brand-font-size:            $font-size-lg !default;\n// Compute the navbar-brand padding-y so the navbar-brand will have the same height as navbar-text and nav-link\n$nav-link-height:                   $font-size-base * $line-height-base + $nav-link-padding-y * 2 !default;\n$navbar-brand-height:               $navbar-brand-font-size * $line-height-base !default;\n$navbar-brand-padding-y:            ($nav-link-height - $navbar-brand-height) * .5 !default;\n$navbar-brand-margin-end:           1rem !default;\n\n$navbar-toggler-padding-y:          .25rem !default;\n$navbar-toggler-padding-x:          .75rem !default;\n$navbar-toggler-font-size:          $font-size-lg !default;\n$navbar-toggler-border-radius:      $btn-border-radius !default;\n$navbar-toggler-focus-width:        $btn-focus-width !default;\n$navbar-toggler-transition:         box-shadow .15s ease-in-out !default;\n\n$navbar-light-color:                rgba(var(--#{$prefix}emphasis-color-rgb), .65) !default;\n$navbar-light-hover-color:          rgba(var(--#{$prefix}emphasis-color-rgb), .8) !default;\n$navbar-light-active-color:         rgba(var(--#{$prefix}emphasis-color-rgb), 1) !default;\n$navbar-light-disabled-color:       rgba(var(--#{$prefix}emphasis-color-rgb), .3) !default;\n$navbar-light-icon-color:           rgba($body-color, .75) !default;\n$navbar-light-toggler-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'><path stroke='#{$navbar-light-icon-color}' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/></svg>\") !default;\n$navbar-light-toggler-border-color: rgba(var(--#{$prefix}emphasis-color-rgb), .15) !default;\n$navbar-light-brand-color:          $navbar-light-active-color !default;\n$navbar-light-brand-hover-color:    $navbar-light-active-color !default;\n// scss-docs-end navbar-variables\n\n// scss-docs-start navbar-dark-variables\n$navbar-dark-color:                 rgba($white, .55) !default;\n$navbar-dark-hover-color:           rgba($white, .75) !default;\n$navbar-dark-active-color:          $white !default;\n$navbar-dark-disabled-color:        rgba($white, .25) !default;\n$navbar-dark-icon-color:            $navbar-dark-color !default;\n$navbar-dark-toggler-icon-bg:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'><path stroke='#{$navbar-dark-icon-color}' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/></svg>\") !default;\n$navbar-dark-toggler-border-color:  rgba($white, .1) !default;\n$navbar-dark-brand-color:           $navbar-dark-active-color !default;\n$navbar-dark-brand-hover-color:     $navbar-dark-active-color !default;\n// scss-docs-end navbar-dark-variables\n\n\n// Dropdowns\n//\n// Dropdown menu container and contents.\n\n// scss-docs-start dropdown-variables\n$dropdown-min-width:                10rem !default;\n$dropdown-padding-x:                0 !default;\n$dropdown-padding-y:                .5rem !default;\n$dropdown-spacer:                   .125rem !default;\n$dropdown-font-size:                $font-size-base !default;\n$dropdown-color:                    var(--#{$prefix}body-color) !default;\n$dropdown-bg:                       var(--#{$prefix}body-bg) !default;\n$dropdown-border-color:             var(--#{$prefix}border-color-translucent) !default;\n$dropdown-border-radius:            var(--#{$prefix}border-radius) !default;\n$dropdown-border-width:             var(--#{$prefix}border-width) !default;\n$dropdown-inner-border-radius:      calc(#{$dropdown-border-radius} - #{$dropdown-border-width}) !default; // stylelint-disable-line function-disallowed-list\n$dropdown-divider-bg:               $dropdown-border-color !default;\n$dropdown-divider-margin-y:         $spacer * .5 !default;\n$dropdown-box-shadow:               var(--#{$prefix}box-shadow) !default;\n\n$dropdown-link-color:               var(--#{$prefix}body-color) !default;\n$dropdown-link-hover-color:         $dropdown-link-color !default;\n$dropdown-link-hover-bg:            var(--#{$prefix}tertiary-bg) !default;\n\n$dropdown-link-active-color:        $component-active-color !default;\n$dropdown-link-active-bg:           $component-active-bg !default;\n\n$dropdown-link-disabled-color:      var(--#{$prefix}tertiary-color) !default;\n\n$dropdown-item-padding-y:           $spacer * .25 !default;\n$dropdown-item-padding-x:           $spacer !default;\n\n$dropdown-header-color:             $gray-600 !default;\n$dropdown-header-padding-x:         $dropdown-item-padding-x !default;\n$dropdown-header-padding-y:         $dropdown-padding-y !default;\n// fusv-disable\n$dropdown-header-padding:           $dropdown-header-padding-y $dropdown-header-padding-x !default; // Deprecated in v5.2.0\n// fusv-enable\n// scss-docs-end dropdown-variables\n\n// scss-docs-start dropdown-dark-variables\n$dropdown-dark-color:               $gray-300 !default;\n$dropdown-dark-bg:                  $gray-800 !default;\n$dropdown-dark-border-color:        $dropdown-border-color !default;\n$dropdown-dark-divider-bg:          $dropdown-divider-bg !default;\n$dropdown-dark-box-shadow:          null !default;\n$dropdown-dark-link-color:          $dropdown-dark-color !default;\n$dropdown-dark-link-hover-color:    $white !default;\n$dropdown-dark-link-hover-bg:       rgba($white, .15) !default;\n$dropdown-dark-link-active-color:   $dropdown-link-active-color !default;\n$dropdown-dark-link-active-bg:      $dropdown-link-active-bg !default;\n$dropdown-dark-link-disabled-color: $gray-500 !default;\n$dropdown-dark-header-color:        $gray-500 !default;\n// scss-docs-end dropdown-dark-variables\n\n\n// Pagination\n\n// scss-docs-start pagination-variables\n$pagination-padding-y:              .375rem !default;\n$pagination-padding-x:              .75rem !default;\n$pagination-padding-y-sm:           .25rem !default;\n$pagination-padding-x-sm:           .5rem !default;\n$pagination-padding-y-lg:           .75rem !default;\n$pagination-padding-x-lg:           1.5rem !default;\n\n$pagination-font-size:              $font-size-base !default;\n\n$pagination-color:                  var(--#{$prefix}link-color) !default;\n$pagination-bg:                     var(--#{$prefix}body-bg) !default;\n$pagination-border-radius:          var(--#{$prefix}border-radius) !default;\n$pagination-border-width:           var(--#{$prefix}border-width) !default;\n$pagination-margin-start:           calc(#{$pagination-border-width} * -1) !default; // stylelint-disable-line function-disallowed-list\n$pagination-border-color:           var(--#{$prefix}border-color) !default;\n\n$pagination-focus-color:            var(--#{$prefix}link-hover-color) !default;\n$pagination-focus-bg:               var(--#{$prefix}secondary-bg) !default;\n$pagination-focus-box-shadow:       $focus-ring-box-shadow !default;\n$pagination-focus-outline:          0 !default;\n\n$pagination-hover-color:            var(--#{$prefix}link-hover-color) !default;\n$pagination-hover-bg:               var(--#{$prefix}tertiary-bg) !default;\n$pagination-hover-border-color:     var(--#{$prefix}border-color) !default; // Todo in v6: remove this?\n\n$pagination-active-color:           $component-active-color !default;\n$pagination-active-bg:              $component-active-bg !default;\n$pagination-active-border-color:    $component-active-bg !default;\n\n$pagination-disabled-color:         var(--#{$prefix}secondary-color) !default;\n$pagination-disabled-bg:            var(--#{$prefix}secondary-bg) !default;\n$pagination-disabled-border-color:  var(--#{$prefix}border-color) !default;\n\n$pagination-transition:              color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$pagination-border-radius-sm:       var(--#{$prefix}border-radius-sm) !default;\n$pagination-border-radius-lg:       var(--#{$prefix}border-radius-lg) !default;\n// scss-docs-end pagination-variables\n\n\n// Placeholders\n\n// scss-docs-start placeholders\n$placeholder-opacity-max:           .5 !default;\n$placeholder-opacity-min:           .2 !default;\n// scss-docs-end placeholders\n\n// Cards\n\n// scss-docs-start card-variables\n$card-spacer-y:                     $spacer !default;\n$card-spacer-x:                     $spacer !default;\n$card-title-spacer-y:               $spacer * .5 !default;\n$card-title-color:                  null !default;\n$card-subtitle-color:               null !default;\n$card-border-width:                 var(--#{$prefix}border-width) !default;\n$card-border-color:                 var(--#{$prefix}border-color-translucent) !default;\n$card-border-radius:                var(--#{$prefix}border-radius) !default;\n$card-box-shadow:                   null !default;\n$card-inner-border-radius:          subtract($card-border-radius, $card-border-width) !default;\n$card-cap-padding-y:                $card-spacer-y * .5 !default;\n$card-cap-padding-x:                $card-spacer-x !default;\n$card-cap-bg:                       rgba(var(--#{$prefix}body-color-rgb), .03) !default;\n$card-cap-color:                    null !default;\n$card-height:                       null !default;\n$card-color:                        null !default;\n$card-bg:                           var(--#{$prefix}body-bg) !default;\n$card-img-overlay-padding:          $spacer !default;\n$card-group-margin:                 $grid-gutter-width * .5 !default;\n// scss-docs-end card-variables\n\n// Accordion\n\n// scss-docs-start accordion-variables\n$accordion-padding-y:                     1rem !default;\n$accordion-padding-x:                     1.25rem !default;\n$accordion-color:                         var(--#{$prefix}body-color) !default;\n$accordion-bg:                            var(--#{$prefix}body-bg) !default;\n$accordion-border-width:                  var(--#{$prefix}border-width) !default;\n$accordion-border-color:                  var(--#{$prefix}border-color) !default;\n$accordion-border-radius:                 var(--#{$prefix}border-radius) !default;\n$accordion-inner-border-radius:           subtract($accordion-border-radius, $accordion-border-width) !default;\n\n$accordion-body-padding-y:                $accordion-padding-y !default;\n$accordion-body-padding-x:                $accordion-padding-x !default;\n\n$accordion-button-padding-y:              $accordion-padding-y !default;\n$accordion-button-padding-x:              $accordion-padding-x !default;\n$accordion-button-color:                  var(--#{$prefix}body-color) !default;\n$accordion-button-bg:                     var(--#{$prefix}accordion-bg) !default;\n$accordion-transition:                    $btn-transition, border-radius .15s ease !default;\n$accordion-button-active-bg:              var(--#{$prefix}primary-bg-subtle) !default;\n$accordion-button-active-color:           var(--#{$prefix}primary-text-emphasis) !default;\n\n// fusv-disable\n$accordion-button-focus-border-color:     $input-focus-border-color !default; // Deprecated in v5.3.3\n// fusv-enable\n$accordion-button-focus-box-shadow:       $btn-focus-box-shadow !default;\n\n$accordion-icon-width:                    1.25rem !default;\n$accordion-icon-color:                    $body-color !default;\n$accordion-icon-active-color:             $primary-text-emphasis !default;\n$accordion-icon-transition:               transform .2s ease-in-out !default;\n$accordion-icon-transform:                rotate(-180deg) !default;\n\n$accordion-button-icon:         url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='none' stroke='#{$accordion-icon-color}' stroke-linecap='round' stroke-linejoin='round'><path d='M2 5L8 11L14 5'/></svg>\") !default;\n$accordion-button-active-icon:  url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='none' stroke='#{$accordion-icon-active-color}' stroke-linecap='round' stroke-linejoin='round'><path d='M2 5L8 11L14 5'/></svg>\") !default;\n// scss-docs-end accordion-variables\n\n// Tooltips\n\n// scss-docs-start tooltip-variables\n$tooltip-font-size:                 $font-size-sm !default;\n$tooltip-max-width:                 200px !default;\n$tooltip-color:                     var(--#{$prefix}body-bg) !default;\n$tooltip-bg:                        var(--#{$prefix}emphasis-color) !default;\n$tooltip-border-radius:             var(--#{$prefix}border-radius) !default;\n$tooltip-opacity:                   .9 !default;\n$tooltip-padding-y:                 $spacer * .25 !default;\n$tooltip-padding-x:                 $spacer * .5 !default;\n$tooltip-margin:                    null !default; // TODO: remove this in v6\n\n$tooltip-arrow-width:               .8rem !default;\n$tooltip-arrow-height:              .4rem !default;\n// fusv-disable\n$tooltip-arrow-color:               null !default; // Deprecated in Bootstrap 5.2.0 for CSS variables\n// fusv-enable\n// scss-docs-end tooltip-variables\n\n// Form tooltips must come after regular tooltips\n// scss-docs-start tooltip-feedback-variables\n$form-feedback-tooltip-padding-y:     $tooltip-padding-y !default;\n$form-feedback-tooltip-padding-x:     $tooltip-padding-x !default;\n$form-feedback-tooltip-font-size:     $tooltip-font-size !default;\n$form-feedback-tooltip-line-height:   null !default;\n$form-feedback-tooltip-opacity:       $tooltip-opacity !default;\n$form-feedback-tooltip-border-radius: $tooltip-border-radius !default;\n// scss-docs-end tooltip-feedback-variables\n\n\n// Popovers\n\n// scss-docs-start popover-variables\n$popover-font-size:                 $font-size-sm !default;\n$popover-bg:                        var(--#{$prefix}body-bg) !default;\n$popover-max-width:                 276px !default;\n$popover-border-width:              var(--#{$prefix}border-width) !default;\n$popover-border-color:              var(--#{$prefix}border-color-translucent) !default;\n$popover-border-radius:             var(--#{$prefix}border-radius-lg) !default;\n$popover-inner-border-radius:       calc(#{$popover-border-radius} - #{$popover-border-width}) !default; // stylelint-disable-line function-disallowed-list\n$popover-box-shadow:                var(--#{$prefix}box-shadow) !default;\n\n$popover-header-font-size:          $font-size-base !default;\n$popover-header-bg:                 var(--#{$prefix}secondary-bg) !default;\n$popover-header-color:              $headings-color !default;\n$popover-header-padding-y:          .5rem !default;\n$popover-header-padding-x:          $spacer !default;\n\n$popover-body-color:                var(--#{$prefix}body-color) !default;\n$popover-body-padding-y:            $spacer !default;\n$popover-body-padding-x:            $spacer !default;\n\n$popover-arrow-width:               1rem !default;\n$popover-arrow-height:              .5rem !default;\n// scss-docs-end popover-variables\n\n// fusv-disable\n// Deprecated in Bootstrap 5.2.0 for CSS variables\n$popover-arrow-color:               $popover-bg !default;\n$popover-arrow-outer-color:         var(--#{$prefix}border-color-translucent) !default;\n// fusv-enable\n\n\n// Toasts\n\n// scss-docs-start toast-variables\n$toast-max-width:                   350px !default;\n$toast-padding-x:                   .75rem !default;\n$toast-padding-y:                   .5rem !default;\n$toast-font-size:                   .875rem !default;\n$toast-color:                       null !default;\n$toast-background-color:            rgba(var(--#{$prefix}body-bg-rgb), .85) !default;\n$toast-border-width:                var(--#{$prefix}border-width) !default;\n$toast-border-color:                var(--#{$prefix}border-color-translucent) !default;\n$toast-border-radius:               var(--#{$prefix}border-radius) !default;\n$toast-box-shadow:                  var(--#{$prefix}box-shadow) !default;\n$toast-spacing:                     $container-padding-x !default;\n\n$toast-header-color:                var(--#{$prefix}secondary-color) !default;\n$toast-header-background-color:     rgba(var(--#{$prefix}body-bg-rgb), .85) !default;\n$toast-header-border-color:         $toast-border-color !default;\n// scss-docs-end toast-variables\n\n\n// Badges\n\n// scss-docs-start badge-variables\n$badge-font-size:                   .75em !default;\n$badge-font-weight:                 $font-weight-bold !default;\n$badge-color:                       $white !default;\n$badge-padding-y:                   .35em !default;\n$badge-padding-x:                   .65em !default;\n$badge-border-radius:               var(--#{$prefix}border-radius) !default;\n// scss-docs-end badge-variables\n\n\n// Modals\n\n// scss-docs-start modal-variables\n$modal-inner-padding:               $spacer !default;\n\n$modal-footer-margin-between:       .5rem !default;\n\n$modal-dialog-margin:               .5rem !default;\n$modal-dialog-margin-y-sm-up:       1.75rem !default;\n\n$modal-title-line-height:           $line-height-base !default;\n\n$modal-content-color:               null !default;\n$modal-content-bg:                  var(--#{$prefix}body-bg) !default;\n$modal-content-border-color:        var(--#{$prefix}border-color-translucent) !default;\n$modal-content-border-width:        var(--#{$prefix}border-width) !default;\n$modal-content-border-radius:       var(--#{$prefix}border-radius-lg) !default;\n$modal-content-inner-border-radius: subtract($modal-content-border-radius, $modal-content-border-width) !default;\n$modal-content-box-shadow-xs:       var(--#{$prefix}box-shadow-sm) !default;\n$modal-content-box-shadow-sm-up:    var(--#{$prefix}box-shadow) !default;\n\n$modal-backdrop-bg:                 $black !default;\n$modal-backdrop-opacity:            .5 !default;\n\n$modal-header-border-color:         var(--#{$prefix}border-color) !default;\n$modal-header-border-width:         $modal-content-border-width !default;\n$modal-header-padding-y:            $modal-inner-padding !default;\n$modal-header-padding-x:            $modal-inner-padding !default;\n$modal-header-padding:              $modal-header-padding-y $modal-header-padding-x !default; // Keep this for backwards compatibility\n\n$modal-footer-bg:                   null !default;\n$modal-footer-border-color:         $modal-header-border-color !default;\n$modal-footer-border-width:         $modal-header-border-width !default;\n\n$modal-sm:                          300px !default;\n$modal-md:                          500px !default;\n$modal-lg:                          800px !default;\n$modal-xl:                          1140px !default;\n\n$modal-fade-transform:              translate(0, -50px) !default;\n$modal-show-transform:              none !default;\n$modal-transition:                  transform .3s ease-out !default;\n$modal-scale-transform:             scale(1.02) !default;\n// scss-docs-end modal-variables\n\n\n// Alerts\n//\n// Define alert colors, border radius, and padding.\n\n// scss-docs-start alert-variables\n$alert-padding-y:               $spacer !default;\n$alert-padding-x:               $spacer !default;\n$alert-margin-bottom:           1rem !default;\n$alert-border-radius:           var(--#{$prefix}border-radius) !default;\n$alert-link-font-weight:        $font-weight-bold !default;\n$alert-border-width:            var(--#{$prefix}border-width) !default;\n$alert-dismissible-padding-r:   $alert-padding-x * 3 !default; // 3x covers width of x plus default padding on either side\n// scss-docs-end alert-variables\n\n// fusv-disable\n$alert-bg-scale:                -80% !default; // Deprecated in v5.2.0, to be removed in v6\n$alert-border-scale:            -70% !default; // Deprecated in v5.2.0, to be removed in v6\n$alert-color-scale:             40% !default; // Deprecated in v5.2.0, to be removed in v6\n// fusv-enable\n\n// Progress bars\n\n// scss-docs-start progress-variables\n$progress-height:                   1rem !default;\n$progress-font-size:                $font-size-base * .75 !default;\n$progress-bg:                       var(--#{$prefix}secondary-bg) !default;\n$progress-border-radius:            var(--#{$prefix}border-radius) !default;\n$progress-box-shadow:               var(--#{$prefix}box-shadow-inset) !default;\n$progress-bar-color:                $white !default;\n$progress-bar-bg:                   $primary !default;\n$progress-bar-animation-timing:     1s linear infinite !default;\n$progress-bar-transition:           width .6s ease !default;\n// scss-docs-end progress-variables\n\n\n// List group\n\n// scss-docs-start list-group-variables\n$list-group-color:                  var(--#{$prefix}body-color) !default;\n$list-group-bg:                     var(--#{$prefix}body-bg) !default;\n$list-group-border-color:           var(--#{$prefix}border-color) !default;\n$list-group-border-width:           var(--#{$prefix}border-width) !default;\n$list-group-border-radius:          var(--#{$prefix}border-radius) !default;\n\n$list-group-item-padding-y:         $spacer * .5 !default;\n$list-group-item-padding-x:         $spacer !default;\n// fusv-disable\n$list-group-item-bg-scale:          -80% !default; // Deprecated in v5.3.0\n$list-group-item-color-scale:       40% !default; // Deprecated in v5.3.0\n// fusv-enable\n\n$list-group-hover-bg:               var(--#{$prefix}tertiary-bg) !default;\n$list-group-active-color:           $component-active-color !default;\n$list-group-active-bg:              $component-active-bg !default;\n$list-group-active-border-color:    $list-group-active-bg !default;\n\n$list-group-disabled-color:         var(--#{$prefix}secondary-color) !default;\n$list-group-disabled-bg:            $list-group-bg !default;\n\n$list-group-action-color:           var(--#{$prefix}secondary-color) !default;\n$list-group-action-hover-color:     var(--#{$prefix}emphasis-color) !default;\n\n$list-group-action-active-color:    var(--#{$prefix}body-color) !default;\n$list-group-action-active-bg:       var(--#{$prefix}secondary-bg) !default;\n// scss-docs-end list-group-variables\n\n\n// Image thumbnails\n\n// scss-docs-start thumbnail-variables\n$thumbnail-padding:                 .25rem !default;\n$thumbnail-bg:                      var(--#{$prefix}body-bg) !default;\n$thumbnail-border-width:            var(--#{$prefix}border-width) !default;\n$thumbnail-border-color:            var(--#{$prefix}border-color) !default;\n$thumbnail-border-radius:           var(--#{$prefix}border-radius) !default;\n$thumbnail-box-shadow:              var(--#{$prefix}box-shadow-sm) !default;\n// scss-docs-end thumbnail-variables\n\n\n// Figures\n\n// scss-docs-start figure-variables\n$figure-caption-font-size:          $small-font-size !default;\n$figure-caption-color:              var(--#{$prefix}secondary-color) !default;\n// scss-docs-end figure-variables\n\n\n// Breadcrumbs\n\n// scss-docs-start breadcrumb-variables\n$breadcrumb-font-size:              null !default;\n$breadcrumb-padding-y:              0 !default;\n$breadcrumb-padding-x:              0 !default;\n$breadcrumb-item-padding-x:         .5rem !default;\n$breadcrumb-margin-bottom:          1rem !default;\n$breadcrumb-bg:                     null !default;\n$breadcrumb-divider-color:          var(--#{$prefix}secondary-color) !default;\n$breadcrumb-active-color:           var(--#{$prefix}secondary-color) !default;\n$breadcrumb-divider:                quote(\"/\") !default;\n$breadcrumb-divider-flipped:        $breadcrumb-divider !default;\n$breadcrumb-border-radius:          null !default;\n// scss-docs-end breadcrumb-variables\n\n// Carousel\n\n// scss-docs-start carousel-variables\n$carousel-control-color:             $white !default;\n$carousel-control-width:             15% !default;\n$carousel-control-opacity:           .5 !default;\n$carousel-control-hover-opacity:     .9 !default;\n$carousel-control-transition:        opacity .15s ease !default;\n\n$carousel-indicator-width:           30px !default;\n$carousel-indicator-height:          3px !default;\n$carousel-indicator-hit-area-height: 10px !default;\n$carousel-indicator-spacer:          3px !default;\n$carousel-indicator-opacity:         .5 !default;\n$carousel-indicator-active-bg:       $white !default;\n$carousel-indicator-active-opacity:  1 !default;\n$carousel-indicator-transition:      opacity .6s ease !default;\n\n$carousel-caption-width:             70% !default;\n$carousel-caption-color:             $white !default;\n$carousel-caption-padding-y:         1.25rem !default;\n$carousel-caption-spacer:            1.25rem !default;\n\n$carousel-control-icon-width:        2rem !default;\n\n$carousel-control-prev-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$carousel-control-color}'><path d='M11.354 1.646a.5.5 0 0 1 0 .708L5.707 8l5.647 5.646a.5.5 0 0 1-.708.708l-6-6a.5.5 0 0 1 0-.708l6-6a.5.5 0 0 1 .708 0z'/></svg>\") !default;\n$carousel-control-next-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$carousel-control-color}'><path d='M4.646 1.646a.5.5 0 0 1 .708 0l6 6a.5.5 0 0 1 0 .708l-6 6a.5.5 0 0 1-.708-.708L10.293 8 4.646 2.354a.5.5 0 0 1 0-.708z'/></svg>\") !default;\n\n$carousel-transition-duration:       .6s !default;\n$carousel-transition:                transform $carousel-transition-duration ease-in-out !default; // Define transform transition first if using multiple transitions (e.g., `transform 2s ease, opacity .5s ease-out`)\n// scss-docs-end carousel-variables\n\n// scss-docs-start carousel-dark-variables\n$carousel-dark-indicator-active-bg:  $black !default;\n$carousel-dark-caption-color:        $black !default;\n$carousel-dark-control-icon-filter:  invert(1) grayscale(100) !default;\n// scss-docs-end carousel-dark-variables\n\n\n// Spinners\n\n// scss-docs-start spinner-variables\n$spinner-width:           2rem !default;\n$spinner-height:          $spinner-width !default;\n$spinner-vertical-align:  -.125em !default;\n$spinner-border-width:    .25em !default;\n$spinner-animation-speed: .75s !default;\n\n$spinner-width-sm:        1rem !default;\n$spinner-height-sm:       $spinner-width-sm !default;\n$spinner-border-width-sm: .2em !default;\n// scss-docs-end spinner-variables\n\n\n// Close\n\n// scss-docs-start close-variables\n$btn-close-width:            1em !default;\n$btn-close-height:           $btn-close-width !default;\n$btn-close-padding-x:        .25em !default;\n$btn-close-padding-y:        $btn-close-padding-x !default;\n$btn-close-color:            $black !default;\n$btn-close-bg:               url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$btn-close-color}'><path d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/></svg>\") !default;\n$btn-close-focus-shadow:     $focus-ring-box-shadow !default;\n$btn-close-opacity:          .5 !default;\n$btn-close-hover-opacity:    .75 !default;\n$btn-close-focus-opacity:    1 !default;\n$btn-close-disabled-opacity: .25 !default;\n$btn-close-white-filter:     invert(1) grayscale(100%) brightness(200%) !default;\n// scss-docs-end close-variables\n\n\n// Offcanvas\n\n// scss-docs-start offcanvas-variables\n$offcanvas-padding-y:               $modal-inner-padding !default;\n$offcanvas-padding-x:               $modal-inner-padding !default;\n$offcanvas-horizontal-width:        400px !default;\n$offcanvas-vertical-height:         30vh !default;\n$offcanvas-transition-duration:     .3s !default;\n$offcanvas-border-color:            $modal-content-border-color !default;\n$offcanvas-border-width:            $modal-content-border-width !default;\n$offcanvas-title-line-height:       $modal-title-line-height !default;\n$offcanvas-bg-color:                var(--#{$prefix}body-bg) !default;\n$offcanvas-color:                   var(--#{$prefix}body-color) !default;\n$offcanvas-box-shadow:              $modal-content-box-shadow-xs !default;\n$offcanvas-backdrop-bg:             $modal-backdrop-bg !default;\n$offcanvas-backdrop-opacity:        $modal-backdrop-opacity !default;\n// scss-docs-end offcanvas-variables\n\n// Code\n\n$code-font-size:                    $small-font-size !default;\n$code-color:                        $pink !default;\n\n$kbd-padding-y:                     .1875rem !default;\n$kbd-padding-x:                     .375rem !default;\n$kbd-font-size:                     $code-font-size !default;\n$kbd-color:                         var(--#{$prefix}body-bg) !default;\n$kbd-bg:                            var(--#{$prefix}body-color) !default;\n$nested-kbd-font-weight:            null !default; // Deprecated in v5.2.0, removing in v6\n\n$pre-color:                         null !default;\n\n@import \"variables-dark\"; // TODO: can be removed safely in v6, only here to avoid breaking changes in v5.3\n","// stylelint-disable property-disallowed-list\n// Single side border-radius\n\n// Helper function to replace negative values with 0\n@function valid-radius($radius) {\n  $return: ();\n  @each $value in $radius {\n    @if type-of($value) == number {\n      $return: append($return, max($value, 0));\n    } @else {\n      $return: append($return, $value);\n    }\n  }\n  @return $return;\n}\n\n// scss-docs-start border-radius-mixins\n@mixin border-radius($radius: $border-radius, $fallback-border-radius: false) {\n  @if $enable-rounded {\n    border-radius: valid-radius($radius);\n  }\n  @else if $fallback-border-radius != false {\n    border-radius: $fallback-border-radius;\n  }\n}\n\n@mixin border-top-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-left-radius: valid-radius($radius);\n    border-top-right-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-end-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-right-radius: valid-radius($radius);\n    border-bottom-right-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-bottom-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-bottom-right-radius: valid-radius($radius);\n    border-bottom-left-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-start-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-left-radius: valid-radius($radius);\n    border-bottom-left-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-top-start-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-left-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-top-end-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-right-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-bottom-end-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-bottom-right-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-bottom-start-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-bottom-left-radius: valid-radius($radius);\n  }\n}\n// scss-docs-end border-radius-mixins\n","//\n// Headings\n//\n.h1 {\n  @extend h1;\n}\n\n.h2 {\n  @extend h2;\n}\n\n.h3 {\n  @extend h3;\n}\n\n.h4 {\n  @extend h4;\n}\n\n.h5 {\n  @extend h5;\n}\n\n.h6 {\n  @extend h6;\n}\n\n\n.lead {\n  @include font-size($lead-font-size);\n  font-weight: $lead-font-weight;\n}\n\n// Type display classes\n@each $display, $font-size in $display-font-sizes {\n  .display-#{$display} {\n    @include font-size($font-size);\n    font-family: $display-font-family;\n    font-style: $display-font-style;\n    font-weight: $display-font-weight;\n    line-height: $display-line-height;\n  }\n}\n\n//\n// Emphasis\n//\n.small {\n  @extend small;\n}\n\n.mark {\n  @extend mark;\n}\n\n//\n// Lists\n//\n\n.list-unstyled {\n  @include list-unstyled();\n}\n\n// Inline turns list items into inline-block\n.list-inline {\n  @include list-unstyled();\n}\n.list-inline-item {\n  display: inline-block;\n\n  &:not(:last-child) {\n    margin-right: $list-inline-padding;\n  }\n}\n\n\n//\n// Misc\n//\n\n// Builds on `abbr`\n.initialism {\n  @include font-size($initialism-font-size);\n  text-transform: uppercase;\n}\n\n// Blockquotes\n.blockquote {\n  margin-bottom: $blockquote-margin-y;\n  @include font-size($blockquote-font-size);\n\n  > :last-child {\n    margin-bottom: 0;\n  }\n}\n\n.blockquote-footer {\n  margin-top: -$blockquote-margin-y;\n  margin-bottom: $blockquote-margin-y;\n  @include font-size($blockquote-footer-font-size);\n  color: $blockquote-footer-color;\n\n  &::before {\n    content: \"\\2014\\00A0\"; // em dash, nbsp\n  }\n}\n","// Lists\n\n// Unstyled keeps list items block level, just removes default browser padding and list-style\n@mixin list-unstyled {\n  padding-left: 0;\n  list-style: none;\n}\n","// Override bootstrap variables\n$spacer: 1rem;\n$container-max-widths: (\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1400px,\n);\n$grid-breakpoints: (\n  xs: 0,\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1200px,\n);\n$dropdown-link-hover-bg: var(--pst-color-surface);\n\n// --pst-color-surface can also be assigned to the dark variant because it is\n// scoped to different values depending on light/dark theme\n$dropdown-dark-link-hover-bg: var(--pst-color-surface);\n$dropdown-link-active-bg: var(--pst-color-surface);\n$dropdown-dark-link-active-bg: var(--pst-color-surface);\n$focus-ring-width: 0.1875rem; // 3px at 100% zoom (0.1875 * 16px base font = 3px)\n$focus-ring-opacity: 1;\n$focus-ring-color: var(--pst-color-accent);\n$focus-ring-blur: 0;\n$focus-ring-box-shadow: 0 0 $focus-ring-blur $focus-ring-width $focus-ring-color;\n\n// outline creates the same style of focus ring, it just uses CSS outline instead of box shadow\n$focus-ring-outline: $focus-ring-color solid $focus-ring-width;\n$btn-focus-box-shadow: $focus-ring-box-shadow;\n","// Responsive images (ensure images don't scale beyond their parents)\n//\n// This is purposefully opt-in via an explicit class rather than being the default for all `<img>`s.\n// We previously tried the \"images are responsive by default\" approach in Bootstrap v2,\n// and abandoned it in Bootstrap v3 because it breaks lots of third-party widgets (including Google Maps)\n// which weren't expecting the images within themselves to be involuntarily resized.\n// See also https://github.com/twbs/bootstrap/issues/18178\n.img-fluid {\n  @include img-fluid();\n}\n\n\n// Image thumbnails\n.img-thumbnail {\n  padding: $thumbnail-padding;\n  background-color: $thumbnail-bg;\n  border: $thumbnail-border-width solid $thumbnail-border-color;\n  @include border-radius($thumbnail-border-radius);\n  @include box-shadow($thumbnail-box-shadow);\n\n  // Keep them at most 100% wide\n  @include img-fluid();\n}\n\n//\n// Figures\n//\n\n.figure {\n  // Ensures the caption's text aligns with the image.\n  display: inline-block;\n}\n\n.figure-img {\n  margin-bottom: $spacer * .5;\n  line-height: 1;\n}\n\n.figure-caption {\n  @include font-size($figure-caption-font-size);\n  color: $figure-caption-color;\n}\n","// Image Mixins\n// - Responsive image\n// - Retina image\n\n\n// Responsive image\n//\n// Keep images from scaling beyond the width of their parents.\n\n@mixin img-fluid {\n  // Part 1: Set a maximum relative to the parent\n  max-width: 100%;\n  // Part 2: Override the height to auto, otherwise images will be stretched\n  // when setting a width and height attribute on the img element.\n  height: auto;\n}\n","// Container widths\n//\n// Set the container width, and override it for fixed navbars in media queries.\n\n@if $enable-container-classes {\n  // Single container class with breakpoint max-widths\n  .container,\n  // 100% wide container at all breakpoints\n  .container-fluid {\n    @include make-container();\n  }\n\n  // Responsive containers that are 100% wide until a breakpoint\n  @each $breakpoint, $container-max-width in $container-max-widths {\n    .container-#{$breakpoint} {\n      @extend .container-fluid;\n    }\n\n    @include media-breakpoint-up($breakpoint, $grid-breakpoints) {\n      %responsive-container-#{$breakpoint} {\n        max-width: $container-max-width;\n      }\n\n      // Extend each breakpoint which is smaller or equal to the current breakpoint\n      $extend-breakpoint: true;\n\n      @each $name, $width in $grid-breakpoints {\n        @if ($extend-breakpoint) {\n          .container#{breakpoint-infix($name, $grid-breakpoints)} {\n            @extend %responsive-container-#{$breakpoint};\n          }\n\n          // Once the current breakpoint is reached, stop extending\n          @if ($breakpoint == $name) {\n            $extend-breakpoint: false;\n          }\n        }\n      }\n    }\n  }\n}\n","// Container mixins\n\n@mixin make-container($gutter: $container-padding-x) {\n  --#{$prefix}gutter-x: #{$gutter};\n  --#{$prefix}gutter-y: 0;\n  width: 100%;\n  padding-right: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  padding-left: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  margin-right: auto;\n  margin-left: auto;\n}\n","// Breakpoint viewport sizes and media queries.\n//\n// Breakpoints are defined as a map of (name: minimum width), order from small to large:\n//\n//    (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px)\n//\n// The map defined in the `$grid-breakpoints` global variable is used as the `$breakpoints` argument by default.\n\n// Name of the next breakpoint, or null for the last breakpoint.\n//\n//    >> breakpoint-next(sm)\n//    md\n//    >> breakpoint-next(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    md\n//    >> breakpoint-next(sm, $breakpoint-names: (xs sm md lg xl xxl))\n//    md\n@function breakpoint-next($name, $breakpoints: $grid-breakpoints, $breakpoint-names: map-keys($breakpoints)) {\n  $n: index($breakpoint-names, $name);\n  @if not $n {\n    @error \"breakpoint `#{$name}` not found in `#{$breakpoints}`\";\n  }\n  @return if($n < length($breakpoint-names), nth($breakpoint-names, $n + 1), null);\n}\n\n// Minimum breakpoint width. Null for the smallest (first) breakpoint.\n//\n//    >> breakpoint-min(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    576px\n@function breakpoint-min($name, $breakpoints: $grid-breakpoints) {\n  $min: map-get($breakpoints, $name);\n  @return if($min != 0, $min, null);\n}\n\n// Maximum breakpoint width.\n// The maximum value is reduced by 0.02px to work around the limitations of\n// `min-` and `max-` prefixes and viewports with fractional widths.\n// See https://www.w3.org/TR/mediaqueries-4/#mq-min-max\n// Uses 0.02px rather than 0.01px to work around a current rounding bug in Safari.\n// See https://bugs.webkit.org/show_bug.cgi?id=178261\n//\n//    >> breakpoint-max(md, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    767.98px\n@function breakpoint-max($name, $breakpoints: $grid-breakpoints) {\n  $max: map-get($breakpoints, $name);\n  @return if($max and $max > 0, $max - .02, null);\n}\n\n// Returns a blank string if smallest breakpoint, otherwise returns the name with a dash in front.\n// Useful for making responsive utilities.\n//\n//    >> breakpoint-infix(xs, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    \"\"  (Returns a blank string)\n//    >> breakpoint-infix(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    \"-sm\"\n@function breakpoint-infix($name, $breakpoints: $grid-breakpoints) {\n  @return if(breakpoint-min($name, $breakpoints) == null, \"\", \"-#{$name}\");\n}\n\n// Media of at least the minimum breakpoint width. No query for the smallest breakpoint.\n// Makes the @content apply to the given breakpoint and wider.\n@mixin media-breakpoint-up($name, $breakpoints: $grid-breakpoints) {\n  $min: breakpoint-min($name, $breakpoints);\n  @if $min {\n    @media (min-width: $min) {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Media of at most the maximum breakpoint width. No query for the largest breakpoint.\n// Makes the @content apply to the given breakpoint and narrower.\n@mixin media-breakpoint-down($name, $breakpoints: $grid-breakpoints) {\n  $max: breakpoint-max($name, $breakpoints);\n  @if $max {\n    @media (max-width: $max) {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Media that spans multiple breakpoint widths.\n// Makes the @content apply between the min and max breakpoints\n@mixin media-breakpoint-between($lower, $upper, $breakpoints: $grid-breakpoints) {\n  $min: breakpoint-min($lower, $breakpoints);\n  $max: breakpoint-max($upper, $breakpoints);\n\n  @if $min != null and $max != null {\n    @media (min-width: $min) and (max-width: $max) {\n      @content;\n    }\n  } @else if $max == null {\n    @include media-breakpoint-up($lower, $breakpoints) {\n      @content;\n    }\n  } @else if $min == null {\n    @include media-breakpoint-down($upper, $breakpoints) {\n      @content;\n    }\n  }\n}\n\n// Media between the breakpoint's minimum and maximum widths.\n// No minimum for the smallest breakpoint, and no maximum for the largest one.\n// Makes the @content apply only to the given breakpoint, not viewports any wider or narrower.\n@mixin media-breakpoint-only($name, $breakpoints: $grid-breakpoints) {\n  $min:  breakpoint-min($name, $breakpoints);\n  $next: breakpoint-next($name, $breakpoints);\n  $max:  breakpoint-max($next, $breakpoints);\n\n  @if $min != null and $max != null {\n    @media (min-width: $min) and (max-width: $max) {\n      @content;\n    }\n  } @else if $max == null {\n    @include media-breakpoint-up($name, $breakpoints) {\n      @content;\n    }\n  } @else if $min == null {\n    @include media-breakpoint-down($next, $breakpoints) {\n      @content;\n    }\n  }\n}\n","// Row\n//\n// Rows contain your columns.\n\n:root {\n  @each $name, $value in $grid-breakpoints {\n    --#{$prefix}breakpoint-#{$name}: #{$value};\n  }\n}\n\n@if $enable-grid-classes {\n  .row {\n    @include make-row();\n\n    > * {\n      @include make-col-ready();\n    }\n  }\n}\n\n@if $enable-cssgrid {\n  .grid {\n    display: grid;\n    grid-template-rows: repeat(var(--#{$prefix}rows, 1), 1fr);\n    grid-template-columns: repeat(var(--#{$prefix}columns, #{$grid-columns}), 1fr);\n    gap: var(--#{$prefix}gap, #{$grid-gutter-width});\n\n    @include make-cssgrid();\n  }\n}\n\n\n// Columns\n//\n// Common styles for small and large grid columns\n\n@if $enable-grid-classes {\n  @include make-grid-columns();\n}\n","// Grid system\n//\n// Generate semantic grid columns with these mixins.\n\n@mixin make-row($gutter: $grid-gutter-width) {\n  --#{$prefix}gutter-x: #{$gutter};\n  --#{$prefix}gutter-y: 0;\n  display: flex;\n  flex-wrap: wrap;\n  // TODO: Revisit calc order after https://github.com/react-bootstrap/react-bootstrap/issues/6039 is fixed\n  margin-top: calc(-1 * var(--#{$prefix}gutter-y)); // stylelint-disable-line function-disallowed-list\n  margin-right: calc(-.5 * var(--#{$prefix}gutter-x)); // stylelint-disable-line function-disallowed-list\n  margin-left: calc(-.5 * var(--#{$prefix}gutter-x)); // stylelint-disable-line function-disallowed-list\n}\n\n@mixin make-col-ready() {\n  // Add box sizing if only the grid is loaded\n  box-sizing: if(variable-exists(include-column-box-sizing) and $include-column-box-sizing, border-box, null);\n  // Prevent columns from becoming too narrow when at smaller grid tiers by\n  // always setting `width: 100%;`. This works because we set the width\n  // later on to override this initial width.\n  flex-shrink: 0;\n  width: 100%;\n  max-width: 100%; // Prevent `.col-auto`, `.col` (& responsive variants) from breaking out the grid\n  padding-right: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  padding-left: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  margin-top: var(--#{$prefix}gutter-y);\n}\n\n@mixin make-col($size: false, $columns: $grid-columns) {\n  @if $size {\n    flex: 0 0 auto;\n    width: percentage(divide($size, $columns));\n\n  } @else {\n    flex: 1 1 0;\n    max-width: 100%;\n  }\n}\n\n@mixin make-col-auto() {\n  flex: 0 0 auto;\n  width: auto;\n}\n\n@mixin make-col-offset($size, $columns: $grid-columns) {\n  $num: divide($size, $columns);\n  margin-left: if($num == 0, 0, percentage($num));\n}\n\n// Row columns\n//\n// Specify on a parent element(e.g., .row) to force immediate children into NN\n// number of columns. Supports wrapping to new lines, but does not do a Masonry\n// style grid.\n@mixin row-cols($count) {\n  > * {\n    flex: 0 0 auto;\n    width: percentage(divide(1, $count));\n  }\n}\n\n// Framework grid generation\n//\n// Used only by Bootstrap to generate the correct number of grid classes given\n// any value of `$grid-columns`.\n\n@mixin make-grid-columns($columns: $grid-columns, $gutter: $grid-gutter-width, $breakpoints: $grid-breakpoints) {\n  @each $breakpoint in map-keys($breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $breakpoints);\n\n    @include media-breakpoint-up($breakpoint, $breakpoints) {\n      // Provide basic `.col-{bp}` classes for equal-width flexbox columns\n      .col#{$infix} {\n        flex: 1 0 0%; // Flexbugs #4: https://github.com/philipwalton/flexbugs#flexbug-4\n      }\n\n      .row-cols#{$infix}-auto > * {\n        @include make-col-auto();\n      }\n\n      @if $grid-row-columns > 0 {\n        @for $i from 1 through $grid-row-columns {\n          .row-cols#{$infix}-#{$i} {\n            @include row-cols($i);\n          }\n        }\n      }\n\n      .col#{$infix}-auto {\n        @include make-col-auto();\n      }\n\n      @if $columns > 0 {\n        @for $i from 1 through $columns {\n          .col#{$infix}-#{$i} {\n            @include make-col($i, $columns);\n          }\n        }\n\n        // `$columns - 1` because offsetting by the width of an entire row isn't possible\n        @for $i from 0 through ($columns - 1) {\n          @if not ($infix == \"\" and $i == 0) { // Avoid emitting useless .offset-0\n            .offset#{$infix}-#{$i} {\n              @include make-col-offset($i, $columns);\n            }\n          }\n        }\n      }\n\n      // Gutters\n      //\n      // Make use of `.g-*`, `.gx-*` or `.gy-*` utilities to change spacing between the columns.\n      @each $key, $value in $gutters {\n        .g#{$infix}-#{$key},\n        .gx#{$infix}-#{$key} {\n          --#{$prefix}gutter-x: #{$value};\n        }\n\n        .g#{$infix}-#{$key},\n        .gy#{$infix}-#{$key} {\n          --#{$prefix}gutter-y: #{$value};\n        }\n      }\n    }\n  }\n}\n\n@mixin make-cssgrid($columns: $grid-columns, $breakpoints: $grid-breakpoints) {\n  @each $breakpoint in map-keys($breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $breakpoints);\n\n    @include media-breakpoint-up($breakpoint, $breakpoints) {\n      @if $columns > 0 {\n        @for $i from 1 through $columns {\n          .g-col#{$infix}-#{$i} {\n            grid-column: auto / span $i;\n          }\n        }\n\n        // Start with `1` because `0` is an invalid value.\n        // Ends with `$columns - 1` because offsetting by the width of an entire row isn't possible.\n        @for $i from 1 through ($columns - 1) {\n          .g-start#{$infix}-#{$i} {\n            grid-column-start: $i;\n          }\n        }\n      }\n    }\n  }\n}\n","//\n// Basic Bootstrap table\n//\n\n.table {\n  // Reset needed for nesting tables\n  --#{$prefix}table-color-type: initial;\n  --#{$prefix}table-bg-type: initial;\n  --#{$prefix}table-color-state: initial;\n  --#{$prefix}table-bg-state: initial;\n  // End of reset\n  --#{$prefix}table-color: #{$table-color};\n  --#{$prefix}table-bg: #{$table-bg};\n  --#{$prefix}table-border-color: #{$table-border-color};\n  --#{$prefix}table-accent-bg: #{$table-accent-bg};\n  --#{$prefix}table-striped-color: #{$table-striped-color};\n  --#{$prefix}table-striped-bg: #{$table-striped-bg};\n  --#{$prefix}table-active-color: #{$table-active-color};\n  --#{$prefix}table-active-bg: #{$table-active-bg};\n  --#{$prefix}table-hover-color: #{$table-hover-color};\n  --#{$prefix}table-hover-bg: #{$table-hover-bg};\n\n  width: 100%;\n  margin-bottom: $spacer;\n  vertical-align: $table-cell-vertical-align;\n  border-color: var(--#{$prefix}table-border-color);\n\n  // Target th & td\n  // We need the child combinator to prevent styles leaking to nested tables which doesn't have a `.table` class.\n  // We use the universal selectors here to simplify the selector (else we would need 6 different selectors).\n  // Another advantage is that this generates less code and makes the selector less specific making it easier to override.\n  // stylelint-disable-next-line selector-max-universal\n  > :not(caption) > * > * {\n    padding: $table-cell-padding-y $table-cell-padding-x;\n    // Following the precept of cascades: https://codepen.io/miriamsuzanne/full/vYNgodb\n    color: var(--#{$prefix}table-color-state, var(--#{$prefix}table-color-type, var(--#{$prefix}table-color)));\n    background-color: var(--#{$prefix}table-bg);\n    border-bottom-width: $table-border-width;\n    box-shadow: inset 0 0 0 9999px var(--#{$prefix}table-bg-state, var(--#{$prefix}table-bg-type, var(--#{$prefix}table-accent-bg)));\n  }\n\n  > tbody {\n    vertical-align: inherit;\n  }\n\n  > thead {\n    vertical-align: bottom;\n  }\n}\n\n.table-group-divider {\n  border-top: calc(#{$table-border-width} * 2) solid $table-group-separator-color; // stylelint-disable-line function-disallowed-list\n}\n\n//\n// Change placement of captions with a class\n//\n\n.caption-top {\n  caption-side: top;\n}\n\n\n//\n// Condensed table w/ half padding\n//\n\n.table-sm {\n  // stylelint-disable-next-line selector-max-universal\n  > :not(caption) > * > * {\n    padding: $table-cell-padding-y-sm $table-cell-padding-x-sm;\n  }\n}\n\n\n// Border versions\n//\n// Add or remove borders all around the table and between all the columns.\n//\n// When borders are added on all sides of the cells, the corners can render odd when\n// these borders do not have the same color or if they are semi-transparent.\n// Therefore we add top and border bottoms to the `tr`s and left and right borders\n// to the `td`s or `th`s\n\n.table-bordered {\n  > :not(caption) > * {\n    border-width: $table-border-width 0;\n\n    // stylelint-disable-next-line selector-max-universal\n    > * {\n      border-width: 0 $table-border-width;\n    }\n  }\n}\n\n.table-borderless {\n  // stylelint-disable-next-line selector-max-universal\n  > :not(caption) > * > * {\n    border-bottom-width: 0;\n  }\n\n  > :not(:first-child) {\n    border-top-width: 0;\n  }\n}\n\n// Zebra-striping\n//\n// Default zebra-stripe styles (alternating gray and transparent backgrounds)\n\n// For rows\n.table-striped {\n  > tbody > tr:nth-of-type(#{$table-striped-order}) > * {\n    --#{$prefix}table-color-type: var(--#{$prefix}table-striped-color);\n    --#{$prefix}table-bg-type: var(--#{$prefix}table-striped-bg);\n  }\n}\n\n// For columns\n.table-striped-columns {\n  > :not(caption) > tr > :nth-child(#{$table-striped-columns-order}) {\n    --#{$prefix}table-color-type: var(--#{$prefix}table-striped-color);\n    --#{$prefix}table-bg-type: var(--#{$prefix}table-striped-bg);\n  }\n}\n\n// Active table\n//\n// The `.table-active` class can be added to highlight rows or cells\n\n.table-active {\n  --#{$prefix}table-color-state: var(--#{$prefix}table-active-color);\n  --#{$prefix}table-bg-state: var(--#{$prefix}table-active-bg);\n}\n\n// Hover effect\n//\n// Placed here since it has to come after the potential zebra striping\n\n.table-hover {\n  > tbody > tr:hover > * {\n    --#{$prefix}table-color-state: var(--#{$prefix}table-hover-color);\n    --#{$prefix}table-bg-state: var(--#{$prefix}table-hover-bg);\n  }\n}\n\n\n// Table variants\n//\n// Table variants set the table cell backgrounds, border colors\n// and the colors of the striped, hovered & active tables\n\n@each $color, $value in $table-variants {\n  @include table-variant($color, $value);\n}\n\n// Responsive tables\n//\n// Generate series of `.table-responsive-*` classes for configuring the screen\n// size of where your table will overflow.\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n  @include media-breakpoint-down($breakpoint) {\n    .table-responsive#{$infix} {\n      overflow-x: auto;\n      -webkit-overflow-scrolling: touch;\n    }\n  }\n}\n","// scss-docs-start table-variant\n@mixin table-variant($state, $background) {\n  .table-#{$state} {\n    $color: color-contrast(opaque($body-bg, $background));\n    $hover-bg: mix($color, $background, percentage($table-hover-bg-factor));\n    $striped-bg: mix($color, $background, percentage($table-striped-bg-factor));\n    $active-bg: mix($color, $background, percentage($table-active-bg-factor));\n    $table-border-color: mix($color, $background, percentage($table-border-factor));\n\n    --#{$prefix}table-color: #{$color};\n    --#{$prefix}table-bg: #{$background};\n    --#{$prefix}table-border-color: #{$table-border-color};\n    --#{$prefix}table-striped-bg: #{$striped-bg};\n    --#{$prefix}table-striped-color: #{color-contrast($striped-bg)};\n    --#{$prefix}table-active-bg: #{$active-bg};\n    --#{$prefix}table-active-color: #{color-contrast($active-bg)};\n    --#{$prefix}table-hover-bg: #{$hover-bg};\n    --#{$prefix}table-hover-color: #{color-contrast($hover-bg)};\n\n    color: var(--#{$prefix}table-color);\n    border-color: var(--#{$prefix}table-border-color);\n  }\n}\n// scss-docs-end table-variant\n","//\n// Labels\n//\n\n.form-label {\n  margin-bottom: $form-label-margin-bottom;\n  @include font-size($form-label-font-size);\n  font-style: $form-label-font-style;\n  font-weight: $form-label-font-weight;\n  color: $form-label-color;\n}\n\n// For use with horizontal and inline forms, when you need the label (or legend)\n// text to align with the form controls.\n.col-form-label {\n  padding-top: add($input-padding-y, $input-border-width);\n  padding-bottom: add($input-padding-y, $input-border-width);\n  margin-bottom: 0; // Override the `<legend>` default\n  @include font-size(inherit); // Override the `<legend>` default\n  font-style: $form-label-font-style;\n  font-weight: $form-label-font-weight;\n  line-height: $input-line-height;\n  color: $form-label-color;\n}\n\n.col-form-label-lg {\n  padding-top: add($input-padding-y-lg, $input-border-width);\n  padding-bottom: add($input-padding-y-lg, $input-border-width);\n  @include font-size($input-font-size-lg);\n}\n\n.col-form-label-sm {\n  padding-top: add($input-padding-y-sm, $input-border-width);\n  padding-bottom: add($input-padding-y-sm, $input-border-width);\n  @include font-size($input-font-size-sm);\n}\n","//\n// Form text\n//\n\n.form-text {\n  margin-top: $form-text-margin-top;\n  @include font-size($form-text-font-size);\n  font-style: $form-text-font-style;\n  font-weight: $form-text-font-weight;\n  color: $form-text-color;\n}\n","//\n// General form controls (plus a few specific high-level interventions)\n//\n\n.form-control {\n  display: block;\n  width: 100%;\n  padding: $input-padding-y $input-padding-x;\n  font-family: $input-font-family;\n  @include font-size($input-font-size);\n  font-weight: $input-font-weight;\n  line-height: $input-line-height;\n  color: $input-color;\n  appearance: none; // Fix appearance for date inputs in Safari\n  background-color: $input-bg;\n  background-clip: padding-box;\n  border: $input-border-width solid $input-border-color;\n\n  // Note: This has no effect on <select>s in some browsers, due to the limited stylability of `<select>`s in CSS.\n  @include border-radius($input-border-radius, 0);\n\n  @include box-shadow($input-box-shadow);\n  @include transition($input-transition);\n\n  &[type=\"file\"] {\n    overflow: hidden; // prevent pseudo element button overlap\n\n    &:not(:disabled):not([readonly]) {\n      cursor: pointer;\n    }\n  }\n\n  // Customize the `:focus` state to imitate native WebKit styles.\n  &:focus {\n    color: $input-focus-color;\n    background-color: $input-focus-bg;\n    border-color: $input-focus-border-color;\n    outline: 0;\n    @if $enable-shadows {\n      @include box-shadow($input-box-shadow, $input-focus-box-shadow);\n    } @else {\n      // Avoid using mixin so we can pass custom focus shadow properly\n      box-shadow: $input-focus-box-shadow;\n    }\n  }\n\n  &::-webkit-date-and-time-value {\n    // On Android Chrome, form-control's \"width: 100%\" makes the input width too small\n    // Tested under Android 11 / Chrome 89, Android 12 / Chrome 100, Android 13 / Chrome 109\n    //\n    // On iOS Safari, form-control's \"appearance: none\" + \"width: 100%\" makes the input width too small\n    // Tested under iOS 16.2 / Safari 16.2\n    min-width: 85px; // Seems to be a good minimum safe width\n\n    // Add some height to date inputs on iOS\n    // https://github.com/twbs/bootstrap/issues/23307\n    // TODO: we can remove this workaround once https://bugs.webkit.org/show_bug.cgi?id=198959 is resolved\n    // Multiply line-height by 1em if it has no unit\n    height: if(unit($input-line-height) == \"\", $input-line-height * 1em, $input-line-height);\n\n    // Android Chrome type=\"date\" is taller than the other inputs\n    // because of \"margin: 1px 24px 1px 4px\" inside the shadow DOM\n    // Tested under Android 11 / Chrome 89, Android 12 / Chrome 100, Android 13 / Chrome 109\n    margin: 0;\n  }\n\n  // Prevent excessive date input height in Webkit\n  // https://github.com/twbs/bootstrap/issues/34433\n  &::-webkit-datetime-edit {\n    display: block;\n    padding: 0;\n  }\n\n  // Placeholder\n  &::placeholder {\n    color: $input-placeholder-color;\n    // Override Firefox's unusual default opacity; see https://github.com/twbs/bootstrap/pull/11526.\n    opacity: 1;\n  }\n\n  // Disabled inputs\n  //\n  // HTML5 says that controls under a fieldset > legend:first-child won't be\n  // disabled if the fieldset is disabled. Due to implementation difficulty, we\n  // don't honor that edge case; we style them as disabled anyway.\n  &:disabled {\n    color: $input-disabled-color;\n    background-color: $input-disabled-bg;\n    border-color: $input-disabled-border-color;\n    // iOS fix for unreadable disabled content; see https://github.com/twbs/bootstrap/issues/11655.\n    opacity: 1;\n  }\n\n  // File input buttons theming\n  &::file-selector-button {\n    padding: $input-padding-y $input-padding-x;\n    margin: (-$input-padding-y) (-$input-padding-x);\n    margin-inline-end: $input-padding-x;\n    color: $form-file-button-color;\n    @include gradient-bg($form-file-button-bg);\n    pointer-events: none;\n    border-color: inherit;\n    border-style: solid;\n    border-width: 0;\n    border-inline-end-width: $input-border-width;\n    border-radius: 0; // stylelint-disable-line property-disallowed-list\n    @include transition($btn-transition);\n  }\n\n  &:hover:not(:disabled):not([readonly])::file-selector-button {\n    background-color: $form-file-button-hover-bg;\n  }\n}\n\n// Readonly controls as plain text\n//\n// Apply class to a readonly input to make it appear like regular plain\n// text (without any border, background color, focus indicator)\n\n.form-control-plaintext {\n  display: block;\n  width: 100%;\n  padding: $input-padding-y 0;\n  margin-bottom: 0; // match inputs if this class comes on inputs with default margins\n  line-height: $input-line-height;\n  color: $input-plaintext-color;\n  background-color: transparent;\n  border: solid transparent;\n  border-width: $input-border-width 0;\n\n  &:focus {\n    outline: 0;\n  }\n\n  &.form-control-sm,\n  &.form-control-lg {\n    padding-right: 0;\n    padding-left: 0;\n  }\n}\n\n// Form control sizing\n//\n// Build on `.form-control` with modifier classes to decrease or increase the\n// height and font-size of form controls.\n//\n// Repeated in `_input_group.scss` to avoid Sass extend issues.\n\n.form-control-sm {\n  min-height: $input-height-sm;\n  padding: $input-padding-y-sm $input-padding-x-sm;\n  @include font-size($input-font-size-sm);\n  @include border-radius($input-border-radius-sm);\n\n  &::file-selector-button {\n    padding: $input-padding-y-sm $input-padding-x-sm;\n    margin: (-$input-padding-y-sm) (-$input-padding-x-sm);\n    margin-inline-end: $input-padding-x-sm;\n  }\n}\n\n.form-control-lg {\n  min-height: $input-height-lg;\n  padding: $input-padding-y-lg $input-padding-x-lg;\n  @include font-size($input-font-size-lg);\n  @include border-radius($input-border-radius-lg);\n\n  &::file-selector-button {\n    padding: $input-padding-y-lg $input-padding-x-lg;\n    margin: (-$input-padding-y-lg) (-$input-padding-x-lg);\n    margin-inline-end: $input-padding-x-lg;\n  }\n}\n\n// Make sure textareas don't shrink too much when resized\n// https://github.com/twbs/bootstrap/pull/29124\n// stylelint-disable selector-no-qualifying-type\ntextarea {\n  &.form-control {\n    min-height: $input-height;\n  }\n\n  &.form-control-sm {\n    min-height: $input-height-sm;\n  }\n\n  &.form-control-lg {\n    min-height: $input-height-lg;\n  }\n}\n// stylelint-enable selector-no-qualifying-type\n\n.form-control-color {\n  width: $form-color-width;\n  height: $input-height;\n  padding: $input-padding-y;\n\n  &:not(:disabled):not([readonly]) {\n    cursor: pointer;\n  }\n\n  &::-moz-color-swatch {\n    border: 0 !important; // stylelint-disable-line declaration-no-important\n    @include border-radius($input-border-radius);\n  }\n\n  &::-webkit-color-swatch {\n    border: 0 !important; // stylelint-disable-line declaration-no-important\n    @include border-radius($input-border-radius);\n  }\n\n  &.form-control-sm { height: $input-height-sm; }\n  &.form-control-lg { height: $input-height-lg; }\n}\n","// stylelint-disable property-disallowed-list\n@mixin transition($transition...) {\n  @if length($transition) == 0 {\n    $transition: $transition-base;\n  }\n\n  @if length($transition) > 1 {\n    @each $value in $transition {\n      @if $value == null or $value == none {\n        @warn \"The keyword 'none' or 'null' must be used as a single argument.\";\n      }\n    }\n  }\n\n  @if $enable-transitions {\n    @if nth($transition, 1) != null {\n      transition: $transition;\n    }\n\n    @if $enable-reduced-motion and nth($transition, 1) != null and nth($transition, 1) != none {\n      @media (prefers-reduced-motion: reduce) {\n        transition: none;\n      }\n    }\n  }\n}\n","// Gradients\n\n// scss-docs-start gradient-bg-mixin\n@mixin gradient-bg($color: null) {\n  background-color: $color;\n\n  @if $enable-gradients {\n    background-image: var(--#{$prefix}gradient);\n  }\n}\n// scss-docs-end gradient-bg-mixin\n\n// scss-docs-start gradient-mixins\n// Horizontal gradient, from left to right\n//\n// Creates two color stops, start and end, by specifying a color and position for each color stop.\n@mixin gradient-x($start-color: $gray-700, $end-color: $gray-800, $start-percent: 0%, $end-percent: 100%) {\n  background-image: linear-gradient(to right, $start-color $start-percent, $end-color $end-percent);\n}\n\n// Vertical gradient, from top to bottom\n//\n// Creates two color stops, start and end, by specifying a color and position for each color stop.\n@mixin gradient-y($start-color: $gray-700, $end-color: $gray-800, $start-percent: null, $end-percent: null) {\n  background-image: linear-gradient(to bottom, $start-color $start-percent, $end-color $end-percent);\n}\n\n@mixin gradient-directional($start-color: $gray-700, $end-color: $gray-800, $deg: 45deg) {\n  background-image: linear-gradient($deg, $start-color, $end-color);\n}\n\n@mixin gradient-x-three-colors($start-color: $blue, $mid-color: $purple, $color-stop: 50%, $end-color: $red) {\n  background-image: linear-gradient(to right, $start-color, $mid-color $color-stop, $end-color);\n}\n\n@mixin gradient-y-three-colors($start-color: $blue, $mid-color: $purple, $color-stop: 50%, $end-color: $red) {\n  background-image: linear-gradient($start-color, $mid-color $color-stop, $end-color);\n}\n\n@mixin gradient-radial($inner-color: $gray-700, $outer-color: $gray-800) {\n  background-image: radial-gradient(circle, $inner-color, $outer-color);\n}\n\n@mixin gradient-striped($color: rgba($white, .15), $angle: 45deg) {\n  background-image: linear-gradient($angle, $color 25%, transparent 25%, transparent 50%, $color 50%, $color 75%, transparent 75%, transparent);\n}\n// scss-docs-end gradient-mixins\n","// Select\n//\n// Replaces the browser default select with a custom one, mostly pulled from\n// https://primer.github.io/.\n\n.form-select {\n  --#{$prefix}form-select-bg-img: #{escape-svg($form-select-indicator)};\n\n  display: block;\n  width: 100%;\n  padding: $form-select-padding-y $form-select-indicator-padding $form-select-padding-y $form-select-padding-x;\n  font-family: $form-select-font-family;\n  @include font-size($form-select-font-size);\n  font-weight: $form-select-font-weight;\n  line-height: $form-select-line-height;\n  color: $form-select-color;\n  appearance: none;\n  background-color: $form-select-bg;\n  background-image: var(--#{$prefix}form-select-bg-img), var(--#{$prefix}form-select-bg-icon, none);\n  background-repeat: no-repeat;\n  background-position: $form-select-bg-position;\n  background-size: $form-select-bg-size;\n  border: $form-select-border-width solid $form-select-border-color;\n  @include border-radius($form-select-border-radius, 0);\n  @include box-shadow($form-select-box-shadow);\n  @include transition($form-select-transition);\n\n  &:focus {\n    border-color: $form-select-focus-border-color;\n    outline: 0;\n    @if $enable-shadows {\n      @include box-shadow($form-select-box-shadow, $form-select-focus-box-shadow);\n    } @else {\n      // Avoid using mixin so we can pass custom focus shadow properly\n      box-shadow: $form-select-focus-box-shadow;\n    }\n  }\n\n  &[multiple],\n  &[size]:not([size=\"1\"]) {\n    padding-right: $form-select-padding-x;\n    background-image: none;\n  }\n\n  &:disabled {\n    color: $form-select-disabled-color;\n    background-color: $form-select-disabled-bg;\n    border-color: $form-select-disabled-border-color;\n  }\n\n  // Remove outline from select box in FF\n  &:-moz-focusring {\n    color: transparent;\n    text-shadow: 0 0 0 $form-select-color;\n  }\n}\n\n.form-select-sm {\n  padding-top: $form-select-padding-y-sm;\n  padding-bottom: $form-select-padding-y-sm;\n  padding-left: $form-select-padding-x-sm;\n  @include font-size($form-select-font-size-sm);\n  @include border-radius($form-select-border-radius-sm);\n}\n\n.form-select-lg {\n  padding-top: $form-select-padding-y-lg;\n  padding-bottom: $form-select-padding-y-lg;\n  padding-left: $form-select-padding-x-lg;\n  @include font-size($form-select-font-size-lg);\n  @include border-radius($form-select-border-radius-lg);\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .form-select {\n      --#{$prefix}form-select-bg-img: #{escape-svg($form-select-indicator-dark)};\n    }\n  }\n}\n","//\n// Check/radio\n//\n\n.form-check {\n  display: block;\n  min-height: $form-check-min-height;\n  padding-left: $form-check-padding-start;\n  margin-bottom: $form-check-margin-bottom;\n\n  .form-check-input {\n    float: left;\n    margin-left: $form-check-padding-start * -1;\n  }\n}\n\n.form-check-reverse {\n  padding-right: $form-check-padding-start;\n  padding-left: 0;\n  text-align: right;\n\n  .form-check-input {\n    float: right;\n    margin-right: $form-check-padding-start * -1;\n    margin-left: 0;\n  }\n}\n\n.form-check-input {\n  --#{$prefix}form-check-bg: #{$form-check-input-bg};\n\n  flex-shrink: 0;\n  width: $form-check-input-width;\n  height: $form-check-input-width;\n  margin-top: ($line-height-base - $form-check-input-width) * .5; // line-height minus check height\n  vertical-align: top;\n  appearance: none;\n  background-color: var(--#{$prefix}form-check-bg);\n  background-image: var(--#{$prefix}form-check-bg-image);\n  background-repeat: no-repeat;\n  background-position: center;\n  background-size: contain;\n  border: $form-check-input-border;\n  print-color-adjust: exact; // Keep themed appearance for print\n  @include transition($form-check-transition);\n\n  &[type=\"checkbox\"] {\n    @include border-radius($form-check-input-border-radius);\n  }\n\n  &[type=\"radio\"] {\n    // stylelint-disable-next-line property-disallowed-list\n    border-radius: $form-check-radio-border-radius;\n  }\n\n  &:active {\n    filter: $form-check-input-active-filter;\n  }\n\n  &:focus {\n    border-color: $form-check-input-focus-border;\n    outline: 0;\n    box-shadow: $form-check-input-focus-box-shadow;\n  }\n\n  &:checked {\n    background-color: $form-check-input-checked-bg-color;\n    border-color: $form-check-input-checked-border-color;\n\n    &[type=\"checkbox\"] {\n      @if $enable-gradients {\n        --#{$prefix}form-check-bg-image: #{escape-svg($form-check-input-checked-bg-image)}, var(--#{$prefix}gradient);\n      } @else {\n        --#{$prefix}form-check-bg-image: #{escape-svg($form-check-input-checked-bg-image)};\n      }\n    }\n\n    &[type=\"radio\"] {\n      @if $enable-gradients {\n        --#{$prefix}form-check-bg-image: #{escape-svg($form-check-radio-checked-bg-image)}, var(--#{$prefix}gradient);\n      } @else {\n        --#{$prefix}form-check-bg-image: #{escape-svg($form-check-radio-checked-bg-image)};\n      }\n    }\n  }\n\n  &[type=\"checkbox\"]:indeterminate {\n    background-color: $form-check-input-indeterminate-bg-color;\n    border-color: $form-check-input-indeterminate-border-color;\n\n    @if $enable-gradients {\n      --#{$prefix}form-check-bg-image: #{escape-svg($form-check-input-indeterminate-bg-image)}, var(--#{$prefix}gradient);\n    } @else {\n      --#{$prefix}form-check-bg-image: #{escape-svg($form-check-input-indeterminate-bg-image)};\n    }\n  }\n\n  &:disabled {\n    pointer-events: none;\n    filter: none;\n    opacity: $form-check-input-disabled-opacity;\n  }\n\n  // Use disabled attribute in addition of :disabled pseudo-class\n  // See: https://github.com/twbs/bootstrap/issues/28247\n  &[disabled],\n  &:disabled {\n    ~ .form-check-label {\n      cursor: default;\n      opacity: $form-check-label-disabled-opacity;\n    }\n  }\n}\n\n.form-check-label {\n  color: $form-check-label-color;\n  cursor: $form-check-label-cursor;\n}\n\n//\n// Switch\n//\n\n.form-switch {\n  padding-left: $form-switch-padding-start;\n\n  .form-check-input {\n    --#{$prefix}form-switch-bg: #{escape-svg($form-switch-bg-image)};\n\n    width: $form-switch-width;\n    margin-left: $form-switch-padding-start * -1;\n    background-image: var(--#{$prefix}form-switch-bg);\n    background-position: left center;\n    @include border-radius($form-switch-border-radius, 0);\n    @include transition($form-switch-transition);\n\n    &:focus {\n      --#{$prefix}form-switch-bg: #{escape-svg($form-switch-focus-bg-image)};\n    }\n\n    &:checked {\n      background-position: $form-switch-checked-bg-position;\n\n      @if $enable-gradients {\n        --#{$prefix}form-switch-bg: #{escape-svg($form-switch-checked-bg-image)}, var(--#{$prefix}gradient);\n      } @else {\n        --#{$prefix}form-switch-bg: #{escape-svg($form-switch-checked-bg-image)};\n      }\n    }\n  }\n\n  &.form-check-reverse {\n    padding-right: $form-switch-padding-start;\n    padding-left: 0;\n\n    .form-check-input {\n      margin-right: $form-switch-padding-start * -1;\n      margin-left: 0;\n    }\n  }\n}\n\n.form-check-inline {\n  display: inline-block;\n  margin-right: $form-check-inline-margin-end;\n}\n\n.btn-check {\n  position: absolute;\n  clip: rect(0, 0, 0, 0);\n  pointer-events: none;\n\n  &[disabled],\n  &:disabled {\n    + .btn {\n      pointer-events: none;\n      filter: none;\n      opacity: $form-check-btn-check-disabled-opacity;\n    }\n  }\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .form-switch .form-check-input:not(:checked):not(:focus) {\n      --#{$prefix}form-switch-bg: #{escape-svg($form-switch-bg-image-dark)};\n    }\n  }\n}\n","// Range\n//\n// Style range inputs the same across browsers. Vendor-specific rules for pseudo\n// elements cannot be mixed. As such, there are no shared styles for focus or\n// active states on prefixed selectors.\n\n.form-range {\n  width: 100%;\n  height: add($form-range-thumb-height, $form-range-thumb-focus-box-shadow-width * 2);\n  padding: 0; // Need to reset padding\n  appearance: none;\n  background-color: transparent;\n\n  &:focus {\n    outline: 0;\n\n    // Pseudo-elements must be split across multiple rulesets to have an effect.\n    // No box-shadow() mixin for focus accessibility.\n    &::-webkit-slider-thumb { box-shadow: $form-range-thumb-focus-box-shadow; }\n    &::-moz-range-thumb     { box-shadow: $form-range-thumb-focus-box-shadow; }\n  }\n\n  &::-moz-focus-outer {\n    border: 0;\n  }\n\n  &::-webkit-slider-thumb {\n    width: $form-range-thumb-width;\n    height: $form-range-thumb-height;\n    margin-top: ($form-range-track-height - $form-range-thumb-height) * .5; // Webkit specific\n    appearance: none;\n    @include gradient-bg($form-range-thumb-bg);\n    border: $form-range-thumb-border;\n    @include border-radius($form-range-thumb-border-radius);\n    @include box-shadow($form-range-thumb-box-shadow);\n    @include transition($form-range-thumb-transition);\n\n    &:active {\n      @include gradient-bg($form-range-thumb-active-bg);\n    }\n  }\n\n  &::-webkit-slider-runnable-track {\n    width: $form-range-track-width;\n    height: $form-range-track-height;\n    color: transparent; // Why?\n    cursor: $form-range-track-cursor;\n    background-color: $form-range-track-bg;\n    border-color: transparent;\n    @include border-radius($form-range-track-border-radius);\n    @include box-shadow($form-range-track-box-shadow);\n  }\n\n  &::-moz-range-thumb {\n    width: $form-range-thumb-width;\n    height: $form-range-thumb-height;\n    appearance: none;\n    @include gradient-bg($form-range-thumb-bg);\n    border: $form-range-thumb-border;\n    @include border-radius($form-range-thumb-border-radius);\n    @include box-shadow($form-range-thumb-box-shadow);\n    @include transition($form-range-thumb-transition);\n\n    &:active {\n      @include gradient-bg($form-range-thumb-active-bg);\n    }\n  }\n\n  &::-moz-range-track {\n    width: $form-range-track-width;\n    height: $form-range-track-height;\n    color: transparent;\n    cursor: $form-range-track-cursor;\n    background-color: $form-range-track-bg;\n    border-color: transparent; // Firefox specific?\n    @include border-radius($form-range-track-border-radius);\n    @include box-shadow($form-range-track-box-shadow);\n  }\n\n  &:disabled {\n    pointer-events: none;\n\n    &::-webkit-slider-thumb {\n      background-color: $form-range-thumb-disabled-bg;\n    }\n\n    &::-moz-range-thumb {\n      background-color: $form-range-thumb-disabled-bg;\n    }\n  }\n}\n",".form-floating {\n  position: relative;\n\n  > .form-control,\n  > .form-control-plaintext,\n  > .form-select {\n    height: $form-floating-height;\n    min-height: $form-floating-height;\n    line-height: $form-floating-line-height;\n  }\n\n  > label {\n    position: absolute;\n    top: 0;\n    left: 0;\n    z-index: 2;\n    height: 100%; // allow textareas\n    padding: $form-floating-padding-y $form-floating-padding-x;\n    overflow: hidden;\n    text-align: start;\n    text-overflow: ellipsis;\n    white-space: nowrap;\n    pointer-events: none;\n    border: $input-border-width solid transparent; // Required for aligning label's text with the input as it affects inner box model\n    transform-origin: 0 0;\n    @include transition($form-floating-transition);\n  }\n\n  > .form-control,\n  > .form-control-plaintext {\n    padding: $form-floating-padding-y $form-floating-padding-x;\n\n    &::placeholder {\n      color: transparent;\n    }\n\n    &:focus,\n    &:not(:placeholder-shown) {\n      padding-top: $form-floating-input-padding-t;\n      padding-bottom: $form-floating-input-padding-b;\n    }\n    // Duplicated because `:-webkit-autofill` invalidates other selectors when grouped\n    &:-webkit-autofill {\n      padding-top: $form-floating-input-padding-t;\n      padding-bottom: $form-floating-input-padding-b;\n    }\n  }\n\n  > .form-select {\n    padding-top: $form-floating-input-padding-t;\n    padding-bottom: $form-floating-input-padding-b;\n  }\n\n  > .form-control:focus,\n  > .form-control:not(:placeholder-shown),\n  > .form-control-plaintext,\n  > .form-select {\n    ~ label {\n      color: rgba(var(--#{$prefix}body-color-rgb), #{$form-floating-label-opacity});\n      transform: $form-floating-label-transform;\n\n      &::after {\n        position: absolute;\n        inset: $form-floating-padding-y ($form-floating-padding-x * .5);\n        z-index: -1;\n        height: $form-floating-label-height;\n        content: \"\";\n        background-color: $input-bg;\n        @include border-radius($input-border-radius);\n      }\n    }\n  }\n  // Duplicated because `:-webkit-autofill` invalidates other selectors when grouped\n  > .form-control:-webkit-autofill {\n    ~ label {\n      color: rgba(var(--#{$prefix}body-color-rgb), #{$form-floating-label-opacity});\n      transform: $form-floating-label-transform;\n    }\n  }\n\n  > .form-control-plaintext {\n    ~ label {\n      border-width: $input-border-width 0; // Required to properly position label text - as explained above\n    }\n  }\n\n  > :disabled ~ label,\n  > .form-control:disabled ~ label { // Required for `.form-control`s because of specificity\n    color: $form-floating-label-disabled-color;\n\n    &::after {\n      background-color: $input-disabled-bg;\n    }\n  }\n}\n","//\n// Base styles\n//\n\n.input-group {\n  position: relative;\n  display: flex;\n  flex-wrap: wrap; // For form validation feedback\n  align-items: stretch;\n  width: 100%;\n\n  > .form-control,\n  > .form-select,\n  > .form-floating {\n    position: relative; // For focus state's z-index\n    flex: 1 1 auto;\n    width: 1%;\n    min-width: 0; // https://stackoverflow.com/questions/36247140/why-dont-flex-items-shrink-past-content-size\n  }\n\n  // Bring the \"active\" form control to the top of surrounding elements\n  > .form-control:focus,\n  > .form-select:focus,\n  > .form-floating:focus-within {\n    z-index: 5;\n  }\n\n  // Ensure buttons are always above inputs for more visually pleasing borders.\n  // This isn't needed for `.input-group-text` since it shares the same border-color\n  // as our inputs.\n  .btn {\n    position: relative;\n    z-index: 2;\n\n    &:focus {\n      z-index: 5;\n    }\n  }\n}\n\n\n// Textual addons\n//\n// Serves as a catch-all element for any text or radio/checkbox input you wish\n// to prepend or append to an input.\n\n.input-group-text {\n  display: flex;\n  align-items: center;\n  padding: $input-group-addon-padding-y $input-group-addon-padding-x;\n  @include font-size($input-font-size); // Match inputs\n  font-weight: $input-group-addon-font-weight;\n  line-height: $input-line-height;\n  color: $input-group-addon-color;\n  text-align: center;\n  white-space: nowrap;\n  background-color: $input-group-addon-bg;\n  border: $input-border-width solid $input-group-addon-border-color;\n  @include border-radius($input-border-radius);\n}\n\n\n// Sizing\n//\n// Remix the default form control sizing classes into new ones for easier\n// manipulation.\n\n.input-group-lg > .form-control,\n.input-group-lg > .form-select,\n.input-group-lg > .input-group-text,\n.input-group-lg > .btn {\n  padding: $input-padding-y-lg $input-padding-x-lg;\n  @include font-size($input-font-size-lg);\n  @include border-radius($input-border-radius-lg);\n}\n\n.input-group-sm > .form-control,\n.input-group-sm > .form-select,\n.input-group-sm > .input-group-text,\n.input-group-sm > .btn {\n  padding: $input-padding-y-sm $input-padding-x-sm;\n  @include font-size($input-font-size-sm);\n  @include border-radius($input-border-radius-sm);\n}\n\n.input-group-lg > .form-select,\n.input-group-sm > .form-select {\n  padding-right: $form-select-padding-x + $form-select-indicator-padding;\n}\n\n\n// Rounded corners\n//\n// These rulesets must come after the sizing ones to properly override sm and lg\n// border-radius values when extending. They're more specific than we'd like\n// with the `.input-group >` part, but without it, we cannot override the sizing.\n\n// stylelint-disable-next-line no-duplicate-selectors\n.input-group {\n  &:not(.has-validation) {\n    > :not(:last-child):not(.dropdown-toggle):not(.dropdown-menu):not(.form-floating),\n    > .dropdown-toggle:nth-last-child(n + 3),\n    > .form-floating:not(:last-child) > .form-control,\n    > .form-floating:not(:last-child) > .form-select {\n      @include border-end-radius(0);\n    }\n  }\n\n  &.has-validation {\n    > :nth-last-child(n + 3):not(.dropdown-toggle):not(.dropdown-menu):not(.form-floating),\n    > .dropdown-toggle:nth-last-child(n + 4),\n    > .form-floating:nth-last-child(n + 3) > .form-control,\n    > .form-floating:nth-last-child(n + 3) > .form-select {\n      @include border-end-radius(0);\n    }\n  }\n\n  $validation-messages: \"\";\n  @each $state in map-keys($form-validation-states) {\n    $validation-messages: $validation-messages + \":not(.\" + unquote($state) + \"-tooltip)\" + \":not(.\" + unquote($state) + \"-feedback)\";\n  }\n\n  > :not(:first-child):not(.dropdown-menu)#{$validation-messages} {\n    margin-left: calc(#{$input-border-width} * -1); // stylelint-disable-line function-disallowed-list\n    @include border-start-radius(0);\n  }\n\n  > .form-floating:not(:first-child) > .form-control,\n  > .form-floating:not(:first-child) > .form-select {\n    @include border-start-radius(0);\n  }\n}\n","// This mixin uses an `if()` technique to be compatible with Dart Sass\n// See https://github.com/sass/sass/issues/1873#issuecomment-152293725 for more details\n\n// scss-docs-start form-validation-mixins\n@mixin form-validation-state-selector($state) {\n  @if ($state == \"valid\" or $state == \"invalid\") {\n    .was-validated #{if(&, \"&\", \"\")}:#{$state},\n    #{if(&, \"&\", \"\")}.is-#{$state} {\n      @content;\n    }\n  } @else {\n    #{if(&, \"&\", \"\")}.is-#{$state} {\n      @content;\n    }\n  }\n}\n\n@mixin form-validation-state(\n  $state,\n  $color,\n  $icon,\n  $tooltip-color: color-contrast($color),\n  $tooltip-bg-color: rgba($color, $form-feedback-tooltip-opacity),\n  $focus-box-shadow: 0 0 $input-btn-focus-blur $input-focus-width rgba($color, $input-btn-focus-color-opacity),\n  $border-color: $color\n) {\n  .#{$state}-feedback {\n    display: none;\n    width: 100%;\n    margin-top: $form-feedback-margin-top;\n    @include font-size($form-feedback-font-size);\n    font-style: $form-feedback-font-style;\n    color: $color;\n  }\n\n  .#{$state}-tooltip {\n    position: absolute;\n    top: 100%;\n    z-index: 5;\n    display: none;\n    max-width: 100%; // Contain to parent when possible\n    padding: $form-feedback-tooltip-padding-y $form-feedback-tooltip-padding-x;\n    margin-top: .1rem;\n    @include font-size($form-feedback-tooltip-font-size);\n    line-height: $form-feedback-tooltip-line-height;\n    color: $tooltip-color;\n    background-color: $tooltip-bg-color;\n    @include border-radius($form-feedback-tooltip-border-radius);\n  }\n\n  @include form-validation-state-selector($state) {\n    ~ .#{$state}-feedback,\n    ~ .#{$state}-tooltip {\n      display: block;\n    }\n  }\n\n  .form-control {\n    @include form-validation-state-selector($state) {\n      border-color: $border-color;\n\n      @if $enable-validation-icons {\n        padding-right: $input-height-inner;\n        background-image: escape-svg($icon);\n        background-repeat: no-repeat;\n        background-position: right $input-height-inner-quarter center;\n        background-size: $input-height-inner-half $input-height-inner-half;\n      }\n\n      &:focus {\n        border-color: $border-color;\n        @if $enable-shadows {\n          @include box-shadow($input-box-shadow, $focus-box-shadow);\n        } @else {\n          // Avoid using mixin so we can pass custom focus shadow properly\n          box-shadow: $focus-box-shadow;\n        }\n      }\n    }\n  }\n\n  // stylelint-disable-next-line selector-no-qualifying-type\n  textarea.form-control {\n    @include form-validation-state-selector($state) {\n      @if $enable-validation-icons {\n        padding-right: $input-height-inner;\n        background-position: top $input-height-inner-quarter right $input-height-inner-quarter;\n      }\n    }\n  }\n\n  .form-select {\n    @include form-validation-state-selector($state) {\n      border-color: $border-color;\n\n      @if $enable-validation-icons {\n        &:not([multiple]):not([size]),\n        &:not([multiple])[size=\"1\"] {\n          --#{$prefix}form-select-bg-icon: #{escape-svg($icon)};\n          padding-right: $form-select-feedback-icon-padding-end;\n          background-position: $form-select-bg-position, $form-select-feedback-icon-position;\n          background-size: $form-select-bg-size, $form-select-feedback-icon-size;\n        }\n      }\n\n      &:focus {\n        border-color: $border-color;\n        @if $enable-shadows {\n          @include box-shadow($form-select-box-shadow, $focus-box-shadow);\n        } @else {\n          // Avoid using mixin so we can pass custom focus shadow properly\n          box-shadow: $focus-box-shadow;\n        }\n      }\n    }\n  }\n\n  .form-control-color {\n    @include form-validation-state-selector($state) {\n      @if $enable-validation-icons {\n        width: add($form-color-width, $input-height-inner);\n      }\n    }\n  }\n\n  .form-check-input {\n    @include form-validation-state-selector($state) {\n      border-color: $border-color;\n\n      &:checked {\n        background-color: $color;\n      }\n\n      &:focus {\n        box-shadow: $focus-box-shadow;\n      }\n\n      ~ .form-check-label {\n        color: $color;\n      }\n    }\n  }\n  .form-check-inline .form-check-input {\n    ~ .#{$state}-feedback {\n      margin-left: .5em;\n    }\n  }\n\n  .input-group {\n    > .form-control:not(:focus),\n    > .form-select:not(:focus),\n    > .form-floating:not(:focus-within) {\n      @include form-validation-state-selector($state) {\n        @if $state == \"valid\" {\n          z-index: 3;\n        } @else if $state == \"invalid\" {\n          z-index: 4;\n        }\n      }\n    }\n  }\n}\n// scss-docs-end form-validation-mixins\n","//\n// Base styles\n//\n\n.btn {\n  // scss-docs-start btn-css-vars\n  --#{$prefix}btn-padding-x: #{$btn-padding-x};\n  --#{$prefix}btn-padding-y: #{$btn-padding-y};\n  --#{$prefix}btn-font-family: #{$btn-font-family};\n  @include rfs($btn-font-size, --#{$prefix}btn-font-size);\n  --#{$prefix}btn-font-weight: #{$btn-font-weight};\n  --#{$prefix}btn-line-height: #{$btn-line-height};\n  --#{$prefix}btn-color: #{$btn-color};\n  --#{$prefix}btn-bg: transparent;\n  --#{$prefix}btn-border-width: #{$btn-border-width};\n  --#{$prefix}btn-border-color: transparent;\n  --#{$prefix}btn-border-radius: #{$btn-border-radius};\n  --#{$prefix}btn-hover-border-color: transparent;\n  --#{$prefix}btn-box-shadow: #{$btn-box-shadow};\n  --#{$prefix}btn-disabled-opacity: #{$btn-disabled-opacity};\n  --#{$prefix}btn-focus-box-shadow: 0 0 0 #{$btn-focus-width} rgba(var(--#{$prefix}btn-focus-shadow-rgb), .5);\n  // scss-docs-end btn-css-vars\n\n  display: inline-block;\n  padding: var(--#{$prefix}btn-padding-y) var(--#{$prefix}btn-padding-x);\n  font-family: var(--#{$prefix}btn-font-family);\n  @include font-size(var(--#{$prefix}btn-font-size));\n  font-weight: var(--#{$prefix}btn-font-weight);\n  line-height: var(--#{$prefix}btn-line-height);\n  color: var(--#{$prefix}btn-color);\n  text-align: center;\n  text-decoration: if($link-decoration == none, null, none);\n  white-space: $btn-white-space;\n  vertical-align: middle;\n  cursor: if($enable-button-pointers, pointer, null);\n  user-select: none;\n  border: var(--#{$prefix}btn-border-width) solid var(--#{$prefix}btn-border-color);\n  @include border-radius(var(--#{$prefix}btn-border-radius));\n  @include gradient-bg(var(--#{$prefix}btn-bg));\n  @include box-shadow(var(--#{$prefix}btn-box-shadow));\n  @include transition($btn-transition);\n\n  &:hover {\n    color: var(--#{$prefix}btn-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n    background-color: var(--#{$prefix}btn-hover-bg);\n    border-color: var(--#{$prefix}btn-hover-border-color);\n  }\n\n  .btn-check + &:hover {\n    // override for the checkbox/radio buttons\n    color: var(--#{$prefix}btn-color);\n    background-color: var(--#{$prefix}btn-bg);\n    border-color: var(--#{$prefix}btn-border-color);\n  }\n\n  &:focus-visible {\n    color: var(--#{$prefix}btn-hover-color);\n    @include gradient-bg(var(--#{$prefix}btn-hover-bg));\n    border-color: var(--#{$prefix}btn-hover-border-color);\n    outline: 0;\n    // Avoid using mixin so we can pass custom focus shadow properly\n    @if $enable-shadows {\n      box-shadow: var(--#{$prefix}btn-box-shadow), var(--#{$prefix}btn-focus-box-shadow);\n    } @else {\n      box-shadow: var(--#{$prefix}btn-focus-box-shadow);\n    }\n  }\n\n  .btn-check:focus-visible + & {\n    border-color: var(--#{$prefix}btn-hover-border-color);\n    outline: 0;\n    // Avoid using mixin so we can pass custom focus shadow properly\n    @if $enable-shadows {\n      box-shadow: var(--#{$prefix}btn-box-shadow), var(--#{$prefix}btn-focus-box-shadow);\n    } @else {\n      box-shadow: var(--#{$prefix}btn-focus-box-shadow);\n    }\n  }\n\n  .btn-check:checked + &,\n  :not(.btn-check) + &:active,\n  &:first-child:active,\n  &.active,\n  &.show {\n    color: var(--#{$prefix}btn-active-color);\n    background-color: var(--#{$prefix}btn-active-bg);\n    // Remove CSS gradients if they're enabled\n    background-image: if($enable-gradients, none, null);\n    border-color: var(--#{$prefix}btn-active-border-color);\n    @include box-shadow(var(--#{$prefix}btn-active-shadow));\n\n    &:focus-visible {\n      // Avoid using mixin so we can pass custom focus shadow properly\n      @if $enable-shadows {\n        box-shadow: var(--#{$prefix}btn-active-shadow), var(--#{$prefix}btn-focus-box-shadow);\n      } @else {\n        box-shadow: var(--#{$prefix}btn-focus-box-shadow);\n      }\n    }\n  }\n\n  .btn-check:checked:focus-visible + & {\n    // Avoid using mixin so we can pass custom focus shadow properly\n    @if $enable-shadows {\n      box-shadow: var(--#{$prefix}btn-active-shadow), var(--#{$prefix}btn-focus-box-shadow);\n    } @else {\n      box-shadow: var(--#{$prefix}btn-focus-box-shadow);\n    }\n  }\n\n  &:disabled,\n  &.disabled,\n  fieldset:disabled & {\n    color: var(--#{$prefix}btn-disabled-color);\n    pointer-events: none;\n    background-color: var(--#{$prefix}btn-disabled-bg);\n    background-image: if($enable-gradients, none, null);\n    border-color: var(--#{$prefix}btn-disabled-border-color);\n    opacity: var(--#{$prefix}btn-disabled-opacity);\n    @include box-shadow(none);\n  }\n}\n\n\n//\n// Alternate buttons\n//\n\n// scss-docs-start btn-variant-loops\n@each $color, $value in $theme-colors {\n  .btn-#{$color} {\n    @if $color == \"light\" {\n      @include button-variant(\n        $value,\n        $value,\n        $hover-background: shade-color($value, $btn-hover-bg-shade-amount),\n        $hover-border: shade-color($value, $btn-hover-border-shade-amount),\n        $active-background: shade-color($value, $btn-active-bg-shade-amount),\n        $active-border: shade-color($value, $btn-active-border-shade-amount)\n      );\n    } @else if $color == \"dark\" {\n      @include button-variant(\n        $value,\n        $value,\n        $hover-background: tint-color($value, $btn-hover-bg-tint-amount),\n        $hover-border: tint-color($value, $btn-hover-border-tint-amount),\n        $active-background: tint-color($value, $btn-active-bg-tint-amount),\n        $active-border: tint-color($value, $btn-active-border-tint-amount)\n      );\n    } @else {\n      @include button-variant($value, $value);\n    }\n  }\n}\n\n@each $color, $value in $theme-colors {\n  .btn-outline-#{$color} {\n    @include button-outline-variant($value);\n  }\n}\n// scss-docs-end btn-variant-loops\n\n\n//\n// Link buttons\n//\n\n// Make a button look and behave like a link\n.btn-link {\n  --#{$prefix}btn-font-weight: #{$font-weight-normal};\n  --#{$prefix}btn-color: #{$btn-link-color};\n  --#{$prefix}btn-bg: transparent;\n  --#{$prefix}btn-border-color: transparent;\n  --#{$prefix}btn-hover-color: #{$btn-link-hover-color};\n  --#{$prefix}btn-hover-border-color: transparent;\n  --#{$prefix}btn-active-color: #{$btn-link-hover-color};\n  --#{$prefix}btn-active-border-color: transparent;\n  --#{$prefix}btn-disabled-color: #{$btn-link-disabled-color};\n  --#{$prefix}btn-disabled-border-color: transparent;\n  --#{$prefix}btn-box-shadow: 0 0 0 #000; // Can't use `none` as keyword negates all values when used with multiple shadows\n  --#{$prefix}btn-focus-shadow-rgb: #{$btn-link-focus-shadow-rgb};\n\n  text-decoration: $link-decoration;\n  @if $enable-gradients {\n    background-image: none;\n  }\n\n  &:hover,\n  &:focus-visible {\n    text-decoration: $link-hover-decoration;\n  }\n\n  &:focus-visible {\n    color: var(--#{$prefix}btn-color);\n  }\n\n  &:hover {\n    color: var(--#{$prefix}btn-hover-color);\n  }\n\n  // No need for an active state here\n}\n\n\n//\n// Button Sizes\n//\n\n.btn-lg {\n  @include button-size($btn-padding-y-lg, $btn-padding-x-lg, $btn-font-size-lg, $btn-border-radius-lg);\n}\n\n.btn-sm {\n  @include button-size($btn-padding-y-sm, $btn-padding-x-sm, $btn-font-size-sm, $btn-border-radius-sm);\n}\n","// Button variants\n//\n// Easily pump out default styles, as well as :hover, :focus, :active,\n// and disabled options for all buttons\n\n// scss-docs-start btn-variant-mixin\n@mixin button-variant(\n  $background,\n  $border,\n  $color: color-contrast($background),\n  $hover-background: if($color == $color-contrast-light, shade-color($background, $btn-hover-bg-shade-amount), tint-color($background, $btn-hover-bg-tint-amount)),\n  $hover-border: if($color == $color-contrast-light, shade-color($border, $btn-hover-border-shade-amount), tint-color($border, $btn-hover-border-tint-amount)),\n  $hover-color: color-contrast($hover-background),\n  $active-background: if($color == $color-contrast-light, shade-color($background, $btn-active-bg-shade-amount), tint-color($background, $btn-active-bg-tint-amount)),\n  $active-border: if($color == $color-contrast-light, shade-color($border, $btn-active-border-shade-amount), tint-color($border, $btn-active-border-tint-amount)),\n  $active-color: color-contrast($active-background),\n  $disabled-background: $background,\n  $disabled-border: $border,\n  $disabled-color: color-contrast($disabled-background)\n) {\n  --#{$prefix}btn-color: #{$color};\n  --#{$prefix}btn-bg: #{$background};\n  --#{$prefix}btn-border-color: #{$border};\n  --#{$prefix}btn-hover-color: #{$hover-color};\n  --#{$prefix}btn-hover-bg: #{$hover-background};\n  --#{$prefix}btn-hover-border-color: #{$hover-border};\n  --#{$prefix}btn-focus-shadow-rgb: #{to-rgb(mix($color, $border, 15%))};\n  --#{$prefix}btn-active-color: #{$active-color};\n  --#{$prefix}btn-active-bg: #{$active-background};\n  --#{$prefix}btn-active-border-color: #{$active-border};\n  --#{$prefix}btn-active-shadow: #{$btn-active-box-shadow};\n  --#{$prefix}btn-disabled-color: #{$disabled-color};\n  --#{$prefix}btn-disabled-bg: #{$disabled-background};\n  --#{$prefix}btn-disabled-border-color: #{$disabled-border};\n}\n// scss-docs-end btn-variant-mixin\n\n// scss-docs-start btn-outline-variant-mixin\n@mixin button-outline-variant(\n  $color,\n  $color-hover: color-contrast($color),\n  $active-background: $color,\n  $active-border: $color,\n  $active-color: color-contrast($active-background)\n) {\n  --#{$prefix}btn-color: #{$color};\n  --#{$prefix}btn-border-color: #{$color};\n  --#{$prefix}btn-hover-color: #{$color-hover};\n  --#{$prefix}btn-hover-bg: #{$active-background};\n  --#{$prefix}btn-hover-border-color: #{$active-border};\n  --#{$prefix}btn-focus-shadow-rgb: #{to-rgb($color)};\n  --#{$prefix}btn-active-color: #{$active-color};\n  --#{$prefix}btn-active-bg: #{$active-background};\n  --#{$prefix}btn-active-border-color: #{$active-border};\n  --#{$prefix}btn-active-shadow: #{$btn-active-box-shadow};\n  --#{$prefix}btn-disabled-color: #{$color};\n  --#{$prefix}btn-disabled-bg: transparent;\n  --#{$prefix}btn-disabled-border-color: #{$color};\n  --#{$prefix}gradient: none;\n}\n// scss-docs-end btn-outline-variant-mixin\n\n// scss-docs-start btn-size-mixin\n@mixin button-size($padding-y, $padding-x, $font-size, $border-radius) {\n  --#{$prefix}btn-padding-y: #{$padding-y};\n  --#{$prefix}btn-padding-x: #{$padding-x};\n  @include rfs($font-size, --#{$prefix}btn-font-size);\n  --#{$prefix}btn-border-radius: #{$border-radius};\n}\n// scss-docs-end btn-size-mixin\n",".fade {\n  @include transition($transition-fade);\n\n  &:not(.show) {\n    opacity: 0;\n  }\n}\n\n// scss-docs-start collapse-classes\n.collapse {\n  &:not(.show) {\n    display: none;\n  }\n}\n\n.collapsing {\n  height: 0;\n  overflow: hidden;\n  @include transition($transition-collapse);\n\n  &.collapse-horizontal {\n    width: 0;\n    height: auto;\n    @include transition($transition-collapse-width);\n  }\n}\n// scss-docs-end collapse-classes\n","// The dropdown wrapper (`<div>`)\n.dropup,\n.dropend,\n.dropdown,\n.dropstart,\n.dropup-center,\n.dropdown-center {\n  position: relative;\n}\n\n.dropdown-toggle {\n  white-space: nowrap;\n\n  // Generate the caret automatically\n  @include caret();\n}\n\n// The dropdown menu\n.dropdown-menu {\n  // scss-docs-start dropdown-css-vars\n  --#{$prefix}dropdown-zindex: #{$zindex-dropdown};\n  --#{$prefix}dropdown-min-width: #{$dropdown-min-width};\n  --#{$prefix}dropdown-padding-x: #{$dropdown-padding-x};\n  --#{$prefix}dropdown-padding-y: #{$dropdown-padding-y};\n  --#{$prefix}dropdown-spacer: #{$dropdown-spacer};\n  @include rfs($dropdown-font-size, --#{$prefix}dropdown-font-size);\n  --#{$prefix}dropdown-color: #{$dropdown-color};\n  --#{$prefix}dropdown-bg: #{$dropdown-bg};\n  --#{$prefix}dropdown-border-color: #{$dropdown-border-color};\n  --#{$prefix}dropdown-border-radius: #{$dropdown-border-radius};\n  --#{$prefix}dropdown-border-width: #{$dropdown-border-width};\n  --#{$prefix}dropdown-inner-border-radius: #{$dropdown-inner-border-radius};\n  --#{$prefix}dropdown-divider-bg: #{$dropdown-divider-bg};\n  --#{$prefix}dropdown-divider-margin-y: #{$dropdown-divider-margin-y};\n  --#{$prefix}dropdown-box-shadow: #{$dropdown-box-shadow};\n  --#{$prefix}dropdown-link-color: #{$dropdown-link-color};\n  --#{$prefix}dropdown-link-hover-color: #{$dropdown-link-hover-color};\n  --#{$prefix}dropdown-link-hover-bg: #{$dropdown-link-hover-bg};\n  --#{$prefix}dropdown-link-active-color: #{$dropdown-link-active-color};\n  --#{$prefix}dropdown-link-active-bg: #{$dropdown-link-active-bg};\n  --#{$prefix}dropdown-link-disabled-color: #{$dropdown-link-disabled-color};\n  --#{$prefix}dropdown-item-padding-x: #{$dropdown-item-padding-x};\n  --#{$prefix}dropdown-item-padding-y: #{$dropdown-item-padding-y};\n  --#{$prefix}dropdown-header-color: #{$dropdown-header-color};\n  --#{$prefix}dropdown-header-padding-x: #{$dropdown-header-padding-x};\n  --#{$prefix}dropdown-header-padding-y: #{$dropdown-header-padding-y};\n  // scss-docs-end dropdown-css-vars\n\n  position: absolute;\n  z-index: var(--#{$prefix}dropdown-zindex);\n  display: none; // none by default, but block on \"open\" of the menu\n  min-width: var(--#{$prefix}dropdown-min-width);\n  padding: var(--#{$prefix}dropdown-padding-y) var(--#{$prefix}dropdown-padding-x);\n  margin: 0; // Override default margin of ul\n  @include font-size(var(--#{$prefix}dropdown-font-size));\n  color: var(--#{$prefix}dropdown-color);\n  text-align: left; // Ensures proper alignment if parent has it changed (e.g., modal footer)\n  list-style: none;\n  background-color: var(--#{$prefix}dropdown-bg);\n  background-clip: padding-box;\n  border: var(--#{$prefix}dropdown-border-width) solid var(--#{$prefix}dropdown-border-color);\n  @include border-radius(var(--#{$prefix}dropdown-border-radius));\n  @include box-shadow(var(--#{$prefix}dropdown-box-shadow));\n\n  &[data-bs-popper] {\n    top: 100%;\n    left: 0;\n    margin-top: var(--#{$prefix}dropdown-spacer);\n  }\n\n  @if $dropdown-padding-y == 0 {\n    > .dropdown-item:first-child,\n    > li:first-child .dropdown-item {\n      @include border-top-radius(var(--#{$prefix}dropdown-inner-border-radius));\n    }\n    > .dropdown-item:last-child,\n    > li:last-child .dropdown-item {\n      @include border-bottom-radius(var(--#{$prefix}dropdown-inner-border-radius));\n    }\n\n  }\n}\n\n// scss-docs-start responsive-breakpoints\n// We deliberately hardcode the `bs-` prefix because we check\n// this custom property in JS to determine Popper's positioning\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  @include media-breakpoint-up($breakpoint) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    .dropdown-menu#{$infix}-start {\n      --bs-position: start;\n\n      &[data-bs-popper] {\n        right: auto;\n        left: 0;\n      }\n    }\n\n    .dropdown-menu#{$infix}-end {\n      --bs-position: end;\n\n      &[data-bs-popper] {\n        right: 0;\n        left: auto;\n      }\n    }\n  }\n}\n// scss-docs-end responsive-breakpoints\n\n// Allow for dropdowns to go bottom up (aka, dropup-menu)\n// Just add .dropup after the standard .dropdown class and you're set.\n.dropup {\n  .dropdown-menu[data-bs-popper] {\n    top: auto;\n    bottom: 100%;\n    margin-top: 0;\n    margin-bottom: var(--#{$prefix}dropdown-spacer);\n  }\n\n  .dropdown-toggle {\n    @include caret(up);\n  }\n}\n\n.dropend {\n  .dropdown-menu[data-bs-popper] {\n    top: 0;\n    right: auto;\n    left: 100%;\n    margin-top: 0;\n    margin-left: var(--#{$prefix}dropdown-spacer);\n  }\n\n  .dropdown-toggle {\n    @include caret(end);\n    &::after {\n      vertical-align: 0;\n    }\n  }\n}\n\n.dropstart {\n  .dropdown-menu[data-bs-popper] {\n    top: 0;\n    right: 100%;\n    left: auto;\n    margin-top: 0;\n    margin-right: var(--#{$prefix}dropdown-spacer);\n  }\n\n  .dropdown-toggle {\n    @include caret(start);\n    &::before {\n      vertical-align: 0;\n    }\n  }\n}\n\n\n// Dividers (basically an `<hr>`) within the dropdown\n.dropdown-divider {\n  height: 0;\n  margin: var(--#{$prefix}dropdown-divider-margin-y) 0;\n  overflow: hidden;\n  border-top: 1px solid var(--#{$prefix}dropdown-divider-bg);\n  opacity: 1; // Revisit in v6 to de-dupe styles that conflict with <hr> element\n}\n\n// Links, buttons, and more within the dropdown menu\n//\n// `<button>`-specific styles are denoted with `// For <button>s`\n.dropdown-item {\n  display: block;\n  width: 100%; // For `<button>`s\n  padding: var(--#{$prefix}dropdown-item-padding-y) var(--#{$prefix}dropdown-item-padding-x);\n  clear: both;\n  font-weight: $font-weight-normal;\n  color: var(--#{$prefix}dropdown-link-color);\n  text-align: inherit; // For `<button>`s\n  text-decoration: if($link-decoration == none, null, none);\n  white-space: nowrap; // prevent links from randomly breaking onto new lines\n  background-color: transparent; // For `<button>`s\n  border: 0; // For `<button>`s\n  @include border-radius(var(--#{$prefix}dropdown-item-border-radius, 0));\n\n  &:hover,\n  &:focus {\n    color: var(--#{$prefix}dropdown-link-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n    @include gradient-bg(var(--#{$prefix}dropdown-link-hover-bg));\n  }\n\n  &.active,\n  &:active {\n    color: var(--#{$prefix}dropdown-link-active-color);\n    text-decoration: none;\n    @include gradient-bg(var(--#{$prefix}dropdown-link-active-bg));\n  }\n\n  &.disabled,\n  &:disabled {\n    color: var(--#{$prefix}dropdown-link-disabled-color);\n    pointer-events: none;\n    background-color: transparent;\n    // Remove CSS gradients if they're enabled\n    background-image: if($enable-gradients, none, null);\n  }\n}\n\n.dropdown-menu.show {\n  display: block;\n}\n\n// Dropdown section headers\n.dropdown-header {\n  display: block;\n  padding: var(--#{$prefix}dropdown-header-padding-y) var(--#{$prefix}dropdown-header-padding-x);\n  margin-bottom: 0; // for use with heading elements\n  @include font-size($font-size-sm);\n  color: var(--#{$prefix}dropdown-header-color);\n  white-space: nowrap; // as with > li > a\n}\n\n// Dropdown text\n.dropdown-item-text {\n  display: block;\n  padding: var(--#{$prefix}dropdown-item-padding-y) var(--#{$prefix}dropdown-item-padding-x);\n  color: var(--#{$prefix}dropdown-link-color);\n}\n\n// Dark dropdowns\n.dropdown-menu-dark {\n  // scss-docs-start dropdown-dark-css-vars\n  --#{$prefix}dropdown-color: #{$dropdown-dark-color};\n  --#{$prefix}dropdown-bg: #{$dropdown-dark-bg};\n  --#{$prefix}dropdown-border-color: #{$dropdown-dark-border-color};\n  --#{$prefix}dropdown-box-shadow: #{$dropdown-dark-box-shadow};\n  --#{$prefix}dropdown-link-color: #{$dropdown-dark-link-color};\n  --#{$prefix}dropdown-link-hover-color: #{$dropdown-dark-link-hover-color};\n  --#{$prefix}dropdown-divider-bg: #{$dropdown-dark-divider-bg};\n  --#{$prefix}dropdown-link-hover-bg: #{$dropdown-dark-link-hover-bg};\n  --#{$prefix}dropdown-link-active-color: #{$dropdown-dark-link-active-color};\n  --#{$prefix}dropdown-link-active-bg: #{$dropdown-dark-link-active-bg};\n  --#{$prefix}dropdown-link-disabled-color: #{$dropdown-dark-link-disabled-color};\n  --#{$prefix}dropdown-header-color: #{$dropdown-dark-header-color};\n  // scss-docs-end dropdown-dark-css-vars\n}\n","// scss-docs-start caret-mixins\n@mixin caret-down($width: $caret-width) {\n  border-top: $width solid;\n  border-right: $width solid transparent;\n  border-bottom: 0;\n  border-left: $width solid transparent;\n}\n\n@mixin caret-up($width: $caret-width) {\n  border-top: 0;\n  border-right: $width solid transparent;\n  border-bottom: $width solid;\n  border-left: $width solid transparent;\n}\n\n@mixin caret-end($width: $caret-width) {\n  border-top: $width solid transparent;\n  border-right: 0;\n  border-bottom: $width solid transparent;\n  border-left: $width solid;\n}\n\n@mixin caret-start($width: $caret-width) {\n  border-top: $width solid transparent;\n  border-right: $width solid;\n  border-bottom: $width solid transparent;\n}\n\n@mixin caret(\n  $direction: down,\n  $width: $caret-width,\n  $spacing: $caret-spacing,\n  $vertical-align: $caret-vertical-align\n) {\n  @if $enable-caret {\n    &::after {\n      display: inline-block;\n      margin-left: $spacing;\n      vertical-align: $vertical-align;\n      content: \"\";\n      @if $direction == down {\n        @include caret-down($width);\n      } @else if $direction == up {\n        @include caret-up($width);\n      } @else if $direction == end {\n        @include caret-end($width);\n      }\n    }\n\n    @if $direction == start {\n      &::after {\n        display: none;\n      }\n\n      &::before {\n        display: inline-block;\n        margin-right: $spacing;\n        vertical-align: $vertical-align;\n        content: \"\";\n        @include caret-start($width);\n      }\n    }\n\n    &:empty::after {\n      margin-left: 0;\n    }\n  }\n}\n// scss-docs-end caret-mixins\n","// Make the div behave like a button\n.btn-group,\n.btn-group-vertical {\n  position: relative;\n  display: inline-flex;\n  vertical-align: middle; // match .btn alignment given font-size hack above\n\n  > .btn {\n    position: relative;\n    flex: 1 1 auto;\n  }\n\n  // Bring the hover, focused, and \"active\" buttons to the front to overlay\n  // the borders properly\n  > .btn-check:checked + .btn,\n  > .btn-check:focus + .btn,\n  > .btn:hover,\n  > .btn:focus,\n  > .btn:active,\n  > .btn.active {\n    z-index: 1;\n  }\n}\n\n// Optional: Group multiple button groups together for a toolbar\n.btn-toolbar {\n  display: flex;\n  flex-wrap: wrap;\n  justify-content: flex-start;\n\n  .input-group {\n    width: auto;\n  }\n}\n\n.btn-group {\n  @include border-radius($btn-border-radius);\n\n  // Prevent double borders when buttons are next to each other\n  > :not(.btn-check:first-child) + .btn,\n  > .btn-group:not(:first-child) {\n    margin-left: calc(#{$btn-border-width} * -1); // stylelint-disable-line function-disallowed-list\n  }\n\n  // Reset rounded corners\n  > .btn:not(:last-child):not(.dropdown-toggle),\n  > .btn.dropdown-toggle-split:first-child,\n  > .btn-group:not(:last-child) > .btn {\n    @include border-end-radius(0);\n  }\n\n  // The left radius should be 0 if the button is:\n  // - the \"third or more\" child\n  // - the second child and the previous element isn't `.btn-check` (making it the first child visually)\n  // - part of a btn-group which isn't the first child\n  > .btn:nth-child(n + 3),\n  > :not(.btn-check) + .btn,\n  > .btn-group:not(:first-child) > .btn {\n    @include border-start-radius(0);\n  }\n}\n\n// Sizing\n//\n// Remix the default button sizing classes into new ones for easier manipulation.\n\n.btn-group-sm > .btn { @extend .btn-sm; }\n.btn-group-lg > .btn { @extend .btn-lg; }\n\n\n//\n// Split button dropdowns\n//\n\n.dropdown-toggle-split {\n  padding-right: $btn-padding-x * .75;\n  padding-left: $btn-padding-x * .75;\n\n  &::after,\n  .dropup &::after,\n  .dropend &::after {\n    margin-left: 0;\n  }\n\n  .dropstart &::before {\n    margin-right: 0;\n  }\n}\n\n.btn-sm + .dropdown-toggle-split {\n  padding-right: $btn-padding-x-sm * .75;\n  padding-left: $btn-padding-x-sm * .75;\n}\n\n.btn-lg + .dropdown-toggle-split {\n  padding-right: $btn-padding-x-lg * .75;\n  padding-left: $btn-padding-x-lg * .75;\n}\n\n\n// The clickable button for toggling the menu\n// Set the same inset shadow as the :active state\n.btn-group.show .dropdown-toggle {\n  @include box-shadow($btn-active-box-shadow);\n\n  // Show no shadow for `.btn-link` since it has no other button styles.\n  &.btn-link {\n    @include box-shadow(none);\n  }\n}\n\n\n//\n// Vertical button groups\n//\n\n.btn-group-vertical {\n  flex-direction: column;\n  align-items: flex-start;\n  justify-content: center;\n\n  > .btn,\n  > .btn-group {\n    width: 100%;\n  }\n\n  > .btn:not(:first-child),\n  > .btn-group:not(:first-child) {\n    margin-top: calc(#{$btn-border-width} * -1); // stylelint-disable-line function-disallowed-list\n  }\n\n  // Reset rounded corners\n  > .btn:not(:last-child):not(.dropdown-toggle),\n  > .btn-group:not(:last-child) > .btn {\n    @include border-bottom-radius(0);\n  }\n\n  > .btn ~ .btn,\n  > .btn-group:not(:first-child) > .btn {\n    @include border-top-radius(0);\n  }\n}\n","// Base class\n//\n// Kickstart any navigation component with a set of style resets. Works with\n// `<nav>`s, `<ul>`s or `<ol>`s.\n\n.nav {\n  // scss-docs-start nav-css-vars\n  --#{$prefix}nav-link-padding-x: #{$nav-link-padding-x};\n  --#{$prefix}nav-link-padding-y: #{$nav-link-padding-y};\n  @include rfs($nav-link-font-size, --#{$prefix}nav-link-font-size);\n  --#{$prefix}nav-link-font-weight: #{$nav-link-font-weight};\n  --#{$prefix}nav-link-color: #{$nav-link-color};\n  --#{$prefix}nav-link-hover-color: #{$nav-link-hover-color};\n  --#{$prefix}nav-link-disabled-color: #{$nav-link-disabled-color};\n  // scss-docs-end nav-css-vars\n\n  display: flex;\n  flex-wrap: wrap;\n  padding-left: 0;\n  margin-bottom: 0;\n  list-style: none;\n}\n\n.nav-link {\n  display: block;\n  padding: var(--#{$prefix}nav-link-padding-y) var(--#{$prefix}nav-link-padding-x);\n  @include font-size(var(--#{$prefix}nav-link-font-size));\n  font-weight: var(--#{$prefix}nav-link-font-weight);\n  color: var(--#{$prefix}nav-link-color);\n  text-decoration: if($link-decoration == none, null, none);\n  background: none;\n  border: 0;\n  @include transition($nav-link-transition);\n\n  &:hover,\n  &:focus {\n    color: var(--#{$prefix}nav-link-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n  }\n\n  &:focus-visible {\n    outline: 0;\n    box-shadow: $nav-link-focus-box-shadow;\n  }\n\n  // Disabled state lightens text\n  &.disabled,\n  &:disabled {\n    color: var(--#{$prefix}nav-link-disabled-color);\n    pointer-events: none;\n    cursor: default;\n  }\n}\n\n//\n// Tabs\n//\n\n.nav-tabs {\n  // scss-docs-start nav-tabs-css-vars\n  --#{$prefix}nav-tabs-border-width: #{$nav-tabs-border-width};\n  --#{$prefix}nav-tabs-border-color: #{$nav-tabs-border-color};\n  --#{$prefix}nav-tabs-border-radius: #{$nav-tabs-border-radius};\n  --#{$prefix}nav-tabs-link-hover-border-color: #{$nav-tabs-link-hover-border-color};\n  --#{$prefix}nav-tabs-link-active-color: #{$nav-tabs-link-active-color};\n  --#{$prefix}nav-tabs-link-active-bg: #{$nav-tabs-link-active-bg};\n  --#{$prefix}nav-tabs-link-active-border-color: #{$nav-tabs-link-active-border-color};\n  // scss-docs-end nav-tabs-css-vars\n\n  border-bottom: var(--#{$prefix}nav-tabs-border-width) solid var(--#{$prefix}nav-tabs-border-color);\n\n  .nav-link {\n    margin-bottom: calc(-1 * var(--#{$prefix}nav-tabs-border-width)); // stylelint-disable-line function-disallowed-list\n    border: var(--#{$prefix}nav-tabs-border-width) solid transparent;\n    @include border-top-radius(var(--#{$prefix}nav-tabs-border-radius));\n\n    &:hover,\n    &:focus {\n      // Prevents active .nav-link tab overlapping focus outline of previous/next .nav-link\n      isolation: isolate;\n      border-color: var(--#{$prefix}nav-tabs-link-hover-border-color);\n    }\n  }\n\n  .nav-link.active,\n  .nav-item.show .nav-link {\n    color: var(--#{$prefix}nav-tabs-link-active-color);\n    background-color: var(--#{$prefix}nav-tabs-link-active-bg);\n    border-color: var(--#{$prefix}nav-tabs-link-active-border-color);\n  }\n\n  .dropdown-menu {\n    // Make dropdown border overlap tab border\n    margin-top: calc(-1 * var(--#{$prefix}nav-tabs-border-width)); // stylelint-disable-line function-disallowed-list\n    // Remove the top rounded corners here since there is a hard edge above the menu\n    @include border-top-radius(0);\n  }\n}\n\n\n//\n// Pills\n//\n\n.nav-pills {\n  // scss-docs-start nav-pills-css-vars\n  --#{$prefix}nav-pills-border-radius: #{$nav-pills-border-radius};\n  --#{$prefix}nav-pills-link-active-color: #{$nav-pills-link-active-color};\n  --#{$prefix}nav-pills-link-active-bg: #{$nav-pills-link-active-bg};\n  // scss-docs-end nav-pills-css-vars\n\n  .nav-link {\n    @include border-radius(var(--#{$prefix}nav-pills-border-radius));\n  }\n\n  .nav-link.active,\n  .show > .nav-link {\n    color: var(--#{$prefix}nav-pills-link-active-color);\n    @include gradient-bg(var(--#{$prefix}nav-pills-link-active-bg));\n  }\n}\n\n\n//\n// Underline\n//\n\n.nav-underline {\n  // scss-docs-start nav-underline-css-vars\n  --#{$prefix}nav-underline-gap: #{$nav-underline-gap};\n  --#{$prefix}nav-underline-border-width: #{$nav-underline-border-width};\n  --#{$prefix}nav-underline-link-active-color: #{$nav-underline-link-active-color};\n  // scss-docs-end nav-underline-css-vars\n\n  gap: var(--#{$prefix}nav-underline-gap);\n\n  .nav-link {\n    padding-right: 0;\n    padding-left: 0;\n    border-bottom: var(--#{$prefix}nav-underline-border-width) solid transparent;\n\n    &:hover,\n    &:focus {\n      border-bottom-color: currentcolor;\n    }\n  }\n\n  .nav-link.active,\n  .show > .nav-link {\n    font-weight: $font-weight-bold;\n    color: var(--#{$prefix}nav-underline-link-active-color);\n    border-bottom-color: currentcolor;\n  }\n}\n\n\n//\n// Justified variants\n//\n\n.nav-fill {\n  > .nav-link,\n  .nav-item {\n    flex: 1 1 auto;\n    text-align: center;\n  }\n}\n\n.nav-justified {\n  > .nav-link,\n  .nav-item {\n    flex-basis: 0;\n    flex-grow: 1;\n    text-align: center;\n  }\n}\n\n.nav-fill,\n.nav-justified {\n  .nav-item .nav-link {\n    width: 100%; // Make sure button will grow\n  }\n}\n\n\n// Tabbable tabs\n//\n// Hide tabbable panes to start, show them when `.active`\n\n.tab-content {\n  > .tab-pane {\n    display: none;\n  }\n  > .active {\n    display: block;\n  }\n}\n","// Navbar\n//\n// Provide a static navbar from which we expand to create full-width, fixed, and\n// other navbar variations.\n\n.navbar {\n  // scss-docs-start navbar-css-vars\n  --#{$prefix}navbar-padding-x: #{if($navbar-padding-x == null, 0, $navbar-padding-x)};\n  --#{$prefix}navbar-padding-y: #{$navbar-padding-y};\n  --#{$prefix}navbar-color: #{$navbar-light-color};\n  --#{$prefix}navbar-hover-color: #{$navbar-light-hover-color};\n  --#{$prefix}navbar-disabled-color: #{$navbar-light-disabled-color};\n  --#{$prefix}navbar-active-color: #{$navbar-light-active-color};\n  --#{$prefix}navbar-brand-padding-y: #{$navbar-brand-padding-y};\n  --#{$prefix}navbar-brand-margin-end: #{$navbar-brand-margin-end};\n  --#{$prefix}navbar-brand-font-size: #{$navbar-brand-font-size};\n  --#{$prefix}navbar-brand-color: #{$navbar-light-brand-color};\n  --#{$prefix}navbar-brand-hover-color: #{$navbar-light-brand-hover-color};\n  --#{$prefix}navbar-nav-link-padding-x: #{$navbar-nav-link-padding-x};\n  --#{$prefix}navbar-toggler-padding-y: #{$navbar-toggler-padding-y};\n  --#{$prefix}navbar-toggler-padding-x: #{$navbar-toggler-padding-x};\n  --#{$prefix}navbar-toggler-font-size: #{$navbar-toggler-font-size};\n  --#{$prefix}navbar-toggler-icon-bg: #{escape-svg($navbar-light-toggler-icon-bg)};\n  --#{$prefix}navbar-toggler-border-color: #{$navbar-light-toggler-border-color};\n  --#{$prefix}navbar-toggler-border-radius: #{$navbar-toggler-border-radius};\n  --#{$prefix}navbar-toggler-focus-width: #{$navbar-toggler-focus-width};\n  --#{$prefix}navbar-toggler-transition: #{$navbar-toggler-transition};\n  // scss-docs-end navbar-css-vars\n\n  position: relative;\n  display: flex;\n  flex-wrap: wrap; // allow us to do the line break for collapsing content\n  align-items: center;\n  justify-content: space-between; // space out brand from logo\n  padding: var(--#{$prefix}navbar-padding-y) var(--#{$prefix}navbar-padding-x);\n  @include gradient-bg();\n\n  // Because flex properties aren't inherited, we need to redeclare these first\n  // few properties so that content nested within behave properly.\n  // The `flex-wrap` property is inherited to simplify the expanded navbars\n  %container-flex-properties {\n    display: flex;\n    flex-wrap: inherit;\n    align-items: center;\n    justify-content: space-between;\n  }\n\n  > .container,\n  > .container-fluid {\n    @extend %container-flex-properties;\n  }\n\n  @each $breakpoint, $container-max-width in $container-max-widths {\n    > .container#{breakpoint-infix($breakpoint, $container-max-widths)} {\n      @extend %container-flex-properties;\n    }\n  }\n}\n\n\n// Navbar brand\n//\n// Used for brand, project, or site names.\n\n.navbar-brand {\n  padding-top: var(--#{$prefix}navbar-brand-padding-y);\n  padding-bottom: var(--#{$prefix}navbar-brand-padding-y);\n  margin-right: var(--#{$prefix}navbar-brand-margin-end);\n  @include font-size(var(--#{$prefix}navbar-brand-font-size));\n  color: var(--#{$prefix}navbar-brand-color);\n  text-decoration: if($link-decoration == none, null, none);\n  white-space: nowrap;\n\n  &:hover,\n  &:focus {\n    color: var(--#{$prefix}navbar-brand-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n  }\n}\n\n\n// Navbar nav\n//\n// Custom navbar navigation (doesn't require `.nav`, but does make use of `.nav-link`).\n\n.navbar-nav {\n  // scss-docs-start navbar-nav-css-vars\n  --#{$prefix}nav-link-padding-x: 0;\n  --#{$prefix}nav-link-padding-y: #{$nav-link-padding-y};\n  @include rfs($nav-link-font-size, --#{$prefix}nav-link-font-size);\n  --#{$prefix}nav-link-font-weight: #{$nav-link-font-weight};\n  --#{$prefix}nav-link-color: var(--#{$prefix}navbar-color);\n  --#{$prefix}nav-link-hover-color: var(--#{$prefix}navbar-hover-color);\n  --#{$prefix}nav-link-disabled-color: var(--#{$prefix}navbar-disabled-color);\n  // scss-docs-end navbar-nav-css-vars\n\n  display: flex;\n  flex-direction: column; // cannot use `inherit` to get the `.navbar`s value\n  padding-left: 0;\n  margin-bottom: 0;\n  list-style: none;\n\n  .nav-link {\n    &.active,\n    &.show {\n      color: var(--#{$prefix}navbar-active-color);\n    }\n  }\n\n  .dropdown-menu {\n    position: static;\n  }\n}\n\n\n// Navbar text\n//\n//\n\n.navbar-text {\n  padding-top: $nav-link-padding-y;\n  padding-bottom: $nav-link-padding-y;\n  color: var(--#{$prefix}navbar-color);\n\n  a,\n  a:hover,\n  a:focus  {\n    color: var(--#{$prefix}navbar-active-color);\n  }\n}\n\n\n// Responsive navbar\n//\n// Custom styles for responsive collapsing and toggling of navbar contents.\n// Powered by the collapse Bootstrap JavaScript plugin.\n\n// When collapsed, prevent the toggleable navbar contents from appearing in\n// the default flexbox row orientation. Requires the use of `flex-wrap: wrap`\n// on the `.navbar` parent.\n.navbar-collapse {\n  flex-basis: 100%;\n  flex-grow: 1;\n  // For always expanded or extra full navbars, ensure content aligns itself\n  // properly vertically. Can be easily overridden with flex utilities.\n  align-items: center;\n}\n\n// Button for toggling the navbar when in its collapsed state\n.navbar-toggler {\n  padding: var(--#{$prefix}navbar-toggler-padding-y) var(--#{$prefix}navbar-toggler-padding-x);\n  @include font-size(var(--#{$prefix}navbar-toggler-font-size));\n  line-height: 1;\n  color: var(--#{$prefix}navbar-color);\n  background-color: transparent; // remove default button style\n  border: var(--#{$prefix}border-width) solid var(--#{$prefix}navbar-toggler-border-color); // remove default button style\n  @include border-radius(var(--#{$prefix}navbar-toggler-border-radius));\n  @include transition(var(--#{$prefix}navbar-toggler-transition));\n\n  &:hover {\n    text-decoration: none;\n  }\n\n  &:focus {\n    text-decoration: none;\n    outline: 0;\n    box-shadow: 0 0 0 var(--#{$prefix}navbar-toggler-focus-width);\n  }\n}\n\n// Keep as a separate element so folks can easily override it with another icon\n// or image file as needed.\n.navbar-toggler-icon {\n  display: inline-block;\n  width: 1.5em;\n  height: 1.5em;\n  vertical-align: middle;\n  background-image: var(--#{$prefix}navbar-toggler-icon-bg);\n  background-repeat: no-repeat;\n  background-position: center;\n  background-size: 100%;\n}\n\n.navbar-nav-scroll {\n  max-height: var(--#{$prefix}scroll-height, 75vh);\n  overflow-y: auto;\n}\n\n// scss-docs-start navbar-expand-loop\n// Generate series of `.navbar-expand-*` responsive classes for configuring\n// where your navbar collapses.\n.navbar-expand {\n  @each $breakpoint in map-keys($grid-breakpoints) {\n    $next: breakpoint-next($breakpoint, $grid-breakpoints);\n    $infix: breakpoint-infix($next, $grid-breakpoints);\n\n    // stylelint-disable-next-line scss/selector-no-union-class-name\n    &#{$infix} {\n      @include media-breakpoint-up($next) {\n        flex-wrap: nowrap;\n        justify-content: flex-start;\n\n        .navbar-nav {\n          flex-direction: row;\n\n          .dropdown-menu {\n            position: absolute;\n          }\n\n          .nav-link {\n            padding-right: var(--#{$prefix}navbar-nav-link-padding-x);\n            padding-left: var(--#{$prefix}navbar-nav-link-padding-x);\n          }\n        }\n\n        .navbar-nav-scroll {\n          overflow: visible;\n        }\n\n        .navbar-collapse {\n          display: flex !important; // stylelint-disable-line declaration-no-important\n          flex-basis: auto;\n        }\n\n        .navbar-toggler {\n          display: none;\n        }\n\n        .offcanvas {\n          // stylelint-disable declaration-no-important\n          position: static;\n          z-index: auto;\n          flex-grow: 1;\n          width: auto !important;\n          height: auto !important;\n          visibility: visible !important;\n          background-color: transparent !important;\n          border: 0 !important;\n          transform: none !important;\n          @include box-shadow(none);\n          @include transition(none);\n          // stylelint-enable declaration-no-important\n\n          .offcanvas-header {\n            display: none;\n          }\n\n          .offcanvas-body {\n            display: flex;\n            flex-grow: 0;\n            padding: 0;\n            overflow-y: visible;\n          }\n        }\n      }\n    }\n  }\n}\n// scss-docs-end navbar-expand-loop\n\n// Navbar themes\n//\n// Styles for switching between navbars with light or dark background.\n\n.navbar-light {\n  @include deprecate(\"`.navbar-light`\", \"v5.2.0\", \"v6.0.0\", true);\n}\n\n.navbar-dark,\n.navbar[data-bs-theme=\"dark\"] {\n  // scss-docs-start navbar-dark-css-vars\n  --#{$prefix}navbar-color: #{$navbar-dark-color};\n  --#{$prefix}navbar-hover-color: #{$navbar-dark-hover-color};\n  --#{$prefix}navbar-disabled-color: #{$navbar-dark-disabled-color};\n  --#{$prefix}navbar-active-color: #{$navbar-dark-active-color};\n  --#{$prefix}navbar-brand-color: #{$navbar-dark-brand-color};\n  --#{$prefix}navbar-brand-hover-color: #{$navbar-dark-brand-hover-color};\n  --#{$prefix}navbar-toggler-border-color: #{$navbar-dark-toggler-border-color};\n  --#{$prefix}navbar-toggler-icon-bg: #{escape-svg($navbar-dark-toggler-icon-bg)};\n  // scss-docs-end navbar-dark-css-vars\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .navbar-toggler-icon {\n      --#{$prefix}navbar-toggler-icon-bg: #{escape-svg($navbar-dark-toggler-icon-bg)};\n    }\n  }\n}\n","//\n// Base styles\n//\n\n.card {\n  // scss-docs-start card-css-vars\n  --#{$prefix}card-spacer-y: #{$card-spacer-y};\n  --#{$prefix}card-spacer-x: #{$card-spacer-x};\n  --#{$prefix}card-title-spacer-y: #{$card-title-spacer-y};\n  --#{$prefix}card-title-color: #{$card-title-color};\n  --#{$prefix}card-subtitle-color: #{$card-subtitle-color};\n  --#{$prefix}card-border-width: #{$card-border-width};\n  --#{$prefix}card-border-color: #{$card-border-color};\n  --#{$prefix}card-border-radius: #{$card-border-radius};\n  --#{$prefix}card-box-shadow: #{$card-box-shadow};\n  --#{$prefix}card-inner-border-radius: #{$card-inner-border-radius};\n  --#{$prefix}card-cap-padding-y: #{$card-cap-padding-y};\n  --#{$prefix}card-cap-padding-x: #{$card-cap-padding-x};\n  --#{$prefix}card-cap-bg: #{$card-cap-bg};\n  --#{$prefix}card-cap-color: #{$card-cap-color};\n  --#{$prefix}card-height: #{$card-height};\n  --#{$prefix}card-color: #{$card-color};\n  --#{$prefix}card-bg: #{$card-bg};\n  --#{$prefix}card-img-overlay-padding: #{$card-img-overlay-padding};\n  --#{$prefix}card-group-margin: #{$card-group-margin};\n  // scss-docs-end card-css-vars\n\n  position: relative;\n  display: flex;\n  flex-direction: column;\n  min-width: 0; // See https://github.com/twbs/bootstrap/pull/22740#issuecomment-305868106\n  height: var(--#{$prefix}card-height);\n  color: var(--#{$prefix}body-color);\n  word-wrap: break-word;\n  background-color: var(--#{$prefix}card-bg);\n  background-clip: border-box;\n  border: var(--#{$prefix}card-border-width) solid var(--#{$prefix}card-border-color);\n  @include border-radius(var(--#{$prefix}card-border-radius));\n  @include box-shadow(var(--#{$prefix}card-box-shadow));\n\n  > hr {\n    margin-right: 0;\n    margin-left: 0;\n  }\n\n  > .list-group {\n    border-top: inherit;\n    border-bottom: inherit;\n\n    &:first-child {\n      border-top-width: 0;\n      @include border-top-radius(var(--#{$prefix}card-inner-border-radius));\n    }\n\n    &:last-child  {\n      border-bottom-width: 0;\n      @include border-bottom-radius(var(--#{$prefix}card-inner-border-radius));\n    }\n  }\n\n  // Due to specificity of the above selector (`.card > .list-group`), we must\n  // use a child selector here to prevent double borders.\n  > .card-header + .list-group,\n  > .list-group + .card-footer {\n    border-top: 0;\n  }\n}\n\n.card-body {\n  // Enable `flex-grow: 1` for decks and groups so that card blocks take up\n  // as much space as possible, ensuring footers are aligned to the bottom.\n  flex: 1 1 auto;\n  padding: var(--#{$prefix}card-spacer-y) var(--#{$prefix}card-spacer-x);\n  color: var(--#{$prefix}card-color);\n}\n\n.card-title {\n  margin-bottom: var(--#{$prefix}card-title-spacer-y);\n  color: var(--#{$prefix}card-title-color);\n}\n\n.card-subtitle {\n  margin-top: calc(-.5 * var(--#{$prefix}card-title-spacer-y)); // stylelint-disable-line function-disallowed-list\n  margin-bottom: 0;\n  color: var(--#{$prefix}card-subtitle-color);\n}\n\n.card-text:last-child {\n  margin-bottom: 0;\n}\n\n.card-link {\n  &:hover {\n    text-decoration: if($link-hover-decoration == underline, none, null);\n  }\n\n  + .card-link {\n    margin-left: var(--#{$prefix}card-spacer-x);\n  }\n}\n\n//\n// Optional textual caps\n//\n\n.card-header {\n  padding: var(--#{$prefix}card-cap-padding-y) var(--#{$prefix}card-cap-padding-x);\n  margin-bottom: 0; // Removes the default margin-bottom of <hN>\n  color: var(--#{$prefix}card-cap-color);\n  background-color: var(--#{$prefix}card-cap-bg);\n  border-bottom: var(--#{$prefix}card-border-width) solid var(--#{$prefix}card-border-color);\n\n  &:first-child {\n    @include border-radius(var(--#{$prefix}card-inner-border-radius) var(--#{$prefix}card-inner-border-radius) 0 0);\n  }\n}\n\n.card-footer {\n  padding: var(--#{$prefix}card-cap-padding-y) var(--#{$prefix}card-cap-padding-x);\n  color: var(--#{$prefix}card-cap-color);\n  background-color: var(--#{$prefix}card-cap-bg);\n  border-top: var(--#{$prefix}card-border-width) solid var(--#{$prefix}card-border-color);\n\n  &:last-child {\n    @include border-radius(0 0 var(--#{$prefix}card-inner-border-radius) var(--#{$prefix}card-inner-border-radius));\n  }\n}\n\n\n//\n// Header navs\n//\n\n.card-header-tabs {\n  margin-right: calc(-.5 * var(--#{$prefix}card-cap-padding-x)); // stylelint-disable-line function-disallowed-list\n  margin-bottom: calc(-1 * var(--#{$prefix}card-cap-padding-y)); // stylelint-disable-line function-disallowed-list\n  margin-left: calc(-.5 * var(--#{$prefix}card-cap-padding-x)); // stylelint-disable-line function-disallowed-list\n  border-bottom: 0;\n\n  .nav-link.active {\n    background-color: var(--#{$prefix}card-bg);\n    border-bottom-color: var(--#{$prefix}card-bg);\n  }\n}\n\n.card-header-pills {\n  margin-right: calc(-.5 * var(--#{$prefix}card-cap-padding-x)); // stylelint-disable-line function-disallowed-list\n  margin-left: calc(-.5 * var(--#{$prefix}card-cap-padding-x)); // stylelint-disable-line function-disallowed-list\n}\n\n// Card image\n.card-img-overlay {\n  position: absolute;\n  top: 0;\n  right: 0;\n  bottom: 0;\n  left: 0;\n  padding: var(--#{$prefix}card-img-overlay-padding);\n  @include border-radius(var(--#{$prefix}card-inner-border-radius));\n}\n\n.card-img,\n.card-img-top,\n.card-img-bottom {\n  width: 100%; // Required because we use flexbox and this inherently applies align-self: stretch\n}\n\n.card-img,\n.card-img-top {\n  @include border-top-radius(var(--#{$prefix}card-inner-border-radius));\n}\n\n.card-img,\n.card-img-bottom {\n  @include border-bottom-radius(var(--#{$prefix}card-inner-border-radius));\n}\n\n\n//\n// Card groups\n//\n\n.card-group {\n  // The child selector allows nested `.card` within `.card-group`\n  // to display properly.\n  > .card {\n    margin-bottom: var(--#{$prefix}card-group-margin);\n  }\n\n  @include media-breakpoint-up(sm) {\n    display: flex;\n    flex-flow: row wrap;\n    // The child selector allows nested `.card` within `.card-group`\n    // to display properly.\n    > .card {\n      // Flexbugs #4: https://github.com/philipwalton/flexbugs#flexbug-4\n      flex: 1 0 0%;\n      margin-bottom: 0;\n\n      + .card {\n        margin-left: 0;\n        border-left: 0;\n      }\n\n      // Handle rounded corners\n      @if $enable-rounded {\n        &:not(:last-child) {\n          @include border-end-radius(0);\n\n          .card-img-top,\n          .card-header {\n            // stylelint-disable-next-line property-disallowed-list\n            border-top-right-radius: 0;\n          }\n          .card-img-bottom,\n          .card-footer {\n            // stylelint-disable-next-line property-disallowed-list\n            border-bottom-right-radius: 0;\n          }\n        }\n\n        &:not(:first-child) {\n          @include border-start-radius(0);\n\n          .card-img-top,\n          .card-header {\n            // stylelint-disable-next-line property-disallowed-list\n            border-top-left-radius: 0;\n          }\n          .card-img-bottom,\n          .card-footer {\n            // stylelint-disable-next-line property-disallowed-list\n            border-bottom-left-radius: 0;\n          }\n        }\n      }\n    }\n  }\n}\n","//\n// Base styles\n//\n\n.accordion {\n  // scss-docs-start accordion-css-vars\n  --#{$prefix}accordion-color: #{$accordion-color};\n  --#{$prefix}accordion-bg: #{$accordion-bg};\n  --#{$prefix}accordion-transition: #{$accordion-transition};\n  --#{$prefix}accordion-border-color: #{$accordion-border-color};\n  --#{$prefix}accordion-border-width: #{$accordion-border-width};\n  --#{$prefix}accordion-border-radius: #{$accordion-border-radius};\n  --#{$prefix}accordion-inner-border-radius: #{$accordion-inner-border-radius};\n  --#{$prefix}accordion-btn-padding-x: #{$accordion-button-padding-x};\n  --#{$prefix}accordion-btn-padding-y: #{$accordion-button-padding-y};\n  --#{$prefix}accordion-btn-color: #{$accordion-button-color};\n  --#{$prefix}accordion-btn-bg: #{$accordion-button-bg};\n  --#{$prefix}accordion-btn-icon: #{escape-svg($accordion-button-icon)};\n  --#{$prefix}accordion-btn-icon-width: #{$accordion-icon-width};\n  --#{$prefix}accordion-btn-icon-transform: #{$accordion-icon-transform};\n  --#{$prefix}accordion-btn-icon-transition: #{$accordion-icon-transition};\n  --#{$prefix}accordion-btn-active-icon: #{escape-svg($accordion-button-active-icon)};\n  --#{$prefix}accordion-btn-focus-box-shadow: #{$accordion-button-focus-box-shadow};\n  --#{$prefix}accordion-body-padding-x: #{$accordion-body-padding-x};\n  --#{$prefix}accordion-body-padding-y: #{$accordion-body-padding-y};\n  --#{$prefix}accordion-active-color: #{$accordion-button-active-color};\n  --#{$prefix}accordion-active-bg: #{$accordion-button-active-bg};\n  // scss-docs-end accordion-css-vars\n}\n\n.accordion-button {\n  position: relative;\n  display: flex;\n  align-items: center;\n  width: 100%;\n  padding: var(--#{$prefix}accordion-btn-padding-y) var(--#{$prefix}accordion-btn-padding-x);\n  @include font-size($font-size-base);\n  color: var(--#{$prefix}accordion-btn-color);\n  text-align: left; // Reset button style\n  background-color: var(--#{$prefix}accordion-btn-bg);\n  border: 0;\n  @include border-radius(0);\n  overflow-anchor: none;\n  @include transition(var(--#{$prefix}accordion-transition));\n\n  &:not(.collapsed) {\n    color: var(--#{$prefix}accordion-active-color);\n    background-color: var(--#{$prefix}accordion-active-bg);\n    box-shadow: inset 0 calc(-1 * var(--#{$prefix}accordion-border-width)) 0 var(--#{$prefix}accordion-border-color); // stylelint-disable-line function-disallowed-list\n\n    &::after {\n      background-image: var(--#{$prefix}accordion-btn-active-icon);\n      transform: var(--#{$prefix}accordion-btn-icon-transform);\n    }\n  }\n\n  // Accordion icon\n  &::after {\n    flex-shrink: 0;\n    width: var(--#{$prefix}accordion-btn-icon-width);\n    height: var(--#{$prefix}accordion-btn-icon-width);\n    margin-left: auto;\n    content: \"\";\n    background-image: var(--#{$prefix}accordion-btn-icon);\n    background-repeat: no-repeat;\n    background-size: var(--#{$prefix}accordion-btn-icon-width);\n    @include transition(var(--#{$prefix}accordion-btn-icon-transition));\n  }\n\n  &:hover {\n    z-index: 2;\n  }\n\n  &:focus {\n    z-index: 3;\n    outline: 0;\n    box-shadow: var(--#{$prefix}accordion-btn-focus-box-shadow);\n  }\n}\n\n.accordion-header {\n  margin-bottom: 0;\n}\n\n.accordion-item {\n  color: var(--#{$prefix}accordion-color);\n  background-color: var(--#{$prefix}accordion-bg);\n  border: var(--#{$prefix}accordion-border-width) solid var(--#{$prefix}accordion-border-color);\n\n  &:first-of-type {\n    @include border-top-radius(var(--#{$prefix}accordion-border-radius));\n\n    > .accordion-header .accordion-button {\n      @include border-top-radius(var(--#{$prefix}accordion-inner-border-radius));\n    }\n  }\n\n  &:not(:first-of-type) {\n    border-top: 0;\n  }\n\n  // Only set a border-radius on the last item if the accordion is collapsed\n  &:last-of-type {\n    @include border-bottom-radius(var(--#{$prefix}accordion-border-radius));\n\n    > .accordion-header .accordion-button {\n      &.collapsed {\n        @include border-bottom-radius(var(--#{$prefix}accordion-inner-border-radius));\n      }\n    }\n\n    > .accordion-collapse {\n      @include border-bottom-radius(var(--#{$prefix}accordion-border-radius));\n    }\n  }\n}\n\n.accordion-body {\n  padding: var(--#{$prefix}accordion-body-padding-y) var(--#{$prefix}accordion-body-padding-x);\n}\n\n\n// Flush accordion items\n//\n// Remove borders and border-radius to keep accordion items edge-to-edge.\n\n.accordion-flush {\n  > .accordion-item {\n    border-right: 0;\n    border-left: 0;\n    @include border-radius(0);\n\n    &:first-child { border-top: 0; }\n    &:last-child { border-bottom: 0; }\n\n    // stylelint-disable selector-max-class\n    > .accordion-header .accordion-button {\n      &,\n      &.collapsed {\n        @include border-radius(0);\n      }\n    }\n    // stylelint-enable selector-max-class\n\n    > .accordion-collapse {\n      @include border-radius(0);\n    }\n  }\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .accordion-button::after {\n      --#{$prefix}accordion-btn-icon: #{escape-svg($accordion-button-icon-dark)};\n      --#{$prefix}accordion-btn-active-icon: #{escape-svg($accordion-button-active-icon-dark)};\n    }\n  }\n}\n",".breadcrumb {\n  // scss-docs-start breadcrumb-css-vars\n  --#{$prefix}breadcrumb-padding-x: #{$breadcrumb-padding-x};\n  --#{$prefix}breadcrumb-padding-y: #{$breadcrumb-padding-y};\n  --#{$prefix}breadcrumb-margin-bottom: #{$breadcrumb-margin-bottom};\n  @include rfs($breadcrumb-font-size, --#{$prefix}breadcrumb-font-size);\n  --#{$prefix}breadcrumb-bg: #{$breadcrumb-bg};\n  --#{$prefix}breadcrumb-border-radius: #{$breadcrumb-border-radius};\n  --#{$prefix}breadcrumb-divider-color: #{$breadcrumb-divider-color};\n  --#{$prefix}breadcrumb-item-padding-x: #{$breadcrumb-item-padding-x};\n  --#{$prefix}breadcrumb-item-active-color: #{$breadcrumb-active-color};\n  // scss-docs-end breadcrumb-css-vars\n\n  display: flex;\n  flex-wrap: wrap;\n  padding: var(--#{$prefix}breadcrumb-padding-y) var(--#{$prefix}breadcrumb-padding-x);\n  margin-bottom: var(--#{$prefix}breadcrumb-margin-bottom);\n  @include font-size(var(--#{$prefix}breadcrumb-font-size));\n  list-style: none;\n  background-color: var(--#{$prefix}breadcrumb-bg);\n  @include border-radius(var(--#{$prefix}breadcrumb-border-radius));\n}\n\n.breadcrumb-item {\n  // The separator between breadcrumbs (by default, a forward-slash: \"/\")\n  + .breadcrumb-item {\n    padding-left: var(--#{$prefix}breadcrumb-item-padding-x);\n\n    &::before {\n      float: left; // Suppress inline spacings and underlining of the separator\n      padding-right: var(--#{$prefix}breadcrumb-item-padding-x);\n      color: var(--#{$prefix}breadcrumb-divider-color);\n      content: var(--#{$prefix}breadcrumb-divider, escape-svg($breadcrumb-divider)) #{\"/* rtl:\"} var(--#{$prefix}breadcrumb-divider, escape-svg($breadcrumb-divider-flipped)) #{\"*/\"};\n    }\n  }\n\n  &.active {\n    color: var(--#{$prefix}breadcrumb-item-active-color);\n  }\n}\n",".pagination {\n  // scss-docs-start pagination-css-vars\n  --#{$prefix}pagination-padding-x: #{$pagination-padding-x};\n  --#{$prefix}pagination-padding-y: #{$pagination-padding-y};\n  @include rfs($pagination-font-size, --#{$prefix}pagination-font-size);\n  --#{$prefix}pagination-color: #{$pagination-color};\n  --#{$prefix}pagination-bg: #{$pagination-bg};\n  --#{$prefix}pagination-border-width: #{$pagination-border-width};\n  --#{$prefix}pagination-border-color: #{$pagination-border-color};\n  --#{$prefix}pagination-border-radius: #{$pagination-border-radius};\n  --#{$prefix}pagination-hover-color: #{$pagination-hover-color};\n  --#{$prefix}pagination-hover-bg: #{$pagination-hover-bg};\n  --#{$prefix}pagination-hover-border-color: #{$pagination-hover-border-color};\n  --#{$prefix}pagination-focus-color: #{$pagination-focus-color};\n  --#{$prefix}pagination-focus-bg: #{$pagination-focus-bg};\n  --#{$prefix}pagination-focus-box-shadow: #{$pagination-focus-box-shadow};\n  --#{$prefix}pagination-active-color: #{$pagination-active-color};\n  --#{$prefix}pagination-active-bg: #{$pagination-active-bg};\n  --#{$prefix}pagination-active-border-color: #{$pagination-active-border-color};\n  --#{$prefix}pagination-disabled-color: #{$pagination-disabled-color};\n  --#{$prefix}pagination-disabled-bg: #{$pagination-disabled-bg};\n  --#{$prefix}pagination-disabled-border-color: #{$pagination-disabled-border-color};\n  // scss-docs-end pagination-css-vars\n\n  display: flex;\n  @include list-unstyled();\n}\n\n.page-link {\n  position: relative;\n  display: block;\n  padding: var(--#{$prefix}pagination-padding-y) var(--#{$prefix}pagination-padding-x);\n  @include font-size(var(--#{$prefix}pagination-font-size));\n  color: var(--#{$prefix}pagination-color);\n  text-decoration: if($link-decoration == none, null, none);\n  background-color: var(--#{$prefix}pagination-bg);\n  border: var(--#{$prefix}pagination-border-width) solid var(--#{$prefix}pagination-border-color);\n  @include transition($pagination-transition);\n\n  &:hover {\n    z-index: 2;\n    color: var(--#{$prefix}pagination-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n    background-color: var(--#{$prefix}pagination-hover-bg);\n    border-color: var(--#{$prefix}pagination-hover-border-color);\n  }\n\n  &:focus {\n    z-index: 3;\n    color: var(--#{$prefix}pagination-focus-color);\n    background-color: var(--#{$prefix}pagination-focus-bg);\n    outline: $pagination-focus-outline;\n    box-shadow: var(--#{$prefix}pagination-focus-box-shadow);\n  }\n\n  &.active,\n  .active > & {\n    z-index: 3;\n    color: var(--#{$prefix}pagination-active-color);\n    @include gradient-bg(var(--#{$prefix}pagination-active-bg));\n    border-color: var(--#{$prefix}pagination-active-border-color);\n  }\n\n  &.disabled,\n  .disabled > & {\n    color: var(--#{$prefix}pagination-disabled-color);\n    pointer-events: none;\n    background-color: var(--#{$prefix}pagination-disabled-bg);\n    border-color: var(--#{$prefix}pagination-disabled-border-color);\n  }\n}\n\n.page-item {\n  &:not(:first-child) .page-link {\n    margin-left: $pagination-margin-start;\n  }\n\n  @if $pagination-margin-start == calc(#{$pagination-border-width} * -1) {\n    &:first-child {\n      .page-link {\n        @include border-start-radius(var(--#{$prefix}pagination-border-radius));\n      }\n    }\n\n    &:last-child {\n      .page-link {\n        @include border-end-radius(var(--#{$prefix}pagination-border-radius));\n      }\n    }\n  } @else {\n    // Add border-radius to all pageLinks in case they have left margin\n    .page-link {\n      @include border-radius(var(--#{$prefix}pagination-border-radius));\n    }\n  }\n}\n\n\n//\n// Sizing\n//\n\n.pagination-lg {\n  @include pagination-size($pagination-padding-y-lg, $pagination-padding-x-lg, $font-size-lg, $pagination-border-radius-lg);\n}\n\n.pagination-sm {\n  @include pagination-size($pagination-padding-y-sm, $pagination-padding-x-sm, $font-size-sm, $pagination-border-radius-sm);\n}\n","// Pagination\n\n// scss-docs-start pagination-mixin\n@mixin pagination-size($padding-y, $padding-x, $font-size, $border-radius) {\n  --#{$prefix}pagination-padding-x: #{$padding-x};\n  --#{$prefix}pagination-padding-y: #{$padding-y};\n  @include rfs($font-size, --#{$prefix}pagination-font-size);\n  --#{$prefix}pagination-border-radius: #{$border-radius};\n}\n// scss-docs-end pagination-mixin\n","// Base class\n//\n// Requires one of the contextual, color modifier classes for `color` and\n// `background-color`.\n\n.badge {\n  // scss-docs-start badge-css-vars\n  --#{$prefix}badge-padding-x: #{$badge-padding-x};\n  --#{$prefix}badge-padding-y: #{$badge-padding-y};\n  @include rfs($badge-font-size, --#{$prefix}badge-font-size);\n  --#{$prefix}badge-font-weight: #{$badge-font-weight};\n  --#{$prefix}badge-color: #{$badge-color};\n  --#{$prefix}badge-border-radius: #{$badge-border-radius};\n  // scss-docs-end badge-css-vars\n\n  display: inline-block;\n  padding: var(--#{$prefix}badge-padding-y) var(--#{$prefix}badge-padding-x);\n  @include font-size(var(--#{$prefix}badge-font-size));\n  font-weight: var(--#{$prefix}badge-font-weight);\n  line-height: 1;\n  color: var(--#{$prefix}badge-color);\n  text-align: center;\n  white-space: nowrap;\n  vertical-align: baseline;\n  @include border-radius(var(--#{$prefix}badge-border-radius));\n  @include gradient-bg();\n\n  // Empty badges collapse automatically\n  &:empty {\n    display: none;\n  }\n}\n\n// Quick fix for badges in buttons\n.btn .badge {\n  position: relative;\n  top: -1px;\n}\n","//\n// Base styles\n//\n\n.alert {\n  // scss-docs-start alert-css-vars\n  --#{$prefix}alert-bg: transparent;\n  --#{$prefix}alert-padding-x: #{$alert-padding-x};\n  --#{$prefix}alert-padding-y: #{$alert-padding-y};\n  --#{$prefix}alert-margin-bottom: #{$alert-margin-bottom};\n  --#{$prefix}alert-color: inherit;\n  --#{$prefix}alert-border-color: transparent;\n  --#{$prefix}alert-border: #{$alert-border-width} solid var(--#{$prefix}alert-border-color);\n  --#{$prefix}alert-border-radius: #{$alert-border-radius};\n  --#{$prefix}alert-link-color: inherit;\n  // scss-docs-end alert-css-vars\n\n  position: relative;\n  padding: var(--#{$prefix}alert-padding-y) var(--#{$prefix}alert-padding-x);\n  margin-bottom: var(--#{$prefix}alert-margin-bottom);\n  color: var(--#{$prefix}alert-color);\n  background-color: var(--#{$prefix}alert-bg);\n  border: var(--#{$prefix}alert-border);\n  @include border-radius(var(--#{$prefix}alert-border-radius));\n}\n\n// Headings for larger alerts\n.alert-heading {\n  // Specified to prevent conflicts of changing $headings-color\n  color: inherit;\n}\n\n// Provide class for links that match alerts\n.alert-link {\n  font-weight: $alert-link-font-weight;\n  color: var(--#{$prefix}alert-link-color);\n}\n\n\n// Dismissible alerts\n//\n// Expand the right padding and account for the close button's positioning.\n\n.alert-dismissible {\n  padding-right: $alert-dismissible-padding-r;\n\n  // Adjust close link position\n  .btn-close {\n    position: absolute;\n    top: 0;\n    right: 0;\n    z-index: $stretched-link-z-index + 1;\n    padding: $alert-padding-y * 1.25 $alert-padding-x;\n  }\n}\n\n\n// scss-docs-start alert-modifiers\n// Generate contextual modifier classes for colorizing the alert\n@each $state in map-keys($theme-colors) {\n  .alert-#{$state} {\n    --#{$prefix}alert-color: var(--#{$prefix}#{$state}-text-emphasis);\n    --#{$prefix}alert-bg: var(--#{$prefix}#{$state}-bg-subtle);\n    --#{$prefix}alert-border-color: var(--#{$prefix}#{$state}-border-subtle);\n    --#{$prefix}alert-link-color: var(--#{$prefix}#{$state}-text-emphasis);\n  }\n}\n// scss-docs-end alert-modifiers\n","// Disable animation if transitions are disabled\n\n// scss-docs-start progress-keyframes\n@if $enable-transitions {\n  @keyframes progress-bar-stripes {\n    0% { background-position-x: $progress-height; }\n  }\n}\n// scss-docs-end progress-keyframes\n\n.progress,\n.progress-stacked {\n  // scss-docs-start progress-css-vars\n  --#{$prefix}progress-height: #{$progress-height};\n  @include rfs($progress-font-size, --#{$prefix}progress-font-size);\n  --#{$prefix}progress-bg: #{$progress-bg};\n  --#{$prefix}progress-border-radius: #{$progress-border-radius};\n  --#{$prefix}progress-box-shadow: #{$progress-box-shadow};\n  --#{$prefix}progress-bar-color: #{$progress-bar-color};\n  --#{$prefix}progress-bar-bg: #{$progress-bar-bg};\n  --#{$prefix}progress-bar-transition: #{$progress-bar-transition};\n  // scss-docs-end progress-css-vars\n\n  display: flex;\n  height: var(--#{$prefix}progress-height);\n  overflow: hidden; // force rounded corners by cropping it\n  @include font-size(var(--#{$prefix}progress-font-size));\n  background-color: var(--#{$prefix}progress-bg);\n  @include border-radius(var(--#{$prefix}progress-border-radius));\n  @include box-shadow(var(--#{$prefix}progress-box-shadow));\n}\n\n.progress-bar {\n  display: flex;\n  flex-direction: column;\n  justify-content: center;\n  overflow: hidden;\n  color: var(--#{$prefix}progress-bar-color);\n  text-align: center;\n  white-space: nowrap;\n  background-color: var(--#{$prefix}progress-bar-bg);\n  @include transition(var(--#{$prefix}progress-bar-transition));\n}\n\n.progress-bar-striped {\n  @include gradient-striped();\n  background-size: var(--#{$prefix}progress-height) var(--#{$prefix}progress-height);\n}\n\n.progress-stacked > .progress {\n  overflow: visible;\n}\n\n.progress-stacked > .progress > .progress-bar {\n  width: 100%;\n}\n\n@if $enable-transitions {\n  .progress-bar-animated {\n    animation: $progress-bar-animation-timing progress-bar-stripes;\n\n    @if $enable-reduced-motion {\n      @media (prefers-reduced-motion: reduce) {\n        animation: none;\n      }\n    }\n  }\n}\n","// Base class\n//\n// Easily usable on <ul>, <ol>, or <div>.\n\n.list-group {\n  // scss-docs-start list-group-css-vars\n  --#{$prefix}list-group-color: #{$list-group-color};\n  --#{$prefix}list-group-bg: #{$list-group-bg};\n  --#{$prefix}list-group-border-color: #{$list-group-border-color};\n  --#{$prefix}list-group-border-width: #{$list-group-border-width};\n  --#{$prefix}list-group-border-radius: #{$list-group-border-radius};\n  --#{$prefix}list-group-item-padding-x: #{$list-group-item-padding-x};\n  --#{$prefix}list-group-item-padding-y: #{$list-group-item-padding-y};\n  --#{$prefix}list-group-action-color: #{$list-group-action-color};\n  --#{$prefix}list-group-action-hover-color: #{$list-group-action-hover-color};\n  --#{$prefix}list-group-action-hover-bg: #{$list-group-hover-bg};\n  --#{$prefix}list-group-action-active-color: #{$list-group-action-active-color};\n  --#{$prefix}list-group-action-active-bg: #{$list-group-action-active-bg};\n  --#{$prefix}list-group-disabled-color: #{$list-group-disabled-color};\n  --#{$prefix}list-group-disabled-bg: #{$list-group-disabled-bg};\n  --#{$prefix}list-group-active-color: #{$list-group-active-color};\n  --#{$prefix}list-group-active-bg: #{$list-group-active-bg};\n  --#{$prefix}list-group-active-border-color: #{$list-group-active-border-color};\n  // scss-docs-end list-group-css-vars\n\n  display: flex;\n  flex-direction: column;\n\n  // No need to set list-style: none; since .list-group-item is block level\n  padding-left: 0; // reset padding because ul and ol\n  margin-bottom: 0;\n  @include border-radius(var(--#{$prefix}list-group-border-radius));\n}\n\n.list-group-numbered {\n  list-style-type: none;\n  counter-reset: section;\n\n  > .list-group-item::before {\n    // Increments only this instance of the section counter\n    content: counters(section, \".\") \". \";\n    counter-increment: section;\n  }\n}\n\n// Interactive list items\n//\n// Use anchor or button elements instead of `li`s or `div`s to create interactive\n// list items. Includes an extra `.active` modifier class for selected items.\n\n.list-group-item-action {\n  width: 100%; // For `<button>`s (anchors become 100% by default though)\n  color: var(--#{$prefix}list-group-action-color);\n  text-align: inherit; // For `<button>`s (anchors inherit)\n\n  // Hover state\n  &:hover,\n  &:focus {\n    z-index: 1; // Place hover/focus items above their siblings for proper border styling\n    color: var(--#{$prefix}list-group-action-hover-color);\n    text-decoration: none;\n    background-color: var(--#{$prefix}list-group-action-hover-bg);\n  }\n\n  &:active {\n    color: var(--#{$prefix}list-group-action-active-color);\n    background-color: var(--#{$prefix}list-group-action-active-bg);\n  }\n}\n\n// Individual list items\n//\n// Use on `li`s or `div`s within the `.list-group` parent.\n\n.list-group-item {\n  position: relative;\n  display: block;\n  padding: var(--#{$prefix}list-group-item-padding-y) var(--#{$prefix}list-group-item-padding-x);\n  color: var(--#{$prefix}list-group-color);\n  text-decoration: if($link-decoration == none, null, none);\n  background-color: var(--#{$prefix}list-group-bg);\n  border: var(--#{$prefix}list-group-border-width) solid var(--#{$prefix}list-group-border-color);\n\n  &:first-child {\n    @include border-top-radius(inherit);\n  }\n\n  &:last-child {\n    @include border-bottom-radius(inherit);\n  }\n\n  &.disabled,\n  &:disabled {\n    color: var(--#{$prefix}list-group-disabled-color);\n    pointer-events: none;\n    background-color: var(--#{$prefix}list-group-disabled-bg);\n  }\n\n  // Include both here for `<a>`s and `<button>`s\n  &.active {\n    z-index: 2; // Place active items above their siblings for proper border styling\n    color: var(--#{$prefix}list-group-active-color);\n    background-color: var(--#{$prefix}list-group-active-bg);\n    border-color: var(--#{$prefix}list-group-active-border-color);\n  }\n\n  // stylelint-disable-next-line scss/selector-no-redundant-nesting-selector\n  & + .list-group-item {\n    border-top-width: 0;\n\n    &.active {\n      margin-top: calc(-1 * var(--#{$prefix}list-group-border-width)); // stylelint-disable-line function-disallowed-list\n      border-top-width: var(--#{$prefix}list-group-border-width);\n    }\n  }\n}\n\n// Horizontal\n//\n// Change the layout of list group items from vertical (default) to horizontal.\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  @include media-breakpoint-up($breakpoint) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    .list-group-horizontal#{$infix} {\n      flex-direction: row;\n\n      > .list-group-item {\n        &:first-child:not(:last-child) {\n          @include border-bottom-start-radius(var(--#{$prefix}list-group-border-radius));\n          @include border-top-end-radius(0);\n        }\n\n        &:last-child:not(:first-child) {\n          @include border-top-end-radius(var(--#{$prefix}list-group-border-radius));\n          @include border-bottom-start-radius(0);\n        }\n\n        &.active {\n          margin-top: 0;\n        }\n\n        + .list-group-item {\n          border-top-width: var(--#{$prefix}list-group-border-width);\n          border-left-width: 0;\n\n          &.active {\n            margin-left: calc(-1 * var(--#{$prefix}list-group-border-width)); // stylelint-disable-line function-disallowed-list\n            border-left-width: var(--#{$prefix}list-group-border-width);\n          }\n        }\n      }\n    }\n  }\n}\n\n\n// Flush list items\n//\n// Remove borders and border-radius to keep list group items edge-to-edge. Most\n// useful within other components (e.g., cards).\n\n.list-group-flush {\n  @include border-radius(0);\n\n  > .list-group-item {\n    border-width: 0 0 var(--#{$prefix}list-group-border-width);\n\n    &:last-child {\n      border-bottom-width: 0;\n    }\n  }\n}\n\n\n// scss-docs-start list-group-modifiers\n// List group contextual variants\n//\n// Add modifier classes to change text and background color on individual items.\n// Organizationally, this must come after the `:hover` states.\n\n@each $state in map-keys($theme-colors) {\n  .list-group-item-#{$state} {\n    --#{$prefix}list-group-color: var(--#{$prefix}#{$state}-text-emphasis);\n    --#{$prefix}list-group-bg: var(--#{$prefix}#{$state}-bg-subtle);\n    --#{$prefix}list-group-border-color: var(--#{$prefix}#{$state}-border-subtle);\n    --#{$prefix}list-group-action-hover-color: var(--#{$prefix}emphasis-color);\n    --#{$prefix}list-group-action-hover-bg: var(--#{$prefix}#{$state}-border-subtle);\n    --#{$prefix}list-group-action-active-color: var(--#{$prefix}emphasis-color);\n    --#{$prefix}list-group-action-active-bg: var(--#{$prefix}#{$state}-border-subtle);\n    --#{$prefix}list-group-active-color: var(--#{$prefix}#{$state}-bg-subtle);\n    --#{$prefix}list-group-active-bg: var(--#{$prefix}#{$state}-text-emphasis);\n    --#{$prefix}list-group-active-border-color: var(--#{$prefix}#{$state}-text-emphasis);\n  }\n}\n// scss-docs-end list-group-modifiers\n","// Transparent background and border properties included for button version.\n// iOS requires the button element instead of an anchor tag.\n// If you want the anchor version, it requires `href=\"#\"`.\n// See https://developer.mozilla.org/en-US/docs/Web/Events/click#Safari_Mobile\n\n.btn-close {\n  // scss-docs-start close-css-vars\n  --#{$prefix}btn-close-color: #{$btn-close-color};\n  --#{$prefix}btn-close-bg: #{ escape-svg($btn-close-bg) };\n  --#{$prefix}btn-close-opacity: #{$btn-close-opacity};\n  --#{$prefix}btn-close-hover-opacity: #{$btn-close-hover-opacity};\n  --#{$prefix}btn-close-focus-shadow: #{$btn-close-focus-shadow};\n  --#{$prefix}btn-close-focus-opacity: #{$btn-close-focus-opacity};\n  --#{$prefix}btn-close-disabled-opacity: #{$btn-close-disabled-opacity};\n  --#{$prefix}btn-close-white-filter: #{$btn-close-white-filter};\n  // scss-docs-end close-css-vars\n\n  box-sizing: content-box;\n  width: $btn-close-width;\n  height: $btn-close-height;\n  padding: $btn-close-padding-y $btn-close-padding-x;\n  color: var(--#{$prefix}btn-close-color);\n  background: transparent var(--#{$prefix}btn-close-bg) center / $btn-close-width auto no-repeat; // include transparent for button elements\n  border: 0; // for button elements\n  @include border-radius();\n  opacity: var(--#{$prefix}btn-close-opacity);\n\n  // Override <a>'s hover style\n  &:hover {\n    color: var(--#{$prefix}btn-close-color);\n    text-decoration: none;\n    opacity: var(--#{$prefix}btn-close-hover-opacity);\n  }\n\n  &:focus {\n    outline: 0;\n    box-shadow: var(--#{$prefix}btn-close-focus-shadow);\n    opacity: var(--#{$prefix}btn-close-focus-opacity);\n  }\n\n  &:disabled,\n  &.disabled {\n    pointer-events: none;\n    user-select: none;\n    opacity: var(--#{$prefix}btn-close-disabled-opacity);\n  }\n}\n\n@mixin btn-close-white() {\n  filter: var(--#{$prefix}btn-close-white-filter);\n}\n\n.btn-close-white {\n  @include btn-close-white();\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .btn-close {\n      @include btn-close-white();\n    }\n  }\n}\n",".toast {\n  // scss-docs-start toast-css-vars\n  --#{$prefix}toast-zindex: #{$zindex-toast};\n  --#{$prefix}toast-padding-x: #{$toast-padding-x};\n  --#{$prefix}toast-padding-y: #{$toast-padding-y};\n  --#{$prefix}toast-spacing: #{$toast-spacing};\n  --#{$prefix}toast-max-width: #{$toast-max-width};\n  @include rfs($toast-font-size, --#{$prefix}toast-font-size);\n  --#{$prefix}toast-color: #{$toast-color};\n  --#{$prefix}toast-bg: #{$toast-background-color};\n  --#{$prefix}toast-border-width: #{$toast-border-width};\n  --#{$prefix}toast-border-color: #{$toast-border-color};\n  --#{$prefix}toast-border-radius: #{$toast-border-radius};\n  --#{$prefix}toast-box-shadow: #{$toast-box-shadow};\n  --#{$prefix}toast-header-color: #{$toast-header-color};\n  --#{$prefix}toast-header-bg: #{$toast-header-background-color};\n  --#{$prefix}toast-header-border-color: #{$toast-header-border-color};\n  // scss-docs-end toast-css-vars\n\n  width: var(--#{$prefix}toast-max-width);\n  max-width: 100%;\n  @include font-size(var(--#{$prefix}toast-font-size));\n  color: var(--#{$prefix}toast-color);\n  pointer-events: auto;\n  background-color: var(--#{$prefix}toast-bg);\n  background-clip: padding-box;\n  border: var(--#{$prefix}toast-border-width) solid var(--#{$prefix}toast-border-color);\n  box-shadow: var(--#{$prefix}toast-box-shadow);\n  @include border-radius(var(--#{$prefix}toast-border-radius));\n\n  &.showing {\n    opacity: 0;\n  }\n\n  &:not(.show) {\n    display: none;\n  }\n}\n\n.toast-container {\n  --#{$prefix}toast-zindex: #{$zindex-toast};\n\n  position: absolute;\n  z-index: var(--#{$prefix}toast-zindex);\n  width: max-content;\n  max-width: 100%;\n  pointer-events: none;\n\n  > :not(:last-child) {\n    margin-bottom: var(--#{$prefix}toast-spacing);\n  }\n}\n\n.toast-header {\n  display: flex;\n  align-items: center;\n  padding: var(--#{$prefix}toast-padding-y) var(--#{$prefix}toast-padding-x);\n  color: var(--#{$prefix}toast-header-color);\n  background-color: var(--#{$prefix}toast-header-bg);\n  background-clip: padding-box;\n  border-bottom: var(--#{$prefix}toast-border-width) solid var(--#{$prefix}toast-header-border-color);\n  @include border-top-radius(calc(var(--#{$prefix}toast-border-radius) - var(--#{$prefix}toast-border-width)));\n\n  .btn-close {\n    margin-right: calc(-.5 * var(--#{$prefix}toast-padding-x)); // stylelint-disable-line function-disallowed-list\n    margin-left: var(--#{$prefix}toast-padding-x);\n  }\n}\n\n.toast-body {\n  padding: var(--#{$prefix}toast-padding-x);\n  word-wrap: break-word;\n}\n","// stylelint-disable function-disallowed-list\n\n// .modal-open      - body class for killing the scroll\n// .modal           - container to scroll within\n// .modal-dialog    - positioning shell for the actual modal\n// .modal-content   - actual modal w/ bg and corners and stuff\n\n\n// Container that the modal scrolls within\n.modal {\n  // scss-docs-start modal-css-vars\n  --#{$prefix}modal-zindex: #{$zindex-modal};\n  --#{$prefix}modal-width: #{$modal-md};\n  --#{$prefix}modal-padding: #{$modal-inner-padding};\n  --#{$prefix}modal-margin: #{$modal-dialog-margin};\n  --#{$prefix}modal-color: #{$modal-content-color};\n  --#{$prefix}modal-bg: #{$modal-content-bg};\n  --#{$prefix}modal-border-color: #{$modal-content-border-color};\n  --#{$prefix}modal-border-width: #{$modal-content-border-width};\n  --#{$prefix}modal-border-radius: #{$modal-content-border-radius};\n  --#{$prefix}modal-box-shadow: #{$modal-content-box-shadow-xs};\n  --#{$prefix}modal-inner-border-radius: #{$modal-content-inner-border-radius};\n  --#{$prefix}modal-header-padding-x: #{$modal-header-padding-x};\n  --#{$prefix}modal-header-padding-y: #{$modal-header-padding-y};\n  --#{$prefix}modal-header-padding: #{$modal-header-padding}; // Todo in v6: Split this padding into x and y\n  --#{$prefix}modal-header-border-color: #{$modal-header-border-color};\n  --#{$prefix}modal-header-border-width: #{$modal-header-border-width};\n  --#{$prefix}modal-title-line-height: #{$modal-title-line-height};\n  --#{$prefix}modal-footer-gap: #{$modal-footer-margin-between};\n  --#{$prefix}modal-footer-bg: #{$modal-footer-bg};\n  --#{$prefix}modal-footer-border-color: #{$modal-footer-border-color};\n  --#{$prefix}modal-footer-border-width: #{$modal-footer-border-width};\n  // scss-docs-end modal-css-vars\n\n  position: fixed;\n  top: 0;\n  left: 0;\n  z-index: var(--#{$prefix}modal-zindex);\n  display: none;\n  width: 100%;\n  height: 100%;\n  overflow-x: hidden;\n  overflow-y: auto;\n  // Prevent Chrome on Windows from adding a focus outline. For details, see\n  // https://github.com/twbs/bootstrap/pull/10951.\n  outline: 0;\n  // We deliberately don't use `-webkit-overflow-scrolling: touch;` due to a\n  // gnarly iOS Safari bug: https://bugs.webkit.org/show_bug.cgi?id=158342\n  // See also https://github.com/twbs/bootstrap/issues/17695\n}\n\n// Shell div to position the modal with bottom padding\n.modal-dialog {\n  position: relative;\n  width: auto;\n  margin: var(--#{$prefix}modal-margin);\n  // allow clicks to pass through for custom click handling to close modal\n  pointer-events: none;\n\n  // When fading in the modal, animate it to slide down\n  .modal.fade & {\n    @include transition($modal-transition);\n    transform: $modal-fade-transform;\n  }\n  .modal.show & {\n    transform: $modal-show-transform;\n  }\n\n  // When trying to close, animate focus to scale\n  .modal.modal-static & {\n    transform: $modal-scale-transform;\n  }\n}\n\n.modal-dialog-scrollable {\n  height: calc(100% - var(--#{$prefix}modal-margin) * 2);\n\n  .modal-content {\n    max-height: 100%;\n    overflow: hidden;\n  }\n\n  .modal-body {\n    overflow-y: auto;\n  }\n}\n\n.modal-dialog-centered {\n  display: flex;\n  align-items: center;\n  min-height: calc(100% - var(--#{$prefix}modal-margin) * 2);\n}\n\n// Actual modal\n.modal-content {\n  position: relative;\n  display: flex;\n  flex-direction: column;\n  width: 100%; // Ensure `.modal-content` extends the full width of the parent `.modal-dialog`\n  // counteract the pointer-events: none; in the .modal-dialog\n  color: var(--#{$prefix}modal-color);\n  pointer-events: auto;\n  background-color: var(--#{$prefix}modal-bg);\n  background-clip: padding-box;\n  border: var(--#{$prefix}modal-border-width) solid var(--#{$prefix}modal-border-color);\n  @include border-radius(var(--#{$prefix}modal-border-radius));\n  @include box-shadow(var(--#{$prefix}modal-box-shadow));\n  // Remove focus outline from opened modal\n  outline: 0;\n}\n\n// Modal background\n.modal-backdrop {\n  // scss-docs-start modal-backdrop-css-vars\n  --#{$prefix}backdrop-zindex: #{$zindex-modal-backdrop};\n  --#{$prefix}backdrop-bg: #{$modal-backdrop-bg};\n  --#{$prefix}backdrop-opacity: #{$modal-backdrop-opacity};\n  // scss-docs-end modal-backdrop-css-vars\n\n  @include overlay-backdrop(var(--#{$prefix}backdrop-zindex), var(--#{$prefix}backdrop-bg), var(--#{$prefix}backdrop-opacity));\n}\n\n// Modal header\n// Top section of the modal w/ title and dismiss\n.modal-header {\n  display: flex;\n  flex-shrink: 0;\n  align-items: center;\n  padding: var(--#{$prefix}modal-header-padding);\n  border-bottom: var(--#{$prefix}modal-header-border-width) solid var(--#{$prefix}modal-header-border-color);\n  @include border-top-radius(var(--#{$prefix}modal-inner-border-radius));\n\n  .btn-close {\n    padding: calc(var(--#{$prefix}modal-header-padding-y) * .5) calc(var(--#{$prefix}modal-header-padding-x) * .5);\n    margin: calc(-.5 * var(--#{$prefix}modal-header-padding-y)) calc(-.5 * var(--#{$prefix}modal-header-padding-x)) calc(-.5 * var(--#{$prefix}modal-header-padding-y)) auto;\n  }\n}\n\n// Title text within header\n.modal-title {\n  margin-bottom: 0;\n  line-height: var(--#{$prefix}modal-title-line-height);\n}\n\n// Modal body\n// Where all modal content resides (sibling of .modal-header and .modal-footer)\n.modal-body {\n  position: relative;\n  // Enable `flex-grow: 1` so that the body take up as much space as possible\n  // when there should be a fixed height on `.modal-dialog`.\n  flex: 1 1 auto;\n  padding: var(--#{$prefix}modal-padding);\n}\n\n// Footer (for actions)\n.modal-footer {\n  display: flex;\n  flex-shrink: 0;\n  flex-wrap: wrap;\n  align-items: center; // vertically center\n  justify-content: flex-end; // Right align buttons with flex property because text-align doesn't work on flex items\n  padding: calc(var(--#{$prefix}modal-padding) - var(--#{$prefix}modal-footer-gap) * .5);\n  background-color: var(--#{$prefix}modal-footer-bg);\n  border-top: var(--#{$prefix}modal-footer-border-width) solid var(--#{$prefix}modal-footer-border-color);\n  @include border-bottom-radius(var(--#{$prefix}modal-inner-border-radius));\n\n  // Place margin between footer elements\n  // This solution is far from ideal because of the universal selector usage,\n  // but is needed to fix https://github.com/twbs/bootstrap/issues/24800\n  > * {\n    margin: calc(var(--#{$prefix}modal-footer-gap) * .5); // Todo in v6: replace with gap on parent class\n  }\n}\n\n// Scale up the modal\n@include media-breakpoint-up(sm) {\n  .modal {\n    --#{$prefix}modal-margin: #{$modal-dialog-margin-y-sm-up};\n    --#{$prefix}modal-box-shadow: #{$modal-content-box-shadow-sm-up};\n  }\n\n  // Automatically set modal's width for larger viewports\n  .modal-dialog {\n    max-width: var(--#{$prefix}modal-width);\n    margin-right: auto;\n    margin-left: auto;\n  }\n\n  .modal-sm {\n    --#{$prefix}modal-width: #{$modal-sm};\n  }\n}\n\n@include media-breakpoint-up(lg) {\n  .modal-lg,\n  .modal-xl {\n    --#{$prefix}modal-width: #{$modal-lg};\n  }\n}\n\n@include media-breakpoint-up(xl) {\n  .modal-xl {\n    --#{$prefix}modal-width: #{$modal-xl};\n  }\n}\n\n// scss-docs-start modal-fullscreen-loop\n@each $breakpoint in map-keys($grid-breakpoints) {\n  $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n  $postfix: if($infix != \"\", $infix + \"-down\", \"\");\n\n  @include media-breakpoint-down($breakpoint) {\n    .modal-fullscreen#{$postfix} {\n      width: 100vw;\n      max-width: none;\n      height: 100%;\n      margin: 0;\n\n      .modal-content {\n        height: 100%;\n        border: 0;\n        @include border-radius(0);\n      }\n\n      .modal-header,\n      .modal-footer {\n        @include border-radius(0);\n      }\n\n      .modal-body {\n        overflow-y: auto;\n      }\n    }\n  }\n}\n// scss-docs-end modal-fullscreen-loop\n","// Shared between modals and offcanvases\n@mixin overlay-backdrop($zindex, $backdrop-bg, $backdrop-opacity) {\n  position: fixed;\n  top: 0;\n  left: 0;\n  z-index: $zindex;\n  width: 100vw;\n  height: 100vh;\n  background-color: $backdrop-bg;\n\n  // Fade for backdrop\n  &.fade { opacity: 0; }\n  &.show { opacity: $backdrop-opacity; }\n}\n","// Base class\n.tooltip {\n  // scss-docs-start tooltip-css-vars\n  --#{$prefix}tooltip-zindex: #{$zindex-tooltip};\n  --#{$prefix}tooltip-max-width: #{$tooltip-max-width};\n  --#{$prefix}tooltip-padding-x: #{$tooltip-padding-x};\n  --#{$prefix}tooltip-padding-y: #{$tooltip-padding-y};\n  --#{$prefix}tooltip-margin: #{$tooltip-margin};\n  @include rfs($tooltip-font-size, --#{$prefix}tooltip-font-size);\n  --#{$prefix}tooltip-color: #{$tooltip-color};\n  --#{$prefix}tooltip-bg: #{$tooltip-bg};\n  --#{$prefix}tooltip-border-radius: #{$tooltip-border-radius};\n  --#{$prefix}tooltip-opacity: #{$tooltip-opacity};\n  --#{$prefix}tooltip-arrow-width: #{$tooltip-arrow-width};\n  --#{$prefix}tooltip-arrow-height: #{$tooltip-arrow-height};\n  // scss-docs-end tooltip-css-vars\n\n  z-index: var(--#{$prefix}tooltip-zindex);\n  display: block;\n  margin: var(--#{$prefix}tooltip-margin);\n  @include deprecate(\"`$tooltip-margin`\", \"v5\", \"v5.x\", true);\n  // Our parent element can be arbitrary since tooltips are by default inserted as a sibling of their target element.\n  // So reset our font and text properties to avoid inheriting weird values.\n  @include reset-text();\n  @include font-size(var(--#{$prefix}tooltip-font-size));\n  // Allow breaking very long words so they don't overflow the tooltip's bounds\n  word-wrap: break-word;\n  opacity: 0;\n\n  &.show { opacity: var(--#{$prefix}tooltip-opacity); }\n\n  .tooltip-arrow {\n    display: block;\n    width: var(--#{$prefix}tooltip-arrow-width);\n    height: var(--#{$prefix}tooltip-arrow-height);\n\n    &::before {\n      position: absolute;\n      content: \"\";\n      border-color: transparent;\n      border-style: solid;\n    }\n  }\n}\n\n.bs-tooltip-top .tooltip-arrow {\n  bottom: calc(-1 * var(--#{$prefix}tooltip-arrow-height)); // stylelint-disable-line function-disallowed-list\n\n  &::before {\n    top: -1px;\n    border-width: var(--#{$prefix}tooltip-arrow-height) calc(var(--#{$prefix}tooltip-arrow-width) * .5) 0; // stylelint-disable-line function-disallowed-list\n    border-top-color: var(--#{$prefix}tooltip-bg);\n  }\n}\n\n/* rtl:begin:ignore */\n.bs-tooltip-end .tooltip-arrow {\n  left: calc(-1 * var(--#{$prefix}tooltip-arrow-height)); // stylelint-disable-line function-disallowed-list\n  width: var(--#{$prefix}tooltip-arrow-height);\n  height: var(--#{$prefix}tooltip-arrow-width);\n\n  &::before {\n    right: -1px;\n    border-width: calc(var(--#{$prefix}tooltip-arrow-width) * .5) var(--#{$prefix}tooltip-arrow-height) calc(var(--#{$prefix}tooltip-arrow-width) * .5) 0; // stylelint-disable-line function-disallowed-list\n    border-right-color: var(--#{$prefix}tooltip-bg);\n  }\n}\n\n/* rtl:end:ignore */\n\n.bs-tooltip-bottom .tooltip-arrow {\n  top: calc(-1 * var(--#{$prefix}tooltip-arrow-height)); // stylelint-disable-line function-disallowed-list\n\n  &::before {\n    bottom: -1px;\n    border-width: 0 calc(var(--#{$prefix}tooltip-arrow-width) * .5) var(--#{$prefix}tooltip-arrow-height); // stylelint-disable-line function-disallowed-list\n    border-bottom-color: var(--#{$prefix}tooltip-bg);\n  }\n}\n\n/* rtl:begin:ignore */\n.bs-tooltip-start .tooltip-arrow {\n  right: calc(-1 * var(--#{$prefix}tooltip-arrow-height)); // stylelint-disable-line function-disallowed-list\n  width: var(--#{$prefix}tooltip-arrow-height);\n  height: var(--#{$prefix}tooltip-arrow-width);\n\n  &::before {\n    left: -1px;\n    border-width: calc(var(--#{$prefix}tooltip-arrow-width) * .5) 0 calc(var(--#{$prefix}tooltip-arrow-width) * .5) var(--#{$prefix}tooltip-arrow-height); // stylelint-disable-line function-disallowed-list\n    border-left-color: var(--#{$prefix}tooltip-bg);\n  }\n}\n\n/* rtl:end:ignore */\n\n.bs-tooltip-auto {\n  &[data-popper-placement^=\"top\"] {\n    @extend .bs-tooltip-top;\n  }\n  &[data-popper-placement^=\"right\"] {\n    @extend .bs-tooltip-end;\n  }\n  &[data-popper-placement^=\"bottom\"] {\n    @extend .bs-tooltip-bottom;\n  }\n  &[data-popper-placement^=\"left\"] {\n    @extend .bs-tooltip-start;\n  }\n}\n\n// Wrapper for the tooltip content\n.tooltip-inner {\n  max-width: var(--#{$prefix}tooltip-max-width);\n  padding: var(--#{$prefix}tooltip-padding-y) var(--#{$prefix}tooltip-padding-x);\n  color: var(--#{$prefix}tooltip-color);\n  text-align: center;\n  background-color: var(--#{$prefix}tooltip-bg);\n  @include border-radius(var(--#{$prefix}tooltip-border-radius));\n}\n","@mixin reset-text {\n  font-family: $font-family-base;\n  // We deliberately do NOT reset font-size or overflow-wrap / word-wrap.\n  font-style: normal;\n  font-weight: $font-weight-normal;\n  line-height: $line-height-base;\n  text-align: left; // Fallback for where `start` is not supported\n  text-align: start;\n  text-decoration: none;\n  text-shadow: none;\n  text-transform: none;\n  letter-spacing: normal;\n  word-break: normal;\n  white-space: normal;\n  word-spacing: normal;\n  line-break: auto;\n}\n",".popover {\n  // scss-docs-start popover-css-vars\n  --#{$prefix}popover-zindex: #{$zindex-popover};\n  --#{$prefix}popover-max-width: #{$popover-max-width};\n  @include rfs($popover-font-size, --#{$prefix}popover-font-size);\n  --#{$prefix}popover-bg: #{$popover-bg};\n  --#{$prefix}popover-border-width: #{$popover-border-width};\n  --#{$prefix}popover-border-color: #{$popover-border-color};\n  --#{$prefix}popover-border-radius: #{$popover-border-radius};\n  --#{$prefix}popover-inner-border-radius: #{$popover-inner-border-radius};\n  --#{$prefix}popover-box-shadow: #{$popover-box-shadow};\n  --#{$prefix}popover-header-padding-x: #{$popover-header-padding-x};\n  --#{$prefix}popover-header-padding-y: #{$popover-header-padding-y};\n  @include rfs($popover-header-font-size, --#{$prefix}popover-header-font-size);\n  --#{$prefix}popover-header-color: #{$popover-header-color};\n  --#{$prefix}popover-header-bg: #{$popover-header-bg};\n  --#{$prefix}popover-body-padding-x: #{$popover-body-padding-x};\n  --#{$prefix}popover-body-padding-y: #{$popover-body-padding-y};\n  --#{$prefix}popover-body-color: #{$popover-body-color};\n  --#{$prefix}popover-arrow-width: #{$popover-arrow-width};\n  --#{$prefix}popover-arrow-height: #{$popover-arrow-height};\n  --#{$prefix}popover-arrow-border: var(--#{$prefix}popover-border-color);\n  // scss-docs-end popover-css-vars\n\n  z-index: var(--#{$prefix}popover-zindex);\n  display: block;\n  max-width: var(--#{$prefix}popover-max-width);\n  // Our parent element can be arbitrary since tooltips are by default inserted as a sibling of their target element.\n  // So reset our font and text properties to avoid inheriting weird values.\n  @include reset-text();\n  @include font-size(var(--#{$prefix}popover-font-size));\n  // Allow breaking very long words so they don't overflow the popover's bounds\n  word-wrap: break-word;\n  background-color: var(--#{$prefix}popover-bg);\n  background-clip: padding-box;\n  border: var(--#{$prefix}popover-border-width) solid var(--#{$prefix}popover-border-color);\n  @include border-radius(var(--#{$prefix}popover-border-radius));\n  @include box-shadow(var(--#{$prefix}popover-box-shadow));\n\n  .popover-arrow {\n    display: block;\n    width: var(--#{$prefix}popover-arrow-width);\n    height: var(--#{$prefix}popover-arrow-height);\n\n    &::before,\n    &::after {\n      position: absolute;\n      display: block;\n      content: \"\";\n      border-color: transparent;\n      border-style: solid;\n      border-width: 0;\n    }\n  }\n}\n\n.bs-popover-top {\n  > .popover-arrow {\n    bottom: calc(-1 * (var(--#{$prefix}popover-arrow-height)) - var(--#{$prefix}popover-border-width)); // stylelint-disable-line function-disallowed-list\n\n    &::before,\n    &::after {\n      border-width: var(--#{$prefix}popover-arrow-height) calc(var(--#{$prefix}popover-arrow-width) * .5) 0; // stylelint-disable-line function-disallowed-list\n    }\n\n    &::before {\n      bottom: 0;\n      border-top-color: var(--#{$prefix}popover-arrow-border);\n    }\n\n    &::after {\n      bottom: var(--#{$prefix}popover-border-width);\n      border-top-color: var(--#{$prefix}popover-bg);\n    }\n  }\n}\n\n/* rtl:begin:ignore */\n.bs-popover-end {\n  > .popover-arrow {\n    left: calc(-1 * (var(--#{$prefix}popover-arrow-height)) - var(--#{$prefix}popover-border-width)); // stylelint-disable-line function-disallowed-list\n    width: var(--#{$prefix}popover-arrow-height);\n    height: var(--#{$prefix}popover-arrow-width);\n\n    &::before,\n    &::after {\n      border-width: calc(var(--#{$prefix}popover-arrow-width) * .5) var(--#{$prefix}popover-arrow-height) calc(var(--#{$prefix}popover-arrow-width) * .5) 0; // stylelint-disable-line function-disallowed-list\n    }\n\n    &::before {\n      left: 0;\n      border-right-color: var(--#{$prefix}popover-arrow-border);\n    }\n\n    &::after {\n      left: var(--#{$prefix}popover-border-width);\n      border-right-color: var(--#{$prefix}popover-bg);\n    }\n  }\n}\n\n/* rtl:end:ignore */\n\n.bs-popover-bottom {\n  > .popover-arrow {\n    top: calc(-1 * (var(--#{$prefix}popover-arrow-height)) - var(--#{$prefix}popover-border-width)); // stylelint-disable-line function-disallowed-list\n\n    &::before,\n    &::after {\n      border-width: 0 calc(var(--#{$prefix}popover-arrow-width) * .5) var(--#{$prefix}popover-arrow-height); // stylelint-disable-line function-disallowed-list\n    }\n\n    &::before {\n      top: 0;\n      border-bottom-color: var(--#{$prefix}popover-arrow-border);\n    }\n\n    &::after {\n      top: var(--#{$prefix}popover-border-width);\n      border-bottom-color: var(--#{$prefix}popover-bg);\n    }\n  }\n\n  // This will remove the popover-header's border just below the arrow\n  .popover-header::before {\n    position: absolute;\n    top: 0;\n    left: 50%;\n    display: block;\n    width: var(--#{$prefix}popover-arrow-width);\n    margin-left: calc(-.5 * var(--#{$prefix}popover-arrow-width)); // stylelint-disable-line function-disallowed-list\n    content: \"\";\n    border-bottom: var(--#{$prefix}popover-border-width) solid var(--#{$prefix}popover-header-bg);\n  }\n}\n\n/* rtl:begin:ignore */\n.bs-popover-start {\n  > .popover-arrow {\n    right: calc(-1 * (var(--#{$prefix}popover-arrow-height)) - var(--#{$prefix}popover-border-width)); // stylelint-disable-line function-disallowed-list\n    width: var(--#{$prefix}popover-arrow-height);\n    height: var(--#{$prefix}popover-arrow-width);\n\n    &::before,\n    &::after {\n      border-width: calc(var(--#{$prefix}popover-arrow-width) * .5) 0 calc(var(--#{$prefix}popover-arrow-width) * .5) var(--#{$prefix}popover-arrow-height); // stylelint-disable-line function-disallowed-list\n    }\n\n    &::before {\n      right: 0;\n      border-left-color: var(--#{$prefix}popover-arrow-border);\n    }\n\n    &::after {\n      right: var(--#{$prefix}popover-border-width);\n      border-left-color: var(--#{$prefix}popover-bg);\n    }\n  }\n}\n\n/* rtl:end:ignore */\n\n.bs-popover-auto {\n  &[data-popper-placement^=\"top\"] {\n    @extend .bs-popover-top;\n  }\n  &[data-popper-placement^=\"right\"] {\n    @extend .bs-popover-end;\n  }\n  &[data-popper-placement^=\"bottom\"] {\n    @extend .bs-popover-bottom;\n  }\n  &[data-popper-placement^=\"left\"] {\n    @extend .bs-popover-start;\n  }\n}\n\n// Offset the popover to account for the popover arrow\n.popover-header {\n  padding: var(--#{$prefix}popover-header-padding-y) var(--#{$prefix}popover-header-padding-x);\n  margin-bottom: 0; // Reset the default from Reboot\n  @include font-size(var(--#{$prefix}popover-header-font-size));\n  color: var(--#{$prefix}popover-header-color);\n  background-color: var(--#{$prefix}popover-header-bg);\n  border-bottom: var(--#{$prefix}popover-border-width) solid var(--#{$prefix}popover-border-color);\n  @include border-top-radius(var(--#{$prefix}popover-inner-border-radius));\n\n  &:empty {\n    display: none;\n  }\n}\n\n.popover-body {\n  padding: var(--#{$prefix}popover-body-padding-y) var(--#{$prefix}popover-body-padding-x);\n  color: var(--#{$prefix}popover-body-color);\n}\n","// Notes on the classes:\n//\n// 1. .carousel.pointer-event should ideally be pan-y (to allow for users to scroll vertically)\n//    even when their scroll action started on a carousel, but for compatibility (with Firefox)\n//    we're preventing all actions instead\n// 2. The .carousel-item-start and .carousel-item-end is used to indicate where\n//    the active slide is heading.\n// 3. .active.carousel-item is the current slide.\n// 4. .active.carousel-item-start and .active.carousel-item-end is the current\n//    slide in its in-transition state. Only one of these occurs at a time.\n// 5. .carousel-item-next.carousel-item-start and .carousel-item-prev.carousel-item-end\n//    is the upcoming slide in transition.\n\n.carousel {\n  position: relative;\n}\n\n.carousel.pointer-event {\n  touch-action: pan-y;\n}\n\n.carousel-inner {\n  position: relative;\n  width: 100%;\n  overflow: hidden;\n  @include clearfix();\n}\n\n.carousel-item {\n  position: relative;\n  display: none;\n  float: left;\n  width: 100%;\n  margin-right: -100%;\n  backface-visibility: hidden;\n  @include transition($carousel-transition);\n}\n\n.carousel-item.active,\n.carousel-item-next,\n.carousel-item-prev {\n  display: block;\n}\n\n.carousel-item-next:not(.carousel-item-start),\n.active.carousel-item-end {\n  transform: translateX(100%);\n}\n\n.carousel-item-prev:not(.carousel-item-end),\n.active.carousel-item-start {\n  transform: translateX(-100%);\n}\n\n\n//\n// Alternate transitions\n//\n\n.carousel-fade {\n  .carousel-item {\n    opacity: 0;\n    transition-property: opacity;\n    transform: none;\n  }\n\n  .carousel-item.active,\n  .carousel-item-next.carousel-item-start,\n  .carousel-item-prev.carousel-item-end {\n    z-index: 1;\n    opacity: 1;\n  }\n\n  .active.carousel-item-start,\n  .active.carousel-item-end {\n    z-index: 0;\n    opacity: 0;\n    @include transition(opacity 0s $carousel-transition-duration);\n  }\n}\n\n\n//\n// Left/right controls for nav\n//\n\n.carousel-control-prev,\n.carousel-control-next {\n  position: absolute;\n  top: 0;\n  bottom: 0;\n  z-index: 1;\n  // Use flex for alignment (1-3)\n  display: flex; // 1. allow flex styles\n  align-items: center; // 2. vertically center contents\n  justify-content: center; // 3. horizontally center contents\n  width: $carousel-control-width;\n  padding: 0;\n  color: $carousel-control-color;\n  text-align: center;\n  background: none;\n  border: 0;\n  opacity: $carousel-control-opacity;\n  @include transition($carousel-control-transition);\n\n  // Hover/focus state\n  &:hover,\n  &:focus {\n    color: $carousel-control-color;\n    text-decoration: none;\n    outline: 0;\n    opacity: $carousel-control-hover-opacity;\n  }\n}\n.carousel-control-prev {\n  left: 0;\n  background-image: if($enable-gradients, linear-gradient(90deg, rgba($black, .25), rgba($black, .001)), null);\n}\n.carousel-control-next {\n  right: 0;\n  background-image: if($enable-gradients, linear-gradient(270deg, rgba($black, .25), rgba($black, .001)), null);\n}\n\n// Icons for within\n.carousel-control-prev-icon,\n.carousel-control-next-icon {\n  display: inline-block;\n  width: $carousel-control-icon-width;\n  height: $carousel-control-icon-width;\n  background-repeat: no-repeat;\n  background-position: 50%;\n  background-size: 100% 100%;\n}\n\n.carousel-control-prev-icon {\n  background-image: escape-svg($carousel-control-prev-icon-bg) #{\"/*rtl:\" + escape-svg($carousel-control-next-icon-bg) + \"*/\"};\n}\n.carousel-control-next-icon {\n  background-image: escape-svg($carousel-control-next-icon-bg) #{\"/*rtl:\" + escape-svg($carousel-control-prev-icon-bg) + \"*/\"};\n}\n\n// Optional indicator pips/controls\n//\n// Add a container (such as a list) with the following class and add an item (ideally a focusable control,\n// like a button) with data-bs-target for each slide your carousel holds.\n\n.carousel-indicators {\n  position: absolute;\n  right: 0;\n  bottom: 0;\n  left: 0;\n  z-index: 2;\n  display: flex;\n  justify-content: center;\n  padding: 0;\n  // Use the .carousel-control's width as margin so we don't overlay those\n  margin-right: $carousel-control-width;\n  margin-bottom: 1rem;\n  margin-left: $carousel-control-width;\n\n  [data-bs-target] {\n    box-sizing: content-box;\n    flex: 0 1 auto;\n    width: $carousel-indicator-width;\n    height: $carousel-indicator-height;\n    padding: 0;\n    margin-right: $carousel-indicator-spacer;\n    margin-left: $carousel-indicator-spacer;\n    text-indent: -999px;\n    cursor: pointer;\n    background-color: $carousel-indicator-active-bg;\n    background-clip: padding-box;\n    border: 0;\n    // Use transparent borders to increase the hit area by 10px on top and bottom.\n    border-top: $carousel-indicator-hit-area-height solid transparent;\n    border-bottom: $carousel-indicator-hit-area-height solid transparent;\n    opacity: $carousel-indicator-opacity;\n    @include transition($carousel-indicator-transition);\n  }\n\n  .active {\n    opacity: $carousel-indicator-active-opacity;\n  }\n}\n\n\n// Optional captions\n//\n//\n\n.carousel-caption {\n  position: absolute;\n  right: (100% - $carousel-caption-width) * .5;\n  bottom: $carousel-caption-spacer;\n  left: (100% - $carousel-caption-width) * .5;\n  padding-top: $carousel-caption-padding-y;\n  padding-bottom: $carousel-caption-padding-y;\n  color: $carousel-caption-color;\n  text-align: center;\n}\n\n// Dark mode carousel\n\n@mixin carousel-dark() {\n  .carousel-control-prev-icon,\n  .carousel-control-next-icon {\n    filter: $carousel-dark-control-icon-filter;\n  }\n\n  .carousel-indicators [data-bs-target] {\n    background-color: $carousel-dark-indicator-active-bg;\n  }\n\n  .carousel-caption {\n    color: $carousel-dark-caption-color;\n  }\n}\n\n.carousel-dark {\n  @include carousel-dark();\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    @if $color-mode-type == \"media-query\" {\n      .carousel {\n        @include carousel-dark();\n      }\n    } @else {\n      .carousel,\n      &.carousel {\n        @include carousel-dark();\n      }\n    }\n  }\n}\n","// scss-docs-start clearfix\n@mixin clearfix() {\n  &::after {\n    display: block;\n    clear: both;\n    content: \"\";\n  }\n}\n// scss-docs-end clearfix\n","//\n// Rotating border\n//\n\n.spinner-grow,\n.spinner-border {\n  display: inline-block;\n  width: var(--#{$prefix}spinner-width);\n  height: var(--#{$prefix}spinner-height);\n  vertical-align: var(--#{$prefix}spinner-vertical-align);\n  // stylelint-disable-next-line property-disallowed-list\n  border-radius: 50%;\n  animation: var(--#{$prefix}spinner-animation-speed) linear infinite var(--#{$prefix}spinner-animation-name);\n}\n\n// scss-docs-start spinner-border-keyframes\n@keyframes spinner-border {\n  to { transform: rotate(360deg) #{\"/* rtl:ignore */\"}; }\n}\n// scss-docs-end spinner-border-keyframes\n\n.spinner-border {\n  // scss-docs-start spinner-border-css-vars\n  --#{$prefix}spinner-width: #{$spinner-width};\n  --#{$prefix}spinner-height: #{$spinner-height};\n  --#{$prefix}spinner-vertical-align: #{$spinner-vertical-align};\n  --#{$prefix}spinner-border-width: #{$spinner-border-width};\n  --#{$prefix}spinner-animation-speed: #{$spinner-animation-speed};\n  --#{$prefix}spinner-animation-name: spinner-border;\n  // scss-docs-end spinner-border-css-vars\n\n  border: var(--#{$prefix}spinner-border-width) solid currentcolor;\n  border-right-color: transparent;\n}\n\n.spinner-border-sm {\n  // scss-docs-start spinner-border-sm-css-vars\n  --#{$prefix}spinner-width: #{$spinner-width-sm};\n  --#{$prefix}spinner-height: #{$spinner-height-sm};\n  --#{$prefix}spinner-border-width: #{$spinner-border-width-sm};\n  // scss-docs-end spinner-border-sm-css-vars\n}\n\n//\n// Growing circle\n//\n\n// scss-docs-start spinner-grow-keyframes\n@keyframes spinner-grow {\n  0% {\n    transform: scale(0);\n  }\n  50% {\n    opacity: 1;\n    transform: none;\n  }\n}\n// scss-docs-end spinner-grow-keyframes\n\n.spinner-grow {\n  // scss-docs-start spinner-grow-css-vars\n  --#{$prefix}spinner-width: #{$spinner-width};\n  --#{$prefix}spinner-height: #{$spinner-height};\n  --#{$prefix}spinner-vertical-align: #{$spinner-vertical-align};\n  --#{$prefix}spinner-animation-speed: #{$spinner-animation-speed};\n  --#{$prefix}spinner-animation-name: spinner-grow;\n  // scss-docs-end spinner-grow-css-vars\n\n  background-color: currentcolor;\n  opacity: 0;\n}\n\n.spinner-grow-sm {\n  --#{$prefix}spinner-width: #{$spinner-width-sm};\n  --#{$prefix}spinner-height: #{$spinner-height-sm};\n}\n\n@if $enable-reduced-motion {\n  @media (prefers-reduced-motion: reduce) {\n    .spinner-border,\n    .spinner-grow {\n      --#{$prefix}spinner-animation-speed: #{$spinner-animation-speed * 2};\n    }\n  }\n}\n","// stylelint-disable function-disallowed-list\n\n%offcanvas-css-vars {\n  // scss-docs-start offcanvas-css-vars\n  --#{$prefix}offcanvas-zindex: #{$zindex-offcanvas};\n  --#{$prefix}offcanvas-width: #{$offcanvas-horizontal-width};\n  --#{$prefix}offcanvas-height: #{$offcanvas-vertical-height};\n  --#{$prefix}offcanvas-padding-x: #{$offcanvas-padding-x};\n  --#{$prefix}offcanvas-padding-y: #{$offcanvas-padding-y};\n  --#{$prefix}offcanvas-color: #{$offcanvas-color};\n  --#{$prefix}offcanvas-bg: #{$offcanvas-bg-color};\n  --#{$prefix}offcanvas-border-width: #{$offcanvas-border-width};\n  --#{$prefix}offcanvas-border-color: #{$offcanvas-border-color};\n  --#{$prefix}offcanvas-box-shadow: #{$offcanvas-box-shadow};\n  --#{$prefix}offcanvas-transition: #{transform $offcanvas-transition-duration ease-in-out};\n  --#{$prefix}offcanvas-title-line-height: #{$offcanvas-title-line-height};\n  // scss-docs-end offcanvas-css-vars\n}\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  $next: breakpoint-next($breakpoint, $grid-breakpoints);\n  $infix: breakpoint-infix($next, $grid-breakpoints);\n\n  .offcanvas#{$infix} {\n    @extend %offcanvas-css-vars;\n  }\n}\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  $next: breakpoint-next($breakpoint, $grid-breakpoints);\n  $infix: breakpoint-infix($next, $grid-breakpoints);\n\n  .offcanvas#{$infix} {\n    @include media-breakpoint-down($next) {\n      position: fixed;\n      bottom: 0;\n      z-index: var(--#{$prefix}offcanvas-zindex);\n      display: flex;\n      flex-direction: column;\n      max-width: 100%;\n      color: var(--#{$prefix}offcanvas-color);\n      visibility: hidden;\n      background-color: var(--#{$prefix}offcanvas-bg);\n      background-clip: padding-box;\n      outline: 0;\n      @include box-shadow(var(--#{$prefix}offcanvas-box-shadow));\n      @include transition(var(--#{$prefix}offcanvas-transition));\n\n      &.offcanvas-start {\n        top: 0;\n        left: 0;\n        width: var(--#{$prefix}offcanvas-width);\n        border-right: var(--#{$prefix}offcanvas-border-width) solid var(--#{$prefix}offcanvas-border-color);\n        transform: translateX(-100%);\n      }\n\n      &.offcanvas-end {\n        top: 0;\n        right: 0;\n        width: var(--#{$prefix}offcanvas-width);\n        border-left: var(--#{$prefix}offcanvas-border-width) solid var(--#{$prefix}offcanvas-border-color);\n        transform: translateX(100%);\n      }\n\n      &.offcanvas-top {\n        top: 0;\n        right: 0;\n        left: 0;\n        height: var(--#{$prefix}offcanvas-height);\n        max-height: 100%;\n        border-bottom: var(--#{$prefix}offcanvas-border-width) solid var(--#{$prefix}offcanvas-border-color);\n        transform: translateY(-100%);\n      }\n\n      &.offcanvas-bottom {\n        right: 0;\n        left: 0;\n        height: var(--#{$prefix}offcanvas-height);\n        max-height: 100%;\n        border-top: var(--#{$prefix}offcanvas-border-width) solid var(--#{$prefix}offcanvas-border-color);\n        transform: translateY(100%);\n      }\n\n      &.showing,\n      &.show:not(.hiding) {\n        transform: none;\n      }\n\n      &.showing,\n      &.hiding,\n      &.show {\n        visibility: visible;\n      }\n    }\n\n    @if not ($infix == \"\") {\n      @include media-breakpoint-up($next) {\n        --#{$prefix}offcanvas-height: auto;\n        --#{$prefix}offcanvas-border-width: 0;\n        background-color: transparent !important; // stylelint-disable-line declaration-no-important\n\n        .offcanvas-header {\n          display: none;\n        }\n\n        .offcanvas-body {\n          display: flex;\n          flex-grow: 0;\n          padding: 0;\n          overflow-y: visible;\n          // Reset `background-color` in case `.bg-*` classes are used in offcanvas\n          background-color: transparent !important; // stylelint-disable-line declaration-no-important\n        }\n      }\n    }\n  }\n}\n\n.offcanvas-backdrop {\n  @include overlay-backdrop($zindex-offcanvas-backdrop, $offcanvas-backdrop-bg, $offcanvas-backdrop-opacity);\n}\n\n.offcanvas-header {\n  display: flex;\n  align-items: center;\n  padding: var(--#{$prefix}offcanvas-padding-y) var(--#{$prefix}offcanvas-padding-x);\n\n  .btn-close {\n    padding: calc(var(--#{$prefix}offcanvas-padding-y) * .5) calc(var(--#{$prefix}offcanvas-padding-x) * .5);\n    margin: calc(-.5 * var(--#{$prefix}offcanvas-padding-y)) calc(-.5 * var(--#{$prefix}offcanvas-padding-x)) calc(-.5 * var(--#{$prefix}offcanvas-padding-y)) auto;\n  }\n}\n\n.offcanvas-title {\n  margin-bottom: 0;\n  line-height: var(--#{$prefix}offcanvas-title-line-height);\n}\n\n.offcanvas-body {\n  flex-grow: 1;\n  padding: var(--#{$prefix}offcanvas-padding-y) var(--#{$prefix}offcanvas-padding-x);\n  overflow-y: auto;\n}\n",".placeholder {\n  display: inline-block;\n  min-height: 1em;\n  vertical-align: middle;\n  cursor: wait;\n  background-color: currentcolor;\n  opacity: $placeholder-opacity-max;\n\n  &.btn::before {\n    display: inline-block;\n    content: \"\";\n  }\n}\n\n// Sizing\n.placeholder-xs {\n  min-height: .6em;\n}\n\n.placeholder-sm {\n  min-height: .8em;\n}\n\n.placeholder-lg {\n  min-height: 1.2em;\n}\n\n// Animation\n.placeholder-glow {\n  .placeholder {\n    animation: placeholder-glow 2s ease-in-out infinite;\n  }\n}\n\n@keyframes placeholder-glow {\n  50% {\n    opacity: $placeholder-opacity-min;\n  }\n}\n\n.placeholder-wave {\n  mask-image: linear-gradient(130deg, $black 55%, rgba(0, 0, 0, (1 - $placeholder-opacity-min)) 75%, $black 95%);\n  mask-size: 200% 100%;\n  animation: placeholder-wave 2s linear infinite;\n}\n\n@keyframes placeholder-wave {\n  100% {\n    mask-position: -200% 0%;\n  }\n}\n","// All-caps `RGBA()` function used because of this Sass bug: https://github.com/sass/node-sass/issues/2251\n@each $color, $value in $theme-colors {\n  .text-bg-#{$color} {\n    color: color-contrast($value) if($enable-important-utilities, !important, null);\n    background-color: RGBA(var(--#{$prefix}#{$color}-rgb), var(--#{$prefix}bg-opacity, 1)) if($enable-important-utilities, !important, null);\n  }\n}\n","// All-caps `RGBA()` function used because of this Sass bug: https://github.com/sass/node-sass/issues/2251\n@each $color, $value in $theme-colors {\n  .link-#{$color} {\n    color: RGBA(var(--#{$prefix}#{$color}-rgb), var(--#{$prefix}link-opacity, 1)) if($enable-important-utilities, !important, null);\n    text-decoration-color: RGBA(var(--#{$prefix}#{$color}-rgb), var(--#{$prefix}link-underline-opacity, 1)) if($enable-important-utilities, !important, null);\n\n    @if $link-shade-percentage != 0 {\n      &:hover,\n      &:focus {\n        $hover-color: if(color-contrast($value) == $color-contrast-light, shade-color($value, $link-shade-percentage), tint-color($value, $link-shade-percentage));\n        color: RGBA(#{to-rgb($hover-color)}, var(--#{$prefix}link-opacity, 1)) if($enable-important-utilities, !important, null);\n        text-decoration-color: RGBA(to-rgb($hover-color), var(--#{$prefix}link-underline-opacity, 1)) if($enable-important-utilities, !important, null);\n      }\n    }\n  }\n}\n\n// One-off special link helper as a bridge until v6\n.link-body-emphasis {\n  color: RGBA(var(--#{$prefix}emphasis-color-rgb), var(--#{$prefix}link-opacity, 1)) if($enable-important-utilities, !important, null);\n  text-decoration-color: RGBA(var(--#{$prefix}emphasis-color-rgb), var(--#{$prefix}link-underline-opacity, 1)) if($enable-important-utilities, !important, null);\n\n  @if $link-shade-percentage != 0 {\n    &:hover,\n    &:focus {\n      color: RGBA(var(--#{$prefix}emphasis-color-rgb), var(--#{$prefix}link-opacity, .75)) if($enable-important-utilities, !important, null);\n      text-decoration-color: RGBA(var(--#{$prefix}emphasis-color-rgb), var(--#{$prefix}link-underline-opacity, .75)) if($enable-important-utilities, !important, null);\n    }\n  }\n}\n",".focus-ring:focus {\n  outline: 0;\n  // By default, there is no `--bs-focus-ring-x`, `--bs-focus-ring-y`, or `--bs-focus-ring-blur`, but we provide CSS variables with fallbacks to initial `0` values\n  box-shadow: var(--#{$prefix}focus-ring-x, 0) var(--#{$prefix}focus-ring-y, 0) var(--#{$prefix}focus-ring-blur, 0) var(--#{$prefix}focus-ring-width) var(--#{$prefix}focus-ring-color);\n}\n",".icon-link {\n  display: inline-flex;\n  gap: $icon-link-gap;\n  align-items: center;\n  text-decoration-color: rgba(var(--#{$prefix}link-color-rgb), var(--#{$prefix}link-opacity, .5));\n  text-underline-offset: $icon-link-underline-offset;\n  backface-visibility: hidden;\n\n  > .bi {\n    flex-shrink: 0;\n    width: $icon-link-icon-size;\n    height: $icon-link-icon-size;\n    fill: currentcolor;\n    @include transition($icon-link-icon-transition);\n  }\n}\n\n.icon-link-hover {\n  &:hover,\n  &:focus-visible {\n    > .bi {\n      transform: var(--#{$prefix}icon-link-transform, $icon-link-icon-transform);\n    }\n  }\n}\n","// Credit: Nicolas Gallagher and SUIT CSS.\n\n.ratio {\n  position: relative;\n  width: 100%;\n\n  &::before {\n    display: block;\n    padding-top: var(--#{$prefix}aspect-ratio);\n    content: \"\";\n  }\n\n  > * {\n    position: absolute;\n    top: 0;\n    left: 0;\n    width: 100%;\n    height: 100%;\n  }\n}\n\n@each $key, $ratio in $aspect-ratios {\n  .ratio-#{$key} {\n    --#{$prefix}aspect-ratio: #{$ratio};\n  }\n}\n","// Shorthand\n\n.fixed-top {\n  position: fixed;\n  top: 0;\n  right: 0;\n  left: 0;\n  z-index: $zindex-fixed;\n}\n\n.fixed-bottom {\n  position: fixed;\n  right: 0;\n  bottom: 0;\n  left: 0;\n  z-index: $zindex-fixed;\n}\n\n// Responsive sticky top and bottom\n@each $breakpoint in map-keys($grid-breakpoints) {\n  @include media-breakpoint-up($breakpoint) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    .sticky#{$infix}-top {\n      position: sticky;\n      top: 0;\n      z-index: $zindex-sticky;\n    }\n\n    .sticky#{$infix}-bottom {\n      position: sticky;\n      bottom: 0;\n      z-index: $zindex-sticky;\n    }\n  }\n}\n","// scss-docs-start stacks\n.hstack {\n  display: flex;\n  flex-direction: row;\n  align-items: center;\n  align-self: stretch;\n}\n\n.vstack {\n  display: flex;\n  flex: 1 1 auto;\n  flex-direction: column;\n  align-self: stretch;\n}\n// scss-docs-end stacks\n","//\n// Visually hidden\n//\n\n.visually-hidden,\n.visually-hidden-focusable:not(:focus):not(:focus-within) {\n  @include visually-hidden();\n}\n","// stylelint-disable declaration-no-important\n\n// Hide content visually while keeping it accessible to assistive technologies\n//\n// See: https://www.a11yproject.com/posts/2013-01-11-how-to-hide-content/\n// See: https://kittygiraudel.com/2016/10/13/css-hide-and-seek/\n\n@mixin visually-hidden() {\n  width: 1px !important;\n  height: 1px !important;\n  padding: 0 !important;\n  margin: -1px !important; // Fix for https://github.com/twbs/bootstrap/issues/25686\n  overflow: hidden !important;\n  clip: rect(0, 0, 0, 0) !important;\n  white-space: nowrap !important;\n  border: 0 !important;\n\n  // Fix for positioned table caption that could become anonymous cells\n  &:not(caption) {\n    position: absolute !important;\n  }\n}\n\n// Use to only display content when it's focused, or one of its child elements is focused\n// (i.e. when focus is within the element/container that the class was applied to)\n//\n// Useful for \"Skip to main content\" links; see https://www.w3.org/TR/2013/NOTE-WCAG20-TECHS-20130905/G1\n\n@mixin visually-hidden-focusable() {\n  &:not(:focus):not(:focus-within) {\n    @include visually-hidden();\n  }\n}\n","//\n// Stretched link\n//\n\n.stretched-link {\n  &::#{$stretched-link-pseudo-element} {\n    position: absolute;\n    top: 0;\n    right: 0;\n    bottom: 0;\n    left: 0;\n    z-index: $stretched-link-z-index;\n    content: \"\";\n  }\n}\n","//\n// Text truncation\n//\n\n.text-truncate {\n  @include text-truncate();\n}\n","// Text truncate\n// Requires inline-block or block for proper styling\n\n@mixin text-truncate() {\n  overflow: hidden;\n  text-overflow: ellipsis;\n  white-space: nowrap;\n}\n",".vr {\n  display: inline-block;\n  align-self: stretch;\n  width: $vr-border-width;\n  min-height: 1em;\n  background-color: currentcolor;\n  opacity: $hr-opacity;\n}\n","// Utility generator\n// Used to generate utilities & print utilities\n@mixin generate-utility($utility, $infix: \"\", $is-rfs-media-query: false) {\n  $values: map-get($utility, values);\n\n  // If the values are a list or string, convert it into a map\n  @if type-of($values) == \"string\" or type-of(nth($values, 1)) != \"list\" {\n    $values: zip($values, $values);\n  }\n\n  @each $key, $value in $values {\n    $properties: map-get($utility, property);\n\n    // Multiple properties are possible, for example with vertical or horizontal margins or paddings\n    @if type-of($properties) == \"string\" {\n      $properties: append((), $properties);\n    }\n\n    // Use custom class if present\n    $property-class: if(map-has-key($utility, class), map-get($utility, class), nth($properties, 1));\n    $property-class: if($property-class == null, \"\", $property-class);\n\n    // Use custom CSS variable name if present, otherwise default to `class`\n    $css-variable-name: if(map-has-key($utility, css-variable-name), map-get($utility, css-variable-name), map-get($utility, class));\n\n    // State params to generate pseudo-classes\n    $state: if(map-has-key($utility, state), map-get($utility, state), ());\n\n    $infix: if($property-class == \"\" and str-slice($infix, 1, 1) == \"-\", str-slice($infix, 2), $infix);\n\n    // Don't prefix if value key is null (e.g. with shadow class)\n    $property-class-modifier: if($key, if($property-class == \"\" and $infix == \"\", \"\", \"-\") + $key, \"\");\n\n    @if map-get($utility, rfs) {\n      // Inside the media query\n      @if $is-rfs-media-query {\n        $val: rfs-value($value);\n\n        // Do not render anything if fluid and non fluid values are the same\n        $value: if($val == rfs-fluid-value($value), null, $val);\n      }\n      @else {\n        $value: rfs-fluid-value($value);\n      }\n    }\n\n    $is-css-var: map-get($utility, css-var);\n    $is-local-vars: map-get($utility, local-vars);\n    $is-rtl: map-get($utility, rtl);\n\n    @if $value != null {\n      @if $is-rtl == false {\n        /* rtl:begin:remove */\n      }\n\n      @if $is-css-var {\n        .#{$property-class + $infix + $property-class-modifier} {\n          --#{$prefix}#{$css-variable-name}: #{$value};\n        }\n\n        @each $pseudo in $state {\n          .#{$property-class + $infix + $property-class-modifier}-#{$pseudo}:#{$pseudo} {\n            --#{$prefix}#{$css-variable-name}: #{$value};\n          }\n        }\n      } @else {\n        .#{$property-class + $infix + $property-class-modifier} {\n          @each $property in $properties {\n            @if $is-local-vars {\n              @each $local-var, $variable in $is-local-vars {\n                --#{$prefix}#{$local-var}: #{$variable};\n              }\n            }\n            #{$property}: $value if($enable-important-utilities, !important, null);\n          }\n        }\n\n        @each $pseudo in $state {\n          .#{$property-class + $infix + $property-class-modifier}-#{$pseudo}:#{$pseudo} {\n            @each $property in $properties {\n              @if $is-local-vars {\n                @each $local-var, $variable in $is-local-vars {\n                  --#{$prefix}#{$local-var}: #{$variable};\n                }\n              }\n              #{$property}: $value if($enable-important-utilities, !important, null);\n            }\n          }\n        }\n      }\n\n      @if $is-rtl == false {\n        /* rtl:end:remove */\n      }\n    }\n  }\n}\n","// Loop over each breakpoint\n@each $breakpoint in map-keys($grid-breakpoints) {\n\n  // Generate media query if needed\n  @include media-breakpoint-up($breakpoint) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    // Loop over each utility property\n    @each $key, $utility in $utilities {\n      // The utility can be disabled with `false`, thus check if the utility is a map first\n      // Only proceed if responsive media queries are enabled or if it's the base media query\n      @if type-of($utility) == \"map\" and (map-get($utility, responsive) or $infix == \"\") {\n        @include generate-utility($utility, $infix);\n      }\n    }\n  }\n}\n\n// RFS rescaling\n@media (min-width: $rfs-mq-value) {\n  @each $breakpoint in map-keys($grid-breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    @if (map-get($grid-breakpoints, $breakpoint) < $rfs-breakpoint) {\n      // Loop over each utility property\n      @each $key, $utility in $utilities {\n        // The utility can be disabled with `false`, thus check if the utility is a map first\n        // Only proceed if responsive media queries are enabled or if it's the base media query\n        @if type-of($utility) == \"map\" and map-get($utility, rfs) and (map-get($utility, responsive) or $infix == \"\") {\n          @include generate-utility($utility, $infix, true);\n        }\n      }\n    }\n  }\n}\n\n\n// Print utilities\n@media print {\n  @each $key, $utility in $utilities {\n    // The utility can be disabled with `false`, thus check if the utility is a map first\n    // Then check if the utility needs print styles\n    @if type-of($utility) == \"map\" and map-get($utility, print) == true {\n      @include generate-utility($utility, \"-print\");\n    }\n  }\n}\n"],"names":[],"sourceRoot":""}
\ No newline at end of file
diff --git a/dev/docs/html/_static/styles/pydata-sphinx-theme.css b/dev/docs/html/_static/styles/pydata-sphinx-theme.css
new file mode 100644
index 0000000..3610bf3
--- /dev/null
+++ b/dev/docs/html/_static/styles/pydata-sphinx-theme.css
@@ -0,0 +1,2 @@
+html{--pst-header-height:4rem;--pst-header-article-height:calc(var(--pst-header-height)*2/3);--pst-sidebar-secondary:17rem;--pst-font-size-base:1rem;--pst-font-size-h1:2.5rem;--pst-font-size-h2:2rem;--pst-font-size-h3:1.75rem;--pst-font-size-h4:1.5rem;--pst-font-size-h5:1.25rem;--pst-font-size-h6:1.1rem;--pst-font-size-milli:0.9rem;--pst-sidebar-font-size:0.9rem;--pst-sidebar-font-size-mobile:1.1rem;--pst-sidebar-header-font-size:1.2rem;--pst-sidebar-header-font-weight:600;--pst-admonition-font-weight-heading:600;--pst-font-weight-caption:300;--pst-font-weight-heading:400;--pst-font-family-base-system:-apple-system,"BlinkMacSystemFont","Segoe UI","Helvetica Neue","Arial",sans-serif,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol";--pst-font-family-monospace-system:"SFMono-Regular","Menlo","Consolas","Monaco","Liberation Mono","Lucida Console",monospace;--pst-font-family-base:var(--pst-font-family-base-system);--pst-font-family-heading:var(--pst-font-family-base-system);--pst-font-family-monospace:var(--pst-font-family-monospace-system);--pst-font-size-icon:1.5rem;--pst-icon-check-circle:"";--pst-icon-info-circle:"";--pst-icon-exclamation-triangle:"";--pst-icon-exclamation-circle:"";--pst-icon-times-circle:"";--pst-icon-lightbulb:"";--pst-icon-download:"";--pst-icon-angle-left:"";--pst-icon-angle-right:"";--pst-icon-external-link:"";--pst-icon-search-minus:"";--pst-icon-github:"";--pst-icon-gitlab:"";--pst-icon-share:"";--pst-icon-bell:"";--pst-icon-pencil:"";--pst-breadcrumb-divider:"";--pst-icon-admonition-default:var(--pst-icon-bell);--pst-icon-admonition-note:var(--pst-icon-info-circle);--pst-icon-admonition-attention:var(--pst-icon-exclamation-circle);--pst-icon-admonition-caution:var(--pst-icon-exclamation-triangle);--pst-icon-admonition-warning:var(--pst-icon-exclamation-triangle);--pst-icon-admonition-danger:var(--pst-icon-exclamation-triangle);--pst-icon-admonition-error:var(--pst-icon-times-circle);--pst-icon-admonition-hint:var(--pst-icon-lightbulb);--pst-icon-admonition-tip:var(--pst-icon-lightbulb);--pst-icon-admonition-important:var(--pst-icon-exclamation-circle);--pst-icon-admonition-seealso:var(--pst-icon-share);--pst-icon-admonition-todo:var(--pst-icon-pencil);--pst-icon-versionmodified-default:var(--pst-icon-exclamation-circle);--pst-icon-versionmodified-added:var(--pst-icon-exclamation-circle);--pst-icon-versionmodified-changed:var(--pst-icon-exclamation-circle);--pst-icon-versionmodified-deprecated:var(--pst-icon-exclamation-circle);font-size:var(--pst-font-size-base);scroll-padding-top:calc(var(--pst-header-height) + 1rem)}body{background-color:var(--pst-color-background);color:var(--pst-color-text-base);display:flex;flex-direction:column;font-family:var(--pst-font-family-base);font-weight:400;line-height:1.65;min-height:100vh}body::-webkit-scrollbar-track{background-color:var(--pst-color-background)}p{font-size:1em;margin-bottom:1.15rem}p.rubric{border-bottom:1px solid var(--pst-color-border)}p.centered{text-align:center}a{word-wrap:break-word;color:var(--pst-color-link);text-decoration:underline;text-decoration-thickness:max(1px,.0625rem);text-underline-offset:.1578em}a:hover{color:var(--pst-color-link-hover);text-decoration-skip-ink:none;text-decoration-thickness:max(3px,.1875rem,.12em)}a:active,a:visited{color:var(--pst-color-link)}a:visited:hover{color:var(--pst-color-link-hover)}a.headerlink{color:var(--pst-color-secondary);font-size:.8em;margin-left:.2em;opacity:.7;padding:0 4px;text-decoration:none;transition:all .2s ease-out;user-select:none}a.headerlink:hover{opacity:1}a.github:before,a.gitlab:before{color:var(--pst-color-text-muted);font:var(--fa-font-brands);margin-right:.25rem}a.github:before{content:var(--pst-icon-github)}a.gitlab:before{content:var(--pst-icon-gitlab)}h1,h2,h3,h4,h5,h6{font-family:var(--pst-font-family-heading);font-weight:var(--pst-font-weight-heading);line-height:1.15;margin:2.75rem 0 1.05rem}h1{font-size:var(--pst-font-size-h1);margin-top:0}h1,h2{color:var(--pst-heading-color)}h2{font-size:var(--pst-font-size-h2)}h3{font-size:var(--pst-font-size-h3)}h3,h4{color:var(--pst-heading-color)}h4{font-size:var(--pst-font-size-h4)}h5{font-size:var(--pst-font-size-h5)}h5,h6{color:var(--pst-color-text-base)}h6{font-size:var(--pst-font-size-h6)}.text_small,small{font-size:var(--pst-font-size-milli)}hr{border:0;border-top:1px solid var(--pst-color-border)}code,kbd,pre,samp{font-family:var(--pst-font-family-monospace)}kbd{background-color:var(--pst-color-on-background);color:var(--pst-color-text-muted)}kbd:not(.compound){border:1px solid var(--pst-color-border);box-shadow:1px 1px 1px var(--pst-color-shadow);margin:0 .1rem;padding:.1rem .4rem}code{color:var(--pst-color-inline-code)}pre{background-color:var(--pst-color-surface);border:1px solid var(--pst-color-border);border-radius:.25rem;color:var(--pst-color-text-base);line-height:1.2em;margin:1.5em 0;padding:1rem}pre .linenos{opacity:.8;padding-right:10px}#pst-back-to-top{background-color:var(--pst-color-secondary);border:none;color:var(--pst-color-secondary-text);display:none;left:50vw;position:fixed;top:90vh;transform:translate(-50%);z-index:1080}#pst-back-to-top 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table.dataframe tbody tr:hover{background-color:var(--pst-color-table-row-hover-bg)}html[data-theme=dark] .bd-content .nboutput .output_area.rendered_html:not(:has(table.dataframe)),html[data-theme=dark] .bd-content .nboutput .output_area .widget-subarea{color:var(--pst-color-on-background);background-color:var(--pst-color-text-base);border-radius:.25rem;padding:.5rem}html[data-theme=dark] .bd-content .nboutput .output_area.stderr{background-color:var(--pst-color-danger)}div.nblast.container{margin-bottom:1rem}div.cell_output .output{max-width:100%;overflow-x:auto}.bd-content div.cell_output table.dataframe{color:var(--pst-color-table);border:1px solid var(--pst-color-table-outer-border)}.bd-content div.cell_output table.dataframe th~th,.bd-content div.cell_output table.dataframe th~td,.bd-content div.cell_output table.dataframe td~th,.bd-content div.cell_output table.dataframe td~td{border-left:1px solid var(--pst-color-table-inner-border)}.bd-content div.cell_output table.dataframe thead tr{background-color:var(--pst-color-table-heading-bg);border-bottom:2px solid var(--pst-color-primary)}.bd-content div.cell_output table.dataframe tbody tr:nth-child(odd){background-color:var(--pst-color-table-row-zebra-low-bg)}.bd-content div.cell_output table.dataframe tbody tr:nth-child(even){background-color:var(--pst-color-table-row-zebra-high-bg)}.bd-content div.cell_output table.dataframe tbody tr:hover{background-color:var(--pst-color-table-row-hover-bg)}html[data-theme=dark] .bd-content div.cell_output img,html[data-theme=dark] .bd-content div.cell_output .text_html:not(:has(table.dataframe)),html[data-theme=dark] .bd-content div.cell_output .widget-subarea{color:var(--pst-color-on-background);background-color:var(--pst-color-text-base);border-radius:.25rem;padding:.5rem}.bd-content div.cell_input{display:flex;flex-direction:column;justify-content:stretch}.bd-content div.cell_input,.bd-content div.output{border-radius:.25rem}.bd-content div.output table{table-layout:auto}html[data-theme=dark] .bd-content img.leaflet-tile.leaflet-tile-loaded{border-radius:0;padding:0}.bd-search-container div#search-results>h2{font-size:var(--pst-font-size-icon);margin-top:1rem}.bd-search-container div#search-results p.search-summary{color:var(--pst-color-text-muted)}.bd-search-container ul.search{margin:0;list-style:none}.bd-search-container ul.search li{background-image:none;padding:1rem 0;margin:1rem 0;border-top:1px solid var(--pst-color-text-muted)}.bd-search-container ul.search li>a{font-size:1.2em}.bd-search-container ul.search li div.context,.bd-search-container ul.search li p.context{color:var(--pst-color-text-base);margin:.5em 0 0}.bd-search-container ul.search li div.context a::before,.bd-search-container ul.search li p.context a::before{content:\"#\";padding-right:.2em;color:var(--pst-color-text-muted)}","html {\n  /*****************************************************************************\n  * Overall Layout Variables\n  */\n\n  // Header height will impact the top offset for many sections\n  // Article header is 66% of Header\n  --pst-header-height: 4rem;\n  --pst-header-article-height: calc(var(--pst-header-height) * 2 / 3);\n  --pst-sidebar-secondary: 17rem;\n}\n\n/*******************************************************************************\n* Breakpoints that trigger UI changes\n*\n* Note that media-breakpoint-down begins at the next highest level!\n* So we should choose a media-breakpoint-down one *lower* than when we want to start\n* example: media-breakpoint-up(md) and media-breakpoint-down(sm) trigger at the same time\n* ref: https://github.com/twbs/bootstrap/issues/31214\n*/\n$breakpoint-sidebar-primary: lg; // When we collapse the primary sidebar\n$breakpoint-sidebar-secondary: xl; // When we collapse the secondary sidebar\n$breakpoint-page-width: 88rem; // taken from sphinx-basic-ng, which we are ultimately going to inherit\n\n/*******************************************************************************\n* Define the animation behaviour\n*/\n$animation-time: 200ms;\n\n/*******************************************************************************\n* UI shaping and padding\n*/\n$admonition-border-radius: 0.25rem;\n\n// In this theme, some focus rings have rounded corners while others do not.\n// This variable sets the border radius for the rounded focus rings.\n$focus-ring-radius: 0.125rem; // 2px at 100% zoom and 16px base font.\n\n$navbar-link-padding-y: 0.25rem;\n","html {\n  /*****************************************************************************\n  * Font features used in this theme\n  */\n\n  // base font size - applied at body/html level\n  --pst-font-size-base: 1rem;\n\n  // heading font sizes based on bootstrap sizing\n  --pst-font-size-h1: 2.5rem;\n  --pst-font-size-h2: 2rem;\n  --pst-font-size-h3: 1.75rem;\n  --pst-font-size-h4: 1.5rem;\n  --pst-font-size-h5: 1.25rem;\n  --pst-font-size-h6: 1.1rem;\n\n  // smaller than heading font sizes\n  --pst-font-size-milli: 0.9rem;\n\n  // Sidebar styles\n  --pst-sidebar-font-size: 0.9rem;\n  --pst-sidebar-font-size-mobile: 1.1rem;\n  --pst-sidebar-header-font-size: 1.2rem;\n  --pst-sidebar-header-font-weight: 600;\n\n  // Admonition styles\n  --pst-admonition-font-weight-heading: 600;\n\n  // Font weights\n  --pst-font-weight-caption: 300;\n  --pst-font-weight-heading: 400;\n\n  // Font family\n  // These are adapted from https://systemfontstack.com/ */\n  --pst-font-family-base-system: -apple-system, \"BlinkMacSystemFont\", \"Segoe UI\",\n    \"Helvetica Neue\", \"Arial\", sans-serif, \"Apple Color Emoji\", \"Segoe UI Emoji\",\n    \"Segoe UI Symbol\";\n  --pst-font-family-monospace-system: \"SFMono-Regular\", \"Menlo\", \"Consolas\",\n    \"Monaco\", \"Liberation Mono\", \"Lucida Console\", monospace;\n  --pst-font-family-base: var(--pst-font-family-base-system);\n  --pst-font-family-heading: var(--pst-font-family-base-system);\n  --pst-font-family-monospace: var(--pst-font-family-monospace-system);\n}\n\n$line-height-body: 1.65;\n","html {\n  /*****************************************************************************\n  * Icon\n  */\n\n  // Font size across all icons\n  --pst-font-size-icon: 1.5rem;\n\n  // Font Awesome default icons\n  --pst-icon-check-circle: \"\\f058\"; // fa-solid fa-circle-check\n  --pst-icon-info-circle: \"\\f05a\"; // fa-solid fa-circle-info\n  --pst-icon-exclamation-triangle: \"\\f071\"; // fa-solid fa-triangle-exclamation\n  --pst-icon-exclamation-circle: \"\\f06a\"; // fa-solid fa-circle-exclamation\n  --pst-icon-times-circle: \"\\f057\"; // fa-solid fa-circle-xmark\n  --pst-icon-lightbulb: \"\\f0eb\"; // fa-solid fa-lightbulb\n  --pst-icon-download: \"\\f019\"; // fa-solid fa-download\n  --pst-icon-angle-left: \"\\f104\"; // fa-solid fa-angle-left\n  --pst-icon-angle-right: \"\\f105\"; // fa-solid fa-angle-right\n  --pst-icon-external-link: \"\\f35d\"; // fa-solid fa-up-right-from-square\n  --pst-icon-search-minus: \"\\f010\"; // fa-solid fa-magnifying-glass-minus\n  --pst-icon-github: \"\\f09b\"; // fa-brands fa-github\n  --pst-icon-gitlab: \"\\f296\"; // fa-brands fa-gitlab\n  --pst-icon-share: \"\\f064\"; // fa-solid fa-share\n  --pst-icon-bell: \"\\f0f3\"; // fa-solid fa-bell\n  --pst-icon-pencil: \"\\f303\"; // fa-solid fa-pencil\n\n  // Bootstrap icons\n  --pst-breadcrumb-divider: \"\\f105\";\n}\n","html {\n  /*****************************************************************************\n  * Admonitions\n  **/\n\n  --pst-icon-admonition-default: var(--pst-icon-bell);\n  --pst-icon-admonition-note: var(--pst-icon-info-circle);\n  --pst-icon-admonition-attention: var(--pst-icon-exclamation-circle);\n  --pst-icon-admonition-caution: var(--pst-icon-exclamation-triangle);\n  --pst-icon-admonition-warning: var(--pst-icon-exclamation-triangle);\n  --pst-icon-admonition-danger: var(--pst-icon-exclamation-triangle);\n  --pst-icon-admonition-error: var(--pst-icon-times-circle);\n  --pst-icon-admonition-hint: var(--pst-icon-lightbulb);\n  --pst-icon-admonition-tip: var(--pst-icon-lightbulb);\n  --pst-icon-admonition-important: var(--pst-icon-exclamation-circle);\n  --pst-icon-admonition-seealso: var(--pst-icon-share);\n  --pst-icon-admonition-todo: var(--pst-icon-pencil);\n}\n","html {\n  /*****************************************************************************\n  * versionmodified\n  **/\n\n  --pst-icon-versionmodified-default: var(--pst-icon-exclamation-circle);\n  --pst-icon-versionmodified-added: var(--pst-icon-exclamation-circle);\n  --pst-icon-versionmodified-changed: var(--pst-icon-exclamation-circle);\n  --pst-icon-versionmodified-deprecated: var(--pst-icon-exclamation-circle);\n}\n","html {\n  font-size: var(--pst-font-size-base);\n  scroll-padding-top: calc(var(--pst-header-height) + 1rem);\n}\n\nbody {\n  background-color: var(--pst-color-background);\n  font-family: var(--pst-font-family-base);\n  font-weight: 400;\n  line-height: $line-height-body;\n  color: var(--pst-color-text-base);\n  min-height: 100vh;\n  display: flex;\n  flex-direction: column;\n\n  // hack to avoid the black background on some browser including Safari\n  &::-webkit-scrollbar-track {\n    background-color: var(--pst-color-background);\n  }\n}\n\np {\n  margin-bottom: 1.15rem;\n  font-size: 1em;\n\n  /* section header in docstring pages */\n  &.rubric {\n    border-bottom: 1px solid var(--pst-color-border);\n  }\n\n  &.centered {\n    text-align: center;\n  }\n}\n\na {\n  @include link-style-default;\n\n  // The # anchor that appears on hover over headings\n  &.headerlink {\n    color: var(--pst-color-secondary);\n    opacity: 0.7;\n    font-size: 0.8em;\n    padding: 0 4px;\n    margin-left: 0.2em;\n    text-decoration: none;\n    transition: all 0.2s ease-out;\n    user-select: none;\n\n    &:hover {\n      opacity: 1;\n    }\n  }\n\n  // set up a icon next to the shorten links from github and gitlab\n  &.github,\n  &.gitlab {\n    &::before {\n      color: var(--pst-color-text-muted);\n      font: var(--fa-font-brands);\n      margin-right: 0.25rem;\n    }\n  }\n\n  &.github::before {\n    content: var(--pst-icon-github);\n  }\n\n  &.gitlab::before {\n    content: var(--pst-icon-gitlab);\n  }\n}\n\n%heading-style {\n  margin: 2.75rem 0 1.05rem;\n  font-family: var(--pst-font-family-heading);\n  font-weight: var(--pst-font-weight-heading);\n  line-height: 1.15;\n}\n\nh1 {\n  @extend %heading-style;\n\n  margin-top: 0;\n  font-size: var(--pst-font-size-h1);\n  color: var(--pst-heading-color);\n}\n\nh2 {\n  @extend %heading-style;\n\n  font-size: var(--pst-font-size-h2);\n  color: var(--pst-heading-color);\n}\n\nh3 {\n  @extend %heading-style;\n\n  font-size: var(--pst-font-size-h3);\n  color: var(--pst-heading-color);\n}\n\nh4 {\n  @extend %heading-style;\n\n  font-size: var(--pst-font-size-h4);\n  color: var(--pst-heading-color);\n}\n\nh5 {\n  @extend %heading-style;\n\n  font-size: var(--pst-font-size-h5);\n  color: var(--pst-color-text-base);\n}\n\nh6 {\n  @extend %heading-style;\n\n  font-size: var(--pst-font-size-h6);\n  color: var(--pst-color-text-base);\n}\n\nsmall,\n.text_small {\n  font-size: var(--pst-font-size-milli);\n}\n\nhr {\n  border: 0;\n  border-top: 1px solid var(--pst-color-border);\n}\n\npre,\ncode,\nkbd,\nsamp {\n  font-family: var(--pst-font-family-monospace);\n}\n\nkbd {\n  // use theme negative\n  background-color: var(--pst-color-on-background);\n  color: var(--pst-color-text-muted);\n\n  // Compound keyboard elements will have nested kbd in them, so this prevents double lines\n  &:not(.compound) {\n    border: 1px solid var(--pst-color-border);\n    margin: 0 0.1rem;\n    padding: 0.1rem 0.4rem;\n    box-shadow: 1px 1px 1px var(--pst-color-shadow);\n  }\n}\n\ncode {\n  color: var(--pst-color-inline-code);\n}\n\npre {\n  margin: 1.5em 0;\n  padding: 1rem;\n  background-color: var(--pst-color-surface);\n  color: var(--pst-color-text-base);\n  line-height: 1.2em;\n  border: 1px solid var(--pst-color-border);\n  border-radius: $admonition-border-radius;\n\n  .linenos {\n    // minimum opacity to make the line numbers WCAG AA conformant\n    opacity: 0.8;\n    padding-right: 10px;\n  }\n}\n\n// the back to top btn\n#pst-back-to-top {\n  z-index: $zindex-tooltip;\n  position: fixed;\n  display: none;\n  top: 90vh;\n  left: 50vw;\n  transform: translate(-50%);\n  color: var(--pst-color-secondary-text);\n  background-color: var(--pst-color-secondary);\n  border: none;\n\n  .fa-arrow-up {\n    // Using margin instead of a space character prevents the space between the\n    // icon and the text from being underlined when the button is hovered.\n    margin-inline-end: 0.5em;\n  }\n\n  &:hover {\n    @include link-style-hover;\n\n    text-decoration-thickness: 1px;\n    background-color: var(--pst-violet-600);\n    color: var(--pst-color-secondary-text);\n  }\n\n  &:focus-visible {\n    box-shadow: none;\n    outline: $focus-ring-outline;\n    outline-color: var(--pst-color-secondary);\n    outline-offset: $focus-ring-width;\n  }\n}\n\n// Focus ring\n// ----------\n// Note: The Bootstrap stylesheet provides the focus ring (customized by this\n// theme via Sass variables in _bootstrap.scss) in some cases. This rule covers\n// all other cases.\n:focus-visible {\n  outline: $focus-ring-outline;\n  box-shadow: none; // override Bootstrap\n}\n","// Consistent styling for links\n// ============================\n\n@use \"sass:string\";\n\n// Define some useful variables for links styling consistency\n\n// The default thickness of the underline for links will be either:\n//  - 1px\n//  - 0.0625rem if it's thicker than 1px because the user has changed the text\n//    size in their browser\n$link-underline-thickness: string.unquote(\"max(1px, .0625rem)\") !default;\n\n// Offset of link underlines from text baseline\n// The default is 3px expressed as ems, as calculated against the default body\n// font size (on desktop).\n$link-underline-offset: 0.1578em !default;\n\n// Thickness of link underlines in hover state\n// The default for each link will be the thickest of the following:\n//  - 3px\n//  - 0.1875rem, if it's thicker than 3px because the user has changed the text\n//    size in their browser\n//  - 0.12em (relative to the link's text size)\n$link-hover-decoration-thickness: string.unquote(\n  \"max(3px, .1875rem, .12em)\"\n) !default;\n\n// Ensures links have an underline decoration by default - needed to meet\n// WCAG SC 1.4.1\n@mixin link-decoration {\n  text-decoration: underline;\n\n  @if $link-underline-thickness {\n    text-decoration-thickness: $link-underline-thickness;\n  }\n\n  @if $link-underline-offset {\n    text-underline-offset: $link-underline-offset;\n  }\n}\n\n// Ensures links have an underline decoration on hover - distinct from the\n// default behaviour\n@mixin link-decoration-hover {\n  @if $link-hover-decoration-thickness {\n    text-decoration-thickness: $link-hover-decoration-thickness;\n\n    // Disable ink skipping on underlines on hover.\n    text-decoration-skip-ink: none;\n  }\n}\n\n// Simple hover style - can be used alone or in conjunction with other mixins\n// Add the text underline and change in thickness on hover.\n// Intended for use with the `:hover` pseudo-class.\n@mixin link-style-hover {\n  @include link-decoration;\n  @include link-decoration-hover;\n\n  color: var(--pst-color-link-hover);\n}\n\n// Default link styles\n// -------------------\n// Defines: default unvisited, visited, hover, and active.\n// TODO: @trallard to improve focus styles in subsequent PR\n@mixin link-style-default {\n  // So that really long links don't spill out of their container\n  word-wrap: break-word;\n  color: var(--pst-color-link);\n\n  @include link-decoration;\n\n  &:hover {\n    color: var(--pst-color-link-hover);\n\n    @include link-decoration-hover;\n  }\n\n  // TODO: @trallard to add active styles in subsequent PR\n  &:active {\n    color: var(--pst-color-link);\n  }\n\n  // Visited should still be hoverable\n  &:visited {\n    color: var(--pst-color-link);\n\n    &:hover {\n      color: var(--pst-color-link-hover);\n    }\n  }\n}\n\n// Text link styles\n// ----------------\n// Makes links use the muted text colour and removes the underline.\n// Use this mixin for navigation bar links.\n@mixin link-style-text {\n  color: var(--pst-color-text-muted);\n  text-decoration: none;\n\n  &:hover {\n    color: var(--pst-color-link-hover);\n\n    @include link-decoration;\n    @include link-decoration-hover;\n  }\n}\n\n// Sidebar and TOC links\n// ---------------------\n// Makes links use the muted text colour and removes the underline.\n// Use this mixin for navigation the primary sidebar and table of contents.\n// Active and hover should work together rather than one overriding the other.\n@mixin link-sidebar {\n  color: var(--pst-color-text-muted);\n  text-decoration: none;\n\n  &:hover {\n    text-decoration: underline;\n    background-color: transparent;\n    color: var(--pst-color-link-hover);\n\n    @include link-decoration-hover;\n  }\n\n  // TODO: @trallard to update active styles in subsequent PR\n  &:active {\n    color: var(--pst-color-link-hover);\n  }\n\n  &:focus-visible {\n    box-shadow: $focus-ring-box-shadow;\n    outline: none;\n    z-index: 10; // keep focus ring on top (prevent the link-sidebar-current notch from lying on top of the ring)\n  }\n}\n\n// Sidebar current page link styles\n// --------------------------------\n// Adds a vertical line on the left hand side of the link to indicate that\n// it's the current page. Note this is distinct from an active state.\n// Used on the primary sidebar and the TOC.\n// We want the side box shadow to have the same thickness as the hover underline\n@mixin link-sidebar-current {\n  font-weight: 600;\n  color: var(--pst-color-primary);\n\n  @if $link-hover-decoration-thickness {\n    $notch-shadow: inset\n      $link-hover-decoration-thickness\n      0\n      0\n      var(--pst-color-primary);\n\n    box-shadow: $notch-shadow;\n\n    &:focus-visible {\n      box-shadow: $notch-shadow, $focus-ring-box-shadow;\n      outline: none;\n    }\n  }\n}\n\n// Heaver navbar text and icon links\n// ---------------------------------\n// (includes light/dark mode button)\n\n// This mixin makes it possible to show hover/underline and focus/ring styles at\n// the same time. The trick is to use:\n//    - a pseudo-element with bottom border for the hover underline\n//    - a CSS outline for the focus ring.\n\n// Normally we use box-shadow for underline and outline for focus ring. But we\n// cannot apply box-shadow and outline together on the same element because the\n// border-radius value that we use to round the outline will also round the\n// box-shadow used for the underline. We also cannot use text-underline because\n// it does not work on non-text links, nor do we want to use it on text links\n// that we want to treat as blocks, such as the header nav links because the\n// underline will wrap across two lines if the link text also wraps across two\n// lines.\n@mixin link-style-block {\n  color: var(--pst-color-text-muted);\n\n  // Set position relative so that the child ::before pseudo-element's absolute\n  // position is relative to this element.\n  position: relative;\n\n  // Set up pseudo-element used for hover underline styles\n  &::before {\n    content: \"\";\n    display: block;\n    position: absolute;\n    inset: 0;\n    background-color: transparent;\n\n    @if $link-hover-decoration-thickness {\n      bottom: calc(-1 * $link-hover-decoration-thickness);\n      margin: $link-hover-decoration-thickness 0;\n    }\n  }\n\n  &:hover {\n    color: var(--pst-color-secondary);\n    text-decoration: none; // override the link-style-hover mixin\n    &::before {\n      @if $link-hover-decoration-thickness {\n        border-bottom: $link-hover-decoration-thickness\n          solid\n          var(--pst-color-secondary);\n      }\n    }\n  }\n\n  &:focus-visible {\n    box-shadow: none; // override Bootstrap\n    outline: 3px solid var(--pst-color-accent);\n    outline-offset: 3px;\n  }\n}\n","// Variables\n//\n// Variables should follow the `$component-state-property-size` formula for\n// consistent naming. Ex: $nav-link-disabled-color and $modal-content-box-shadow-xs.\n\n// Color system\n\n// scss-docs-start gray-color-variables\n$white:    #fff !default;\n$gray-100: #f8f9fa !default;\n$gray-200: #e9ecef !default;\n$gray-300: #dee2e6 !default;\n$gray-400: #ced4da !default;\n$gray-500: #adb5bd !default;\n$gray-600: #6c757d !default;\n$gray-700: #495057 !default;\n$gray-800: #343a40 !default;\n$gray-900: #212529 !default;\n$black:    #000 !default;\n// scss-docs-end gray-color-variables\n\n// fusv-disable\n// scss-docs-start gray-colors-map\n$grays: (\n  \"100\": $gray-100,\n  \"200\": $gray-200,\n  \"300\": $gray-300,\n  \"400\": $gray-400,\n  \"500\": $gray-500,\n  \"600\": $gray-600,\n  \"700\": $gray-700,\n  \"800\": $gray-800,\n  \"900\": $gray-900\n) !default;\n// scss-docs-end gray-colors-map\n// fusv-enable\n\n// scss-docs-start color-variables\n$blue:    #0d6efd !default;\n$indigo:  #6610f2 !default;\n$purple:  #6f42c1 !default;\n$pink:    #d63384 !default;\n$red:     #dc3545 !default;\n$orange:  #fd7e14 !default;\n$yellow:  #ffc107 !default;\n$green:   #198754 !default;\n$teal:    #20c997 !default;\n$cyan:    #0dcaf0 !default;\n// scss-docs-end color-variables\n\n// scss-docs-start colors-map\n$colors: (\n  \"blue\":       $blue,\n  \"indigo\":     $indigo,\n  \"purple\":     $purple,\n  \"pink\":       $pink,\n  \"red\":        $red,\n  \"orange\":     $orange,\n  \"yellow\":     $yellow,\n  \"green\":      $green,\n  \"teal\":       $teal,\n  \"cyan\":       $cyan,\n  \"black\":      $black,\n  \"white\":      $white,\n  \"gray\":       $gray-600,\n  \"gray-dark\":  $gray-800\n) !default;\n// scss-docs-end colors-map\n\n// The contrast ratio to reach against white, to determine if color changes from \"light\" to \"dark\". Acceptable values for WCAG 2.0 are 3, 4.5 and 7.\n// See https://www.w3.org/TR/WCAG20/#visual-audio-contrast-contrast\n$min-contrast-ratio:   4.5 !default;\n\n// Customize the light and dark text colors for use in our color contrast function.\n$color-contrast-dark:      $black !default;\n$color-contrast-light:     $white !default;\n\n// fusv-disable\n$blue-100: tint-color($blue, 80%) !default;\n$blue-200: tint-color($blue, 60%) !default;\n$blue-300: tint-color($blue, 40%) !default;\n$blue-400: tint-color($blue, 20%) !default;\n$blue-500: $blue !default;\n$blue-600: shade-color($blue, 20%) !default;\n$blue-700: shade-color($blue, 40%) !default;\n$blue-800: shade-color($blue, 60%) !default;\n$blue-900: shade-color($blue, 80%) !default;\n\n$indigo-100: tint-color($indigo, 80%) !default;\n$indigo-200: tint-color($indigo, 60%) !default;\n$indigo-300: tint-color($indigo, 40%) !default;\n$indigo-400: tint-color($indigo, 20%) !default;\n$indigo-500: $indigo !default;\n$indigo-600: shade-color($indigo, 20%) !default;\n$indigo-700: shade-color($indigo, 40%) !default;\n$indigo-800: shade-color($indigo, 60%) !default;\n$indigo-900: shade-color($indigo, 80%) !default;\n\n$purple-100: tint-color($purple, 80%) !default;\n$purple-200: tint-color($purple, 60%) !default;\n$purple-300: tint-color($purple, 40%) !default;\n$purple-400: tint-color($purple, 20%) !default;\n$purple-500: $purple !default;\n$purple-600: shade-color($purple, 20%) !default;\n$purple-700: shade-color($purple, 40%) !default;\n$purple-800: shade-color($purple, 60%) !default;\n$purple-900: shade-color($purple, 80%) !default;\n\n$pink-100: tint-color($pink, 80%) !default;\n$pink-200: tint-color($pink, 60%) !default;\n$pink-300: tint-color($pink, 40%) !default;\n$pink-400: tint-color($pink, 20%) !default;\n$pink-500: $pink !default;\n$pink-600: shade-color($pink, 20%) !default;\n$pink-700: shade-color($pink, 40%) !default;\n$pink-800: shade-color($pink, 60%) !default;\n$pink-900: shade-color($pink, 80%) !default;\n\n$red-100: tint-color($red, 80%) !default;\n$red-200: tint-color($red, 60%) !default;\n$red-300: tint-color($red, 40%) !default;\n$red-400: tint-color($red, 20%) !default;\n$red-500: $red !default;\n$red-600: shade-color($red, 20%) !default;\n$red-700: shade-color($red, 40%) !default;\n$red-800: shade-color($red, 60%) !default;\n$red-900: shade-color($red, 80%) !default;\n\n$orange-100: tint-color($orange, 80%) !default;\n$orange-200: tint-color($orange, 60%) !default;\n$orange-300: tint-color($orange, 40%) !default;\n$orange-400: tint-color($orange, 20%) !default;\n$orange-500: $orange !default;\n$orange-600: shade-color($orange, 20%) !default;\n$orange-700: shade-color($orange, 40%) !default;\n$orange-800: shade-color($orange, 60%) !default;\n$orange-900: shade-color($orange, 80%) !default;\n\n$yellow-100: tint-color($yellow, 80%) !default;\n$yellow-200: tint-color($yellow, 60%) !default;\n$yellow-300: tint-color($yellow, 40%) !default;\n$yellow-400: tint-color($yellow, 20%) !default;\n$yellow-500: $yellow !default;\n$yellow-600: shade-color($yellow, 20%) !default;\n$yellow-700: shade-color($yellow, 40%) !default;\n$yellow-800: shade-color($yellow, 60%) !default;\n$yellow-900: shade-color($yellow, 80%) !default;\n\n$green-100: tint-color($green, 80%) !default;\n$green-200: tint-color($green, 60%) !default;\n$green-300: tint-color($green, 40%) !default;\n$green-400: tint-color($green, 20%) !default;\n$green-500: $green !default;\n$green-600: shade-color($green, 20%) !default;\n$green-700: shade-color($green, 40%) !default;\n$green-800: shade-color($green, 60%) !default;\n$green-900: shade-color($green, 80%) !default;\n\n$teal-100: tint-color($teal, 80%) !default;\n$teal-200: tint-color($teal, 60%) !default;\n$teal-300: tint-color($teal, 40%) !default;\n$teal-400: tint-color($teal, 20%) !default;\n$teal-500: $teal !default;\n$teal-600: shade-color($teal, 20%) !default;\n$teal-700: shade-color($teal, 40%) !default;\n$teal-800: shade-color($teal, 60%) !default;\n$teal-900: shade-color($teal, 80%) !default;\n\n$cyan-100: tint-color($cyan, 80%) !default;\n$cyan-200: tint-color($cyan, 60%) !default;\n$cyan-300: tint-color($cyan, 40%) !default;\n$cyan-400: tint-color($cyan, 20%) !default;\n$cyan-500: $cyan !default;\n$cyan-600: shade-color($cyan, 20%) !default;\n$cyan-700: shade-color($cyan, 40%) !default;\n$cyan-800: shade-color($cyan, 60%) !default;\n$cyan-900: shade-color($cyan, 80%) !default;\n\n$blues: (\n  \"blue-100\": $blue-100,\n  \"blue-200\": $blue-200,\n  \"blue-300\": $blue-300,\n  \"blue-400\": $blue-400,\n  \"blue-500\": $blue-500,\n  \"blue-600\": $blue-600,\n  \"blue-700\": $blue-700,\n  \"blue-800\": $blue-800,\n  \"blue-900\": $blue-900\n) !default;\n\n$indigos: (\n  \"indigo-100\": $indigo-100,\n  \"indigo-200\": $indigo-200,\n  \"indigo-300\": $indigo-300,\n  \"indigo-400\": $indigo-400,\n  \"indigo-500\": $indigo-500,\n  \"indigo-600\": $indigo-600,\n  \"indigo-700\": $indigo-700,\n  \"indigo-800\": $indigo-800,\n  \"indigo-900\": $indigo-900\n) !default;\n\n$purples: (\n  \"purple-100\": $purple-100,\n  \"purple-200\": $purple-200,\n  \"purple-300\": $purple-300,\n  \"purple-400\": $purple-400,\n  \"purple-500\": $purple-500,\n  \"purple-600\": $purple-600,\n  \"purple-700\": $purple-700,\n  \"purple-800\": $purple-800,\n  \"purple-900\": $purple-900\n) !default;\n\n$pinks: (\n  \"pink-100\": $pink-100,\n  \"pink-200\": $pink-200,\n  \"pink-300\": $pink-300,\n  \"pink-400\": $pink-400,\n  \"pink-500\": $pink-500,\n  \"pink-600\": $pink-600,\n  \"pink-700\": $pink-700,\n  \"pink-800\": $pink-800,\n  \"pink-900\": $pink-900\n) !default;\n\n$reds: (\n  \"red-100\": $red-100,\n  \"red-200\": $red-200,\n  \"red-300\": $red-300,\n  \"red-400\": $red-400,\n  \"red-500\": $red-500,\n  \"red-600\": $red-600,\n  \"red-700\": $red-700,\n  \"red-800\": $red-800,\n  \"red-900\": $red-900\n) !default;\n\n$oranges: (\n  \"orange-100\": $orange-100,\n  \"orange-200\": $orange-200,\n  \"orange-300\": $orange-300,\n  \"orange-400\": $orange-400,\n  \"orange-500\": $orange-500,\n  \"orange-600\": $orange-600,\n  \"orange-700\": $orange-700,\n  \"orange-800\": $orange-800,\n  \"orange-900\": $orange-900\n) !default;\n\n$yellows: (\n  \"yellow-100\": $yellow-100,\n  \"yellow-200\": $yellow-200,\n  \"yellow-300\": $yellow-300,\n  \"yellow-400\": $yellow-400,\n  \"yellow-500\": $yellow-500,\n  \"yellow-600\": $yellow-600,\n  \"yellow-700\": $yellow-700,\n  \"yellow-800\": $yellow-800,\n  \"yellow-900\": $yellow-900\n) !default;\n\n$greens: (\n  \"green-100\": $green-100,\n  \"green-200\": $green-200,\n  \"green-300\": $green-300,\n  \"green-400\": $green-400,\n  \"green-500\": $green-500,\n  \"green-600\": $green-600,\n  \"green-700\": $green-700,\n  \"green-800\": $green-800,\n  \"green-900\": $green-900\n) !default;\n\n$teals: (\n  \"teal-100\": $teal-100,\n  \"teal-200\": $teal-200,\n  \"teal-300\": $teal-300,\n  \"teal-400\": $teal-400,\n  \"teal-500\": $teal-500,\n  \"teal-600\": $teal-600,\n  \"teal-700\": $teal-700,\n  \"teal-800\": $teal-800,\n  \"teal-900\": $teal-900\n) !default;\n\n$cyans: (\n  \"cyan-100\": $cyan-100,\n  \"cyan-200\": $cyan-200,\n  \"cyan-300\": $cyan-300,\n  \"cyan-400\": $cyan-400,\n  \"cyan-500\": $cyan-500,\n  \"cyan-600\": $cyan-600,\n  \"cyan-700\": $cyan-700,\n  \"cyan-800\": $cyan-800,\n  \"cyan-900\": $cyan-900\n) !default;\n// fusv-enable\n\n// scss-docs-start theme-color-variables\n$primary:       $blue !default;\n$secondary:     $gray-600 !default;\n$success:       $green !default;\n$info:          $cyan !default;\n$warning:       $yellow !default;\n$danger:        $red !default;\n$light:         $gray-100 !default;\n$dark:          $gray-900 !default;\n// scss-docs-end theme-color-variables\n\n// scss-docs-start theme-colors-map\n$theme-colors: (\n  \"primary\":    $primary,\n  \"secondary\":  $secondary,\n  \"success\":    $success,\n  \"info\":       $info,\n  \"warning\":    $warning,\n  \"danger\":     $danger,\n  \"light\":      $light,\n  \"dark\":       $dark\n) !default;\n// scss-docs-end theme-colors-map\n\n// scss-docs-start theme-text-variables\n$primary-text-emphasis:   shade-color($primary, 60%) !default;\n$secondary-text-emphasis: shade-color($secondary, 60%) !default;\n$success-text-emphasis:   shade-color($success, 60%) !default;\n$info-text-emphasis:      shade-color($info, 60%) !default;\n$warning-text-emphasis:   shade-color($warning, 60%) !default;\n$danger-text-emphasis:    shade-color($danger, 60%) !default;\n$light-text-emphasis:     $gray-700 !default;\n$dark-text-emphasis:      $gray-700 !default;\n// scss-docs-end theme-text-variables\n\n// scss-docs-start theme-bg-subtle-variables\n$primary-bg-subtle:       tint-color($primary, 80%) !default;\n$secondary-bg-subtle:     tint-color($secondary, 80%) !default;\n$success-bg-subtle:       tint-color($success, 80%) !default;\n$info-bg-subtle:          tint-color($info, 80%) !default;\n$warning-bg-subtle:       tint-color($warning, 80%) !default;\n$danger-bg-subtle:        tint-color($danger, 80%) !default;\n$light-bg-subtle:         mix($gray-100, $white) !default;\n$dark-bg-subtle:          $gray-400 !default;\n// scss-docs-end theme-bg-subtle-variables\n\n// scss-docs-start theme-border-subtle-variables\n$primary-border-subtle:   tint-color($primary, 60%) !default;\n$secondary-border-subtle: tint-color($secondary, 60%) !default;\n$success-border-subtle:   tint-color($success, 60%) !default;\n$info-border-subtle:      tint-color($info, 60%) !default;\n$warning-border-subtle:   tint-color($warning, 60%) !default;\n$danger-border-subtle:    tint-color($danger, 60%) !default;\n$light-border-subtle:     $gray-200 !default;\n$dark-border-subtle:      $gray-500 !default;\n// scss-docs-end theme-border-subtle-variables\n\n// Characters which are escaped by the escape-svg function\n$escaped-characters: (\n  (\"<\", \"%3c\"),\n  (\">\", \"%3e\"),\n  (\"#\", \"%23\"),\n  (\"(\", \"%28\"),\n  (\")\", \"%29\"),\n) !default;\n\n// Options\n//\n// Quickly modify global styling by enabling or disabling optional features.\n\n$enable-caret:                true !default;\n$enable-rounded:              true !default;\n$enable-shadows:              false !default;\n$enable-gradients:            false !default;\n$enable-transitions:          true !default;\n$enable-reduced-motion:       true !default;\n$enable-smooth-scroll:        true !default;\n$enable-grid-classes:         true !default;\n$enable-container-classes:    true !default;\n$enable-cssgrid:              false !default;\n$enable-button-pointers:      true !default;\n$enable-rfs:                  true !default;\n$enable-validation-icons:     true !default;\n$enable-negative-margins:     false !default;\n$enable-deprecation-messages: true !default;\n$enable-important-utilities:  true !default;\n\n$enable-dark-mode:            true !default;\n$color-mode-type:             data !default; // `data` or `media-query`\n\n// Prefix for :root CSS variables\n\n$variable-prefix:             bs- !default; // Deprecated in v5.2.0 for the shorter `$prefix`\n$prefix:                      $variable-prefix !default;\n\n// Gradient\n//\n// The gradient which is added to components if `$enable-gradients` is `true`\n// This gradient is also added to elements with `.bg-gradient`\n// scss-docs-start variable-gradient\n$gradient: linear-gradient(180deg, rgba($white, .15), rgba($white, 0)) !default;\n// scss-docs-end variable-gradient\n\n// Spacing\n//\n// Control the default styling of most Bootstrap elements by modifying these\n// variables. Mostly focused on spacing.\n// You can add more entries to the $spacers map, should you need more variation.\n\n// scss-docs-start spacer-variables-maps\n$spacer: 1rem !default;\n$spacers: (\n  0: 0,\n  1: $spacer * .25,\n  2: $spacer * .5,\n  3: $spacer,\n  4: $spacer * 1.5,\n  5: $spacer * 3,\n) !default;\n// scss-docs-end spacer-variables-maps\n\n// Position\n//\n// Define the edge positioning anchors of the position utilities.\n\n// scss-docs-start position-map\n$position-values: (\n  0: 0,\n  50: 50%,\n  100: 100%\n) !default;\n// scss-docs-end position-map\n\n// Body\n//\n// Settings for the `<body>` element.\n\n$body-text-align:           null !default;\n$body-color:                $gray-900 !default;\n$body-bg:                   $white !default;\n\n$body-secondary-color:      rgba($body-color, .75) !default;\n$body-secondary-bg:         $gray-200 !default;\n\n$body-tertiary-color:       rgba($body-color, .5) !default;\n$body-tertiary-bg:          $gray-100 !default;\n\n$body-emphasis-color:       $black !default;\n\n// Links\n//\n// Style anchor elements.\n\n$link-color:                              $primary !default;\n$link-decoration:                         underline !default;\n$link-shade-percentage:                   20% !default;\n$link-hover-color:                        shift-color($link-color, $link-shade-percentage) !default;\n$link-hover-decoration:                   null !default;\n\n$stretched-link-pseudo-element:           after !default;\n$stretched-link-z-index:                  1 !default;\n\n// Icon links\n// scss-docs-start icon-link-variables\n$icon-link-gap:               .375rem !default;\n$icon-link-underline-offset:  .25em !default;\n$icon-link-icon-size:         1em !default;\n$icon-link-icon-transition:   .2s ease-in-out transform !default;\n$icon-link-icon-transform:    translate3d(.25em, 0, 0) !default;\n// scss-docs-end icon-link-variables\n\n// Paragraphs\n//\n// Style p element.\n\n$paragraph-margin-bottom:   1rem !default;\n\n\n// Grid breakpoints\n//\n// Define the minimum dimensions at which your layout will change,\n// adapting to different screen sizes, for use in media queries.\n\n// scss-docs-start grid-breakpoints\n$grid-breakpoints: (\n  xs: 0,\n  sm: 576px,\n  md: 768px,\n  lg: 992px,\n  xl: 1200px,\n  xxl: 1400px\n) !default;\n// scss-docs-end grid-breakpoints\n\n@include _assert-ascending($grid-breakpoints, \"$grid-breakpoints\");\n@include _assert-starts-at-zero($grid-breakpoints, \"$grid-breakpoints\");\n\n\n// Grid containers\n//\n// Define the maximum width of `.container` for different screen sizes.\n\n// scss-docs-start container-max-widths\n$container-max-widths: (\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1140px,\n  xxl: 1320px\n) !default;\n// scss-docs-end container-max-widths\n\n@include _assert-ascending($container-max-widths, \"$container-max-widths\");\n\n\n// Grid columns\n//\n// Set the number of columns and specify the width of the gutters.\n\n$grid-columns:                12 !default;\n$grid-gutter-width:           1.5rem !default;\n$grid-row-columns:            6 !default;\n\n// Container padding\n\n$container-padding-x: $grid-gutter-width !default;\n\n\n// Components\n//\n// Define common padding and border radius sizes and more.\n\n// scss-docs-start border-variables\n$border-width:                1px !default;\n$border-widths: (\n  1: 1px,\n  2: 2px,\n  3: 3px,\n  4: 4px,\n  5: 5px\n) !default;\n$border-style:                solid !default;\n$border-color:                $gray-300 !default;\n$border-color-translucent:    rgba($black, .175) !default;\n// scss-docs-end border-variables\n\n// scss-docs-start border-radius-variables\n$border-radius:               .375rem !default;\n$border-radius-sm:            .25rem !default;\n$border-radius-lg:            .5rem !default;\n$border-radius-xl:            1rem !default;\n$border-radius-xxl:           2rem !default;\n$border-radius-pill:          50rem !default;\n// scss-docs-end border-radius-variables\n// fusv-disable\n$border-radius-2xl:           $border-radius-xxl !default; // Deprecated in v5.3.0\n// fusv-enable\n\n// scss-docs-start box-shadow-variables\n$box-shadow:                  0 .5rem 1rem rgba($black, .15) !default;\n$box-shadow-sm:               0 .125rem .25rem rgba($black, .075) !default;\n$box-shadow-lg:               0 1rem 3rem rgba($black, .175) !default;\n$box-shadow-inset:            inset 0 1px 2px rgba($black, .075) !default;\n// scss-docs-end box-shadow-variables\n\n$component-active-color:      $white !default;\n$component-active-bg:         $primary !default;\n\n// scss-docs-start focus-ring-variables\n$focus-ring-width:      .25rem !default;\n$focus-ring-opacity:    .25 !default;\n$focus-ring-color:      rgba($primary, $focus-ring-opacity) !default;\n$focus-ring-blur:       0 !default;\n$focus-ring-box-shadow: 0 0 $focus-ring-blur $focus-ring-width $focus-ring-color !default;\n// scss-docs-end focus-ring-variables\n\n// scss-docs-start caret-variables\n$caret-width:                 .3em !default;\n$caret-vertical-align:        $caret-width * .85 !default;\n$caret-spacing:               $caret-width * .85 !default;\n// scss-docs-end caret-variables\n\n$transition-base:             all .2s ease-in-out !default;\n$transition-fade:             opacity .15s linear !default;\n// scss-docs-start collapse-transition\n$transition-collapse:         height .35s ease !default;\n$transition-collapse-width:   width .35s ease !default;\n// scss-docs-end collapse-transition\n\n// stylelint-disable function-disallowed-list\n// scss-docs-start aspect-ratios\n$aspect-ratios: (\n  \"1x1\": 100%,\n  \"4x3\": calc(3 / 4 * 100%),\n  \"16x9\": calc(9 / 16 * 100%),\n  \"21x9\": calc(9 / 21 * 100%)\n) !default;\n// scss-docs-end aspect-ratios\n// stylelint-enable function-disallowed-list\n\n// Typography\n//\n// Font, line-height, and color for body text, headings, and more.\n\n// scss-docs-start font-variables\n// stylelint-disable value-keyword-case\n$font-family-sans-serif:      system-ui, -apple-system, \"Segoe UI\", Roboto, \"Helvetica Neue\", \"Noto Sans\", \"Liberation Sans\", Arial, sans-serif, \"Apple Color Emoji\", \"Segoe UI Emoji\", \"Segoe UI Symbol\", \"Noto Color Emoji\" !default;\n$font-family-monospace:       SFMono-Regular, Menlo, Monaco, Consolas, \"Liberation Mono\", \"Courier New\", monospace !default;\n// stylelint-enable value-keyword-case\n$font-family-base:            var(--#{$prefix}font-sans-serif) !default;\n$font-family-code:            var(--#{$prefix}font-monospace) !default;\n\n// $font-size-root affects the value of `rem`, which is used for as well font sizes, paddings, and margins\n// $font-size-base affects the font size of the body text\n$font-size-root:              null !default;\n$font-size-base:              1rem !default; // Assumes the browser default, typically `16px`\n$font-size-sm:                $font-size-base * .875 !default;\n$font-size-lg:                $font-size-base * 1.25 !default;\n\n$font-weight-lighter:         lighter !default;\n$font-weight-light:           300 !default;\n$font-weight-normal:          400 !default;\n$font-weight-medium:          500 !default;\n$font-weight-semibold:        600 !default;\n$font-weight-bold:            700 !default;\n$font-weight-bolder:          bolder !default;\n\n$font-weight-base:            $font-weight-normal !default;\n\n$line-height-base:            1.5 !default;\n$line-height-sm:              1.25 !default;\n$line-height-lg:              2 !default;\n\n$h1-font-size:                $font-size-base * 2.5 !default;\n$h2-font-size:                $font-size-base * 2 !default;\n$h3-font-size:                $font-size-base * 1.75 !default;\n$h4-font-size:                $font-size-base * 1.5 !default;\n$h5-font-size:                $font-size-base * 1.25 !default;\n$h6-font-size:                $font-size-base !default;\n// scss-docs-end font-variables\n\n// scss-docs-start font-sizes\n$font-sizes: (\n  1: $h1-font-size,\n  2: $h2-font-size,\n  3: $h3-font-size,\n  4: $h4-font-size,\n  5: $h5-font-size,\n  6: $h6-font-size\n) !default;\n// scss-docs-end font-sizes\n\n// scss-docs-start headings-variables\n$headings-margin-bottom:      $spacer * .5 !default;\n$headings-font-family:        null !default;\n$headings-font-style:         null !default;\n$headings-font-weight:        500 !default;\n$headings-line-height:        1.2 !default;\n$headings-color:              inherit !default;\n// scss-docs-end headings-variables\n\n// scss-docs-start display-headings\n$display-font-sizes: (\n  1: 5rem,\n  2: 4.5rem,\n  3: 4rem,\n  4: 3.5rem,\n  5: 3rem,\n  6: 2.5rem\n) !default;\n\n$display-font-family: null !default;\n$display-font-style:  null !default;\n$display-font-weight: 300 !default;\n$display-line-height: $headings-line-height !default;\n// scss-docs-end display-headings\n\n// scss-docs-start type-variables\n$lead-font-size:              $font-size-base * 1.25 !default;\n$lead-font-weight:            300 !default;\n\n$small-font-size:             .875em !default;\n\n$sub-sup-font-size:           .75em !default;\n\n// fusv-disable\n$text-muted:                  var(--#{$prefix}secondary-color) !default; // Deprecated in 5.3.0\n// fusv-enable\n\n$initialism-font-size:        $small-font-size !default;\n\n$blockquote-margin-y:         $spacer !default;\n$blockquote-font-size:        $font-size-base * 1.25 !default;\n$blockquote-footer-color:     $gray-600 !default;\n$blockquote-footer-font-size: $small-font-size !default;\n\n$hr-margin-y:                 $spacer !default;\n$hr-color:                    inherit !default;\n\n// fusv-disable\n$hr-bg-color:                 null !default; // Deprecated in v5.2.0\n$hr-height:                   null !default; // Deprecated in v5.2.0\n// fusv-enable\n\n$hr-border-color:             null !default; // Allows for inherited colors\n$hr-border-width:             var(--#{$prefix}border-width) !default;\n$hr-opacity:                  .25 !default;\n\n// scss-docs-start vr-variables\n$vr-border-width:             var(--#{$prefix}border-width) !default;\n// scss-docs-end vr-variables\n\n$legend-margin-bottom:        .5rem !default;\n$legend-font-size:            1.5rem !default;\n$legend-font-weight:          null !default;\n\n$dt-font-weight:              $font-weight-bold !default;\n\n$list-inline-padding:         .5rem !default;\n\n$mark-padding:                .1875em !default;\n$mark-color:                  $body-color !default;\n$mark-bg:                     $yellow-100 !default;\n// scss-docs-end type-variables\n\n\n// Tables\n//\n// Customizes the `.table` component with basic values, each used across all table variations.\n\n// scss-docs-start table-variables\n$table-cell-padding-y:        .5rem !default;\n$table-cell-padding-x:        .5rem !default;\n$table-cell-padding-y-sm:     .25rem !default;\n$table-cell-padding-x-sm:     .25rem !default;\n\n$table-cell-vertical-align:   top !default;\n\n$table-color:                 var(--#{$prefix}emphasis-color) !default;\n$table-bg:                    var(--#{$prefix}body-bg) !default;\n$table-accent-bg:             transparent !default;\n\n$table-th-font-weight:        null !default;\n\n$table-striped-color:         $table-color !default;\n$table-striped-bg-factor:     .05 !default;\n$table-striped-bg:            rgba(var(--#{$prefix}emphasis-color-rgb), $table-striped-bg-factor) !default;\n\n$table-active-color:          $table-color !default;\n$table-active-bg-factor:      .1 !default;\n$table-active-bg:             rgba(var(--#{$prefix}emphasis-color-rgb), $table-active-bg-factor) !default;\n\n$table-hover-color:           $table-color !default;\n$table-hover-bg-factor:       .075 !default;\n$table-hover-bg:              rgba(var(--#{$prefix}emphasis-color-rgb), $table-hover-bg-factor) !default;\n\n$table-border-factor:         .2 !default;\n$table-border-width:          var(--#{$prefix}border-width) !default;\n$table-border-color:          var(--#{$prefix}border-color) !default;\n\n$table-striped-order:         odd !default;\n$table-striped-columns-order: even !default;\n\n$table-group-separator-color: currentcolor !default;\n\n$table-caption-color:         var(--#{$prefix}secondary-color) !default;\n\n$table-bg-scale:              -80% !default;\n// scss-docs-end table-variables\n\n// scss-docs-start table-loop\n$table-variants: (\n  \"primary\":    shift-color($primary, $table-bg-scale),\n  \"secondary\":  shift-color($secondary, $table-bg-scale),\n  \"success\":    shift-color($success, $table-bg-scale),\n  \"info\":       shift-color($info, $table-bg-scale),\n  \"warning\":    shift-color($warning, $table-bg-scale),\n  \"danger\":     shift-color($danger, $table-bg-scale),\n  \"light\":      $light,\n  \"dark\":       $dark,\n) !default;\n// scss-docs-end table-loop\n\n\n// Buttons + Forms\n//\n// Shared variables that are reassigned to `$input-` and `$btn-` specific variables.\n\n// scss-docs-start input-btn-variables\n$input-btn-padding-y:         .375rem !default;\n$input-btn-padding-x:         .75rem !default;\n$input-btn-font-family:       null !default;\n$input-btn-font-size:         $font-size-base !default;\n$input-btn-line-height:       $line-height-base !default;\n\n$input-btn-focus-width:         $focus-ring-width !default;\n$input-btn-focus-color-opacity: $focus-ring-opacity !default;\n$input-btn-focus-color:         $focus-ring-color !default;\n$input-btn-focus-blur:          $focus-ring-blur !default;\n$input-btn-focus-box-shadow:    $focus-ring-box-shadow !default;\n\n$input-btn-padding-y-sm:      .25rem !default;\n$input-btn-padding-x-sm:      .5rem !default;\n$input-btn-font-size-sm:      $font-size-sm !default;\n\n$input-btn-padding-y-lg:      .5rem !default;\n$input-btn-padding-x-lg:      1rem !default;\n$input-btn-font-size-lg:      $font-size-lg !default;\n\n$input-btn-border-width:      var(--#{$prefix}border-width) !default;\n// scss-docs-end input-btn-variables\n\n\n// Buttons\n//\n// For each of Bootstrap's buttons, define text, background, and border color.\n\n// scss-docs-start btn-variables\n$btn-color:                   var(--#{$prefix}body-color) !default;\n$btn-padding-y:               $input-btn-padding-y !default;\n$btn-padding-x:               $input-btn-padding-x !default;\n$btn-font-family:             $input-btn-font-family !default;\n$btn-font-size:               $input-btn-font-size !default;\n$btn-line-height:             $input-btn-line-height !default;\n$btn-white-space:             null !default; // Set to `nowrap` to prevent text wrapping\n\n$btn-padding-y-sm:            $input-btn-padding-y-sm !default;\n$btn-padding-x-sm:            $input-btn-padding-x-sm !default;\n$btn-font-size-sm:            $input-btn-font-size-sm !default;\n\n$btn-padding-y-lg:            $input-btn-padding-y-lg !default;\n$btn-padding-x-lg:            $input-btn-padding-x-lg !default;\n$btn-font-size-lg:            $input-btn-font-size-lg !default;\n\n$btn-border-width:            $input-btn-border-width !default;\n\n$btn-font-weight:             $font-weight-normal !default;\n$btn-box-shadow:              inset 0 1px 0 rgba($white, .15), 0 1px 1px rgba($black, .075) !default;\n$btn-focus-width:             $input-btn-focus-width !default;\n$btn-focus-box-shadow:        $input-btn-focus-box-shadow !default;\n$btn-disabled-opacity:        .65 !default;\n$btn-active-box-shadow:       inset 0 3px 5px rgba($black, .125) !default;\n\n$btn-link-color:              var(--#{$prefix}link-color) !default;\n$btn-link-hover-color:        var(--#{$prefix}link-hover-color) !default;\n$btn-link-disabled-color:     $gray-600 !default;\n$btn-link-focus-shadow-rgb:   to-rgb(mix(color-contrast($link-color), $link-color, 15%)) !default;\n\n// Allows for customizing button radius independently from global border radius\n$btn-border-radius:           var(--#{$prefix}border-radius) !default;\n$btn-border-radius-sm:        var(--#{$prefix}border-radius-sm) !default;\n$btn-border-radius-lg:        var(--#{$prefix}border-radius-lg) !default;\n\n$btn-transition:              color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$btn-hover-bg-shade-amount:       15% !default;\n$btn-hover-bg-tint-amount:        15% !default;\n$btn-hover-border-shade-amount:   20% !default;\n$btn-hover-border-tint-amount:    10% !default;\n$btn-active-bg-shade-amount:      20% !default;\n$btn-active-bg-tint-amount:       20% !default;\n$btn-active-border-shade-amount:  25% !default;\n$btn-active-border-tint-amount:   10% !default;\n// scss-docs-end btn-variables\n\n\n// Forms\n\n// scss-docs-start form-text-variables\n$form-text-margin-top:                  .25rem !default;\n$form-text-font-size:                   $small-font-size !default;\n$form-text-font-style:                  null !default;\n$form-text-font-weight:                 null !default;\n$form-text-color:                       var(--#{$prefix}secondary-color) !default;\n// scss-docs-end form-text-variables\n\n// scss-docs-start form-label-variables\n$form-label-margin-bottom:              .5rem !default;\n$form-label-font-size:                  null !default;\n$form-label-font-style:                 null !default;\n$form-label-font-weight:                null !default;\n$form-label-color:                      null !default;\n// scss-docs-end form-label-variables\n\n// scss-docs-start form-input-variables\n$input-padding-y:                       $input-btn-padding-y !default;\n$input-padding-x:                       $input-btn-padding-x !default;\n$input-font-family:                     $input-btn-font-family !default;\n$input-font-size:                       $input-btn-font-size !default;\n$input-font-weight:                     $font-weight-base !default;\n$input-line-height:                     $input-btn-line-height !default;\n\n$input-padding-y-sm:                    $input-btn-padding-y-sm !default;\n$input-padding-x-sm:                    $input-btn-padding-x-sm !default;\n$input-font-size-sm:                    $input-btn-font-size-sm !default;\n\n$input-padding-y-lg:                    $input-btn-padding-y-lg !default;\n$input-padding-x-lg:                    $input-btn-padding-x-lg !default;\n$input-font-size-lg:                    $input-btn-font-size-lg !default;\n\n$input-bg:                              var(--#{$prefix}body-bg) !default;\n$input-disabled-color:                  null !default;\n$input-disabled-bg:                     var(--#{$prefix}secondary-bg) !default;\n$input-disabled-border-color:           null !default;\n\n$input-color:                           var(--#{$prefix}body-color) !default;\n$input-border-color:                    var(--#{$prefix}border-color) !default;\n$input-border-width:                    $input-btn-border-width !default;\n$input-box-shadow:                      var(--#{$prefix}box-shadow-inset) !default;\n\n$input-border-radius:                   var(--#{$prefix}border-radius) !default;\n$input-border-radius-sm:                var(--#{$prefix}border-radius-sm) !default;\n$input-border-radius-lg:                var(--#{$prefix}border-radius-lg) !default;\n\n$input-focus-bg:                        $input-bg !default;\n$input-focus-border-color:              tint-color($component-active-bg, 50%) !default;\n$input-focus-color:                     $input-color !default;\n$input-focus-width:                     $input-btn-focus-width !default;\n$input-focus-box-shadow:                $input-btn-focus-box-shadow !default;\n\n$input-placeholder-color:               var(--#{$prefix}secondary-color) !default;\n$input-plaintext-color:                 var(--#{$prefix}body-color) !default;\n\n$input-height-border:                   calc(#{$input-border-width} * 2) !default; // stylelint-disable-line function-disallowed-list\n\n$input-height-inner:                    add($input-line-height * 1em, $input-padding-y * 2) !default;\n$input-height-inner-half:               add($input-line-height * .5em, $input-padding-y) !default;\n$input-height-inner-quarter:            add($input-line-height * .25em, $input-padding-y * .5) !default;\n\n$input-height:                          add($input-line-height * 1em, add($input-padding-y * 2, $input-height-border, false)) !default;\n$input-height-sm:                       add($input-line-height * 1em, add($input-padding-y-sm * 2, $input-height-border, false)) !default;\n$input-height-lg:                       add($input-line-height * 1em, add($input-padding-y-lg * 2, $input-height-border, false)) !default;\n\n$input-transition:                      border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$form-color-width:                      3rem !default;\n// scss-docs-end form-input-variables\n\n// scss-docs-start form-check-variables\n$form-check-input-width:                  1em !default;\n$form-check-min-height:                   $font-size-base * $line-height-base !default;\n$form-check-padding-start:                $form-check-input-width + .5em !default;\n$form-check-margin-bottom:                .125rem !default;\n$form-check-label-color:                  null !default;\n$form-check-label-cursor:                 null !default;\n$form-check-transition:                   null !default;\n\n$form-check-input-active-filter:          brightness(90%) !default;\n\n$form-check-input-bg:                     $input-bg !default;\n$form-check-input-border:                 var(--#{$prefix}border-width) solid var(--#{$prefix}border-color) !default;\n$form-check-input-border-radius:          .25em !default;\n$form-check-radio-border-radius:          50% !default;\n$form-check-input-focus-border:           $input-focus-border-color !default;\n$form-check-input-focus-box-shadow:       $focus-ring-box-shadow !default;\n\n$form-check-input-checked-color:          $component-active-color !default;\n$form-check-input-checked-bg-color:       $component-active-bg !default;\n$form-check-input-checked-border-color:   $form-check-input-checked-bg-color !default;\n$form-check-input-checked-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 20 20'><path fill='none' stroke='#{$form-check-input-checked-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='3' d='m6 10 3 3 6-6'/></svg>\") !default;\n$form-check-radio-checked-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='2' fill='#{$form-check-input-checked-color}'/></svg>\") !default;\n\n$form-check-input-indeterminate-color:          $component-active-color !default;\n$form-check-input-indeterminate-bg-color:       $component-active-bg !default;\n$form-check-input-indeterminate-border-color:   $form-check-input-indeterminate-bg-color !default;\n$form-check-input-indeterminate-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 20 20'><path fill='none' stroke='#{$form-check-input-indeterminate-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='3' d='M6 10h8'/></svg>\") !default;\n\n$form-check-input-disabled-opacity:        .5 !default;\n$form-check-label-disabled-opacity:        $form-check-input-disabled-opacity !default;\n$form-check-btn-check-disabled-opacity:    $btn-disabled-opacity !default;\n\n$form-check-inline-margin-end:    1rem !default;\n// scss-docs-end form-check-variables\n\n// scss-docs-start form-switch-variables\n$form-switch-color:               rgba($black, .25) !default;\n$form-switch-width:               2em !default;\n$form-switch-padding-start:       $form-switch-width + .5em !default;\n$form-switch-bg-image:            url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-color}'/></svg>\") !default;\n$form-switch-border-radius:       $form-switch-width !default;\n$form-switch-transition:          background-position .15s ease-in-out !default;\n\n$form-switch-focus-color:         $input-focus-border-color !default;\n$form-switch-focus-bg-image:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-focus-color}'/></svg>\") !default;\n\n$form-switch-checked-color:       $component-active-color !default;\n$form-switch-checked-bg-image:    url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-checked-color}'/></svg>\") !default;\n$form-switch-checked-bg-position: right center !default;\n// scss-docs-end form-switch-variables\n\n// scss-docs-start input-group-variables\n$input-group-addon-padding-y:           $input-padding-y !default;\n$input-group-addon-padding-x:           $input-padding-x !default;\n$input-group-addon-font-weight:         $input-font-weight !default;\n$input-group-addon-color:               $input-color !default;\n$input-group-addon-bg:                  var(--#{$prefix}tertiary-bg) !default;\n$input-group-addon-border-color:        $input-border-color !default;\n// scss-docs-end input-group-variables\n\n// scss-docs-start form-select-variables\n$form-select-padding-y:             $input-padding-y !default;\n$form-select-padding-x:             $input-padding-x !default;\n$form-select-font-family:           $input-font-family !default;\n$form-select-font-size:             $input-font-size !default;\n$form-select-indicator-padding:     $form-select-padding-x * 3 !default; // Extra padding for background-image\n$form-select-font-weight:           $input-font-weight !default;\n$form-select-line-height:           $input-line-height !default;\n$form-select-color:                 $input-color !default;\n$form-select-bg:                    $input-bg !default;\n$form-select-disabled-color:        null !default;\n$form-select-disabled-bg:           $input-disabled-bg !default;\n$form-select-disabled-border-color: $input-disabled-border-color !default;\n$form-select-bg-position:           right $form-select-padding-x center !default;\n$form-select-bg-size:               16px 12px !default; // In pixels because image dimensions\n$form-select-indicator-color:       $gray-800 !default;\n$form-select-indicator:             url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16'><path fill='none' stroke='#{$form-select-indicator-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='2' d='m2 5 6 6 6-6'/></svg>\") !default;\n\n$form-select-feedback-icon-padding-end: $form-select-padding-x * 2.5 + $form-select-indicator-padding !default;\n$form-select-feedback-icon-position:    center right $form-select-indicator-padding !default;\n$form-select-feedback-icon-size:        $input-height-inner-half $input-height-inner-half !default;\n\n$form-select-border-width:        $input-border-width !default;\n$form-select-border-color:        $input-border-color !default;\n$form-select-border-radius:       $input-border-radius !default;\n$form-select-box-shadow:          var(--#{$prefix}box-shadow-inset) !default;\n\n$form-select-focus-border-color:  $input-focus-border-color !default;\n$form-select-focus-width:         $input-focus-width !default;\n$form-select-focus-box-shadow:    0 0 0 $form-select-focus-width $input-btn-focus-color !default;\n\n$form-select-padding-y-sm:        $input-padding-y-sm !default;\n$form-select-padding-x-sm:        $input-padding-x-sm !default;\n$form-select-font-size-sm:        $input-font-size-sm !default;\n$form-select-border-radius-sm:    $input-border-radius-sm !default;\n\n$form-select-padding-y-lg:        $input-padding-y-lg !default;\n$form-select-padding-x-lg:        $input-padding-x-lg !default;\n$form-select-font-size-lg:        $input-font-size-lg !default;\n$form-select-border-radius-lg:    $input-border-radius-lg !default;\n\n$form-select-transition:          $input-transition !default;\n// scss-docs-end form-select-variables\n\n// scss-docs-start form-range-variables\n$form-range-track-width:          100% !default;\n$form-range-track-height:         .5rem !default;\n$form-range-track-cursor:         pointer !default;\n$form-range-track-bg:             var(--#{$prefix}secondary-bg) !default;\n$form-range-track-border-radius:  1rem !default;\n$form-range-track-box-shadow:     var(--#{$prefix}box-shadow-inset) !default;\n\n$form-range-thumb-width:                   1rem !default;\n$form-range-thumb-height:                  $form-range-thumb-width !default;\n$form-range-thumb-bg:                      $component-active-bg !default;\n$form-range-thumb-border:                  0 !default;\n$form-range-thumb-border-radius:           1rem !default;\n$form-range-thumb-box-shadow:              0 .1rem .25rem rgba($black, .1) !default;\n$form-range-thumb-focus-box-shadow:        0 0 0 1px $body-bg, $input-focus-box-shadow !default;\n$form-range-thumb-focus-box-shadow-width:  $input-focus-width !default; // For focus box shadow issue in Edge\n$form-range-thumb-active-bg:               tint-color($component-active-bg, 70%) !default;\n$form-range-thumb-disabled-bg:             var(--#{$prefix}secondary-color) !default;\n$form-range-thumb-transition:              background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n// scss-docs-end form-range-variables\n\n// scss-docs-start form-file-variables\n$form-file-button-color:          $input-color !default;\n$form-file-button-bg:             var(--#{$prefix}tertiary-bg) !default;\n$form-file-button-hover-bg:       var(--#{$prefix}secondary-bg) !default;\n// scss-docs-end form-file-variables\n\n// scss-docs-start form-floating-variables\n$form-floating-height:                  add(3.5rem, $input-height-border) !default;\n$form-floating-line-height:             1.25 !default;\n$form-floating-padding-x:               $input-padding-x !default;\n$form-floating-padding-y:               1rem !default;\n$form-floating-input-padding-t:         1.625rem !default;\n$form-floating-input-padding-b:         .625rem !default;\n$form-floating-label-height:            1.5em !default;\n$form-floating-label-opacity:           .65 !default;\n$form-floating-label-transform:         scale(.85) translateY(-.5rem) translateX(.15rem) !default;\n$form-floating-label-disabled-color:    $gray-600 !default;\n$form-floating-transition:              opacity .1s ease-in-out, transform .1s ease-in-out !default;\n// scss-docs-end form-floating-variables\n\n// Form validation\n\n// scss-docs-start form-feedback-variables\n$form-feedback-margin-top:          $form-text-margin-top !default;\n$form-feedback-font-size:           $form-text-font-size !default;\n$form-feedback-font-style:          $form-text-font-style !default;\n$form-feedback-valid-color:         $success !default;\n$form-feedback-invalid-color:       $danger !default;\n\n$form-feedback-icon-valid-color:    $form-feedback-valid-color !default;\n$form-feedback-icon-valid:          url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 8 8'><path fill='#{$form-feedback-icon-valid-color}' d='M2.3 6.73.6 4.53c-.4-1.04.46-1.4 1.1-.8l1.1 1.4 3.4-3.8c.6-.63 1.6-.27 1.2.7l-4 4.6c-.43.5-.8.4-1.1.1z'/></svg>\") !default;\n$form-feedback-icon-invalid-color:  $form-feedback-invalid-color !default;\n$form-feedback-icon-invalid:        url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 12 12' width='12' height='12' fill='none' stroke='#{$form-feedback-icon-invalid-color}'><circle cx='6' cy='6' r='4.5'/><path stroke-linejoin='round' d='M5.8 3.6h.4L6 6.5z'/><circle cx='6' cy='8.2' r='.6' fill='#{$form-feedback-icon-invalid-color}' stroke='none'/></svg>\") !default;\n// scss-docs-end form-feedback-variables\n\n// scss-docs-start form-validation-colors\n$form-valid-color:                  $form-feedback-valid-color !default;\n$form-valid-border-color:           $form-feedback-valid-color !default;\n$form-invalid-color:                $form-feedback-invalid-color !default;\n$form-invalid-border-color:         $form-feedback-invalid-color !default;\n// scss-docs-end form-validation-colors\n\n// scss-docs-start form-validation-states\n$form-validation-states: (\n  \"valid\": (\n    \"color\": var(--#{$prefix}form-valid-color),\n    \"icon\": $form-feedback-icon-valid,\n    \"tooltip-color\": #fff,\n    \"tooltip-bg-color\": var(--#{$prefix}success),\n    \"focus-box-shadow\": 0 0 $input-btn-focus-blur $input-focus-width rgba(var(--#{$prefix}success-rgb), $input-btn-focus-color-opacity),\n    \"border-color\": var(--#{$prefix}form-valid-border-color),\n  ),\n  \"invalid\": (\n    \"color\": var(--#{$prefix}form-invalid-color),\n    \"icon\": $form-feedback-icon-invalid,\n    \"tooltip-color\": #fff,\n    \"tooltip-bg-color\": var(--#{$prefix}danger),\n    \"focus-box-shadow\": 0 0 $input-btn-focus-blur $input-focus-width rgba(var(--#{$prefix}danger-rgb), $input-btn-focus-color-opacity),\n    \"border-color\": var(--#{$prefix}form-invalid-border-color),\n  )\n) !default;\n// scss-docs-end form-validation-states\n\n// Z-index master list\n//\n// Warning: Avoid customizing these values. They're used for a bird's eye view\n// of components dependent on the z-axis and are designed to all work together.\n\n// scss-docs-start zindex-stack\n$zindex-dropdown:                   1000 !default;\n$zindex-sticky:                     1020 !default;\n$zindex-fixed:                      1030 !default;\n$zindex-offcanvas-backdrop:         1040 !default;\n$zindex-offcanvas:                  1045 !default;\n$zindex-modal-backdrop:             1050 !default;\n$zindex-modal:                      1055 !default;\n$zindex-popover:                    1070 !default;\n$zindex-tooltip:                    1080 !default;\n$zindex-toast:                      1090 !default;\n// scss-docs-end zindex-stack\n\n// scss-docs-start zindex-levels-map\n$zindex-levels: (\n  n1: -1,\n  0: 0,\n  1: 1,\n  2: 2,\n  3: 3\n) !default;\n// scss-docs-end zindex-levels-map\n\n\n// Navs\n\n// scss-docs-start nav-variables\n$nav-link-padding-y:                .5rem !default;\n$nav-link-padding-x:                1rem !default;\n$nav-link-font-size:                null !default;\n$nav-link-font-weight:              null !default;\n$nav-link-color:                    var(--#{$prefix}link-color) !default;\n$nav-link-hover-color:              var(--#{$prefix}link-hover-color) !default;\n$nav-link-transition:               color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out !default;\n$nav-link-disabled-color:           var(--#{$prefix}secondary-color) !default;\n$nav-link-focus-box-shadow:         $focus-ring-box-shadow !default;\n\n$nav-tabs-border-color:             var(--#{$prefix}border-color) !default;\n$nav-tabs-border-width:             var(--#{$prefix}border-width) !default;\n$nav-tabs-border-radius:            var(--#{$prefix}border-radius) !default;\n$nav-tabs-link-hover-border-color:  var(--#{$prefix}secondary-bg) var(--#{$prefix}secondary-bg) $nav-tabs-border-color !default;\n$nav-tabs-link-active-color:        var(--#{$prefix}emphasis-color) !default;\n$nav-tabs-link-active-bg:           var(--#{$prefix}body-bg) !default;\n$nav-tabs-link-active-border-color: var(--#{$prefix}border-color) var(--#{$prefix}border-color) $nav-tabs-link-active-bg !default;\n\n$nav-pills-border-radius:           var(--#{$prefix}border-radius) !default;\n$nav-pills-link-active-color:       $component-active-color !default;\n$nav-pills-link-active-bg:          $component-active-bg !default;\n\n$nav-underline-gap:                 1rem !default;\n$nav-underline-border-width:        .125rem !default;\n$nav-underline-link-active-color:   var(--#{$prefix}emphasis-color) !default;\n// scss-docs-end nav-variables\n\n\n// Navbar\n\n// scss-docs-start navbar-variables\n$navbar-padding-y:                  $spacer * .5 !default;\n$navbar-padding-x:                  null !default;\n\n$navbar-nav-link-padding-x:         .5rem !default;\n\n$navbar-brand-font-size:            $font-size-lg !default;\n// Compute the navbar-brand padding-y so the navbar-brand will have the same height as navbar-text and nav-link\n$nav-link-height:                   $font-size-base * $line-height-base + $nav-link-padding-y * 2 !default;\n$navbar-brand-height:               $navbar-brand-font-size * $line-height-base !default;\n$navbar-brand-padding-y:            ($nav-link-height - $navbar-brand-height) * .5 !default;\n$navbar-brand-margin-end:           1rem !default;\n\n$navbar-toggler-padding-y:          .25rem !default;\n$navbar-toggler-padding-x:          .75rem !default;\n$navbar-toggler-font-size:          $font-size-lg !default;\n$navbar-toggler-border-radius:      $btn-border-radius !default;\n$navbar-toggler-focus-width:        $btn-focus-width !default;\n$navbar-toggler-transition:         box-shadow .15s ease-in-out !default;\n\n$navbar-light-color:                rgba(var(--#{$prefix}emphasis-color-rgb), .65) !default;\n$navbar-light-hover-color:          rgba(var(--#{$prefix}emphasis-color-rgb), .8) !default;\n$navbar-light-active-color:         rgba(var(--#{$prefix}emphasis-color-rgb), 1) !default;\n$navbar-light-disabled-color:       rgba(var(--#{$prefix}emphasis-color-rgb), .3) !default;\n$navbar-light-icon-color:           rgba($body-color, .75) !default;\n$navbar-light-toggler-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'><path stroke='#{$navbar-light-icon-color}' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/></svg>\") !default;\n$navbar-light-toggler-border-color: rgba(var(--#{$prefix}emphasis-color-rgb), .15) !default;\n$navbar-light-brand-color:          $navbar-light-active-color !default;\n$navbar-light-brand-hover-color:    $navbar-light-active-color !default;\n// scss-docs-end navbar-variables\n\n// scss-docs-start navbar-dark-variables\n$navbar-dark-color:                 rgba($white, .55) !default;\n$navbar-dark-hover-color:           rgba($white, .75) !default;\n$navbar-dark-active-color:          $white !default;\n$navbar-dark-disabled-color:        rgba($white, .25) !default;\n$navbar-dark-icon-color:            $navbar-dark-color !default;\n$navbar-dark-toggler-icon-bg:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'><path stroke='#{$navbar-dark-icon-color}' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/></svg>\") !default;\n$navbar-dark-toggler-border-color:  rgba($white, .1) !default;\n$navbar-dark-brand-color:           $navbar-dark-active-color !default;\n$navbar-dark-brand-hover-color:     $navbar-dark-active-color !default;\n// scss-docs-end navbar-dark-variables\n\n\n// Dropdowns\n//\n// Dropdown menu container and contents.\n\n// scss-docs-start dropdown-variables\n$dropdown-min-width:                10rem !default;\n$dropdown-padding-x:                0 !default;\n$dropdown-padding-y:                .5rem !default;\n$dropdown-spacer:                   .125rem !default;\n$dropdown-font-size:                $font-size-base !default;\n$dropdown-color:                    var(--#{$prefix}body-color) !default;\n$dropdown-bg:                       var(--#{$prefix}body-bg) !default;\n$dropdown-border-color:             var(--#{$prefix}border-color-translucent) !default;\n$dropdown-border-radius:            var(--#{$prefix}border-radius) !default;\n$dropdown-border-width:             var(--#{$prefix}border-width) !default;\n$dropdown-inner-border-radius:      calc(#{$dropdown-border-radius} - #{$dropdown-border-width}) !default; // stylelint-disable-line function-disallowed-list\n$dropdown-divider-bg:               $dropdown-border-color !default;\n$dropdown-divider-margin-y:         $spacer * .5 !default;\n$dropdown-box-shadow:               var(--#{$prefix}box-shadow) !default;\n\n$dropdown-link-color:               var(--#{$prefix}body-color) !default;\n$dropdown-link-hover-color:         $dropdown-link-color !default;\n$dropdown-link-hover-bg:            var(--#{$prefix}tertiary-bg) !default;\n\n$dropdown-link-active-color:        $component-active-color !default;\n$dropdown-link-active-bg:           $component-active-bg !default;\n\n$dropdown-link-disabled-color:      var(--#{$prefix}tertiary-color) !default;\n\n$dropdown-item-padding-y:           $spacer * .25 !default;\n$dropdown-item-padding-x:           $spacer !default;\n\n$dropdown-header-color:             $gray-600 !default;\n$dropdown-header-padding-x:         $dropdown-item-padding-x !default;\n$dropdown-header-padding-y:         $dropdown-padding-y !default;\n// fusv-disable\n$dropdown-header-padding:           $dropdown-header-padding-y $dropdown-header-padding-x !default; // Deprecated in v5.2.0\n// fusv-enable\n// scss-docs-end dropdown-variables\n\n// scss-docs-start dropdown-dark-variables\n$dropdown-dark-color:               $gray-300 !default;\n$dropdown-dark-bg:                  $gray-800 !default;\n$dropdown-dark-border-color:        $dropdown-border-color !default;\n$dropdown-dark-divider-bg:          $dropdown-divider-bg !default;\n$dropdown-dark-box-shadow:          null !default;\n$dropdown-dark-link-color:          $dropdown-dark-color !default;\n$dropdown-dark-link-hover-color:    $white !default;\n$dropdown-dark-link-hover-bg:       rgba($white, .15) !default;\n$dropdown-dark-link-active-color:   $dropdown-link-active-color !default;\n$dropdown-dark-link-active-bg:      $dropdown-link-active-bg !default;\n$dropdown-dark-link-disabled-color: $gray-500 !default;\n$dropdown-dark-header-color:        $gray-500 !default;\n// scss-docs-end dropdown-dark-variables\n\n\n// Pagination\n\n// scss-docs-start pagination-variables\n$pagination-padding-y:              .375rem !default;\n$pagination-padding-x:              .75rem !default;\n$pagination-padding-y-sm:           .25rem !default;\n$pagination-padding-x-sm:           .5rem !default;\n$pagination-padding-y-lg:           .75rem !default;\n$pagination-padding-x-lg:           1.5rem !default;\n\n$pagination-font-size:              $font-size-base !default;\n\n$pagination-color:                  var(--#{$prefix}link-color) !default;\n$pagination-bg:                     var(--#{$prefix}body-bg) !default;\n$pagination-border-radius:          var(--#{$prefix}border-radius) !default;\n$pagination-border-width:           var(--#{$prefix}border-width) !default;\n$pagination-margin-start:           calc(#{$pagination-border-width} * -1) !default; // stylelint-disable-line function-disallowed-list\n$pagination-border-color:           var(--#{$prefix}border-color) !default;\n\n$pagination-focus-color:            var(--#{$prefix}link-hover-color) !default;\n$pagination-focus-bg:               var(--#{$prefix}secondary-bg) !default;\n$pagination-focus-box-shadow:       $focus-ring-box-shadow !default;\n$pagination-focus-outline:          0 !default;\n\n$pagination-hover-color:            var(--#{$prefix}link-hover-color) !default;\n$pagination-hover-bg:               var(--#{$prefix}tertiary-bg) !default;\n$pagination-hover-border-color:     var(--#{$prefix}border-color) !default; // Todo in v6: remove this?\n\n$pagination-active-color:           $component-active-color !default;\n$pagination-active-bg:              $component-active-bg !default;\n$pagination-active-border-color:    $component-active-bg !default;\n\n$pagination-disabled-color:         var(--#{$prefix}secondary-color) !default;\n$pagination-disabled-bg:            var(--#{$prefix}secondary-bg) !default;\n$pagination-disabled-border-color:  var(--#{$prefix}border-color) !default;\n\n$pagination-transition:              color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$pagination-border-radius-sm:       var(--#{$prefix}border-radius-sm) !default;\n$pagination-border-radius-lg:       var(--#{$prefix}border-radius-lg) !default;\n// scss-docs-end pagination-variables\n\n\n// Placeholders\n\n// scss-docs-start placeholders\n$placeholder-opacity-max:           .5 !default;\n$placeholder-opacity-min:           .2 !default;\n// scss-docs-end placeholders\n\n// Cards\n\n// scss-docs-start card-variables\n$card-spacer-y:                     $spacer !default;\n$card-spacer-x:                     $spacer !default;\n$card-title-spacer-y:               $spacer * .5 !default;\n$card-title-color:                  null !default;\n$card-subtitle-color:               null !default;\n$card-border-width:                 var(--#{$prefix}border-width) !default;\n$card-border-color:                 var(--#{$prefix}border-color-translucent) !default;\n$card-border-radius:                var(--#{$prefix}border-radius) !default;\n$card-box-shadow:                   null !default;\n$card-inner-border-radius:          subtract($card-border-radius, $card-border-width) !default;\n$card-cap-padding-y:                $card-spacer-y * .5 !default;\n$card-cap-padding-x:                $card-spacer-x !default;\n$card-cap-bg:                       rgba(var(--#{$prefix}body-color-rgb), .03) !default;\n$card-cap-color:                    null !default;\n$card-height:                       null !default;\n$card-color:                        null !default;\n$card-bg:                           var(--#{$prefix}body-bg) !default;\n$card-img-overlay-padding:          $spacer !default;\n$card-group-margin:                 $grid-gutter-width * .5 !default;\n// scss-docs-end card-variables\n\n// Accordion\n\n// scss-docs-start accordion-variables\n$accordion-padding-y:                     1rem !default;\n$accordion-padding-x:                     1.25rem !default;\n$accordion-color:                         var(--#{$prefix}body-color) !default;\n$accordion-bg:                            var(--#{$prefix}body-bg) !default;\n$accordion-border-width:                  var(--#{$prefix}border-width) !default;\n$accordion-border-color:                  var(--#{$prefix}border-color) !default;\n$accordion-border-radius:                 var(--#{$prefix}border-radius) !default;\n$accordion-inner-border-radius:           subtract($accordion-border-radius, $accordion-border-width) !default;\n\n$accordion-body-padding-y:                $accordion-padding-y !default;\n$accordion-body-padding-x:                $accordion-padding-x !default;\n\n$accordion-button-padding-y:              $accordion-padding-y !default;\n$accordion-button-padding-x:              $accordion-padding-x !default;\n$accordion-button-color:                  var(--#{$prefix}body-color) !default;\n$accordion-button-bg:                     var(--#{$prefix}accordion-bg) !default;\n$accordion-transition:                    $btn-transition, border-radius .15s ease !default;\n$accordion-button-active-bg:              var(--#{$prefix}primary-bg-subtle) !default;\n$accordion-button-active-color:           var(--#{$prefix}primary-text-emphasis) !default;\n\n// fusv-disable\n$accordion-button-focus-border-color:     $input-focus-border-color !default; // Deprecated in v5.3.3\n// fusv-enable\n$accordion-button-focus-box-shadow:       $btn-focus-box-shadow !default;\n\n$accordion-icon-width:                    1.25rem !default;\n$accordion-icon-color:                    $body-color !default;\n$accordion-icon-active-color:             $primary-text-emphasis !default;\n$accordion-icon-transition:               transform .2s ease-in-out !default;\n$accordion-icon-transform:                rotate(-180deg) !default;\n\n$accordion-button-icon:         url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='none' stroke='#{$accordion-icon-color}' stroke-linecap='round' stroke-linejoin='round'><path d='M2 5L8 11L14 5'/></svg>\") !default;\n$accordion-button-active-icon:  url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='none' stroke='#{$accordion-icon-active-color}' stroke-linecap='round' stroke-linejoin='round'><path d='M2 5L8 11L14 5'/></svg>\") !default;\n// scss-docs-end accordion-variables\n\n// Tooltips\n\n// scss-docs-start tooltip-variables\n$tooltip-font-size:                 $font-size-sm !default;\n$tooltip-max-width:                 200px !default;\n$tooltip-color:                     var(--#{$prefix}body-bg) !default;\n$tooltip-bg:                        var(--#{$prefix}emphasis-color) !default;\n$tooltip-border-radius:             var(--#{$prefix}border-radius) !default;\n$tooltip-opacity:                   .9 !default;\n$tooltip-padding-y:                 $spacer * .25 !default;\n$tooltip-padding-x:                 $spacer * .5 !default;\n$tooltip-margin:                    null !default; // TODO: remove this in v6\n\n$tooltip-arrow-width:               .8rem !default;\n$tooltip-arrow-height:              .4rem !default;\n// fusv-disable\n$tooltip-arrow-color:               null !default; // Deprecated in Bootstrap 5.2.0 for CSS variables\n// fusv-enable\n// scss-docs-end tooltip-variables\n\n// Form tooltips must come after regular tooltips\n// scss-docs-start tooltip-feedback-variables\n$form-feedback-tooltip-padding-y:     $tooltip-padding-y !default;\n$form-feedback-tooltip-padding-x:     $tooltip-padding-x !default;\n$form-feedback-tooltip-font-size:     $tooltip-font-size !default;\n$form-feedback-tooltip-line-height:   null !default;\n$form-feedback-tooltip-opacity:       $tooltip-opacity !default;\n$form-feedback-tooltip-border-radius: $tooltip-border-radius !default;\n// scss-docs-end tooltip-feedback-variables\n\n\n// Popovers\n\n// scss-docs-start popover-variables\n$popover-font-size:                 $font-size-sm !default;\n$popover-bg:                        var(--#{$prefix}body-bg) !default;\n$popover-max-width:                 276px !default;\n$popover-border-width:              var(--#{$prefix}border-width) !default;\n$popover-border-color:              var(--#{$prefix}border-color-translucent) !default;\n$popover-border-radius:             var(--#{$prefix}border-radius-lg) !default;\n$popover-inner-border-radius:       calc(#{$popover-border-radius} - #{$popover-border-width}) !default; // stylelint-disable-line function-disallowed-list\n$popover-box-shadow:                var(--#{$prefix}box-shadow) !default;\n\n$popover-header-font-size:          $font-size-base !default;\n$popover-header-bg:                 var(--#{$prefix}secondary-bg) !default;\n$popover-header-color:              $headings-color !default;\n$popover-header-padding-y:          .5rem !default;\n$popover-header-padding-x:          $spacer !default;\n\n$popover-body-color:                var(--#{$prefix}body-color) !default;\n$popover-body-padding-y:            $spacer !default;\n$popover-body-padding-x:            $spacer !default;\n\n$popover-arrow-width:               1rem !default;\n$popover-arrow-height:              .5rem !default;\n// scss-docs-end popover-variables\n\n// fusv-disable\n// Deprecated in Bootstrap 5.2.0 for CSS variables\n$popover-arrow-color:               $popover-bg !default;\n$popover-arrow-outer-color:         var(--#{$prefix}border-color-translucent) !default;\n// fusv-enable\n\n\n// Toasts\n\n// scss-docs-start toast-variables\n$toast-max-width:                   350px !default;\n$toast-padding-x:                   .75rem !default;\n$toast-padding-y:                   .5rem !default;\n$toast-font-size:                   .875rem !default;\n$toast-color:                       null !default;\n$toast-background-color:            rgba(var(--#{$prefix}body-bg-rgb), .85) !default;\n$toast-border-width:                var(--#{$prefix}border-width) !default;\n$toast-border-color:                var(--#{$prefix}border-color-translucent) !default;\n$toast-border-radius:               var(--#{$prefix}border-radius) !default;\n$toast-box-shadow:                  var(--#{$prefix}box-shadow) !default;\n$toast-spacing:                     $container-padding-x !default;\n\n$toast-header-color:                var(--#{$prefix}secondary-color) !default;\n$toast-header-background-color:     rgba(var(--#{$prefix}body-bg-rgb), .85) !default;\n$toast-header-border-color:         $toast-border-color !default;\n// scss-docs-end toast-variables\n\n\n// Badges\n\n// scss-docs-start badge-variables\n$badge-font-size:                   .75em !default;\n$badge-font-weight:                 $font-weight-bold !default;\n$badge-color:                       $white !default;\n$badge-padding-y:                   .35em !default;\n$badge-padding-x:                   .65em !default;\n$badge-border-radius:               var(--#{$prefix}border-radius) !default;\n// scss-docs-end badge-variables\n\n\n// Modals\n\n// scss-docs-start modal-variables\n$modal-inner-padding:               $spacer !default;\n\n$modal-footer-margin-between:       .5rem !default;\n\n$modal-dialog-margin:               .5rem !default;\n$modal-dialog-margin-y-sm-up:       1.75rem !default;\n\n$modal-title-line-height:           $line-height-base !default;\n\n$modal-content-color:               null !default;\n$modal-content-bg:                  var(--#{$prefix}body-bg) !default;\n$modal-content-border-color:        var(--#{$prefix}border-color-translucent) !default;\n$modal-content-border-width:        var(--#{$prefix}border-width) !default;\n$modal-content-border-radius:       var(--#{$prefix}border-radius-lg) !default;\n$modal-content-inner-border-radius: subtract($modal-content-border-radius, $modal-content-border-width) !default;\n$modal-content-box-shadow-xs:       var(--#{$prefix}box-shadow-sm) !default;\n$modal-content-box-shadow-sm-up:    var(--#{$prefix}box-shadow) !default;\n\n$modal-backdrop-bg:                 $black !default;\n$modal-backdrop-opacity:            .5 !default;\n\n$modal-header-border-color:         var(--#{$prefix}border-color) !default;\n$modal-header-border-width:         $modal-content-border-width !default;\n$modal-header-padding-y:            $modal-inner-padding !default;\n$modal-header-padding-x:            $modal-inner-padding !default;\n$modal-header-padding:              $modal-header-padding-y $modal-header-padding-x !default; // Keep this for backwards compatibility\n\n$modal-footer-bg:                   null !default;\n$modal-footer-border-color:         $modal-header-border-color !default;\n$modal-footer-border-width:         $modal-header-border-width !default;\n\n$modal-sm:                          300px !default;\n$modal-md:                          500px !default;\n$modal-lg:                          800px !default;\n$modal-xl:                          1140px !default;\n\n$modal-fade-transform:              translate(0, -50px) !default;\n$modal-show-transform:              none !default;\n$modal-transition:                  transform .3s ease-out !default;\n$modal-scale-transform:             scale(1.02) !default;\n// scss-docs-end modal-variables\n\n\n// Alerts\n//\n// Define alert colors, border radius, and padding.\n\n// scss-docs-start alert-variables\n$alert-padding-y:               $spacer !default;\n$alert-padding-x:               $spacer !default;\n$alert-margin-bottom:           1rem !default;\n$alert-border-radius:           var(--#{$prefix}border-radius) !default;\n$alert-link-font-weight:        $font-weight-bold !default;\n$alert-border-width:            var(--#{$prefix}border-width) !default;\n$alert-dismissible-padding-r:   $alert-padding-x * 3 !default; // 3x covers width of x plus default padding on either side\n// scss-docs-end alert-variables\n\n// fusv-disable\n$alert-bg-scale:                -80% !default; // Deprecated in v5.2.0, to be removed in v6\n$alert-border-scale:            -70% !default; // Deprecated in v5.2.0, to be removed in v6\n$alert-color-scale:             40% !default; // Deprecated in v5.2.0, to be removed in v6\n// fusv-enable\n\n// Progress bars\n\n// scss-docs-start progress-variables\n$progress-height:                   1rem !default;\n$progress-font-size:                $font-size-base * .75 !default;\n$progress-bg:                       var(--#{$prefix}secondary-bg) !default;\n$progress-border-radius:            var(--#{$prefix}border-radius) !default;\n$progress-box-shadow:               var(--#{$prefix}box-shadow-inset) !default;\n$progress-bar-color:                $white !default;\n$progress-bar-bg:                   $primary !default;\n$progress-bar-animation-timing:     1s linear infinite !default;\n$progress-bar-transition:           width .6s ease !default;\n// scss-docs-end progress-variables\n\n\n// List group\n\n// scss-docs-start list-group-variables\n$list-group-color:                  var(--#{$prefix}body-color) !default;\n$list-group-bg:                     var(--#{$prefix}body-bg) !default;\n$list-group-border-color:           var(--#{$prefix}border-color) !default;\n$list-group-border-width:           var(--#{$prefix}border-width) !default;\n$list-group-border-radius:          var(--#{$prefix}border-radius) !default;\n\n$list-group-item-padding-y:         $spacer * .5 !default;\n$list-group-item-padding-x:         $spacer !default;\n// fusv-disable\n$list-group-item-bg-scale:          -80% !default; // Deprecated in v5.3.0\n$list-group-item-color-scale:       40% !default; // Deprecated in v5.3.0\n// fusv-enable\n\n$list-group-hover-bg:               var(--#{$prefix}tertiary-bg) !default;\n$list-group-active-color:           $component-active-color !default;\n$list-group-active-bg:              $component-active-bg !default;\n$list-group-active-border-color:    $list-group-active-bg !default;\n\n$list-group-disabled-color:         var(--#{$prefix}secondary-color) !default;\n$list-group-disabled-bg:            $list-group-bg !default;\n\n$list-group-action-color:           var(--#{$prefix}secondary-color) !default;\n$list-group-action-hover-color:     var(--#{$prefix}emphasis-color) !default;\n\n$list-group-action-active-color:    var(--#{$prefix}body-color) !default;\n$list-group-action-active-bg:       var(--#{$prefix}secondary-bg) !default;\n// scss-docs-end list-group-variables\n\n\n// Image thumbnails\n\n// scss-docs-start thumbnail-variables\n$thumbnail-padding:                 .25rem !default;\n$thumbnail-bg:                      var(--#{$prefix}body-bg) !default;\n$thumbnail-border-width:            var(--#{$prefix}border-width) !default;\n$thumbnail-border-color:            var(--#{$prefix}border-color) !default;\n$thumbnail-border-radius:           var(--#{$prefix}border-radius) !default;\n$thumbnail-box-shadow:              var(--#{$prefix}box-shadow-sm) !default;\n// scss-docs-end thumbnail-variables\n\n\n// Figures\n\n// scss-docs-start figure-variables\n$figure-caption-font-size:          $small-font-size !default;\n$figure-caption-color:              var(--#{$prefix}secondary-color) !default;\n// scss-docs-end figure-variables\n\n\n// Breadcrumbs\n\n// scss-docs-start breadcrumb-variables\n$breadcrumb-font-size:              null !default;\n$breadcrumb-padding-y:              0 !default;\n$breadcrumb-padding-x:              0 !default;\n$breadcrumb-item-padding-x:         .5rem !default;\n$breadcrumb-margin-bottom:          1rem !default;\n$breadcrumb-bg:                     null !default;\n$breadcrumb-divider-color:          var(--#{$prefix}secondary-color) !default;\n$breadcrumb-active-color:           var(--#{$prefix}secondary-color) !default;\n$breadcrumb-divider:                quote(\"/\") !default;\n$breadcrumb-divider-flipped:        $breadcrumb-divider !default;\n$breadcrumb-border-radius:          null !default;\n// scss-docs-end breadcrumb-variables\n\n// Carousel\n\n// scss-docs-start carousel-variables\n$carousel-control-color:             $white !default;\n$carousel-control-width:             15% !default;\n$carousel-control-opacity:           .5 !default;\n$carousel-control-hover-opacity:     .9 !default;\n$carousel-control-transition:        opacity .15s ease !default;\n\n$carousel-indicator-width:           30px !default;\n$carousel-indicator-height:          3px !default;\n$carousel-indicator-hit-area-height: 10px !default;\n$carousel-indicator-spacer:          3px !default;\n$carousel-indicator-opacity:         .5 !default;\n$carousel-indicator-active-bg:       $white !default;\n$carousel-indicator-active-opacity:  1 !default;\n$carousel-indicator-transition:      opacity .6s ease !default;\n\n$carousel-caption-width:             70% !default;\n$carousel-caption-color:             $white !default;\n$carousel-caption-padding-y:         1.25rem !default;\n$carousel-caption-spacer:            1.25rem !default;\n\n$carousel-control-icon-width:        2rem !default;\n\n$carousel-control-prev-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$carousel-control-color}'><path d='M11.354 1.646a.5.5 0 0 1 0 .708L5.707 8l5.647 5.646a.5.5 0 0 1-.708.708l-6-6a.5.5 0 0 1 0-.708l6-6a.5.5 0 0 1 .708 0z'/></svg>\") !default;\n$carousel-control-next-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$carousel-control-color}'><path d='M4.646 1.646a.5.5 0 0 1 .708 0l6 6a.5.5 0 0 1 0 .708l-6 6a.5.5 0 0 1-.708-.708L10.293 8 4.646 2.354a.5.5 0 0 1 0-.708z'/></svg>\") !default;\n\n$carousel-transition-duration:       .6s !default;\n$carousel-transition:                transform $carousel-transition-duration ease-in-out !default; // Define transform transition first if using multiple transitions (e.g., `transform 2s ease, opacity .5s ease-out`)\n// scss-docs-end carousel-variables\n\n// scss-docs-start carousel-dark-variables\n$carousel-dark-indicator-active-bg:  $black !default;\n$carousel-dark-caption-color:        $black !default;\n$carousel-dark-control-icon-filter:  invert(1) grayscale(100) !default;\n// scss-docs-end carousel-dark-variables\n\n\n// Spinners\n\n// scss-docs-start spinner-variables\n$spinner-width:           2rem !default;\n$spinner-height:          $spinner-width !default;\n$spinner-vertical-align:  -.125em !default;\n$spinner-border-width:    .25em !default;\n$spinner-animation-speed: .75s !default;\n\n$spinner-width-sm:        1rem !default;\n$spinner-height-sm:       $spinner-width-sm !default;\n$spinner-border-width-sm: .2em !default;\n// scss-docs-end spinner-variables\n\n\n// Close\n\n// scss-docs-start close-variables\n$btn-close-width:            1em !default;\n$btn-close-height:           $btn-close-width !default;\n$btn-close-padding-x:        .25em !default;\n$btn-close-padding-y:        $btn-close-padding-x !default;\n$btn-close-color:            $black !default;\n$btn-close-bg:               url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$btn-close-color}'><path d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/></svg>\") !default;\n$btn-close-focus-shadow:     $focus-ring-box-shadow !default;\n$btn-close-opacity:          .5 !default;\n$btn-close-hover-opacity:    .75 !default;\n$btn-close-focus-opacity:    1 !default;\n$btn-close-disabled-opacity: .25 !default;\n$btn-close-white-filter:     invert(1) grayscale(100%) brightness(200%) !default;\n// scss-docs-end close-variables\n\n\n// Offcanvas\n\n// scss-docs-start offcanvas-variables\n$offcanvas-padding-y:               $modal-inner-padding !default;\n$offcanvas-padding-x:               $modal-inner-padding !default;\n$offcanvas-horizontal-width:        400px !default;\n$offcanvas-vertical-height:         30vh !default;\n$offcanvas-transition-duration:     .3s !default;\n$offcanvas-border-color:            $modal-content-border-color !default;\n$offcanvas-border-width:            $modal-content-border-width !default;\n$offcanvas-title-line-height:       $modal-title-line-height !default;\n$offcanvas-bg-color:                var(--#{$prefix}body-bg) !default;\n$offcanvas-color:                   var(--#{$prefix}body-color) !default;\n$offcanvas-box-shadow:              $modal-content-box-shadow-xs !default;\n$offcanvas-backdrop-bg:             $modal-backdrop-bg !default;\n$offcanvas-backdrop-opacity:        $modal-backdrop-opacity !default;\n// scss-docs-end offcanvas-variables\n\n// Code\n\n$code-font-size:                    $small-font-size !default;\n$code-color:                        $pink !default;\n\n$kbd-padding-y:                     .1875rem !default;\n$kbd-padding-x:                     .375rem !default;\n$kbd-font-size:                     $code-font-size !default;\n$kbd-color:                         var(--#{$prefix}body-bg) !default;\n$kbd-bg:                            var(--#{$prefix}body-color) !default;\n$nested-kbd-font-weight:            null !default; // Deprecated in v5.2.0, removing in v6\n\n$pre-color:                         null !default;\n\n@import \"variables-dark\"; // TODO: can be removed safely in v6, only here to avoid breaking changes in v5.3\n","// Override bootstrap variables\n$spacer: 1rem;\n$container-max-widths: (\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1400px,\n);\n$grid-breakpoints: (\n  xs: 0,\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1200px,\n);\n$dropdown-link-hover-bg: var(--pst-color-surface);\n\n// --pst-color-surface can also be assigned to the dark variant because it is\n// scoped to different values depending on light/dark theme\n$dropdown-dark-link-hover-bg: var(--pst-color-surface);\n$dropdown-link-active-bg: var(--pst-color-surface);\n$dropdown-dark-link-active-bg: var(--pst-color-surface);\n$focus-ring-width: 0.1875rem; // 3px at 100% zoom (0.1875 * 16px base font = 3px)\n$focus-ring-opacity: 1;\n$focus-ring-color: var(--pst-color-accent);\n$focus-ring-blur: 0;\n$focus-ring-box-shadow: 0 0 $focus-ring-blur $focus-ring-width $focus-ring-color;\n\n// outline creates the same style of focus ring, it just uses CSS outline instead of box shadow\n$focus-ring-outline: $focus-ring-color solid $focus-ring-width;\n$btn-focus-box-shadow: $focus-ring-box-shadow;\n","/***\n * Rules for the UX/UI of skip navigation link btn.\n * It's only visible to people\n * navigating with keyboard for accessibility purposes\n * ref: https://www.youtube.com/watch?v=VUR0I5mqq7I\n ***/\n\n.skip-link {\n  position: fixed;\n  top: 0;\n  left: 0;\n  right: 0;\n  text-align: center;\n  background-color: var(--pst-color-warning);\n  padding: 0.5rem;\n  z-index: $zindex-modal;\n  border-bottom: 1px solid var(--pst-color-border);\n\n  // This shows / hides the button\n  transform: translateY(-100%);\n  transition: transform 150ms ease-in-out;\n\n  &:focus-within {\n    transform: translateY(0%);\n  }\n\n  a {\n    // Ensure we are using a WCAG conformant colour\n    color: var(--pst-color-warning-text) !important;\n\n    &:focus-visible {\n      // use color with sufficient contrast\n      outline-color: $foundation-black;\n    }\n  }\n}\n","/*******************************************************************************\n* master color map. Only the colors that actually differ between light and dark\n* themes are specified separately.\n*\n* To see the full list of colors see https://www.figma.com/file/rUrrHGhUBBIAAjQ82x6pz9/PyData-Design-system---proposal-for-implementation-(2)?node-id=1234%3A765&t=ifcFT1JtnrSshGfi-1\n*/\n\n@use \"sass:map\";\n@use \"sass:meta\";\n@use \"sass:string\";\n\n/**\n* Function to get items from nested maps\n*/\n// @param {Map} $map - Map\n// @param {Arglist} $keys - Keys to fetch\n// @return {*}\n@function map-deep-get($map, $keys...) {\n  @each $key in $keys {\n    $map: map.get($map, $key);\n  }\n\n  @return $map;\n}\n\n/* Assign base colors for the PyData theme */\n$color-palette: (\n  // Primary color\n  \"teal\":\n    (\n      \"50\": #f4fbfc,\n      \"100\": #e9f6f8,\n      \"200\": #d0ecf1,\n      \"300\": #abdde6,\n      \"400\": #3fb1c5,\n      \"500\": #0a7d91,\n      \"600\": #085d6c,\n      \"700\": #064752,\n      \"800\": #042c33,\n      \"900\": #021b1f,\n    ),\n  // Secondary color\n  \"violet\":\n    (\n      \"50\": #f4eefb,\n      \"100\": #e0c7ff,\n      \"200\": #d5b4fd,\n      \"300\": #b780ff,\n      \"400\": #9c5ffd,\n      \"500\": #8045e5,\n      \"600\": #6432bd,\n      \"700\": #4b258f,\n      \"800\": #341a61,\n      \"900\": #1e0e39,\n    ),\n  // Neutrals\n  \"gray\":\n    (\n      \"50\": #f9f9fa,\n      \"100\": #f3f4f5,\n      \"200\": #e5e7ea,\n      \"300\": #d1d5da,\n      \"400\": #9ca4af,\n      \"500\": #677384,\n      \"600\": #48566b,\n      \"700\": #29313d,\n      \"800\": #222832,\n      \"900\": #14181e,\n    ),\n  // Accent color\n  \"pink\":\n    (\n      \"50\": #fcf8fd,\n      \"100\": #fcf0fa,\n      \"200\": #f8dff5,\n      \"300\": #f3c7ee,\n      \"400\": #e47fd7,\n      \"500\": #c132af,\n      \"600\": #912583,\n      \"700\": #6e1c64,\n      \"800\": #46123f,\n      \"900\": #2b0b27,\n    ),\n  \"foundation\": (\n    \"white\": #ffffff,\n    // gray-900\n    \"black\": #14181e,\n  )\n);\n\n:root {\n  // Add theme colours to the html root element\n  @each $group-color, $color in $color-palette {\n    @each $color-name, $definition in $color {\n      --pst-#{$group-color}-#{$color-name}: #{$definition};\n    }\n  }\n}\n\n// Static SCSS variables used thoroughout the theme\n// Minimum contrast ratio used for the theme.\n// Acceptable values for WCAG 2.0 are 3, 4.5 and 7.\n// See https://www.w3.org/TR/WCAG20/#visual-audio-contrast-contrast\n// 4.5 - is for text that is 14pt or less\n$min-contrast-ratio-4: 4.5;\n\n// 3 is for text that is 18pt or bold, or for non-text elements\n$min-contrast-ratio-3: 3;\n\n// Customize the light and dark text colors for use in our color contrast function.\n$foundation-black: #14181e;\n$foundation-white: #fff;\n\n// This is a custom - calculated  color between gray 100 and 200 - to reduce\n// the contrast ratio (avoid a jarring effect)\n$base-light-text: #ced6dd;\n\n// used in sphinx_design - gray 100\n$foundation-light-gray: #f3f4f5;\n\n// used in sphinx_design - gray 700\n$foundation-muted-gray: #29313d;\n\n// used in sphinx_design - gray 800\n$foundation-dark-gray: #222832;\n$pst-semantic-colors: (\n  \"primary\": (\n    \"light\": #{map-deep-get($color-palette, \"teal\", \"500\")},\n    \"bg-light\": #{map-deep-get($color-palette, \"teal\", \"200\")},\n    \"dark\": #{map-deep-get($color-palette, \"teal\", \"400\")},\n    \"bg-dark\": #{map-deep-get($color-palette, \"teal\", \"800\")},\n  ),\n  \"secondary\": (\n    \"light\": #{map-deep-get($color-palette, \"violet\", \"500\")},\n    \"bg-light\": #{map-deep-get($color-palette, \"violet\", \"100\")},\n    \"dark\": #{map-deep-get($color-palette, \"violet\", \"400\")},\n    \"bg-dark\": #{map-deep-get($color-palette, \"violet\", \"800\")},\n  ),\n  \"accent\": (\n    \"light\": #{map-deep-get($color-palette, \"pink\", \"500\")},\n    \"bg-light\": #{map-deep-get($color-palette, \"pink\", \"200\")},\n    \"dark\": #{map-deep-get($color-palette, \"pink\", \"400\")},\n    \"bg-dark\": #{map-deep-get($color-palette, \"pink\", \"800\")},\n  ),\n  \"info\": (\n    \"light\": #276be9,\n    \"bg-light\": #dce7fc,\n    \"dark\": #79a3f2,\n    \"bg-dark\": #06245d,\n  ),\n  \"warning\": (\n    \"light\": #f66a0a,\n    \"bg-light\": #f8e3d0,\n    \"dark\": #ff9245,\n    \"bg-dark\": #652a02,\n  ),\n  \"success\": (\n    \"light\": #00843f,\n    \"bg-light\": #d6ece1,\n    \"dark\": #5fb488,\n    \"bg-dark\": #002f17,\n  ),\n  // This is based on the warning color\n  \"attention\":\n    (\n      \"light\": var(--pst-color-warning),\n      \"bg-light\": var(--pst-color-warning-bg),\n      \"dark\": var(--pst-color-warning),\n      \"bg-dark\": var(--pst-color-warning-bg),\n    ),\n  \"danger\": (\n    \"light\": #d72d47,\n    \"bg-light\": #f9e1e4,\n    \"dark\": #e78894,\n    \"bg-dark\": #4e111b,\n  ),\n  \"text-base\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"800\")},\n    \"dark\": $base-light-text,\n  ),\n  \"text-muted\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"600\")},\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"400\")},\n  ),\n  \"heading-color\": (\n    \"light\": #{$foundation-white},\n    \"dark\": #{$foundation-black},\n  ),\n  \"shadow\": (\n    \"light\": rgba(0, 0, 0, 0.1),\n    \"dark\": rgba(0, 0, 0, 0.2),\n  ),\n  \"border\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"300\")},\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"600\")},\n  ),\n  \"border-muted\": (\n    \"light\": rgba(23, 23, 26, 0.2),\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"700\")},\n  ),\n  \"blockquote-notch\": (\n    // These colors have a contrast ratio > 3.0 against both the background and\n    // surface colors that the notch is sandwiched between\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"500\")},\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"400\")},\n  ),\n  \"inline-code\": (\n    \"light\": #{map-deep-get($color-palette, \"pink\", \"600\")},\n    \"dark\": #{map-deep-get($color-palette, \"pink\", \"300\")},\n  ),\n  \"inline-code-links\": (\n    // need to make sure there is enough contrast against the code bg\n    \"light\": #{map-deep-get($color-palette, \"teal\", \"600\")},\n    // keep primary color for dark mode\n    \"dark\": #{map-deep-get($color-palette, \"teal\", \"400\")},\n  ),\n  \"target\": (\n    \"light\": #f3cf95,\n    \"dark\": #675c04,\n  ),\n  \"table\": (\n    \"light\": #{map-deep-get($color-palette, \"foundation\", \"black\")},\n    \"dark\": #{map-deep-get($color-palette, \"foundation\", \"white\")},\n  ),\n  \"table-row-hover\": (\n    \"bg-light\": #{map-deep-get($color-palette, \"violet\", \"300\")},\n    \"bg-dark\": #{map-deep-get($color-palette, \"violet\", \"600\")},\n  ),\n  \"table-inner-border\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"200\")},\n    \"dark\": #364150,\n  ),\n  // DEPTH COLORS - you can see the elevation colours and shades\n  // in the Figma file https://www.figma.com/file/rUrrHGhUBBIAAjQ82x6pz9/PyData-Design-system---proposal-for-implementation-(2)?node-id=1492%3A922&t=sQeQZehkOzposYEg-1\n  // background: color of the canvas / the furthest back layer\n  \"background\":\n    (\n      \"light\": #{map-deep-get($color-palette, \"foundation\", \"white\")},\n      \"dark\": #{map-deep-get($color-palette, \"foundation\", \"black\")},\n    ),\n  // on-background: provides slight contrast against background\n  // (by use of shadows in light theme)\n  \"on-background\":\n    (\n      \"light\": #{map-deep-get($color-palette, \"foundation\", \"white\")},\n      \"dark\": #{map-deep-get($color-palette, \"gray\", \"800\")},\n    ),\n  \"surface\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"100\")},\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"700\")},\n  ),\n  // on_surface: object on top of surface object (without shadows)\n  \"on-surface\":\n    (\n      \"light\": #{map-deep-get($color-palette, \"gray\", \"800\")},\n      \"dark\": $foundation-light-gray,\n    ),\n);\n\n/*******************************************************************************\n* write the color rules for each theme (light/dark)\n*/\n\n/* NOTE:\n * Mixins enable us to reuse the same definitions for the different modes\n * https://sass-lang.com/documentation/at-rules/mixin\n * #{something} inserts a variable into a CSS selector or property name\n * https://sass-lang.com/documentation/interpolation\n */\n@mixin theme-colors($mode) {\n  // check if this color is defined differently for light/dark\n  @each $col-name, $definition in $pst-semantic-colors {\n    @if meta.type-of($definition) == map {\n      @each $key, $val in $definition {\n        @if string.index($key, $mode) {\n          // since now we define the bg colours in the semantic colours and not\n          // by changing opacity, we need to check if the key contains bg and the\n          // correct mode (light/dark)\n          @if string.index($key, \"bg\") {\n            --pst-color-#{$col-name}-bg: #{$val};\n          } @else {\n            --pst-color-#{$col-name}: #{$val};\n          }\n        }\n      }\n    } @else {\n      --pst-color-#{$col-name}: #{$definition};\n    }\n  }\n\n  // assign the \"duplicate\" colors (ones that just reference other variables)\n  & {\n    --pst-color-link: var(--pst-color-primary);\n    --pst-color-link-hover: var(--pst-color-secondary);\n    --pst-color-table-outer-border: var(--pst-color-surface);\n    --pst-color-table-heading-bg: var(--pst-color-surface);\n    --pst-color-table-row-zebra-high-bg: var(--pst-color-on-background);\n    --pst-color-table-row-zebra-low-bg: var(--pst-color-surface);\n  }\n\n  // adapt to light/dark-specific content\n  @if $mode == \"light\" {\n    .only-dark,\n    .only-dark ~ figcaption {\n      display: none !important;\n    }\n  } @else {\n    .only-light,\n    .only-light ~ figcaption {\n      display: none !important;\n    }\n\n    /* Adjust images in dark mode (unless they have class .only-dark or\n     * .dark-light, in which case assume they're already optimized for dark\n     * mode).\n     */\n    img:not(.only-dark, .dark-light) {\n      filter: brightness(0.8) contrast(1.2);\n    }\n\n    /* Give images a light background in dark mode in case they have\n    *  transparency and black text (unless they have class .only-dark or .dark-light, in\n    *  which case assume they're already optimized for dark mode).\n    */\n    .bd-content img:not(.only-dark, .dark-light) {\n      background-color: rgb(255 255 255);\n      border-radius: 0.25rem;\n    }\n\n    // MathJax SVG outputs should be filled to same color as text.\n    .MathJax_SVG * {\n      fill: var(--pst-color-text-base);\n    }\n  }\n}\n\n/* Defaults to light mode if data-theme is not set */\nhtml:not([data-theme]) {\n  @include theme-colors(\"light\");\n}\n\n/* NOTE: @each {...} is like a for-loop\n * https://sass-lang.com/documentation/at-rules/control/each\n */\n@each $mode in (light, dark) {\n  html[data-theme=\"#{$mode}\"] {\n    @include theme-colors($mode);\n\n    color-scheme: $mode;\n  }\n}\n\n// assign classes too, for runtime use of theme colors\n@each $col-name, $definition in $pst-semantic-colors {\n  .pst-color-#{$col-name} {\n    color: var(--pst-color-#{$col-name});\n  }\n}\n",".bd-container {\n  flex-grow: 1;\n  display: flex;\n  justify-content: center;\n\n  .bd-container__inner {\n    display: flex;\n  }\n}\n\n.bd-page-width {\n  width: 100%;\n\n  @include media-breakpoint-up(lg) {\n    max-width: $breakpoint-page-width;\n  }\n}\n","// Breakpoint viewport sizes and media queries.\n//\n// Breakpoints are defined as a map of (name: minimum width), order from small to large:\n//\n//    (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px)\n//\n// The map defined in the `$grid-breakpoints` global variable is used as the `$breakpoints` argument by default.\n\n// Name of the next breakpoint, or null for the last breakpoint.\n//\n//    >> breakpoint-next(sm)\n//    md\n//    >> breakpoint-next(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    md\n//    >> breakpoint-next(sm, $breakpoint-names: (xs sm md lg xl xxl))\n//    md\n@function breakpoint-next($name, $breakpoints: $grid-breakpoints, $breakpoint-names: map-keys($breakpoints)) {\n  $n: index($breakpoint-names, $name);\n  @if not $n {\n    @error \"breakpoint `#{$name}` not found in `#{$breakpoints}`\";\n  }\n  @return if($n < length($breakpoint-names), nth($breakpoint-names, $n + 1), null);\n}\n\n// Minimum breakpoint width. Null for the smallest (first) breakpoint.\n//\n//    >> breakpoint-min(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    576px\n@function breakpoint-min($name, $breakpoints: $grid-breakpoints) {\n  $min: map-get($breakpoints, $name);\n  @return if($min != 0, $min, null);\n}\n\n// Maximum breakpoint width.\n// The maximum value is reduced by 0.02px to work around the limitations of\n// `min-` and `max-` prefixes and viewports with fractional widths.\n// See https://www.w3.org/TR/mediaqueries-4/#mq-min-max\n// Uses 0.02px rather than 0.01px to work around a current rounding bug in Safari.\n// See https://bugs.webkit.org/show_bug.cgi?id=178261\n//\n//    >> breakpoint-max(md, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    767.98px\n@function breakpoint-max($name, $breakpoints: $grid-breakpoints) {\n  $max: map-get($breakpoints, $name);\n  @return if($max and $max > 0, $max - .02, null);\n}\n\n// Returns a blank string if smallest breakpoint, otherwise returns the name with a dash in front.\n// Useful for making responsive utilities.\n//\n//    >> breakpoint-infix(xs, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    \"\"  (Returns a blank string)\n//    >> breakpoint-infix(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    \"-sm\"\n@function breakpoint-infix($name, $breakpoints: $grid-breakpoints) {\n  @return if(breakpoint-min($name, $breakpoints) == null, \"\", \"-#{$name}\");\n}\n\n// Media of at least the minimum breakpoint width. No query for the smallest breakpoint.\n// Makes the @content apply to the given breakpoint and wider.\n@mixin media-breakpoint-up($name, $breakpoints: $grid-breakpoints) {\n  $min: breakpoint-min($name, $breakpoints);\n  @if $min {\n    @media (min-width: $min) {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Media of at most the maximum breakpoint width. No query for the largest breakpoint.\n// Makes the @content apply to the given breakpoint and narrower.\n@mixin media-breakpoint-down($name, $breakpoints: $grid-breakpoints) {\n  $max: breakpoint-max($name, $breakpoints);\n  @if $max {\n    @media (max-width: $max) {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Media that spans multiple breakpoint widths.\n// Makes the @content apply between the min and max breakpoints\n@mixin media-breakpoint-between($lower, $upper, $breakpoints: $grid-breakpoints) {\n  $min: breakpoint-min($lower, $breakpoints);\n  $max: breakpoint-max($upper, $breakpoints);\n\n  @if $min != null and $max != null {\n    @media (min-width: $min) and (max-width: $max) {\n      @content;\n    }\n  } @else if $max == null {\n    @include media-breakpoint-up($lower, $breakpoints) {\n      @content;\n    }\n  } @else if $min == null {\n    @include media-breakpoint-down($upper, $breakpoints) {\n      @content;\n    }\n  }\n}\n\n// Media between the breakpoint's minimum and maximum widths.\n// No minimum for the smallest breakpoint, and no maximum for the largest one.\n// Makes the @content apply only to the given breakpoint, not viewports any wider or narrower.\n@mixin media-breakpoint-only($name, $breakpoints: $grid-breakpoints) {\n  $min:  breakpoint-min($name, $breakpoints);\n  $next: breakpoint-next($name, $breakpoints);\n  $max:  breakpoint-max($next, $breakpoints);\n\n  @if $min != null and $max != null {\n    @media (min-width: $min) and (max-width: $max) {\n      @content;\n    }\n  } @else if $max == null {\n    @include media-breakpoint-up($name, $breakpoints) {\n      @content;\n    }\n  } @else if $min == null {\n    @include media-breakpoint-down($next, $breakpoints) {\n      @content;\n    }\n  }\n}\n",".pst-async-banner-revealer {\n  // Setting height to 0 and overflow to hidden allows us to add up the heights\n  // of this element's children before revealing them.\n  height: 0;\n  overflow: hidden;\n\n  // Height to be set by JavaScript, which should trigger the following\n  // transition rule (unless the user has set their system to reduce motion).\n  transition: height 300ms ease-in-out;\n\n  @media (prefers-reduced-motion) {\n    transition: none;\n  }\n}\n\n#bd-header-version-warning,\n.bd-header-announcement {\n  min-height: 3rem;\n  width: 100%;\n  display: flex;\n  position: relative;\n  align-items: center;\n  justify-content: center;\n  text-align: center;\n  padding: 0.5rem 12.5%; // Horizontal padding so the width is 75%\n  // One breakpoint less than $breakpoint-sidebar-primary. See variables/_layout.scss for more info.\n  @include media-breakpoint-down(lg) {\n    // Announcements can take a bit more width on mobile\n    padding: 0.5rem 2%;\n  }\n\n  p {\n    font-weight: bold;\n    margin: 0;\n  }\n\n  // Ensure there is enough contrast against the background\n  a {\n    color: var(--pst-color-inline-code-links);\n  }\n\n  // The \"Switch to stable version\" link (styled like a button)\n  .pst-button-link-to-stable-version {\n    @include box-shadow;\n\n    $background-color: var(--pst-color-danger);\n    $hover-background-color: var(--pst-color-danger-highlight);\n    $color: var(--pst-color-danger-text);\n\n    background-color: $background-color;\n    border-color: $background-color;\n    color: $color;\n    border-radius: 0.25rem;\n\n    &:hover {\n      background-color: $hover-background-color;\n      border-color: $hover-background-color;\n      color: $color;\n    }\n\n    &:focus-visible {\n      outline: $focus-ring-width solid $background-color;\n      outline-offset: $focus-ring-width;\n    }\n  }\n}\n\n// Bg color is now defined in the theme color palette - using our secondary color\n.bd-header-announcement {\n  background-color: var(--pst-color-secondary-bg);\n}\n\n#bd-header-version-warning {\n  background-color: var(--pst-color-danger-bg);\n}\n","/*********************************************\n* SASS Mixins\n*********************************************/\n\n/**\n * A consistent box shadow style we apply across elements.\n */\n@mixin box-shadow() {\n  box-shadow:\n    0 0.2rem 0.5rem var(--pst-color-shadow),\n    0 0 0.0625rem var(--pst-color-shadow) !important;\n}\n\n/**\n   * Set background of some cell outputs to white-ish to make sure colors work\n   * This is because many libraries make output that only looks good on white\n   */\n@mixin cell-output-background {\n  color: var(--pst-color-on-background);\n  background-color: var(--pst-color-text-base);\n  border-radius: 0.25rem;\n  padding: 0.5rem;\n}\n\n@mixin table-colors {\n  color: var(--pst-color-table);\n  border: 1px solid var(--pst-color-table-outer-border);\n\n  th,\n  td {\n    ~ th,\n    ~ td {\n      border-left: 1px solid var(--pst-color-table-inner-border);\n    }\n  }\n\n  thead {\n    tr {\n      background-color: var(--pst-color-table-heading-bg);\n      border-bottom: 2px solid var(--pst-color-primary);\n    }\n  }\n\n  tbody {\n    tr {\n      &:nth-child(odd) {\n        background-color: var(--pst-color-table-row-zebra-low-bg);\n      }\n\n      &:nth-child(even) {\n        background-color: var(--pst-color-table-row-zebra-high-bg);\n      }\n\n      &:hover {\n        background-color: var(--pst-color-table-row-hover-bg);\n      }\n    }\n  }\n}\n\n// Minimum mouse hit area\n// ----------------------\n// Ensures that the element has a minimum hit area that conforms to\n// accessibility guidelines. For WCAG AA, we need 24px x 24px, see:\n// https://www.w3.org/WAI/WCAG22/Understanding/target-size-minimum.html\n@mixin min-hit-area() {\n  box-sizing: border-box;\n  min-width: 24px;\n  min-height: 24px;\n}\n","/**\n * Main content area\n */\n.bd-main {\n  flex-grow: 1;\n  flex-direction: column;\n  display: flex;\n  min-width: 0;\n\n  .bd-content {\n    display: flex;\n    justify-content: center;\n    height: 100%;\n\n    .bd-article-container {\n      justify-content: start;\n      display: flex;\n      flex-direction: column;\n\n      // Max-width is slightly more than the W3 since our docs often have images.\n      // We shoot for about 100 characters per line instead of 80.\n      // ref: https://www.w3.org/WAI/tutorials/page-structure/styling/#line-length\n      width: 100%;\n      max-width: 60em;\n      overflow-x: auto; // Prevent wide content from pushing off the secondary sidebar\n      padding: 1rem;\n\n      .bd-article {\n        // Give a bit more verticle spacing on wide screens\n        @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n          padding-top: 1.5rem;\n          padding-left: 2rem;\n        }\n      }\n    }\n  }\n}\n",".bd-footer {\n  width: 100%;\n  border-top: 1px solid var(--pst-color-border);\n\n  .bd-footer__inner {\n    display: flex;\n    flex-grow: 1;\n    padding: 1rem;\n    margin: auto;\n  }\n\n  .footer-items__start,\n  .footer-items__center,\n  .footer-items__end {\n    display: flex;\n    flex-direction: column;\n    gap: 0.5rem;\n    justify-content: center;\n    flex-grow: 1;\n  }\n\n  .footer-items__center {\n    text-align: center;\n  }\n\n  .footer-items__end {\n    text-align: end;\n  }\n\n  // So that paragraphs don't take up extra room\n  .footer-item p {\n    margin-bottom: 0;\n  }\n}\n",".bd-footer-article {\n  margin-top: auto;\n\n  .footer-article-items {\n    display: flex;\n    flex-direction: column;\n  }\n}\n",".bd-footer-content {\n  .footer-content-items {\n    display: flex;\n    flex-direction: column;\n    margin-top: auto;\n  }\n}\n","/**\n * Header at the top of the page\n * It includes the announcement bar and the navigation bar.\n */\n\n// Styling for the Icon links can be found in components/_icon-links.scss\n\n// If we want the shadow to only point downward in the future, set\n// box-shadow to: 0 0.125rem 0.25rem -0.125rem rgba(0, 0, 0, 0.11);\n.bd-header {\n  position: sticky;\n  top: 0;\n  z-index: $zindex-fixed;\n\n  // Overrides bootstrap\n  background-color: var(--pst-color-on-background) !important;\n  box-shadow: 0 0.125rem 0.25rem 0 var(--pst-color-shadow);\n  width: 100%;\n  padding: 0;\n  max-width: 100vw;\n  justify-content: center;\n\n  .bd-header__inner {\n    display: flex;\n    align-items: center;\n    height: fit-content;\n    padding-left: 1rem;\n    padding-right: 1rem;\n  }\n\n  :focus-visible {\n    border-radius: $focus-ring-radius;\n  }\n\n  // These items will define the height of the header\n  .navbar-item {\n    height: var(--pst-header-height);\n    max-height: var(--pst-header-height);\n    display: flex;\n    align-items: center;\n  }\n\n  // Hide the navbar header items on mobile because they're in the sidebar\n  .navbar-header-items {\n    display: none;\n    flex-shrink: 1;\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      display: inherit;\n      flex-grow: 1;\n      padding: 0 0 0 0.5rem;\n    }\n  }\n\n  .navbar-header-items__end,\n  .navbar-header-items__center,\n  .navbar-header-items__start {\n    display: flex;\n    align-items: center;\n    flex-flow: wrap;\n\n    // In case we wrap our items to multiple rows on small screens\n    row-gap: 0;\n  }\n\n  .navbar-header-items__end,\n  .navbar-header-items__center {\n    column-gap: 1rem;\n  }\n\n  // A little smaller because this is displayed by default on mobile\n  .navbar-header-items__start {\n    flex-shrink: 0;\n    margin-right: auto;\n    gap: 0.5rem;\n  }\n\n  .navbar-header-items__end {\n    // End navbar items should snap to the right\n    justify-content: end;\n  }\n\n  // Contains the navigation links within the navbar\n  ul.navbar-nav {\n    display: flex;\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      // Align on wide screens so the dropdown button is centered properly\n      align-items: baseline;\n    }\n\n    > li.nav-item {\n      margin-inline: 2px; // breathing room so hover and focus styles do not overlap\n\n      > .nav-link {\n        @include link-style-block;\n\n        padding-inline: 6px;\n      }\n\n      &.current {\n        > .nav-link {\n          color: var(--pst-color-primary);\n\n          // Underline the current navbar item\n          &::before {\n            border-bottom: 3px solid var(--pst-color-primary);\n          }\n        }\n      }\n\n      &.dropdown {\n        margin-inline: 4px;\n\n        button {\n          padding-inline: 8px;\n        }\n\n        > .dropdown-toggle {\n          border-radius: $focus-ring-radius; // make border radius the same for both hover ring and focus ring\n          color: var(--pst-color-text-muted);\n\n          &:focus-visible {\n            box-shadow: $focus-ring-box-shadow;\n          }\n\n          &:hover {\n            text-decoration: none;\n            box-shadow: 0 0 0 $focus-ring-width var(--pst-color-link-hover); // purple focus ring\n            // Brighten the text on hover (muted -> base)\n            color: var(--pst-color-text-base);\n          }\n        }\n      }\n    }\n\n    li a.nav-link.dropdown-item {\n      @include link-style-text;\n    }\n\n    // Dropdowns for the extra links\n    .dropdown {\n      button {\n        display: unset;\n        border: none;\n\n        &:hover {\n          @include link-style-hover;\n        }\n      }\n\n      .dropdown-menu {\n        z-index: $zindex-popover;\n        border: 1px solid var(--pst-color-border);\n        box-shadow: 0 0 0.3rem 0.1rem var(--pst-color-shadow);\n        background-color: var(--pst-color-on-background);\n        padding: 0.5rem 0;\n        margin: 0.5rem 0;\n        min-width: 20rem;\n\n        .dropdown-item {\n          // Give the items in the dropdown some breathing room but let the hit\n          // and hover area of the items extend to the edges of the menu\n          padding: 0.25rem 1.5rem;\n\n          // Override Bootstrap\n          &:focus:not(:hover, :active) {\n            background-color: inherit;\n          }\n\n          &:focus-visible {\n            z-index: 10; // keep focus ring on top (prevent the hover background of the next dropdown item from covering the ring)\n          }\n        }\n\n        // Hide the menu unless show has been clicked\n        &:not(.show) {\n          display: none;\n        }\n      }\n    }\n  }\n\n  // **************************************************************\n  // Showing and hiding the sidebar toggle buttons and header items\n  // **************************************************************\n\n  // Toggle buttons\n  button.sidebar-toggle {\n    font-size: var(--pst-font-size-icon);\n    color: var(--pst-color-muted);\n    margin-bottom: 0;\n    background-color: inherit;\n    padding: 0.5rem;\n  }\n\n  button.primary-toggle {\n    margin-right: 1rem;\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      display: none;\n    }\n  }\n\n  button.secondary-toggle {\n    margin-left: 1rem;\n\n    @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n      display: none;\n    }\n  }\n}\n\n// inline the element in the navbar as long as they fit and use display block when collapsing\n@include media-breakpoint-up($breakpoint-sidebar-primary) {\n  .navbar-center-items .navbar-item {\n    display: inline-block;\n  }\n}\n\n.nav-link {\n  &:hover {\n    @include link-style-hover;\n  }\n\n  // Override Bootstrap\n  transition: none;\n\n  &.nav-external::after {\n    font: var(--fa-font-solid);\n    content: var(--pst-icon-external-link);\n    font-size: 0.75em;\n    margin-left: 0.3em;\n  }\n}\n\n.bd-navbar-elements li.nav-item i {\n  font-size: 0.7rem;\n  padding-left: 2px;\n  vertical-align: middle;\n}\n\n// THe elements next to the hamburger menu only show on narrow screens\n.navbar-persistent--mobile {\n  margin-left: auto;\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    display: none;\n  }\n}\n\n// The navbar-persistent content should only show on wide screens\n.navbar-persistent--container {\n  display: none;\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    display: flex;\n  }\n}\n",".header-article__inner {\n  display: flex;\n  padding: 0 0.5rem;\n\n  // The items define the height so that it disappears if there are no items\n  .header-article-item {\n    min-height: var(--pst-header-article-height);\n    height: var(--pst-header-article-height);\n  }\n\n  .header-article-items__start,\n  .header-article-items__end {\n    display: flex;\n    align-items: start;\n    gap: 0.5rem;\n  }\n\n  .header-article-items__end {\n    margin-left: auto;\n  }\n}\n","/**\n * The primary sidebar on the left.\n * e.g., between-pages navigation.\n */\n\n$sidebar-padding-right: 1rem;\n\n.bd-sidebar-primary {\n  display: flex;\n  flex-direction: column;\n  gap: 1rem;\n  max-height: calc(100vh - var(--pst-header-height));\n  position: sticky;\n  top: var(--pst-header-height);\n\n  @include make-col(3);\n\n  // Borders padding and whitespace\n  padding: 2rem $sidebar-padding-right 1rem 1rem;\n  border-right: 1px solid var(--pst-color-border);\n  background-color: var(--pst-color-background);\n  overflow-y: auto;\n  font-size: var(--pst-sidebar-font-size-mobile);\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    font-size: var(--pst-sidebar-font-size);\n  }\n\n  :focus-visible {\n    border-radius: $focus-ring-radius;\n  }\n\n  // override bootstrap when navlink are displayed in the sidebar\n  .nav-link {\n    font-size: var(--pst-sidebar-font-size-mobile);\n  }\n\n  &.no-sidebar {\n    border-right: 0;\n  }\n\n  &.hide-on-wide {\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      display: none;\n    }\n  }\n\n  // Headers shouldn't be colored in the sidebars and some extensions add headers\n  h1,\n  h2,\n  h3,\n  h4 {\n    color: var(--pst-color-text-base);\n  }\n\n  .sidebar-primary-items__start,\n  .sidebar-primary-items__end {\n    .sidebar-primary-item {\n      padding: 0.5rem 0;\n    }\n  }\n\n  // Hide the sidebar header items on widescreen since they are visible in the header\n  .sidebar-header-items {\n    display: flex;\n    flex-direction: column;\n\n    .sidebar-header-items__title {\n      font-weight: var(--pst-sidebar-header-font-weight);\n      font-size: var(--pst-sidebar-header-font-size);\n      color: var(--pst-color-text-base);\n      margin-bottom: 0.5rem;\n    }\n\n    // The dropdown toggle for extra links just shows them all instead.\n    .nav-item.dropdown {\n      // On mobile, the dropdown behaves like any other link, no hiding\n      button {\n        display: none;\n      }\n\n      .dropdown-menu {\n        display: flex;\n        flex-direction: column;\n        padding: 0;\n        margin: 0;\n        border: none;\n        background-color: inherit;\n        font-size: inherit;\n\n        .dropdown-item {\n          &:hover,\n          &:focus {\n            // In the mobile sidebar, the dropdown menu is inlined with the\n            // other links, which do not have background-color changes on hover\n            // and focus\n            background-color: unset;\n          }\n        }\n      }\n    }\n\n    .bd-navbar-elements {\n      .nav-link {\n        &:focus-visible {\n          box-shadow: none; // Override Bootstrap\n          outline: $focus-ring-outline;\n          outline-offset: $focus-ring-width;\n        }\n      }\n    }\n\n    // Center header items get displayed vertically, end items are displayed horizontally\n    .sidebar-header-items__center {\n      display: flex;\n      flex-direction: column;\n    }\n\n    // Positioning end items\n    .sidebar-header-items__end {\n      display: flex;\n      align-items: center;\n      gap: 1rem;\n    }\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      display: none;\n    }\n  }\n\n  .sidebar-primary-items__start {\n    // Add a border on mobile to separate it from the header sidebar area\n    border-top: 1px solid var(--pst-color-border);\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      border-top: none;\n    }\n  }\n\n  .sidebar-primary-items__end {\n    margin-top: auto;\n    margin-bottom: 1em;\n  }\n\n  .list-caption {\n    list-style: none;\n    padding-left: 0;\n\n    // Level 0 TOC heading is put inside the <summary> tag\n    // so let the <summary> tag take up more space\n    li.toctree-l0.has-children {\n      > details {\n        > summary {\n          position: relative;\n          height: auto;\n          width: auto;\n          display: flex;\n          justify-content: space-between;\n          align-items: baseline;\n\n          .toctree-toggle {\n            // Prevent toggle icon from getting squished by summary being a\n            // flexbox\n            flex: 0 0 auto;\n\n            // Make the level 0 chevron icon slightly bigger than descendant\n            // levels\n            .fa-chevron-down {\n              font-size: 1rem;\n            }\n          }\n        }\n      }\n    }\n  }\n\n  li.has-children {\n    $toctree-toggle-width: 30px;\n\n    position: relative;\n\n    > .reference,\n    .caption {\n      margin-right: calc(\n        $toctree-toggle-width + $focus-ring-width\n      ); // keep clear of the toggle icon\n\n      padding-top: 0.25rem; // align caption text with toggle chevron\n    }\n\n    > details {\n      > summary {\n        // Remove browser default toggle icon\n        list-style: none;\n\n        &::-webkit-details-marker {\n          display: none;\n        }\n\n        // The summary element is natively focusable, but delegate the focus state to the toggle icon\n        &:focus-visible {\n          outline: none;\n\n          > .toctree-toggle {\n            outline: $focus-ring-outline;\n            outline-offset: -$focus-ring-width; // Prevent right side of focus ring from disappearing underneath the sidebar's right edge\n          }\n        }\n\n        // Container for expand/collapse chevron icon\n        .toctree-toggle {\n          cursor: pointer;\n\n          // Position it so that it's aligned with the top right corner of the\n          // last positioned element, in this case the li.has-children\n          position: absolute;\n          top: 0;\n          right: 0;\n\n          // Give it dimensions\n          width: $toctree-toggle-width;\n          height: $toctree-toggle-width; // make it square\n\n          // Vertically and horizontally center the icon within the container\n          display: inline-flex;\n          justify-content: center;\n          align-items: center;\n\n          .fa-chevron-down {\n            font-size: 0.75rem;\n          }\n        }\n      }\n\n      // The section is open/expanded, rotate the toggle icon (chevron) so it\n      // points up instead of down\n      &[open] {\n        > summary {\n          .fa-chevron-down {\n            transform: rotate(180deg);\n          }\n        }\n      }\n    }\n  }\n}\n\n/* Between-page links and captions */\nnav.bd-links {\n  margin-right: -$sidebar-padding-right; // align toctree toggle chevrons with right edge of sidebar and allow text to flow closer to the right edge\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    display: block;\n  }\n\n  ul {\n    display: block;\n    list-style: none;\n\n    // Reduce padding of nested `ul` items a bit\n    ul {\n      padding: 0 0 0 1rem;\n    }\n  }\n\n  li > a {\n    display: block;\n    padding: 0.25rem 0.65rem;\n\n    @include link-sidebar;\n\n    box-shadow: none;\n    margin-right: $focus-ring-width; // prevent the right side focus ring from disappearing under the sidebar right edge\n\n    &.reference.external {\n      &::after {\n        font: var(--fa-font-solid);\n        content: var(--pst-icon-external-link);\n        font-size: 0.75em;\n        margin-left: 0.3em;\n      }\n    }\n  }\n\n  .current > a {\n    @include link-sidebar-current;\n\n    background-color: transparent;\n  }\n\n  // Title\n  p.bd-links__title {\n    font-size: var(--pst-sidebar-header-font-size);\n    font-weight: var(--pst-sidebar-header-font-weight);\n    margin-bottom: 0.5rem;\n  }\n\n  // Toctree captions\n  p.caption {\n    font-weight: var(--pst-sidebar-header-font-weight);\n    position: relative;\n    margin-top: 1.25rem;\n    margin-bottom: 0.5rem;\n    color: var(--pst-color-text-base);\n\n    &:first-child {\n      margin-top: 0;\n    }\n\n    font-size: var(--pst-sidebar-font-size-mobile);\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      font-size: var(--pst-sidebar-font-size);\n    }\n  }\n}\n","// Grid system\n//\n// Generate semantic grid columns with these mixins.\n\n@mixin make-row($gutter: $grid-gutter-width) {\n  --#{$prefix}gutter-x: #{$gutter};\n  --#{$prefix}gutter-y: 0;\n  display: flex;\n  flex-wrap: wrap;\n  // TODO: Revisit calc order after https://github.com/react-bootstrap/react-bootstrap/issues/6039 is fixed\n  margin-top: calc(-1 * var(--#{$prefix}gutter-y)); // stylelint-disable-line function-disallowed-list\n  margin-right: calc(-.5 * var(--#{$prefix}gutter-x)); // stylelint-disable-line function-disallowed-list\n  margin-left: calc(-.5 * var(--#{$prefix}gutter-x)); // stylelint-disable-line function-disallowed-list\n}\n\n@mixin make-col-ready() {\n  // Add box sizing if only the grid is loaded\n  box-sizing: if(variable-exists(include-column-box-sizing) and $include-column-box-sizing, border-box, null);\n  // Prevent columns from becoming too narrow when at smaller grid tiers by\n  // always setting `width: 100%;`. This works because we set the width\n  // later on to override this initial width.\n  flex-shrink: 0;\n  width: 100%;\n  max-width: 100%; // Prevent `.col-auto`, `.col` (& responsive variants) from breaking out the grid\n  padding-right: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  padding-left: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  margin-top: var(--#{$prefix}gutter-y);\n}\n\n@mixin make-col($size: false, $columns: $grid-columns) {\n  @if $size {\n    flex: 0 0 auto;\n    width: percentage(divide($size, $columns));\n\n  } @else {\n    flex: 1 1 0;\n    max-width: 100%;\n  }\n}\n\n@mixin make-col-auto() {\n  flex: 0 0 auto;\n  width: auto;\n}\n\n@mixin make-col-offset($size, $columns: $grid-columns) {\n  $num: divide($size, $columns);\n  margin-left: if($num == 0, 0, percentage($num));\n}\n\n// Row columns\n//\n// Specify on a parent element(e.g., .row) to force immediate children into NN\n// number of columns. Supports wrapping to new lines, but does not do a Masonry\n// style grid.\n@mixin row-cols($count) {\n  > * {\n    flex: 0 0 auto;\n    width: percentage(divide(1, $count));\n  }\n}\n\n// Framework grid generation\n//\n// Used only by Bootstrap to generate the correct number of grid classes given\n// any value of `$grid-columns`.\n\n@mixin make-grid-columns($columns: $grid-columns, $gutter: $grid-gutter-width, $breakpoints: $grid-breakpoints) {\n  @each $breakpoint in map-keys($breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $breakpoints);\n\n    @include media-breakpoint-up($breakpoint, $breakpoints) {\n      // Provide basic `.col-{bp}` classes for equal-width flexbox columns\n      .col#{$infix} {\n        flex: 1 0 0%; // Flexbugs #4: https://github.com/philipwalton/flexbugs#flexbug-4\n      }\n\n      .row-cols#{$infix}-auto > * {\n        @include make-col-auto();\n      }\n\n      @if $grid-row-columns > 0 {\n        @for $i from 1 through $grid-row-columns {\n          .row-cols#{$infix}-#{$i} {\n            @include row-cols($i);\n          }\n        }\n      }\n\n      .col#{$infix}-auto {\n        @include make-col-auto();\n      }\n\n      @if $columns > 0 {\n        @for $i from 1 through $columns {\n          .col#{$infix}-#{$i} {\n            @include make-col($i, $columns);\n          }\n        }\n\n        // `$columns - 1` because offsetting by the width of an entire row isn't possible\n        @for $i from 0 through ($columns - 1) {\n          @if not ($infix == \"\" and $i == 0) { // Avoid emitting useless .offset-0\n            .offset#{$infix}-#{$i} {\n              @include make-col-offset($i, $columns);\n            }\n          }\n        }\n      }\n\n      // Gutters\n      //\n      // Make use of `.g-*`, `.gx-*` or `.gy-*` utilities to change spacing between the columns.\n      @each $key, $value in $gutters {\n        .g#{$infix}-#{$key},\n        .gx#{$infix}-#{$key} {\n          --#{$prefix}gutter-x: #{$value};\n        }\n\n        .g#{$infix}-#{$key},\n        .gy#{$infix}-#{$key} {\n          --#{$prefix}gutter-y: #{$value};\n        }\n      }\n    }\n  }\n}\n\n@mixin make-cssgrid($columns: $grid-columns, $breakpoints: $grid-breakpoints) {\n  @each $breakpoint in map-keys($breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $breakpoints);\n\n    @include media-breakpoint-up($breakpoint, $breakpoints) {\n      @if $columns > 0 {\n        @for $i from 1 through $columns {\n          .g-col#{$infix}-#{$i} {\n            grid-column: auto / span $i;\n          }\n        }\n\n        // Start with `1` because `0` is an invalid value.\n        // Ends with `$columns - 1` because offsetting by the width of an entire row isn't possible.\n        @for $i from 1 through ($columns - 1) {\n          .g-start#{$infix}-#{$i} {\n            grid-column-start: $i;\n          }\n        }\n      }\n    }\n  }\n}\n","/**\n * Secondary sidebar on the right.\n * e.g., in-page table of contents.\n */\n.bd-sidebar-secondary {\n  display: flex;\n  order: 2;\n  flex-shrink: 0;\n  flex-direction: column;\n  position: sticky;\n  top: var(--pst-header-height);\n  max-height: calc(100vh - var(--pst-header-height));\n  padding: 2rem 1rem 1rem;\n  width: var(--pst-sidebar-secondary);\n  font-size: var(--pst-sidebar-font-size-mobile);\n\n  @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n    font-size: var(--pst-sidebar-font-size);\n  }\n\n  // Color and border\n  background-color: var(--pst-color-background);\n  overflow-y: auto;\n}\n\n.sidebar-secondary-item {\n  padding: 0.5rem;\n\n  @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n    border-left: 1px solid var(--pst-color-border);\n    padding-left: 1rem;\n  }\n\n  i {\n    padding-right: 0.5rem;\n  }\n}\n","/*******************************************************************************\n* Rules for the UX/UI of sidebar sliding drawers on mobile\n* Note that this sheet controls styles across many parts of the theme\n* It is aggregated into this one sheet instead of being split across\n* components in order to keep it easier to debug in one place.\n* It is broken up into major sections below.\n*/\n\n/*******************************************************************************\n* Buttons and overlays\n*/\ninput.sidebar-toggle {\n  display: none;\n}\n\n// Background overlays\nlabel.overlay {\n  background-color: black;\n  opacity: 0.5;\n  height: 0;\n  width: 0;\n  position: fixed;\n  top: 0;\n  left: 0;\n  transition: opacity $animation-time ease-out;\n  z-index: $zindex-modal-backdrop;\n}\n\ninput {\n  // Show the correct overlay when its input is checked\n  &#pst-primary-sidebar-checkbox:checked + label.overlay.overlay-primary,\n  &#pst-secondary-sidebar-checkbox:checked + label.overlay.overlay-secondary {\n    height: 100vh;\n    width: 100vw;\n  }\n\n  // Primary sidebar slides in from the left\n  &#pst-primary-sidebar-checkbox:checked ~ .bd-container .bd-sidebar-primary {\n    visibility: visible;\n    margin-left: 0;\n  }\n\n  // Secondary sidebar slides in from the right\n  &#pst-secondary-sidebar-checkbox:checked\n    ~ .bd-container\n    .bd-sidebar-secondary {\n    visibility: visible;\n    margin-right: 0;\n  }\n}\n\n/*******************************************************************************\n* Sidebar drawer behavior\n*/\n\n/**\n * Behavior for sliding drawer elements that will be toggled with an input\n *\n * NOTE: We use this mixin to define the toggle behavior on narrow screens,\n * And the wide-screen behavior of the sections is defined in their own section\n * .scss files.\n */\n@mixin sliding-drawer($side: \"left\") {\n  position: fixed;\n  top: 0;\n  z-index: $zindex-modal;\n  height: 100vh;\n  max-height: 100vh;\n  width: 75%;\n  flex-grow: 0.75;\n  max-width: 350px;\n  transition:\n    visibility $animation-time ease-out,\n    margin $animation-time ease-out;\n  visibility: hidden;\n\n  @if $side == \"right\" {\n    margin-right: -75%;\n    right: 0;\n  } @else {\n    margin-left: -75%;\n    left: 0;\n  }\n}\n\n// Primary sidebar hides/shows at earlier widths\n@include media-breakpoint-up($breakpoint-sidebar-primary) {\n  .sidebar-toggle.primary-toggle {\n    display: none;\n  }\n\n  input#pst-primary-sidebar-checkbox {\n    &:checked + label.overlay.overlay-primary {\n      height: 0;\n      width: 0;\n    }\n  }\n\n  .bd-sidebar-primary {\n    margin-left: 0;\n    visibility: visible;\n  }\n}\n\n.bd-sidebar-primary {\n  @include media-breakpoint-down($breakpoint-sidebar-primary) {\n    @include sliding-drawer(\"left\");\n  }\n}\n\n.bd-sidebar-secondary {\n  @include media-breakpoint-down($breakpoint-sidebar-secondary) {\n    @include sliding-drawer(\"right\");\n  }\n}\n","/**\n * Breadcrumbs for parent pages meant for the article header\n */\nul.bd-breadcrumbs {\n  list-style: none;\n  padding-left: 0;\n  display: flex;\n  flex-wrap: wrap;\n\n  // Font size slightly smaller to avoid cluttering up space too much\n  font-size: 0.8rem;\n\n  li.breadcrumb-item {\n    display: flex;\n    align-items: center;\n\n    // Style should look like heavier in-page links\n    // keeping visited in the default link colour\n    font-weight: bold;\n\n    a {\n      @include link-style-text;\n    }\n\n    // Items that aren't the home have a caret to the left\n    &:not(.breadcrumb-home)::before {\n      font: var(--fa-font-solid);\n      font-size: 0.8rem;\n      content: var(--pst-breadcrumb-divider);\n      color: var(--pst-color-text-muted);\n      padding: 0 0.5rem;\n    }\n  }\n}\n","/**\n * Icon links in the navbar\n */\n\n.pst-navbar-icon {\n  // Extra specificity needed for overrides\n  html & {\n    @include min-hit-area;\n    @include link-style-block;\n\n    display: flex;\n    align-items: center;\n    justify-content: center;\n\n    // Bootstrap overrides\n    border-radius: 0;\n    border: none;\n    font-size: 1rem;\n    line-height: $line-height-body; // Override Boostrap, which defines a separate line-height for buttons\n    padding: $navbar-link-padding-y 0; // Horizontal white space in nav bar between items is controlled via column gap rule on the container.\n\n    // Make the navbar icon links have the same size as the navbar text links\n    height: calc(2 * $navbar-link-padding-y + $line-height-body * 1rem);\n  }\n}\n\nul.navbar-icon-links {\n  display: flex;\n  flex-flow: row wrap;\n  column-gap: 1rem;\n  justify-content: space-evenly;\n  align-items: center;\n  padding-left: 0;\n  margin-bottom: 0;\n  list-style: none;\n\n  // Icons styling\n  i {\n    &.fa-brands,\n    &.fa-regular,\n    &.fa-solid {\n      vertical-align: middle;\n      font-style: normal;\n      font-size: var(--pst-font-size-icon);\n    }\n\n    /* Social media buttons hard-code the brand color */\n    &.fa-square-twitter::before {\n      color: #55acee;\n    }\n\n    &.fa-square-gitlab::before {\n      color: #548;\n    }\n\n    &.fa-bitbucket::before {\n      color: #0052cc;\n    }\n  }\n\n  // Force images to be icon-sized\n  img.icon-link-image {\n    height: 1.5em;\n    border-radius: 0.2rem;\n  }\n\n  .fa-pydata {\n    stroke: var(--pst-color-background);\n    stroke-linejoin: round;\n    stroke-width: 0.35;\n  }\n}\n","/**\n * Logo in the navbar\n */\n\n.navbar-brand {\n  position: relative;\n  height: var(--pst-header-height);\n  max-height: var(--pst-header-height);\n  padding: 0.5rem 0;\n  width: auto;\n  margin: 0;\n  display: flex;\n\n  // Ensure that the logo stays the same length while other content shrinks\n  flex-shrink: 0;\n  align-items: center;\n  gap: 0.5rem;\n\n  // If there's no logo image, we use a p element w/ the site title\n  p {\n    color: var(--pst-color-text-base);\n    margin-bottom: 0;\n  }\n\n  // If there's a logo, it'll be in an img block\n  img {\n    max-width: 100%;\n    height: 100%;\n    width: auto;\n  }\n\n  &:hover,\n  &:visited:hover {\n    @include link-style-hover;\n\n    color: var(--pst-color-text-base);\n  }\n}\n","/**\n * Navigation links in the navbar and icon links\n */\nul.navbar-nav {\n  // Reduce padding of nested `ul` items a bit\n  ul {\n    padding: 0 0 0 1rem;\n  }\n\n  // Navbar links - do not have an underline by default\n  li {\n    display: flex;\n    flex-direction: column;\n\n    a {\n      display: flex;\n      align-items: center;\n      height: 100%;\n      padding-top: $navbar-link-padding-y;\n      padding-bottom: $navbar-link-padding-y;\n\n      @include link-style-text;\n    }\n  }\n}\n","/**\n * The list of in-page TOC links\n */\n.page-toc {\n  .section-nav {\n    padding-left: 0;\n    border-bottom: none;\n\n    ul {\n      padding-left: 1rem;\n    }\n  }\n\n  // override bootstrap settings\n  .nav-link {\n    font-size: var(--pst-sidebar-font-size-mobile);\n\n    @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n      font-size: var(--pst-sidebar-font-size);\n    }\n  }\n\n  .onthispage {\n    color: var(--pst-color-text-base);\n    font-weight: var(--pst-sidebar-header-font-weight);\n    margin-bottom: 0.5rem;\n  }\n}\n","/**\n* Previous / Next navigation buttons\n**/\n.prev-next-area {\n  width: 100%;\n\n  p {\n    color: var(--pst-color-text-muted);\n    margin: 0 0.3em;\n    line-height: 1.3em;\n  }\n\n  i {\n    font-size: 1.2em;\n  }\n\n  a {\n    // So that buttons align with icons\n    display: flex;\n    align-items: center;\n    border: none;\n    padding: 10px;\n    max-width: 45%;\n    overflow-x: hidden;\n    color: var(--pst-color-text-muted);\n    text-decoration: none;\n\n    p.prev-next-title {\n      @include link-style-default;\n\n      font-weight: var(--pst-admonition-font-weight-heading);\n      font-size: 1.1em;\n    }\n\n    &:hover,\n    &:visited:hover {\n      p.prev-next-title {\n        @include link-style-hover;\n      }\n    }\n\n    .prev-next-info {\n      flex-direction: column;\n      margin: 0 0.5em;\n\n      .prev-next-subtitle {\n        text-transform: capitalize;\n      }\n    }\n\n    &.left-prev {\n      float: left;\n    }\n\n    &.right-next {\n      float: right;\n\n      div.prev-next-info {\n        text-align: right;\n      }\n    }\n  }\n}\n","/**\n * Search field\n **/\n.bd-search {\n  position: relative;\n  padding-left: 0.5rem;\n  gap: 0.5rem;\n  background-color: var(--pst-color-background);\n  border-radius: $admonition-border-radius;\n  border: 1px solid var(--pst-color-border);\n  color: var(--pst-color-text-base);\n\n  // Background should always be same color regardless of active or not\n  &:active {\n    background-color: var(--pst-color-background);\n    color: var(--pst-color-text-muted);\n  }\n\n  .icon {\n    position: absolute;\n    color: var(--pst-color-border);\n    left: 25px;\n  }\n\n  .fa-solid.fa-magnifying-glass {\n    position: absolute;\n    left: calc((2.5rem - 0.7em) / 2);\n    color: var(--pst-color-text-muted);\n  }\n\n  input {\n    // Inner-text of the search bar\n    &::placeholder {\n      color: var(--pst-color-text-muted);\n    }\n\n    // Remove the little \"x\" that pops up when you start typing\n    &::-webkit-search-cancel-button,\n    &::-webkit-search-decoration {\n      appearance: none;\n    }\n  }\n\n  // Shows off the keyboard shortcuts for the button\n  .search-button__kbd-shortcut {\n    display: flex;\n    position: absolute;\n    right: 0.5rem;\n    color: var(--pst-color-border);\n  }\n}\n\n.form-control {\n  background-color: var(--pst-color-background);\n  color: var(--pst-color-text-base);\n\n  &:focus,\n  &:focus-visible {\n    border: none;\n    background-color: var(--pst-color-background);\n    color: var(--pst-color-text-muted);\n  }\n}\n\n/**\n * Search button - located in the navbar\n */\n\n// Search link icon should be a bit bigger since it is separate from icon links\n.search-button i {\n  font-size: 1.3rem;\n}\n\n// __search-container will only show up when we use the search pop-up bar\n.search-button__search-container,\n.search-button__overlay {\n  display: none;\n}\n\n.search-button__wrapper.show {\n  .search-button__search-container {\n    display: flex;\n\n    // Center in middle of screen just underneath header\n    position: fixed;\n    z-index: $zindex-modal;\n    top: 30%;\n    left: 50%;\n    transform: translate(-50%, -50%);\n    right: 1rem;\n    margin-top: 0.5rem;\n    width: 90%;\n    max-width: 800px;\n  }\n\n  .search-button__overlay {\n    display: flex;\n    position: fixed;\n    z-index: $zindex-modal-backdrop;\n    background-color: black;\n    opacity: 0.5;\n    width: 100%;\n    height: 100%;\n    top: 0;\n    left: 0;\n  }\n\n  form.bd-search {\n    flex-grow: 1;\n    padding-top: 0;\n    padding-bottom: 0;\n  }\n\n  // Font and input text a bit bigger\n  svg,\n  input {\n    font-size: var(--pst-font-size-icon);\n  }\n}\n\n/**\n * The search button component that looks like a field.\n * Lives at components/search-button-field.html\n */\n.search-button-field {\n  $search-button-border-radius: 1.5em;\n\n  display: inline-flex;\n  align-items: center;\n  border: var(--pst-color-border) solid 1px;\n  border-radius: $search-button-border-radius;\n  color: var(--pst-color-text-muted);\n  padding: 0.5em;\n  background-color: var(--pst-color-surface);\n\n  &:hover {\n    box-shadow: 0 0 0 $focus-ring-width var(--pst-color-link-hover);\n  }\n\n  &:focus-visible {\n    border-radius: $search-button-border-radius;\n  }\n\n  // The keyboard shotcut text\n  .search-button__default-text {\n    font-size: var(--bs-nav-link-font-size);\n    font-weight: var(--bs-nav-link-font-weight);\n    margin-right: 0.5em;\n    margin-left: 0.5em;\n  }\n\n  .kbd-shortcut__modifier {\n    font-size: 0.75em;\n  }\n\n  // Ensures that all the text lines up in the middle\n  > * {\n    align-items: center;\n  }\n\n  // Only the icon should be visible on narrow screens\n  > :not(svg) {\n    display: none;\n\n    @include media-breakpoint-up(lg) {\n      display: flex;\n    }\n  }\n}\n","/**\n * The 'Hide Search Matches' button.\n * This only shows up when a person lands on a page after clicking a search result.\n * Clicking it removes the highlighting of the search term from the page.\n * We want it to behave like a button.\n */\ndiv#searchbox {\n  // Leave `#searchbox` rules empty so that it doesn't show at all when it is empty\n  p.highlight-link {\n    margin: 1rem 0;\n    width: fit-content;\n\n    // A bit more margin on wide screens to mimic article behavior\n    @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n      margin-left: 2rem;\n    }\n\n    // Put outer shadow on this one so that we can darken the link w/ an inner shadow\n    @include box-shadow;\n\n    // Style the button to look like a Sphinx Design button\n    a {\n      border-radius: 0.25rem;\n      font-size: 1.25rem;\n      padding: 0.75rem;\n      background-color: var(--pst-color-primary);\n      color: var(--pst-color-primary-text);\n      text-decoration: none;\n\n      // The box shadow is inset so that it darkens the button on hover\n      transition: box-shadow 0.25s ease-out;\n\n      &:hover {\n        box-shadow: inset 0 0 50px 50px rgb(0 0 0 / 25%);\n      }\n\n      &::before {\n        content: var(--pst-icon-search-minus);\n        color: unset;\n        font: var(--fa-font-solid);\n        margin-right: 0.5rem;\n      }\n    }\n  }\n}\n","/**\n * Light/dark theme switcher\n */\n\n.theme-switch-button {\n  .theme-switch {\n    display: none;\n\n    &:active {\n      text-decoration: none;\n      color: var(--pst-color-link-hover);\n    }\n\n    .fa-lg {\n      aspect-ratio: 1 / 1;\n    }\n  }\n}\n\n@each $mode in auto, light, dark {\n  html[data-mode=\"#{$mode}\"]\n    .theme-switch-button\n    .theme-switch[data-mode=\"#{$mode}\"] {\n    display: inline; // inline needed for span height to be calculated using inherited font size and line height\n  }\n}\n","button.btn.version-switcher__button {\n  border-color: var(--pst-color-border);\n  color: var(--pst-color-text-base);\n\n  // Add a margin on narrow screens to avoid feeling cramped\n  margin-bottom: 1em;\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    margin-bottom: unset;\n  }\n\n  &:hover {\n    box-shadow: 0 0 0 $focus-ring-width var(--pst-color-secondary);\n    border-color: transparent;\n  }\n\n  &:active {\n    color: var(--pst-color-text-base);\n    border-color: var(--pst-color-border);\n  }\n\n  &:focus-visible {\n    border-color: transparent;\n  }\n}\n\n.version-switcher__menu {\n  border-color: var(--pst-color-border);\n  border-radius: var(--bs-dropdown-border-radius);\n\n  a.list-group-item {\n    background-color: var(--pst-color-on-background);\n    color: var(--pst-color-text-base);\n    padding: 0.75rem 1.25rem;\n\n    &:not(:last-child) {\n      border-bottom: 1px solid var(--pst-color-border);\n    }\n\n    &:hover {\n      @include link-style-hover;\n\n      background-color: var(--pst-color-surface);\n    }\n\n    &.active {\n      @include link-sidebar-current;\n\n      position: relative;\n      z-index: 1;\n\n      span::before {\n        content: \"\";\n        width: 100%;\n        height: 100%;\n        position: absolute;\n        z-index: -1;\n        left: 0;\n        top: 0;\n      }\n    }\n\n    &:focus-visible {\n      z-index: 10; // keep focus ring on top (prevent the hover background of the next dropdown item from covering the ring)\n    }\n  }\n}\n\n// Font behavior on mobile\nbutton.version-switcher__button,\n.version-switcher__menu {\n  font-size: 1.1em; // A bit smaller than other menu font\n  z-index: $zindex-modal; // higher than the sidebars\n\n  // Make sure it meets WCAG target size requirement no matter the version\n  // string displayed in the button\n  @include min-hit-area;\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    font-size: unset;\n  }\n}\n","/* Collapsing of the TOC sidebar while scrolling */\n\n/* Nav: hide second level (shown on .active) */\n\nnav.page-toc {\n  // A little extra space before the buttons\n  margin-bottom: 1rem;\n}\n\n.bd-toc .nav {\n  .nav {\n    display: none;\n\n    // So we can manually specify a level as visible in the config\n    &.visible {\n      display: block;\n    }\n  }\n\n  > .active > ul {\n    display: block;\n  }\n}\n\n// Each entry of the in-page TOC\n.toc-entry {\n  display: block;\n\n  a > code {\n    color: var(--pst-color-text-muted);\n  }\n\n  a.nav-link {\n    display: block;\n    padding: 0.125rem 0;\n\n    // Padding w/ negative margin so the top TOC item highlight overlaps w/ the TOC border\n    padding-left: 1rem;\n    margin-left: -1rem;\n\n    @include link-sidebar;\n\n    &.active {\n      @include link-sidebar-current;\n\n      background-color: transparent;\n\n      &:hover {\n        color: var(--pst-color-link-hover);\n      }\n    }\n\n    &:focus-visible {\n      border-radius: $focus-ring-radius;\n    }\n  }\n}\n","div.versionadded,\ndiv.versionchanged,\ndiv.deprecated {\n  vertical-align: middle;\n  margin: 1.5625em auto;\n  padding: 0 0.6rem;\n  overflow: hidden;\n\n  /* break-inside has replaced page-break-inside and is widely usable since 2019 */\n  page-break-inside: avoid;\n  break-inside: avoid;\n  border-left: 0.2rem solid;\n  border-color: var(--pst-color-info);\n  border-radius: $admonition-border-radius;\n  background-color: var(--pst-color-on-background);\n\n  @include box-shadow;\n\n  position: relative;\n\n  > p {\n    margin-bottom: 0.6rem;\n    margin-top: 0.6rem;\n  }\n}\n\ndiv.versionadded {\n  border-color: var(--pst-color-success);\n  background-color: var(--pst-color-success-bg);\n}\n\ndiv.versionchanged {\n  border-color: var(--pst-color-warning);\n  background-color: var(--pst-color-warning-bg);\n}\n\ndiv.deprecated {\n  border-color: var(--pst-color-danger);\n  background-color: var(--pst-color-danger-bg);\n}\n\nspan.versionmodified {\n  font-weight: 600;\n\n  &::before {\n    margin-right: 0.6rem;\n    color: var(--pst-color-info);\n    font: var(--fa-font-solid);\n    content: var(--pst-icon-versionmodified-default);\n  }\n}\n\nspan.versionmodified.added {\n  &::before {\n    color: var(--pst-color-success);\n    content: var(--pst-icon-versionmodified-added);\n  }\n}\n\nspan.versionmodified.changed {\n  &::before {\n    color: var(--pst-color-warning);\n    content: var(--pst-icon-versionmodified-changed);\n  }\n}\n\nspan.versionmodified.deprecated {\n  &::before {\n    color: var(--pst-color-danger);\n    content: var(--pst-icon-versionmodified-deprecated);\n  }\n}\n",".sidebar-indices-items {\n  display: flex;\n  flex-direction: column;\n  border-top: 1px solid var(--pst-color-border);\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    border-top: none;\n  }\n\n  .sidebar-indices-items__title {\n    font-weight: var(--pst-sidebar-header-font-weight);\n    font-size: var(--pst-sidebar-header-font-size);\n    color: var(--pst-color-text-base);\n    margin-bottom: 0.5rem;\n  }\n\n  ul.indices-link {\n    margin-right: -1rem;\n    list-style: none;\n    padding: 0;\n\n    li > a {\n      display: block;\n      padding: 0.25rem 0;\n      color: var(--pst-color-text-muted);\n\n      &:hover {\n        color: var(--pst-color-primary);\n        text-decoration: none;\n        background-color: transparent;\n      }\n    }\n  }\n}\n",".bd-sidebar-primary div#rtd-footer-container {\n  position: sticky;\n  bottom: -1rem;\n  margin: -1rem; // ignore sidebar padding\n\n  .rst-versions.rst-badge {\n    position: unset;\n    font-size: 0.9em;\n    font-family: var(--pst-font-family-base);\n    max-width: unset;\n\n    .rst-current-version {\n      display: flex;\n      align-items: center;\n      gap: 0.2rem;\n      height: 2.5rem;\n      transition: background-color 0.2s ease-out;\n      background-color: var(--pst-color-background);\n      color: var(--pst-color-success);\n      border-top: 1px solid var(--pst-color-border);\n    }\n\n    .fa.fa-book {\n      color: var(--pst-color-text-muted);\n      margin-right: auto;\n\n      &::after {\n        color: var(--pst-color-text-base);\n        content: \"Read The Docs\";\n        font-family: var(--pst-font-family-base);\n        font-weight: var(--pst-admonition-font-weight-heading);\n      }\n    }\n\n    .fa.fa-caret-down {\n      color: var(--pst-color-text-muted);\n    }\n  }\n\n  .rst-versions.rst-badge.shift-up {\n    .rst-current-version {\n      border-bottom: 1px solid var(--pst-color-border);\n    }\n  }\n\n  .rst-other-versions {\n    background-color: var(--pst-color-surface);\n    color: var(--pst-color-text-base);\n\n    dl {\n      dd a {\n        color: var(--pst-color-text-muted);\n      }\n    }\n\n    hr {\n      background-color: var(--pst-color-border);\n    }\n\n    small a {\n      color: var(--pst-color-link);\n    }\n\n    input {\n      padding-left: 0.5rem;\n      border: 1px solid var(--pst-color-border);\n      background-color: var(--pst-color-surface);\n    }\n  }\n}\n","/**\n * Admonitions and blocks of styled content.\n * Admonitions CSS originally inspired by https://squidfunk.github.io/mkdocs-material/getting-started/\n */\n$admonition-border-radius: 0.25rem;\n$admonition-left-border-width: 0.2rem;\n\ndiv.admonition,\n.admonition {\n  margin: 1.5625em auto;\n  padding: 0 0.6rem 0.8rem;\n  overflow: hidden;\n\n  /* break-inside has replaced page-break-inside and is widely usable since 2019 */\n  page-break-inside: avoid;\n  break-inside: avoid;\n  border-left: $admonition-left-border-width solid;\n  border-color: var(--pst-color-info);\n  border-radius: $admonition-border-radius;\n  background-color: var(--pst-color-on-background);\n\n  @include box-shadow;\n\n  // Last item should have no spacing since we'll control that w/ padding\n  *:last-child {\n    margin-bottom: 0;\n  }\n\n  // Items after the title should be indented\n  p.admonition-title ~ * {\n    margin-left: 1.4rem;\n    margin-right: 1.4rem;\n  }\n\n  // Lists need to have left margin so they don't spill into it\n  > ol,\n  > ul {\n    margin-left: 1em;\n  }\n\n  // Defaults for all admonitions\n  > .admonition-title {\n    margin: 0 -0.6rem;\n    padding: 0.4rem 0.6rem 0.4rem 2rem;\n    font-weight: var(--pst-admonition-font-weight-heading);\n    position: relative;\n\n    @include legacy-backdrop-placeholder;\n\n    background-color: var(--pst-color-info-bg);\n\n    // now that we use solid colors we want the title on top\n    z-index: 1;\n\n    &::after {\n      position: absolute;\n      left: 0.5rem;\n      width: 1rem;\n      height: 1rem;\n      color: var(--pst-color-info);\n      font: var(--fa-font-solid);\n      line-height: inherit;\n      content: var(--pst-icon-admonition-default);\n      opacity: 1;\n    }\n\n    // Next element after title needs some extra upper-space\n    + * {\n      margin-top: 0.4em;\n    }\n  }\n\n  &.attention {\n    border-color: var(--pst-color-attention);\n\n    > .admonition-title {\n      background-color: var(--pst-color-attention-bg);\n\n      &::after {\n        color: var(--pst-color-attention);\n        content: var(--pst-icon-admonition-attention);\n      }\n    }\n  }\n\n  &.caution {\n    border-color: var(--pst-color-warning);\n\n    > .admonition-title {\n      background-color: var(--pst-color-warning-bg);\n\n      &::after {\n        color: var(--pst-color-warning);\n        content: var(--pst-icon-admonition-caution);\n      }\n    }\n  }\n\n  &.warning {\n    border-color: var(--pst-color-warning);\n\n    > .admonition-title {\n      background-color: var(--pst-color-warning-bg);\n\n      &::after {\n        color: var(--pst-color-warning);\n        content: var(--pst-icon-admonition-warning);\n      }\n    }\n  }\n\n  &.danger {\n    border-color: var(--pst-color-danger);\n\n    > .admonition-title {\n      background-color: var(--pst-color-danger-bg);\n\n      &::after {\n        color: var(--pst-color-danger);\n        content: var(--pst-icon-admonition-danger);\n      }\n    }\n  }\n\n  &.error {\n    border-color: var(--pst-color-danger);\n\n    > .admonition-title {\n      background-color: var(--pst-color-danger-bg);\n\n      &::after {\n        color: var(--pst-color-danger);\n        content: var(--pst-icon-admonition-error);\n      }\n    }\n  }\n\n  &.hint {\n    border-color: var(--pst-color-success);\n\n    > .admonition-title {\n      background-color: var(--pst-color-success-bg);\n\n      &::after {\n        color: var(--pst-color-success);\n        content: var(--pst-icon-admonition-hint);\n      }\n    }\n  }\n\n  &.tip {\n    border-color: var(--pst-color-success);\n\n    > .admonition-title {\n      background-color: var(--pst-color-success-bg);\n\n      &::after {\n        color: var(--pst-color-success);\n        content: var(--pst-icon-admonition-tip);\n      }\n    }\n  }\n\n  &.important {\n    border-color: var(--pst-color-attention);\n\n    > .admonition-title {\n      background-color: var(--pst-color-attention-bg);\n\n      &::after {\n        color: var(--pst-color-attention);\n        content: var(--pst-icon-admonition-important);\n      }\n    }\n  }\n\n  &.note {\n    border-color: var(--pst-color-info);\n\n    > .admonition-title {\n      background-color: var(--pst-color-info-bg);\n\n      &::after {\n        color: var(--pst-color-info);\n        content: var(--pst-icon-admonition-note);\n      }\n    }\n  }\n\n  &.seealso {\n    border-color: var(--pst-color-success);\n\n    > .admonition-title {\n      background-color: var(--pst-color-success-bg);\n\n      &::after {\n        color: var(--pst-color-success);\n        content: var(--pst-icon-admonition-seealso);\n      }\n    }\n  }\n\n  &.admonition-todo {\n    border-color: var(--pst-color-secondary);\n\n    > .admonition-title {\n      background-color: var(--pst-color-secondary-bg);\n\n      &::after {\n        color: var(--pst-color-secondary);\n        content: var(--pst-icon-admonition-todo);\n      }\n    }\n  }\n\n  /**\n   * Special-case for a `sidebar` class that makes the admonition float to\n   * the right like the {sidebar} directive.\n   */\n  &.sidebar {\n    max-width: 40%;\n    float: right;\n    clear: both;\n    margin-left: 0.5rem;\n    margin-top: 0;\n\n    // Undo the .sidebar directive border\n    border-width: 0 0 0 $admonition-left-border-width;\n\n    // TODO: these semantic-color-names border-color rules might no longer be\n    //       needed when we drop support for Sphinx 4 / docutils 0.17\n    &.attention,\n    &.important {\n      border-color: var(--pst-color-attention);\n    }\n\n    &.caution,\n    &.warning {\n      border-color: var(--pst-color-warning);\n    }\n\n    &.danger,\n    &.error {\n      border-color: var(--pst-color-danger);\n    }\n\n    &.hint,\n    &.tip,\n    &.seealso {\n      border-color: var(--pst-color-success);\n    }\n\n    &.note,\n    &.todo {\n      border-color: var(--pst-color-info);\n    }\n\n    // No inner margin since we have less horizontal space w/ the sidebar\n    p.admonition-title ~ * {\n      margin-left: 0;\n      margin-right: 0;\n    }\n  }\n}\n\n/**************************************************************\n * Similar content blocks that are not technically admonitions.\n */\n\n/**\n * Topics and the {contents} directive\n */\n// Docutils <= 0.17\ndiv.topic,\ndiv.topic.contents,\n// Docutils >= 0.18\nnav.contents,\naside.topic {\n  display: flex;\n  flex-direction: column;\n  background-color: var(--pst-color-surface);\n  border-color: var(--pst-color-border);\n  border-radius: $admonition-border-radius;\n  padding: 1rem 1.25rem;\n\n  @include box-shadow;\n\n  .topic-title {\n    margin: 0 0 0.5rem;\n  }\n\n  // Over-ride text color to ensure enough contrast\n  p {\n    color: var(--pst-color-on-surface) !important;\n  }\n\n  // Over-ride large default padding\n  ul.simple {\n    padding-left: 1rem;\n\n    ul {\n      // So that sub-lists will have a bit less padding\n      padding-left: 2em;\n    }\n  }\n}\n\n/**\n * Sidebar directive\n */\naside.sidebar {\n  border: 1px solid var(--pst-color-border);\n  background-color: var(--pst-color-surface);\n  border-radius: $admonition-border-radius;\n\n  // to match the admonition-styled sidebars:\n  margin-left: 0.5rem;\n  padding: 0;\n\n  > *:last-child {\n    padding-bottom: 1rem;\n  }\n\n  p.sidebar-title {\n    position: relative;\n    margin-bottom: 0;\n    padding-top: 0.5rem;\n    padding-bottom: 0.5rem;\n    border-bottom: 1px solid var(--pst-color-border);\n    font-family: var(--pst-font-family-heading);\n    font-weight: var(--pst-admonition-font-weight-heading);\n  }\n\n  // Add margin to the first non-`.sidebar-title` item\n  > *:not(.sidebar-title):first-child,\n  > p.sidebar-title + * {\n    margin-top: 1rem;\n  }\n\n  > * {\n    padding-left: 1rem;\n    padding-right: 1rem;\n  }\n}\n\n/**\n * Rubrics look kind of like section headers\n */\np.rubric {\n  display: flex;\n  flex-direction: column;\n}\n\n/**\n * Seealso is kind of like a vertically-collapsed admonition\n */\n.seealso dd {\n  margin-top: 0;\n  margin-bottom: 0;\n}\n","/**\n * Miscellaneous color functions and mixins\n**/\n\n@use \"sass:list\";\n@use \"sass:map\";\n@use \"sass:meta\";\n@use \"sass:math\";\n@use \"sass:string\";\n\n// We must add ::before pseudo-element to some theme components (such as admonitions)\n// because users were instructed to customize the background color this way.\n@mixin legacy-backdrop-placeholder {\n  &::before {\n    content: \"\";\n    width: 100%;\n    height: 100%;\n    position: absolute;\n    left: 0;\n    top: 0;\n    z-index: -1;\n\n    // So that hovering over the text within background is still possible.\n    // Otherwise the background overlays the text and you cannot click or select easily.\n    // ref: https://developer.mozilla.org/en-US/docs/Web/CSS/pointer-events\n    pointer-events: none;\n  }\n}\n\n/**\n* Function to get items from nested maps\n*/\n// @param {Map} $map - Map\n// @param {Arglist} $keys - Keys to fetc\n// @return {*}\n@function map-deep-get($map, $keys...) {\n  @each $key in $keys {\n    $map: map.get($map, $key);\n  }\n\n  @return $map;\n}\n\n/**\n  * Function to check if the color needs converting to a \"color\" type\n  * if it is a string we cannot use other color manipulation functions\n  * It is used to create the sphinx-design colours as these are often interpolated\n  */\n// @param {String/Color} $color - Color definition from map\n// @return {Color} - Color type (in hex)\n@function check-color($color) {\n  @if meta.type-of($color) == string {\n    $color: from-hex($color);\n  }\n\n  @return $color;\n}\n\n/**\n  * Function to convert the string representation of a color to a color type (hex)\n  */\n// @param {String} $string - String representation of a color\n@function from-hex($string) {\n  $string-lower: string.to-lower-case($string);\n  $r: \"\";\n  $g: \"\";\n  $b: \"\";\n  $hex: \"0\" \"1\" \"2\" \"3\" \"4\" \"5\" \"6\" \"7\" \"8\" \"9\" \"a\" \"b\" \"c\" \"d\" \"e\" \"f\";\n  $length: string.length($string);\n  $max: if($length == 4, 1, 2);\n\n  // Check for length accuracy\n  @if $length != 4 and $length != 7 {\n    @return $string;\n  }\n\n  // Loop from the second character (omitting #)\n  @for $i from 2 through $length {\n    $c: string.slice($string-lower, $i, $i);\n\n    // If wrong character, return\n    @if not list.index($hex, $c) {\n      @return $string;\n    }\n\n    @if string.length($r) < $max {\n      $r: $r + $c;\n    } @else if string.length($g) < $max {\n      $g: $g + $c;\n    } @else if string.length($b) < $max {\n      $b: $b + $c;\n    }\n  }\n\n  @if $length == 4 {\n    $r: $r + $r;\n    $g: $g + $g;\n    $b: $b + $b;\n  }\n\n  @return rgb(hex-to-dec($r), hex-to-dec($g), hex-to-dec($b));\n}\n\n@function hex-to-dec($string) {\n  $hex: \"0\" \"1\" \"2\" \"3\" \"4\" \"5\" \"6\" \"7\" \"8\" \"9\" \"a\" \"b\" \"c\" \"d\" \"e\" \"f\";\n  $string: string.to-lower-case($string);\n  $length: string.length($string);\n  $dec: 0;\n\n  @for $i from 1 through $length {\n    $factor: 1 + (15 * ($length - $i));\n    $index: list.index($hex, string.slice($string, $i, $i));\n    $dec: $dec + $factor * ($index - 1);\n  }\n\n  @return $dec;\n}\n","// Style API docs from sphinx' autodoc / autosummary\n\n/*******************************************************************************\n* Styling for field lists\n*/\n\n/* grey highlighting of 'parameter' and 'returns' field */\ntable.field-list {\n  border-collapse: separate;\n  border-spacing: 10px;\n  margin-left: 1px;\n\n  th.field-name {\n    padding: 1px 8px 1px 5px;\n    white-space: nowrap;\n    background-color: var(--pst-color-surface);\n  }\n\n  /* italic font for parameter types */\n  td.field-body {\n    p {\n      font-style: italic;\n\n      > strong {\n        font-style: normal;\n      }\n    }\n\n    /* reduced space around parameter description */\n    blockquote {\n      border-left: none;\n      margin: 0 0 0.3em;\n      padding-left: 30px;\n    }\n  }\n}\n\n/*******************************************************************************\n* Styling for autosummary tables\n*/\n\n.table.autosummary {\n  // The first column (with the signature) should not wrap\n  td:first-child {\n    white-space: nowrap;\n  }\n}\n\n/* overriding basic.css to use our own monospace font */\n.sig {\n  font-family: var(--pst-font-family-monospace);\n}\n\n/* C++ specific styling - overriding the basic.css to avoid custom colors */\n\n.sig-inline.c-texpr,\n.sig-inline.cpp-texpr {\n  font-family: unset;\n}\n\n.sig.c .k,\n.sig.c .kt,\n.sig.cpp .k,\n.sig.cpp .kt {\n  color: var(--pst-color-text-base);\n}\n\n.sig.c .m,\n.sig.cpp .m {\n  color: var(--pst-color-text-base);\n}\n\n.sig.c .s,\n.sig.c .sc,\n.sig.cpp .s,\n.sig.cpp .sc {\n  color: var(--pst-color-text-base);\n}\n\n// addition\n\n// .sig.c .sig-name .n,\n// .sig.cpp .sig-name .n {\n//   color: var(--pst-color-inline-code);\n// }\n\n.sig-name {\n  color: var(--pst-color-inline-code);\n}\n\n.sig-param .o,\n.sig-param .default_value {\n  color: var(--pst-color-text-muted);\n  font-weight: normal;\n}\n\n// change target color for dark theme\ndt:target,\nspan.highlighted {\n  background-color: var(--pst-color-target);\n}\n\n.viewcode-back {\n  font-family: var(--pst-font-family-base);\n}\n\n.viewcode-block:target {\n  border-top: 1px solid var(--pst-color-border);\n  border-bottom: 1px solid var(--pst-color-border);\n  position: relative;\n  background-color: var(--pst-color-target);\n}\n\ndl > dt > a:has(.viewcode-link) {\n  // Sphinx applies a `float:right` rule to the .viewcode-line span, which\n  // exposes a browser glitch in the focus ring. It seems the browser creates\n  // two separate boxes, an empty box where the anchor element gets laid out and\n  // then another box around the anchor's contents that have been floated right.\n  // Firefox draws the focus ring around the empty anchor element box. Chrome\n  // draws two focus rings: one around the empty anchor and one around the\n  // floated-right element. To fix the glitch, we apply the float rule on the\n  // parent rather than the child.\n  float: right;\n\n  .viewcode-link {\n    float: none;\n  }\n}\n\n/*******************************************************************************\n* Styling for autosummary titles like \"parameters\" and \"returns\"\n*/\n\n// the API selector\n// from https://github.com/pradyunsg/furo/blob/main/src/furo/assets/styles/content/_api.sass#L6)\ndl[class]:not(.option-list, .field-list, .footnote, .glossary, .simple) {\n  // increase margin bottom after the dl elements\n  margin-bottom: 3rem;\n\n  dd {\n    margin-left: 2rem;\n\n    // Fix until this will be solved to Sphinx https://github.com/sphinx-doc/sphinx/issues/10815\n    & > dl.simple > dt {\n      display: flex;\n    }\n  }\n\n  dl.field-list {\n    display: grid;\n    grid-template-columns: unset;\n  }\n\n  dt.field-odd,\n  dt.field-even {\n    margin-top: 0.2rem;\n    margin-bottom: 0.1rem;\n    background-color: var(--pst-color-surface);\n  }\n}\n","/**\n * Code block styling\n * Note that we inherit a lot of styling from Bootstrap so not many rules here.\n */\n\n// General code block behavior\n// Unset bootstrap behavior\ndiv[class*=\"highlight-\"],\ndiv.highlight,\ndiv.literal-block-wrapper {\n  display: flex;\n  flex-direction: column;\n  width: unset;\n  border-radius: $admonition-border-radius;\n  break-inside: avoid;\n}\n\n// Code blocks with captions\n// There's a wrapper when the code block has a title\ndiv.literal-block-wrapper {\n  border: 1px solid var(--pst-color-border);\n  border-radius: $admonition-border-radius;\n\n  // This is where the title goes\n  div.code-block-caption {\n    margin: 0;\n    border-bottom: 1px solid var(--pst-color-border);\n    padding: 0.5rem;\n    font-size: 1rem;\n    font-weight: var(--pst-font-weight-caption);\n\n    a.headerlink {\n      font-size: inherit;\n    }\n  }\n\n  // Remove the upper border radius since we want it to connect with the title\n  // Remove the box shadow so the wrapper gets the shadow instead\n  div[class*=\"highlight-\"] {\n    margin: 0;\n    border-radius: 0;\n\n    pre {\n      border: none;\n      box-shadow: none;\n    }\n  }\n}\n\n/**\n * In-line code\n */\ncode.literal {\n  padding: 0.1rem 0.25rem;\n  background-color: var(--pst-color-surface);\n  border: 1px solid var(--pst-color-border);\n  border-radius: 0.25rem;\n}\n\na > code {\n  color: var(--pst-color-inline-code-links);\n}\n\n// Fix for Sphinx's \"highlight\" directive - this is an issue with our accessible pygments theme\n// and the colour we are using for the background of the code blocks.\nhtml[data-theme=\"light\"] .highlight .nf {\n  color: #0078a1 !important;\n}\n\n// Minimum opacity needed for linenos to be WCAG AA conformant\nspan.linenos {\n  opacity: 0.8 !important;\n}\n","figure > a,\nfigure > a > img,\nfigure > img,\nfigure > video {\n  display: block;\n  margin-left: auto;\n  margin-right: auto;\n}\n\nfigure {\n  a.headerlink {\n    // So that header link doesn't push caption to be off-center.\n    position: absolute;\n    font-size: inherit;\n  }\n\n  // Default headerlink hover doesn't trigger on figures\n  &:hover a.headerlink {\n    visibility: visible;\n  }\n\n  figcaption {\n    font-family: var(--pst-font-family-heading);\n    font-weight: var(--pst-font-weight-caption);\n    color: var(--pst-color-text-muted);\n    margin-left: auto;\n    margin-right: auto;\n    margin-top: 0.3rem;\n    text-align: center;\n\n    & > p:last-child {\n      // Don't add extra margin to already existing figure bottom margin\n      margin-bottom: 0;\n    }\n\n    p {\n      text-align: start;\n      display: inline-block;\n    }\n\n    table.table {\n      width: fit-content;\n      margin-left: auto;\n      margin-right: auto;\n    }\n  }\n}\n","// For consistency, add bracket around footnotes/citations which are\n// cited more than once. E.g. [Newton](1,2) instead of Newton(1,2)\ndt.label > span.brackets:not(:only-child)::before {\n  content: \"[\";\n}\n\ndt.label > span.brackets:not(:only-child)::after {\n  content: \"]\";\n}\n\n// Make footnote as a superscript\na.footnote-reference {\n  vertical-align: super;\n  font-size: small;\n}\n\n// Docutils 0.18 uses an `aside.footnote` container with different internal structure\naside.footnote {\n  margin-bottom: 0.5rem;\n\n  &:last-child {\n    margin-bottom: 1rem;\n  }\n\n  span.label,\n  span.backrefs {\n    font-weight: bold;\n  }\n\n  &:target {\n    background-color: var(--pst-color-target);\n  }\n}\n","/**\n * Hacky fixes that don't fit cleanly into other sections\n */\n\n// Ensure user highlighting/selecting behaves properly\n// From https://stackoverflow.com/a/34372191\ntable.highlighttable td.linenos,\nspan.linenos,\ndiv.doctest > div.highlight span.gp {\n  /* gp: Generic.Prompt */\n  user-select: none;\n}\n","// Override bootstrap by restoring the basic theme default.\ndd {\n  margin-top: 3px;\n  margin-bottom: 10px;\n  margin-left: 30px;\n}\n\nol,\nul {\n  padding-inline-start: 2rem;\n\n  li > p:first-child {\n    margin-bottom: 0.25rem;\n    margin-top: 0.25rem;\n  }\n}\n","// GitHub blockquote style\nblockquote {\n  padding: 1em;\n  color: var(--pst-color-text-muted);\n  border-left: 0.25em solid var(--pst-color-blockquote-notch);\n  border-radius: $admonition-border-radius;\n  position: relative;\n\n  p {\n    color: var(--pst-color-text-base);\n  }\n\n  // remove padding from included line-block to avoid duplication\n  .line-block {\n    margin: 0;\n  }\n\n  // remove margin bottom for the last p\n  p:last-child {\n    margin-bottom: 0;\n  }\n\n  @include legacy-backdrop-placeholder;\n\n  background-color: var(--pst-color-surface);\n\n  // Ensure there is enough contrast against the background\n  a {\n    color: var(--pst-color-inline-code-links);\n  }\n\n  // hack to make the text in the blockquote selectable\n  &::before {\n    z-index: -1;\n  }\n}\n","/**\n * Span-level styling within content\n */\n\nspan.guilabel {\n  border: 1px solid var(--pst-color-info);\n  font-size: 80%;\n  font-weight: 700;\n  border-radius: 4px;\n  padding: 2.4px 6px;\n  margin: auto 2px;\n  position: relative;\n\n  @include legacy-backdrop-placeholder;\n\n  background-color: var(--pst-color-info-bg);\n}\n\na.reference.download::before {\n  content: var(--pst-icon-download);\n  font: var(--fa-font-solid);\n  font-size: 0.8em;\n  padding: 0 0.25em;\n  color: var(--pst-color-text-muted);\n}\n","/**\n * Tables\n */\n\ntable {\n  // default to table-center\n  margin-left: auto;\n  margin-right: auto;\n\n  &.table-right {\n    margin-right: 0;\n  }\n\n  &.table-left {\n    margin-left: 0;\n  }\n}\n\n// customize table caption from bootstrap\n// to display them on top and centered\ntable caption {\n  text-align: center;\n  caption-side: top;\n  color: var(--pst-color-text-muted);\n}\n\n// MyST Markdown tables use these classes to control alignment\nth,\ntd {\n  &.text-left {\n    text-align: left;\n  }\n\n  &.text-right {\n    text-align: right;\n  }\n\n  &.text-center {\n    text-align: center;\n  }\n}\n\n// override bootstrap table colors\n.table {\n  @include table-colors;\n\n  --bs-table-bg: transparent; // background\n  --bs-table-color: var(\n    --pst-color-text-base\n  ); // ensure text and bullets are visible\n}\n\n.pst-scrollable-table-container {\n  // Put a horizontal scrollbar just below tables that are too wide to fit\n  // within the main column\n  overflow-x: auto;\n}\n","/**\n * Style the toctree component in pages (avoid modifying the navbars)\n */\n.toctree-wrapper {\n  p.caption {\n    font-size: 1.5em;\n    margin-bottom: 0;\n  }\n\n  & > ul {\n    padding-left: 0;\n  }\n\n  li[class^=\"toctree-l\"] {\n    list-style: none;\n    margin-bottom: 0.2em;\n\n    & > a {\n      list-style: none;\n      font-size: 1.1em;\n    }\n\n    & > ul {\n      list-style: none;\n      padding-inline-start: 1.5em;\n    }\n  }\n\n  // slightly bigger font for l1\n  .toctree-l1 > a {\n    font-size: 1.3em;\n  }\n}\n\ndiv.topic.contents,  // Docutils <= 0.17\nnav.contents  // Docutils >= 0.18\n{\n  // Style similarly to toctree\n  ul.simple {\n    list-style: none;\n    padding-left: 0;\n  }\n}\n","/**\n * Mathematics via MathJax.\n *\n * This is designed for MathJax v3\n * ref: https://www.sphinx-doc.org/en/master/usage/extensions/math.html#module-sphinx.ext.mathjax\n */\n\n// Applies to all math elements\nspan.math,\ndiv.math {\n  align-items: center;\n  display: flex;\n  max-width: 100%;\n\n  // This will be over-ridden for the y-direction and divs\n  overflow: hidden;\n}\n\n// Inline-only\nspan.math {\n  display: inline-flex;\n}\n\n// Block-level only\ndiv.math {\n  gap: 0.5em;\n\n  // So that the eqno shows up after the equation\n  flex-direction: row-reverse;\n\n  // The equation number / link\n  span.eqno a.headerlink {\n    position: relative;\n    font-size: 1em;\n  }\n\n  // The math container\n  mjx-container {\n    flex-grow: 1;\n    padding-bottom: 0.2rem;\n    overflow: auto;\n\n    // Set height to 0 so that it does not cause scrollbars to appear\n    // ref: https://github.com/mathjax/MathJax/issues/2521\n    mjx-assistive-mml {\n      height: 0;\n    }\n  }\n}\n","/**\n * ABlog\n * ref: https://ablog.readthedocs.io/\n */\n\n/**\n * Sidebar template components\n */\n.ablog-sidebar-item {\n  h2,\n  h3 {\n    font-size: var(--pst-sidebar-header-font-size);\n\n    // Remove unnecessary vertical whitespace\n    margin-top: 0.5rem;\n\n    // The headers are all links, but this makes them hard to parse\n    // So we change the colors to make them look like headers\n    a {\n      color: var(--pst-color-text-base);\n    }\n  }\n\n  ul {\n    // No bullet points for the primary sidebar items\n    list-style: none;\n    padding-left: 0;\n\n    // Otherwise a scrollbar randomly shows up\n    overflow-y: hidden;\n\n    // List of recent post items\n    display: flex;\n    flex-direction: column;\n    gap: 0.5em;\n    margin-bottom: 0;\n\n    // The ablog cloud should move horizontally\n    &.ablog-cloud {\n      flex-flow: row wrap;\n      gap: 0.5rem;\n\n      // Vertical-align tag clouds\n      li {\n        // Center the tag cloud items\n        display: flex;\n        align-items: center;\n      }\n    }\n  }\n}\n\n/**\n * Previous / next buttons at the bottom\n */\n.ablog__prev-next {\n  font-size: 1.2em;\n  display: flex;\n  padding: 1rem 0;\n\n  // The bottom previous / next arrows\n  > span {\n    // To ensure that the whole thing fits on one line even if there are long titles\n    display: flex;\n    max-width: 45%;\n\n    // Links within each span have the collection of icon + text\n    a {\n      display: flex;\n      align-items: center;\n      margin-left: auto;\n      gap: 1rem;\n      line-height: 1.5rem;\n\n      i::before {\n        color: var(--pst-color-text-base);\n      }\n    }\n  }\n\n  // The first span is for the previous page and aligns to the left\n  span.ablog__prev {\n    i.fa-arrow-circle-left::before {\n      content: var(--pst-icon-angle-left);\n    }\n  }\n\n  // The second span is just an empty space so we remove it because we're\n  // positioning with flex\n  span.ablog__spacer {\n    display: none;\n  }\n\n  // The third span is aligned to the right\n  span.ablog__next {\n    margin-left: auto;\n    text-align: right;\n\n    i.fa-arrow-circle-right::before {\n      content: var(--pst-icon-angle-right);\n    }\n  }\n}\n\n/**\n * {postlist} directive and posts page\n */\n.ablog__collection,\n.postlist {\n  padding-left: 0;\n\n  .ablog-post {\n    list-style: none;\n\n    // Post metadata tags (author, links ,etc) should be a bit smaller\n    .ablog-archive {\n      display: flex;\n      flex-flow: row wrap;\n      gap: 1rem;\n      list-style: none;\n      font-size: 0.75rem;\n      padding-left: 0;\n    }\n\n    // Title line should be a bit bigger and bold to stand out\n    .ablog-post-title {\n      margin-top: 0;\n      font-size: 1.25rem;\n\n      a {\n        font-weight: bold;\n      }\n    }\n\n    // Read more button should be a bit bigger\n    .ablog-post-expand {\n      margin-bottom: 0.5rem;\n    }\n  }\n}\n","/**\n * Special cases for Bootstrap functionality\n */\n\n// Bootstrap adds margin to their general container class. However, sphinx/docutils\n// can also generate output with the container class, but in those cases we should\n// not add the margin from bootstrap. Same for max-width.\n.docutils.container {\n  padding-left: unset;\n  padding-right: unset;\n  margin-left: unset;\n  margin-right: unset;\n  max-width: unset;\n  width: unset;\n}\n\n.btn {\n  --bs-btn-focus-box-shadow: #{$btn-focus-box-shadow};\n}\n","/**\n * Sphinx Copybutton\n * ref: https://sphinx-copybutton.readthedocs.io/\n */\n\ndiv.highlight button.copybtn {\n  // Nicer spacing\n  display: flex;\n  align-items: center;\n  justify-content: center;\n\n  // Don't over-ride the success color\n  &:not(.success) {\n    color: var(--pst-color-muted);\n  }\n\n  border: none;\n  background-color: var(--pst-color-surface);\n\n  &:hover {\n    &:not(.success) {\n      color: var(--pst-color-text);\n      background-color: var(--pst-color-shadow);\n    }\n  }\n\n  // Tooltip styling\n  &.o-tooltip--left::after {\n    color: var(--pst-color-text);\n    background-color: var(--pst-color-surface);\n  }\n\n  &:focus {\n    // For keyboard users, make the copy button visible when focussed.\n    opacity: 1;\n  }\n\n  &:focus-visible {\n    outline: $focus-ring-outline;\n  }\n}\n\ndiv.highlight:has(button.copybtn) {\n  // Make sure the code block has enough height for the copy button.\n  // Sphinx-copybutton sets 0.3em top offset plus 1.7em height:\n  // https://github.com/executablebooks/sphinx-copybutton/blob/master/sphinx_copybutton/_static/copybutton.css\n  min-height: 2em;\n}\n","// adapt ethical ad to the theme\n#ethical-ad-placement {\n  .ethical-sidebar a,\n  .ethical-sidebar a:visited,\n  .ethical-sidebar a:hover,\n  .ethical-sidebar a:active,\n  .ethical-footer a,\n  .ethical-footer a:visited,\n  .ethical-footer a:hover,\n  .ethical-footer a:active {\n    color: var(--pst-color-text-base);\n  }\n\n  .ethical-sidebar,\n  .ethical-footer {\n    background-color: var(--pst-color-background);\n    border: 1px solid var(--pst-color-border);\n    border-radius: 5px;\n    color: var(--pst-color-text-base);\n    font-size: 14px;\n    line-height: 20px;\n  }\n}\n","/**\n * Styles for various Sphinx execution libraries.\n * For now, where these define output sections, we simply revert their background\n * to be a \"light theme\" background. This ensures that inputs/outputs behave similarly,\n * because the CSS is often controlled by each package.\n * In the future, we might add dark theme support for specific packages.\n */\n\n/******************************************************************************\n * Jupyter Sphinx\n */\n\n.bd-content div.jupyter_container {\n  // We don't want borders around the whole container, just around code cells\n  border: none;\n  background-color: unset;\n  box-shadow: none;\n\n  // Code cells should have the same style as our other code objects\n  div.output,\n  div.highlight {\n    border-radius: 0.25rem;\n  }\n\n  div.highlight {\n    background-color: var(--pst-color-surface);\n  }\n\n  // Ensure the style is the same as our code cells. Jupyter Sphinx makes it tiny.\n  .cell_input,\n  .cell_output {\n    border-radius: 0.25rem;\n\n    pre {\n      padding: 1rem;\n    }\n  }\n}\n","/* Styles for graphviz generated output from Sphinx */\n\n/* Style the inheritance diagram such that it has a dark mode */\nhtml[data-theme=\"dark\"] div.graphviz > object.inheritance {\n  filter: brightness(0.8) invert(0.82) contrast(1.2);\n  color-scheme: normal;\n}\n","/**\n * Special-cases for packages in the PyData ecosystem\n */\n\n// xarray output display in bootstrap\n.xr-wrap[hidden] {\n  display: block !important;\n}\n\n// ipywidgets\n.jp-OutputArea-output.lm-Widget {\n  // override overflow:hidden rule from Lumino (.lm-Widget) to allow scrolling\n  overflow: auto;\n}\n","/*******************************************************************************\n * Special-cases for the sphinx-design library, mainly to make it compatible\n * with the dark/light themes of pydata-sphinx-theme.\n *\n * NOTE: sphinx-design uses !important quite liberally, so here we must do the\n * same for our overrides to have any effect.\n */\n@use \"../variables/color\" as pst-color;\n@use \"sass:color\";\n@use \"sass:map\";\n@use \"sass:meta\";\n@use \"sass:string\";\n\n/*******************************************************************************\n  * Color and variables\n  *\n  * This is a list of the semantic color names from sphinx-design (we only\n  * need to override variables that sphinx-design has actually defined).\n  * https://github.com/executablebooks/sphinx-design/blob/9226a12a/style/_colors.scss#L31-L43\n  */\n$sd-semantic-color-names: (\n  \"primary\",\n  \"secondary\",\n  \"success\",\n  \"info\",\n  \"warning\",\n  \"danger\",\n  \"light\",\n  \"muted\",\n  \"dark\",\n  \"black\",\n  \"white\"\n);\n\n/**\n  * Here we create some extra --pst-color-* variables and use\n  * them to override the value of the corresponding sphinx-design variables.\n  * This is easier than re-writing the sphinx-design rules. Even easier would be\n  * directly assigning our values to the --sd-color-* variables, but then our\n  * downstream users couldn't override *our* colors and have it affect buttons\n  * and badges.\n  *\n  * First, define the extra keys needed to cover the full range of semantic\n  * color names used in sphinx-design, then merge them with the names we\n  * already define for our own needs.\n  * see https://sphinx-design.readthedocs.io/en/latest/css_variables.html\n  */\n$extra-semantic-colors: (\n  \"white\": $foundation-white,\n  \"light\": (\n    light: $foundation-light-gray,\n    bg-light: color.scale($foundation-light-gray, $lightness: 30%),\n    dark: $foundation-light-gray,\n    bg-dark: color.scale($foundation-light-gray, $lightness: -30%),\n  ),\n  \"muted\": (\n    light: $foundation-muted-gray,\n    bg-light: color.scale($foundation-muted-gray, $lightness: 30%),\n    dark: $foundation-light-gray,\n    bg-dark: color.scale($foundation-muted-gray, $lightness: -30%),\n  ),\n  \"dark\": $foundation-dark-gray,\n  \"black\": $foundation-black,\n);\n$all-colors: map.merge($pst-semantic-colors, $extra-semantic-colors);\n\n@mixin create-sd-colors($value, $name) {\n  // define the pst variables, so that downstream user overrides will work\n  --pst-color-#{$name}: #{$value};\n\n  // we are now using a11y-combination to calculate the text color - this is based\n  // on the WCAG color contrast guidelines\n  --pst-color-#{$name}-text: #{a11y-combination($value)};\n\n  // TODO: highlight seems to be used for buttons @trallard to fix on a11y follow-up work\n  --pst-color-#{$name}-highlight: #{color.adjust($value, $lightness: -15%)};\n\n  // override the sphinx-design variables\n  --sd-color-#{$name}: var(--pst-color-#{$name});\n  --sd-color-#{$name}-text: var(--pst-color-#{$name}-text);\n\n  // TODO: highlight seems to be used for buttons @trallard to fix on a11y follow-up work\n  --sd-color-#{$name}-highlight: var(--pst-color-#{$name}-highlight);\n}\n\n// Now we override the --sd-color-* variables.\n@each $mode in (light, dark) {\n  html[data-theme=\"#{$mode}\"] {\n    // check if this color is defined differently for light/dark\n    @each $name in $sd-semantic-color-names {\n      $definition: map.get($all-colors, $name);\n\n      @if meta.type-of($definition) == map {\n        @each $key, $value in $definition {\n          @if string.index($key, $mode) {\n            // since now we define the bg colours in the semantic colours and not\n            // by changing opacity, we need to check if the key contains bg and the\n            // correct mode (light/dark)\n            @if string.index($key, \"bg\") {\n              --sd-color-#{$name}-bg: #{$value};\n\n              // create local variable\n              $value: check-color($value);\n\n              --sd-color-#{$name}-bg-text: #{a11y-combination($value)};\n            } @else {\n              $value: check-color($value);\n\n              @include create-sd-colors($value, $name);\n            }\n          }\n        }\n      } @else {\n        $value: map.get($all-colors, $name);\n\n        @include create-sd-colors($value, $name);\n      }\n    }\n  }\n}\n\n// Make sure the color border variables are set using our variables\n@each $mode in (light, dark) {\n  html[data-theme=\"#{$mode}\"] {\n    --sd-color-card-border: var(--pst-color-border);\n  }\n}\n\n/*******************************************************************************\n  * shadows\n  */\nhtml[data-theme=\"light\"] {\n  .sd-shadow-xs,\n  .sd-shadow-sm,\n  .sd-shadow-md,\n  .sd-shadow-lg {\n    @include box-shadow;\n  }\n}\n\n/*******************************************************************************\n  * cards\n  */\n\n.bd-content .sd-card {\n  border: 1px solid var(--pst-color-border);\n\n  // TODO - --pst-color-panel-background is not defined... where is this coming from?\n  .sd-card-header {\n    background-color: var(--pst-color-panel-background);\n    border-bottom: 1px solid var(--pst-color-border);\n  }\n\n  .sd-card-footer {\n    background-color: var(--pst-color-panel-background);\n    border-top: 1px solid var(--pst-color-border);\n  }\n\n  .sd-card-body {\n    background-color: var(--pst-color-panel-background);\n  }\n\n  // Focus ring for link-cards\n  .sd-stretched-link:focus-visible {\n    // Don't put the focus ring on the <a> element (it has zero height in Sphinx Design cards)\n    outline: none;\n\n    // Put the focus ring on the <a> element's ::after pseudo-element\n    &::after {\n      outline: $focus-ring-outline;\n      border-radius: 0.25rem; // copied from Sphinx Design CSS for .sd-card\n    }\n  }\n\n  &.sd-card-hover:hover {\n    border-color: var(--pst-color-link-hover);\n  }\n}\n\n/*******************************************************************************\n  * tabs\n  */\n\n.bd-content .sd-tab-set {\n  > input {\n    // Active tab label\n    &:checked + label {\n      border-style: solid solid none;\n      border-color: var(--pst-color-primary) var(--pst-color-primary)\n        transparent; // top LR bottom\n\n      border-width: 0.125rem 0.125rem 0;\n      border-radius: 0.125rem 0.125rem 0 0;\n      background-color: var(--pst-color-on-background);\n      transform: translateY(0.125rem);\n      color: var(--pst-color-primary);\n    }\n\n    &:focus-visible + label {\n      border: 0.125rem solid var(--pst-color-accent);\n      border-radius: 0.125rem;\n      background-color: var(--pst-color-accent-bg);\n      color: var(--pst-color-on-surface);\n    }\n\n    // Hover label\n    &:not(:checked, :focus-visible) + label:hover {\n      border-color: transparent;\n      color: var(--pst-color-secondary);\n      text-decoration-line: underline;\n      text-decoration-thickness: $link-hover-decoration-thickness;\n    }\n  }\n\n  // Tab label\n  > label {\n    color: var(--pst-color-on-surface);\n    border: 0.125rem solid transparent;\n    border-radius: 0.125rem 0.125rem 0 0;\n    background-color: var(--pst-color-surface);\n    padding: 0 0.75em;\n    margin-inline-end: 0.25rem;\n    line-height: 1.95;\n\n    html[data-theme=\"dark\"] & {\n      background-color: var(--pst-color-on-background);\n    }\n  }\n\n  // panel\n  .sd-tab-content {\n    border: 0.125rem solid var(--pst-color-primary);\n    border-radius: 0.1875rem;\n    box-shadow: unset;\n    padding: 0.625rem;\n    background-color: var(--pst-color-on-background);\n  }\n}\n\n/*******************************************************************************\n* Dropdowns\n*/\n\ndetails.sd-dropdown {\n  // Remove all borders to over-ride SD behavior, and we'll add our own later\n  border: 0 !important;\n\n  // Drop shadow should behave same as admonitions\n  @include box-shadow;\n\n  // Header is where the \"clickable\" box goes\n  summary.sd-card-header {\n    border: 0 !important;\n    display: flex;\n    align-items: center;\n    position: relative; // So background color works\n    font-weight: 600;\n    padding-top: 0.5rem;\n    padding-bottom: 0.5rem;\n\n    // Set a variable that we can re-use for colors later\n    // We must set this in the current and content sibling container\n    // so that it is defined in both places\n    --pst-sd-dropdown-color: var(--pst-gray-500);\n    --pst-sd-dropdown-bg-color: var(--pst-color-surface);\n\n    & + div.sd-summary-content {\n      border: 0;\n\n      --pst-sd-dropdown-color: var(--sd-color-card-border);\n    }\n\n    @each $name in $sd-semantic-color-names {\n      &.sd-bg-#{$name} {\n        --pst-sd-dropdown-color: var(--sd-color-#{$name});\n        --pst-sd-dropdown-bg-color: var(--sd-color-#{$name}-bg);\n\n        // Otherwise it won't be defined in the sibling element\n        & + div.sd-summary-content {\n          --pst-sd-dropdown-color: var(--sd-color-#{$name});\n          --pst-sd-dropdown-bg-color: var(--sd-color-#{$name}-bg);\n        }\n      }\n      &.sd-bg-text-#{$name} {\n        // Use the WCAG conformant text color\n        color: var(--sd-color-#{$name}-bg-text) !important;\n      }\n    }\n\n    @include legacy-backdrop-placeholder;\n\n    background-color: var(--pst-sd-dropdown-bg-color) !important;\n\n    // Add a left border with the same structure as our admonitions\n    border-left: 0.2rem solid var(--pst-sd-dropdown-color) !important;\n\n    // stylelint-disable-next-line no-duplicate-selectors\n    & + div.sd-summary-content {\n      border-left: 0.2rem solid var(--pst-sd-dropdown-color) !important;\n      border-bottom-left-radius: calc(0.25rem - 1px);\n      background-color: var(--pst-color-on-background);\n    }\n\n    span.sd-summary-icon {\n      display: inline-flex;\n      align-items: center;\n      color: var(--pst-sd-dropdown-color) !important;\n\n      svg {\n        opacity: 1;\n      }\n    }\n\n    // Positioning of the caret\n    .sd-summary-up,\n    .sd-summary-down {\n      top: 0.7rem;\n    }\n\n    // Focus ring\n    &:focus-visible {\n      outline: $focus-ring-outline;\n      outline-offset: -$focus-ring-width;\n    }\n  }\n}\n\n/*******************************************************************************\n* Buttons (which in Sphinx Design are actually links that look like buttons)\n* ref: https://sphinx-design.readthedocs.io/en/pydata-theme/badges_buttons.html#buttons\n*/\nhtml {\n  .sd-btn {\n    min-width: 2.25rem;\n    padding: 0.3125rem 0.75rem 0.4375rem; // 5px 12px 7px\n\n    &:hover {\n      @include link-style-hover; // override Sphinx Design\n\n      text-decoration-thickness: 1px;\n    }\n  }\n\n  @each $name in $sd-semantic-color-names {\n    .sd-btn-#{$name},\n    .sd-btn-outline-#{$name} {\n      &:focus-visible {\n        // Override Sphinx Design's use of -highlight colors. The -highlight\n        // colors are 15% darker, so this would create the effect of darkening\n        // the button when focused but we just want the button to have a focus\n        // ring of the same (non-highlight) color.\n        background-color: var(--sd-color-#{$name}) !important;\n        border-color: var(--sd-color-#{$name}) !important;\n        outline: var(--sd-color-#{$name}) solid $focus-ring-width;\n        outline-offset: $focus-ring-width;\n      }\n    }\n  }\n}\n","/**\n * Sphinx togglebutton\n */\n\n.bd-content {\n  @mixin chevron-down {\n    .toggle-chevron-right {\n      transform: rotate(90deg); // point chevron down\n      transition: none; // match non-animated behavior of other chevrons on site\n    }\n  }\n\n  @mixin chevron-up {\n    .toggle-chevron-right {\n      transform: rotate(-90deg); // point chevron up\n      transition: none; // match non-animated behavior of other chevrons on site\n    }\n  }\n\n  // Admonition toggles\n  .admonition {\n    button.toggle-button {\n      color: inherit;\n\n      // When disclosure widget is closed\n      &.toggle-button-hidden {\n        @include chevron-down;\n      }\n\n      // When open\n      @include chevron-up;\n    }\n\n    // Focus ring\n    // ----------\n    // Sphinx-togglebutton makes the entire admonition header clickable, but\n    // only the button within the header is focusable. We want the entire\n    // clickable area to be surrounded with a focus ring, so that's why we use\n    // the :focus-within selector, rather than a :focus-visible selector on the\n    // button.\n    &:focus-within {\n      overflow: visible;\n\n      // The complicated focus ring styles here are a consequence of the markup\n      // and border styles for this particular admonition class. (For the other\n      // type of admonition on this site, the focus ring style is achieved with\n      // simple `outline` and `outline-offset` rules on the admonition's\n      // header.) The problem is that Sphinx-togglebutton puts the admonition's\n      // left border on the outermost container (rather than separately setting\n      // the left border on the container's children). This makes it complicated\n      // to get the focus ring to simultaneously cover the left border in the\n      // header and align perfectly on the right with the body.\n      .admonition-title:focus-within::before {\n        content: \"\";\n        transform: translateX(\n          -$admonition-left-border-width\n        ); // align left edges of admonition and ring\n\n        width: calc(100% + $admonition-left-border-width); // align right edges\n        height: 100%;\n        border: $focus-ring-outline;\n        border-radius: $focus-ring-width;\n      }\n\n      // When expanded, sharpen the bottom left and right corners of the focus ring\n      &:not(.toggle-hidden) .admonition-title:focus-within::before {\n        border-bottom-left-radius: 0;\n        border-bottom-right-radius: 0;\n      }\n    }\n  }\n\n  // Details buttons\n  details.toggle-details {\n    // Over-ride border color to re-use our primary color\n    summary {\n      border-left: 3px solid var(--pst-color-primary);\n\n      @include chevron-down;\n    }\n\n    // When expanded, sharpen the bottom left and right corners of the focus ring\n    &[open] {\n      @include chevron-up;\n\n      :focus-visible {\n        border-bottom-left-radius: 0;\n        border-bottom-right-radius: 0;\n      }\n    }\n  }\n}\n","/**\n * Styles for various Sphinx execution libraries to display pre-executed notebooks.\n * For now, where these define output sections, we simply revert their background\n * to be a \"light theme\" background. This ensures that inputs/outputs behave similarly,\n * because the CSS is often controlled by each package.\n * In the future, we might add dark theme support for specific packages.\n */\n\n/*******************************************************************************\n * nbsphinx\n */\nhtml div.rendered_html,\n// NBsphinx ipywidgets output selector\nhtml .jp-RenderedHTMLCommon {\n  // Add some margin around the element box for the focus ring. Otherwise the\n  // focus ring gets clipped because the containing elements have `overflow:\n  // hidden` applied to them (via the `.lm-Widget` selector)\n  margin: $focus-ring-width;\n\n  table {\n    table-layout: auto;\n  }\n}\n\n.bd-content .nboutput {\n  .output_area {\n    &.rendered_html,\n    .jp-RenderedHTMLCommon {\n      // pandas\n      table.dataframe {\n        @include table-colors;\n      }\n    }\n\n    // Dark theme special-cases\n    html[data-theme=\"dark\"] & {\n      &.rendered_html:not(:has(table.dataframe)),\n      // ipywidgets\n      .widget-subarea {\n        @include cell-output-background;\n      }\n\n      &.stderr {\n        background-color: var(--pst-color-danger);\n      }\n    }\n  }\n}\n\n// Add extra padding to the final item in an nbsphinx container\ndiv.nblast.container {\n  margin-bottom: 1rem;\n}\n\n/*******************************************************************************\n * myst NB\n */\n\ndiv.cell_output .output {\n  max-width: 100%;\n  overflow-x: auto;\n}\n\n.bd-content div.cell_output {\n  // pandas\n  table.dataframe {\n    @include table-colors;\n  }\n\n  html[data-theme=\"dark\"] & {\n    img,\n    .text_html:not(:has(table.dataframe)),\n    // ipywidgets\n    .widget-subarea {\n      @include cell-output-background;\n    }\n  }\n}\n\n// Prevent tables from scrunching together\n.bd-content {\n  div.cell_input {\n    display: flex;\n    flex-direction: column;\n    justify-content: stretch;\n  }\n\n  div.cell_input,\n  div.output {\n    border-radius: $admonition-border-radius;\n  }\n\n  div.output {\n    table {\n      table-layout: auto;\n    }\n  }\n}\n","/**\n * style for the various mapping libs based on leaflet (folium, geemap, ipyleaflet)\n * mainly ensure the good display of the maps in both themes and avoid the customization\n * of the tiles\n */\n\n/**\n * avoid border override from pydata-sphinx-theme\n * minimal selctor to get the priority\n */\nhtml[data-theme=\"dark\"] .bd-content img.leaflet-tile.leaflet-tile-loaded {\n  border-radius: 0;\n  padding: 0;\n}\n","/**\n * /search.html page special-cases\n */\n\n.bd-search-container {\n  /*******************************************\n  * Search results\n  */\n  // Whitespace\n  div#search-results {\n    > h2 {\n      font-size: var(--pst-font-size-icon);\n      margin-top: 1rem;\n    }\n\n    p.search-summary {\n      color: var(--pst-color-text-muted);\n    }\n  }\n\n  ul.search {\n    margin: 0;\n    list-style: none;\n\n    li {\n      background-image: none;\n      padding: 1rem 0;\n      margin: 1rem 0;\n      border-top: 1px solid var(--pst-color-text-muted);\n\n      // First link is the page title, it should be a bit bigger\n      > a {\n        font-size: 1.2em;\n      }\n\n      div.context,\n      p.context {\n        color: var(--pst-color-text-base);\n        margin: 0.5em 0 0;\n\n        // Add a # before page section titles to make it clear they are sections\n        a::before {\n          content: \"#\";\n          padding-right: 0.2em;\n          color: var(--pst-color-text-muted);\n        }\n      }\n    }\n  }\n}\n"],"names":[],"sourceRoot":""}
\ No newline at end of file
diff --git a/dev/docs/html/_static/styles/sphinx-book-theme.css b/dev/docs/html/_static/styles/sphinx-book-theme.css
new file mode 100644
index 0000000..5197540
--- /dev/null
+++ b/dev/docs/html/_static/styles/sphinx-book-theme.css
@@ -0,0 +1,9 @@
+/*! sphinx-book-theme CSS
+ * BSD 3-Clause License
+ * Copyright (c) 2020, EBP
+ * All rights reserved.
+ *
+ * This follows the 7-1 pattern described here:
+ * https://sass-guidelin.es/#architecture
+ */html[data-theme=dark],html[data-theme=light]{--pst-font-size-base:none;--pst-color-secondary:#e89217}html[data-theme=light]{--sbt-color-announcement:#616161;--pst-color-primary:#176de8}html[data-theme=dark]{--pst-color-primary:#528fe4;--sbt-color-announcement:#616161;--pst-color-background:#121212}html{scroll-padding-top:4rem}.sbt-scroll-pixel-helper{height:0;left:0;position:absolute;top:0;width:0}.d-n,.onlyprint{display:none}@media print{.onlyprint{display:block!important}.noprint{display:none!important}}.bd-article-container h1,.bd-article-container h2,.bd-article-container h3,.bd-article-container h4,.bd-article-container h5,.bd-article-container p.caption{color:var(--pst-color-muted)}.bd-article-container h1,.bd-article-container h2{font-weight:500}a.brackets:before{color:inherit;font-family:inherit;margin-right:0}table{position:relative}@media print{.bd-main .bd-content{height:auto;margin-left:2rem}.bd-main .bd-content #jb-print-docs-body{margin-left:0}.bd-main .bd-content #jb-print-docs-body h1{font-size:3em;margin-bottom:0;text-align:center}.bd-main .bd-content .bd-article{padding-top:0}.bd-main .bd-content .bd-article h1:first-of-type{display:none}.bd-main .bd-content .container{min-width:0!important}.bd-main .bd-content h1{margin-bottom:1em;margin-top:1em}.bd-main .bd-content h1,.bd-main .bd-content h2,.bd-main .bd-content h3,.bd-main .bd-content h4{break-after:avoid;color:#000}.bd-main .bd-content table{break-inside:avoid}.bd-main .bd-content pre{word-wrap:break-word}.bd-main .bd-content a.headerlink{display:none}.bd-main .bd-content aside.margin,.bd-main .bd-content aside.sidebar,.bd-main .bd-content blockquote.epigraph{border:none}.bd-main .bd-content .footer{margin-top:1em}.bd-main .bd-content #jb-print-toc{margin-bottom:1.5rem;margin-left:0}.bd-main .bd-content #jb-print-toc .section-nav{border-left:0!important;list-style-type:disc!important;margin-left:3em!important}.bd-main .bd-content #jb-print-toc .section-nav 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var(--pst-color-shadow),\n    0 0 0.0625rem var(--pst-color-shadow) !important;\n}\n",".bd-header {\n  // Turn off sticky positioning so that it scrolls\n  position: inherit;\n\n  // Turn these off because we'll show it in the article header\n  // The primary sidebar toggle\n  button.sidebar-toggle {\n    display: none;\n  }\n}\n","/*********************************************\n* Left Nav Bar *\n*********************************************/\n\n.bd-sidebar-primary {\n  // So that it sticks to the top of the page instead of the header height\n  top: 0;\n  max-height: 100vh;\n\n  // Tighter padding so the spacing is more equal\n  padding: 1rem;\n\n  // Draw open/close animation\n  transition:\n    margin-left $animation-time ease 0s,\n    opacity $animation-time ease 0s,\n    visibility $animation-time ease 0s;\n\n  @media (max-width: $breakpoint-md) {\n    // So that tooltips don't overlap\n    z-index: $zindex-tooltip + 1;\n  }\n\n  // Remove the top border since there's often no header items above\n  .sidebar-primary-items__start {\n    border-top: none;\n  }\n\n  // On wide screens we have a smaller sidebar width than PST\n  @media (min-width: $breakpoint-lg) {\n    flex-basis: $sidebar-primary-width-widescreen;\n  }\n}\n\n// Default visibility for left sidebar is the reverse of what it is on mobile.\n// It is shown by default, and hidden with a click.\n// Mobile behavior is defined in the pydata sphinx theme\n@media (min-width: $breakpoint-lg) {\n  input#pst-primary-sidebar-checkbox:checked\n    ~ .bd-container\n    .bd-sidebar-primary {\n    margin-left: -$sidebar-primary-width-widescreen;\n    visibility: hidden;\n    opacity: 0;\n  }\n}\n","/*********************************************\n* Table of Contents (right nav bar)          *\n*********************************************/\n\n.bd-sidebar-secondary {\n  // So that it sticks to the top of the page instead of the header height\n  top: 0;\n\n  @media (max-width: $breakpoint-xl) {\n    // So that tooltips don't overlap\n    z-index: $zindex-tooltip + 1;\n  }\n\n  // So that we can center the TOC button with the article header buttons\n  .sidebar-secondary-items {\n    padding: 0;\n    display: flex;\n    gap: 0.5rem;\n\n    // Remove the manual padding since we'll use `gap`\n    .sidebar-secondary-item {\n      padding-top: 0;\n      padding-bottom: 0;\n    }\n  }\n\n  // The 'on this page' title div should take up the full header\n  .onthispage {\n    height: $header-article-height;\n    min-height: $header-article-height;\n    display: flex;\n    gap: 0.5rem;\n    align-items: center;\n    margin: 0;\n    color: var(--pst-color-muted);\n  }\n\n  // Wide screen behavior\n  @media (min-width: $breakpoint-xl) {\n    background: var(--pst-color-background);\n    height: fit-content;\n    transition: max-height 0.4s ease;\n\n    // To make sure it shows above the page content\n    z-index: $zindex-middle;\n\n    // Remove padding so that it's flush on the top and left\n    padding: 0;\n\n    // This is the container div for the secondary sidebar nav\n    .toc-item {\n      // Border is a bit less prominent\n      border-left-color: var(--pst-color-surface);\n      // Flush with the top so that we can fix the height of the toc header div\n      padding-top: 0;\n\n      // The table of contents <ul> after the title div\n      nav.page-toc {\n        // Remove extra margin the PST adds for other sidebar items\n        margin-bottom: 0;\n\n        // Items after \"on this page' should fade in and out\n        transition: opacity 0.4s ease;\n      }\n    }\n\n    // Rules for when sidebar is hidden\n    // the `not:hover` means that hovering over the sidebar will show it again.\n    &.hide:not(:hover) {\n      // Cap the height, and remove the scrollbar\n      max-height: $header-article-height;\n      overflow-y: hidden;\n\n      // Shadow matches what we have in the article header\n      .scrolled & {\n        box-shadow: 0 6px 6px -6px rgba(0, 0, 0, 0.3);\n      }\n\n      // Show a down carot\n      .onthispage:after {\n        opacity: 1;\n        content: \"\\f107\";\n        font-family: \"Font Awesome 5 Free\";\n        font-weight: 900;\n        padding-left: 0.5em;\n        transition: opacity 0.3s ease;\n      }\n\n      // The page-toc list is hidden\n      nav.page-toc {\n        opacity: 0;\n      }\n    }\n  }\n}\n","/*********************************************\n* Footer - content *\n*********************************************/\nfooter {\n  font-size: var(--sbt-font-size-small-1);\n  &.bd-footer-content {\n    display: flex;\n    flex-wrap: wrap;\n    padding: 15px;\n    border-top: 1px solid #ccc;\n    font-size: $sbt-font-size-small-1;\n    .bd-footer-content__inner {\n      padding-left: 0px;\n      p {\n        margin-bottom: 0px;\n      }\n    }\n  }\n}\n",".bd-footer-article {\n  padding: 0rem 1rem;\n  @media (max-width: $breakpoint-md) {\n    // More room for text on mobile\n    padding: 0rem 0.5rem;\n  }\n}\n","#pst-back-to-top {\n  // We should only display the back-to-top button when the TOC is not available.\n  // AKA, on narrow-ish screens.\n  // This over-rides a PST default\n  @media (min-width: $breakpoint-xl) {\n    display: none !important;\n  }\n\n  // Reduce size to be slightly less intrusive on the page\n  font-size: 0.8rem;\n}\n","/**\n * Icon links design for the primary sidebar, where it defaults in this theme.\n */\n.bd-sidebar-primary {\n  .navbar-icon-links {\n    column-gap: 0.5rem;\n\n    .nav-link {\n      // There are few kinds of elements that can be icon links and each is different\n      i,\n      span {\n        font-size: 1.2rem;\n      }\n\n      // Images usually fill more vertical space so we make them a bit smaller\n      img {\n        font-size: 0.8rem;\n      }\n    }\n  }\n}\n",".navbar-brand {\n  height: unset;\n  max-height: unset;\n\n  // Auto-center the image and title and make items vertical\n  flex-direction: column;\n  justify-content: center;\n  gap: 0.25rem;\n\n  // To prevent site text from having underlines\n  &:hover {\n    text-decoration: none;\n  }\n  .logo__title {\n    font-size: 1.25rem;\n    white-space: normal;\n    text-align: center;\n  }\n  // Prevent images from stretching vertically\n  .logo__image {\n    height: unset;\n  }\n}\n","// Clean up the search page so that it has less unnecessary text\n.bd-search-container {\n  margin: 2em;\n\n  #search-results {\n    h2:first-child {\n      display: none;\n    }\n  }\n}\n\n// Sidebar search button field\n.search-button-field {\n  width: 100%;\n  font-size: 0.9rem;\n  // Hide by default and we'll add a rule to show it on wide sreen below\n  display: none;\n\n  .search-button__kbd-shortcut {\n    margin-left: auto;\n  }\n}\n\n// Rules to switch off visibility of the field button and the header button\n@media (min-width: $breakpoint-lg) {\n  .search-button {\n    display: none !important;\n  }\n\n  .search-button-field {\n    display: flex;\n  }\n}\n\n// The \"clear search results\" button\n// Make it line up with our content better\ndiv#searchbox {\n  padding-right: 2rem;\n  padding-left: 2rem;\n  @media (max-width: $breakpoint-md) {\n    padding-right: 1rem;\n    padding-left: 1rem;\n  }\n\n  p.highlight-link {\n    // Remove the extra margin that is added\n    @media (min-width: $breakpoint-md) {\n      margin-left: 0;\n      a {\n        font-size: 1rem;\n      }\n    }\n  }\n}\n","// Images\nimg {\n  max-width: 100%;\n\n  &.align-center {\n    margin-left: auto;\n    margin-right: auto;\n    display: block;\n  }\n\n  &.align-left {\n    clear: left;\n    float: left;\n    margin-right: 1em;\n  }\n\n  &.align-right {\n    clear: right;\n    float: right;\n    margin-left: 1em;\n  }\n}\n\n// Figures\ndiv.figure {\n  width: 100%;\n  margin-bottom: 1em;\n  text-align: center;\n\n  &.align-left {\n    text-align: left;\n    p.caption {\n      margin-left: 0;\n    }\n  }\n  &.align-right {\n    text-align: right;\n    p.caption {\n      margin-right: 0;\n    }\n  }\n\n  p.caption {\n    margin: 0.5em 10%;\n  }\n\n  &.margin,\n  &.margin-caption {\n    p.caption {\n      margin: 0.5em 0;\n    }\n  }\n\n  &.margin-caption p.caption {\n    text-align: left;\n  }\n\n  span.caption-number {\n    font-weight: bold;\n  }\n\n  span {\n    font-size: 0.9rem;\n  }\n}\n","/**\n * Margin content\n * This is the area to the right of the main content\n * normally covered by the in-page table of contents\n * but if `.margin` classes are present, it shows up to the right in the margin\n * and the in-page TOC is hidden.\n * On narrow screens, margin content behaves like a sidebar\n */\n\n/**\n * Variables for calculating margin widths\n */\n// We want our margin content to be 1/3 the width of the main page content\n// after factoring in a small margin\n$margin-width-relative-to-content: 36%;\n$margin-gutter: 3%;\n$margin-width-width-gutter: $margin-width-relative-to-content - $margin-gutter;\n\n// re-subtract the gutter width because we want it to leave a white-space\n// This is how much to offset the margin content.\n$margin-offset: -$margin-width-relative-to-content;\n\n/**\n * This mixin will cause something to \"pop out\" to the margin on wide screens\n */\n@mixin margin-content() {\n  // On narrow screens this is the width of margin content\n  width: 40%;\n  float: right;\n  background-color: unset;\n  font-size: 0.9em;\n  margin-left: 0.5rem;\n\n  // Wide screens and printing should force margin behavior\n  @media (min-width: $breakpoint-lg), print {\n    width: $margin-width-width-gutter;\n    margin: 0 $margin-offset 0 0;\n\n    // Prevent sequential margin content from overlapping\n    clear: right;\n\n    p.sidebar-title {\n      margin-bottom: -1rem;\n      border-bottom: none;\n      padding-left: 0px;\n\n      // Content of admonitions has fewer horizontal white space to save space\n      ~ * {\n        padding-left: 0px;\n        padding-right: 0px;\n      }\n    }\n  }\n}\n\n/**\n * Sidenotes and marginnotes\n * over-rides the \"footnote\" behavior to pop out to the margin instead.\n */\nlabel.margin-toggle {\n  margin-bottom: 0em;\n  &.marginnote-label {\n    display: none;\n  }\n  sup {\n    user-select: none;\n  }\n  @media (max-width: $breakpoint-lg) {\n    cursor: pointer;\n    color: rgb(0, 113, 188);\n    &.marginnote-label {\n      display: inline;\n      &:after {\n        content: \"\\2295\";\n      }\n    }\n  }\n}\n\ninput.margin-toggle {\n  display: none;\n  @media (max-width: $breakpoint-lg) {\n    &:checked + .sidenote,\n    &:checked + .marginnote {\n      display: block;\n      float: left;\n      left: 1rem;\n      clear: both;\n      width: 95%;\n      margin: 1rem 2.5%;\n      position: relative;\n    }\n  }\n}\n\nspan.sidenote,\nspan.marginnote {\n  z-index: $zindex-middle;\n  position: relative;\n  sup {\n    user-select: none;\n  }\n  @include margin-content();\n  border-left: none;\n  @media (max-width: $breakpoint-lg) {\n    display: none;\n  }\n}\n\naside.sidebar {\n  .note {\n    // styling for notes inside sidebars\n    margin: 1rem;\n    padding: 0rem 0rem 1rem 0rem;\n  }\n  // The titles sticking\n  .admonition-title {\n    margin: 0rem -1rem 0rem 0rem;\n  }\n}\n\n/**\n * Sidebar content in the margin.\n * This will be added by the `margin` directive or the `sidebar` directive\n */\naside.sidebar.margin {\n  // Margin content can have an empty title so this prevents an empty title block\n  // from showing up on mobile\n  .sidebar-title:empty {\n    display: none;\n  }\n\n  // On narrow screens padding is added to all sidebar content\n  // This replaces padding w/ margin for admonitions because padding is used\n  // for the title background color\n  .admonition {\n    margin: 0.5rem;\n    padding-left: 0;\n    padding-right: 0;\n\n    // Remove the title margin so it\n    .admonition-title {\n      margin-left: 0;\n      margin-right: 0;\n    }\n  }\n\n  @media (min-width: $breakpoint-lg) {\n    // No border for margin sidebars on wide screen\n    border: none;\n\n    // Admonitions don't need the extra whitespace in the margin\n    .admonition {\n      margin: 1rem 0rem;\n      padding: 0rem 0rem 1rem 0rem;\n    }\n  }\n}\n\ndiv.margin,\nfigure.margin,\naside.margin,\n.margin.docutils.container,\n.cell.tag_popout,\n.cell.tag_margin {\n  z-index: $zindex-middle;\n  position: relative;\n  @include margin-content();\n\n  // Make cell outputs take up more space if they're in the margin\n  div.cell.tag_margin .cell_output {\n    padding-left: 0;\n  }\n}\n\n// A few permutations because docutils 0.18 switched to semantic tags\ndiv.figure.margin-caption p.caption,\ndiv.figure.margin-caption figcaption,\nfigure.margin-caption figcaption {\n  z-index: $zindex-middle;\n  position: relative;\n  @include margin-content();\n}\n\n// Margin captions\n.margin-caption figcaption {\n  text-align: left;\n}\n\n/**\n * Full width content\n */\n\n// Grow 100% by the ratio of the margin to the content width.\n$content-fullwidth-width: 100% + $margin-width-relative-to-content;\n\ndiv.cell.tag_full-width,\ndiv.cell.tag_full_width,\ndiv.full_width,\ndiv.full-width {\n  z-index: $zindex-middle;\n  position: relative;\n  @media (min-width: $breakpoint-lg) {\n    max-width: $content-fullwidth-width;\n    width: $content-fullwidth-width;\n  }\n}\n","/**\n * Epigraphs and other \"special\" quote blocks\n */\n\nblockquote {\n  &.pull-quote,\n  &.epigraph,\n  &.highlights {\n    font-size: 1.25em;\n    border-left: none;\n    background-color: var(--pst-color-background);\n  }\n\n  div > p + p.attribution {\n    font-style: normal;\n    font-size: 0.9em;\n    text-align: right;\n    color: #6c757d;\n    padding-right: 2em;\n  }\n}\n","// Over-ride Sphinx default so that content fills whitespace w/ margin content\npre,\ndiv[class*=\"highlight-\"] {\n  clear: none;\n}\n","// MyST-NB and Jupyter Notebooks\ndiv.cell.tag_output_scroll div.cell_output,\ndiv.cell.tag_scroll-output div.cell_output {\n  max-height: 24em;\n  overflow-y: auto;\n}\n\ndiv.cell.tag_scroll-input div.cell_input {\n  max-height: 24em;\n  overflow-y: auto;\n}\n","/**\n * Comment JavaScript libraries\n  * Libraries supported here are:\n  * - Hypothes.is : https://web.hypothes.is/\n  * - Utteranc.es : https://utteranc.es/\n  */\n// In general we want to hide any comment libraries on print\n@media only print {\n  hypothesis-sidebar,\n  div.utterances {\n    display: none;\n  }\n}\n","/*********************************************\n* Thebe *\n* ref: https://github.com/executablebooks/thebe\n*********************************************/\n.thebelab-cell {\n  border: none !important;\n  margin-right: 0.5em !important;\n}\n\n.thebelab-cell .thebelab-input {\n  padding-left: 10px !important;\n}\n\n.cell.docutils.container {\n  padding-right: 0px !important;\n}\n\nbutton.thebe-launch-button {\n  height: 2.5em;\n  font-size: 1em;\n}\n"],"names":[],"sourceRoot":""}
\ No newline at end of file
diff --git a/dev/docs/html/_static/styles/theme.css b/dev/docs/html/_static/styles/theme.css
new file mode 100644
index 0000000..4519dd9
--- /dev/null
+++ b/dev/docs/html/_static/styles/theme.css
@@ -0,0 +1,2 @@
+/* Provided by Sphinx's 'basic' theme, and included in the final set of assets */
+@import "../basic.css";
diff --git a/dev/docs/html/_static/table1.png b/dev/docs/html/_static/table1.png
new file mode 100644
index 0000000..c02f860
Binary files /dev/null and b/dev/docs/html/_static/table1.png differ
diff --git a/dev/docs/html/_static/vendor/fontawesome/6.5.2/LICENSE.txt b/dev/docs/html/_static/vendor/fontawesome/6.5.2/LICENSE.txt
new file mode 100644
index 0000000..e69c5e3
--- /dev/null
+++ b/dev/docs/html/_static/vendor/fontawesome/6.5.2/LICENSE.txt
@@ -0,0 +1,165 @@
+Fonticons, Inc. (https://fontawesome.com)
+
+--------------------------------------------------------------------------------
+
+Font Awesome Free License
+
+Font Awesome Free is free, open source, and GPL friendly. You can use it for
+commercial projects, open source projects, or really almost whatever you want.
+Full Font Awesome Free license: https://fontawesome.com/license/free.
+
+--------------------------------------------------------------------------------
+
+# Icons: CC BY 4.0 License (https://creativecommons.org/licenses/by/4.0/)
+
+The Font Awesome Free download is licensed under a Creative Commons
+Attribution 4.0 International License and applies to all icons packaged
+as SVG and JS file types.
+
+--------------------------------------------------------------------------------
+
+# Fonts: SIL OFL 1.1 License
+
+In the Font Awesome Free download, the SIL OFL license applies to all icons
+packaged as web and desktop font files.
+
+Copyright (c) 2024 Fonticons, Inc. (https://fontawesome.com)
+with Reserved Font Name: "Font Awesome".
+
+This Font Software is licensed under the SIL Open Font License, Version 1.1.
+This license is copied below, and is also available with a FAQ at:
+http://scripts.sil.org/OFL
+
+SIL OPEN FONT LICENSE
+Version 1.1 - 26 February 2007
+
+PREAMBLE
+The goals of the Open Font License (OFL) are to stimulate worldwide
+development of collaborative font projects, to support the font creation
+efforts of academic and linguistic communities, and to provide a free and
+open framework in which fonts may be shared and improved in partnership
+with others.
+
+The OFL allows the licensed fonts to be used, studied, modified and
+redistributed freely as long as they are not sold by themselves. The
+fonts, including any derivative works, can be bundled, embedded,
+redistributed and/or sold with any software provided that any reserved
+names are not used by derivative works. The fonts and derivatives,
+however, cannot be released under any other type of license. The
+requirement for fonts to remain under this license does not apply
+to any document created using the fonts or their derivatives.
+
+DEFINITIONS
+"Font Software" refers to the set of files released by the Copyright
+Holder(s) under this license and clearly marked as such. This may
+include source files, build scripts and documentation.
+
+"Reserved Font Name" refers to any names specified as such after the
+copyright statement(s).
+
+"Original Version" refers to the collection of Font Software components as
+distributed by the Copyright Holder(s).
+
+"Modified Version" refers to any derivative made by adding to, deleting,
+or substituting — in part or in whole — any of the components of the
+Original Version, by changing formats or by porting the Font Software to a
+new environment.
+
+"Author" refers to any designer, engineer, programmer, technical
+writer or other person who contributed to the Font Software.
+
+PERMISSION & CONDITIONS
+Permission is hereby granted, free of charge, to any person obtaining
+a copy of the Font Software, to use, study, copy, merge, embed, modify,
+redistribute, and sell modified and unmodified copies of the Font
+Software, subject to the following conditions:
+
+1) Neither the Font Software nor any of its individual components,
+in Original or Modified Versions, may be sold by itself.
+
+2) Original or Modified Versions of the Font Software may be bundled,
+redistributed and/or sold with any software, provided that each copy
+contains the above copyright notice and this license. These can be
+included either as stand-alone text files, human-readable headers or
+in the appropriate machine-readable metadata fields within text or
+binary files as long as those fields can be easily viewed by the user.
+
+3) No Modified Version of the Font Software may use the Reserved Font
+Name(s) unless explicit written permission is granted by the corresponding
+Copyright Holder. This restriction only applies to the primary font name as
+presented to the users.
+
+4) The name(s) of the Copyright Holder(s) or the Author(s) of the Font
+Software shall not be used to promote, endorse or advertise any
+Modified Version, except to acknowledge the contribution(s) of the
+Copyright Holder(s) and the Author(s) or with their explicit written
+permission.
+
+5) The Font Software, modified or unmodified, in part or in whole,
+must be distributed entirely under this license, and must not be
+distributed under any other license. The requirement for fonts to
+remain under this license does not apply to any document created
+using the Font Software.
+
+TERMINATION
+This license becomes null and void if any of the above conditions are
+not met.
+
+DISCLAIMER
+THE FONT SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTIES OF
+MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT
+OF COPYRIGHT, PATENT, TRADEMARK, OR OTHER RIGHT. IN NO EVENT SHALL THE
+COPYRIGHT HOLDER BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
+INCLUDING ANY GENERAL, SPECIAL, INDIRECT, INCIDENTAL, OR CONSEQUENTIAL
+DAMAGES, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
+FROM, OUT OF THE USE OR INABILITY TO USE THE FONT SOFTWARE OR FROM
+OTHER DEALINGS IN THE FONT SOFTWARE.
+
+--------------------------------------------------------------------------------
+
+# Code: MIT License (https://opensource.org/licenses/MIT)
+
+In the Font Awesome Free download, the MIT license applies to all non-font and
+non-icon files.
+
+Copyright 2024 Fonticons, Inc.
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of
+this software and associated documentation files (the "Software"), to deal in the
+Software without restriction, including without limitation the rights to use, copy,
+modify, merge, publish, distribute, sublicense, and/or sell copies of the Software,
+and to permit persons to whom the Software is furnished to do so, subject to the
+following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED,
+INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A
+PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
+HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
+OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
+SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+
+--------------------------------------------------------------------------------
+
+# Attribution
+
+Attribution is required by MIT, SIL OFL, and CC BY licenses. Downloaded Font
+Awesome Free files already contain embedded comments with sufficient
+attribution, so you shouldn't need to do anything additional when using these
+files normally.
+
+We've kept attribution comments terse, so we ask that you do not actively work
+to remove them from files, especially code. They're a great way for folks to
+learn about Font Awesome.
+
+--------------------------------------------------------------------------------
+
+# Brand Icons
+
+All brand icons are trademarks of their respective owners. The use of these
+trademarks does not indicate endorsement of the trademark holder by Font
+Awesome, nor vice versa. **Please do not use brand logos for any purpose except
+to represent the company, product, or service to which they refer.**
diff --git a/dev/docs/html/_static/vendor/fontawesome/6.5.2/css/all.min.css b/dev/docs/html/_static/vendor/fontawesome/6.5.2/css/all.min.css
new file mode 100644
index 0000000..628b2cc
--- /dev/null
+++ b/dev/docs/html/_static/vendor/fontawesome/6.5.2/css/all.min.css
@@ -0,0 +1,5 @@
+/*!
+ * Font Awesome Free 6.5.2 by @fontawesome - https://fontawesome.com
+ * License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License)
+ * Copyright 2024 Fonticons, Inc.
+ */.fa{font-family:var(--fa-style-family,"Font Awesome 6 Free");font-weight:var(--fa-style,900)}.fa,.fa-brands,.fa-classic,.fa-regular,.fa-sharp,.fa-solid,.fab,.far,.fas{-moz-osx-font-smoothing:grayscale;-webkit-font-smoothing:antialiased;display:var(--fa-display,inline-block);font-style:normal;font-variant:normal;line-height:1;text-rendering:auto}.fa-classic,.fa-regular,.fa-solid,.far,.fas{font-family:Font Awesome\ 6 Free}.fa-brands,.fab{font-family:Font Awesome\ 6 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+/*!
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+ * Copyright 2024 Fonticons, Inc.
+ */
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new file mode 100644
index 0000000..72d229f
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+++ b/dev/docs/html/_static/webpack-macros.html
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+<!--
+  AUTO-GENERATED from webpack.config.js, do **NOT** edit by hand.
+  These are re-used in layout.html
+-->
+{# Load FontAwesome icons #}
+{% macro head_pre_icons() %}
+  <link href="{{ pathto('_static/vendor/fontawesome/6.5.2/css/all.min.css', 1) }}?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
+  <link rel="preload" as="font" type="font/woff2" crossorigin href="{{ pathto('_static/vendor/fontawesome/6.5.2/webfonts/fa-solid-900.woff2', 1) }}" />
+<link rel="preload" as="font" type="font/woff2" crossorigin href="{{ pathto('_static/vendor/fontawesome/6.5.2/webfonts/fa-brands-400.woff2', 1) }}" />
+<link rel="preload" as="font" type="font/woff2" crossorigin href="{{ pathto('_static/vendor/fontawesome/6.5.2/webfonts/fa-regular-400.woff2', 1) }}" />
+{% endmacro %}
+
+{% macro head_pre_assets() %}
+  <!-- Loaded before other Sphinx assets -->
+  <link href="{{ pathto('_static/styles/theme.css', 1) }}?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
+<link href="{{ pathto('_static/styles/bootstrap.css', 1) }}?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
+<link href="{{ pathto('_static/styles/pydata-sphinx-theme.css', 1) }}?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
+{% endmacro %}
+
+{% macro head_js_preload() %}
+  <!-- Pre-loaded scripts that we'll load fully later -->
+  <link rel="preload" as="script" href="{{ pathto('_static/scripts/bootstrap.js', 1) }}?digest=dfe6caa3a7d634c4db9b" />
+<link rel="preload" as="script" href="{{ pathto('_static/scripts/pydata-sphinx-theme.js', 1) }}?digest=dfe6caa3a7d634c4db9b" />
+  <script src="{{ pathto('_static/vendor/fontawesome/6.5.2/js/all.min.js', 1) }}?digest=dfe6caa3a7d634c4db9b"></script>
+{% endmacro %}
+
+{% macro body_post() %}
+  <!-- Scripts loaded after <body> so the DOM is not blocked -->
+  <script src="{{ pathto('_static/scripts/bootstrap.js', 1) }}?digest=dfe6caa3a7d634c4db9b"></script>
+<script src="{{ pathto('_static/scripts/pydata-sphinx-theme.js', 1) }}?digest=dfe6caa3a7d634c4db9b"></script>
+{% endmacro %}
\ No newline at end of file
diff --git a/dev/docs/html/acknowledgements.html b/dev/docs/html/acknowledgements.html
new file mode 100644
index 0000000..8bb228e
--- /dev/null
+++ b/dev/docs/html/acknowledgements.html
@@ -0,0 +1,488 @@
+
+<!DOCTYPE html>
+
+
+<html lang="en" data-content_root="./" >
+
+  <head>
+    <meta charset="utf-8" />
+    <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
+
+    <title>Acknowledgements &#8212; Addressing Uncertainty in MultiSector Dynamics Research  documentation</title>
+  
+  
+  
+  <script data-cfasync="false">
+    document.documentElement.dataset.mode = localStorage.getItem("mode") || "";
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+  <!-- Loaded before other Sphinx assets -->
+  <link href="_static/styles/theme.css?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
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+<link href="_static/styles/pydata-sphinx-theme.css?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
+
+  
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+<h1>Acknowledgements<a class="headerlink" href="#acknowledgements" title="Link to this heading">#</a></h1>
+<p>This e-book was developed by the <a class="reference external" href="https://im3.pnnl.gov">Integrated Multisector, Multiscale Modeling (IM3) project</a>, supported by the <a class="reference external" href="https://www.energy.gov/">U.S. Department of Energy</a>, <a class="reference external" href="https://www.energy.gov/science/office-science">Office of Science</a>, as part of research in the <a class="reference external" href="https://climatemodeling.science.energy.gov/program/multisector-dynamics">MultiSector Dynamics</a>, Earth and Environmental System Modeling Program.</p>
+<p>The authors would like to thank Casey Burleyson (PNNL) and Isaac Thompson (PNNL) for their insightful feedback on drafts of the text and tutorials.</p>
+<p>Open source under license <a class="reference external" href="https://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND-4.0</a>.</p>
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+<h2>Using mathjax<a class="headerlink" href="#using-mathjax" title="Link to this heading">#</a></h2>
+<p>The <strong>first-order sensitivity index</strong> indicates the percent of model output variance contributed by a factor individually (i.e., the effect of varying <em>x</em><sub>i</sub>alone) and is obtained using the following (Saltelli, 2002a; Sobol, 2001):</p>
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+\[S_i^1 = \frac{V_{x_i} [E_{x\sim_i} (x_i)]}{V(y)}\]</div>
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+<li class="toctree-l1"><a class="reference internal" href="preface.html">Preface</a></li>
+<li class="toctree-l1"><a class="reference internal" href="citation.html">Suggested Citation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="acknowledgements.html">Acknowledgements</a></li>
+<li class="toctree-l1"><a class="reference internal" href="code_of_conduct.html">Code of Conduct</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html">2. Diagnostic Modeling Overview and Perspectives</a></li>
+<li class="toctree-l1"><a class="reference internal" href="3_sensitivity_analysis_the_basics.html">3. Sensitivity Analysis: The Basics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html">4. Sensitivity Analysis: Diagnostic &amp; Exploratory Modeling</a></li>
+<li class="toctree-l1"><a class="reference internal" href="5_conclusion.html">5. Conclusion</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
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+    <h1>Addressing Uncertainty in MultiSector Dynamics Research</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
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+  <section id="addressing-uncertainty-in-multisector-dynamics-research">
+<h1>Addressing Uncertainty in MultiSector Dynamics Research<a class="headerlink" href="#addressing-uncertainty-in-multisector-dynamics-research" title="Link to this heading">#</a></h1>
+<blockquote class="epigraph">
+<div><p>Patrick M. Reed, Antonia Hadjimichael, Keyvan Malek,
+Tina Karimi, Chris R. Vernon, Vivek Srikrishnan, Rohini S. Gupta,
+David F. Gold, Ben Lee, Klaus Keller, Travis B. Thurber, Jennie S. Rice</p>
+</div></blockquote>
+<p>A practical guide to sensitivity analysis and diagnostic model evaluation techniques for confronting the computational and conceptual challenges of multi-model, transdisciplinary workflows.</p>
+<p><strong>Topics in the Book</strong></p>
+<ul class="simple">
+<li><p><a class="reference external" href="preface.html">Preface</a></p></li>
+<li><p><a class="reference external" href="1_introduction.html">Introduction</a></p></li>
+<li><p><a class="reference external" href="2_diagnostic_modeling_overview_and_perspectives.html">An overview of diagnostic modeling and perspectives on model evaluation</a></p></li>
+<li><p><a class="reference external" href="3_sensitivity_analysis_the_basics.html">A framework for the basic methods and concepts used in sensitivity analysis</a></p></li>
+<li><p><a class="reference external" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html">Technical applications supporting diagnostic model evaluation and exploration</a></p></li>
+</ul>
+<p><strong>Interactive Tutorials</strong></p>
+<ul class="simple">
+<li><p><a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/fishery_dynamics.ipynb">Factor Discovery</a></p></li>
+<li><p><a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">Model Calibration</a></p></li>
+<li><p><a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol Sensitivity Analysis</a></p></li>
+<li><p><a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/basin_users_logistic_regression.ipynb">Factor Mapping using Logistic Regression</a></p></li>
+<li><p><a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/Bedford_Greene_SD_tutorial.ipynb">Time-evolving scenario discovery for infrastructure pathways</a></p></li>
+<li><p><a class="reference external" href="https://uc-ebook.dev.msdlive.org/user-redirect/lab/tree/notebooks/Hidden-Markov_Modeling_Approaches_to_Creating_Synthetic_Streamflow_Scenarios.ipynb">A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios</a></p></li>
+</ul>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>
+    Use the sidebar on the left to quickly navigate the eBook!
+    <br/>
+    Click or tap the <i style="padding: 0 0.25rem;" class="fas fa-bars"></i> icon to show and hide the sidebar.
+</p></div>
+<div class="admonition-info admonition">
+<p class="admonition-title">Info</p>
+<p>
+    Report a typo or just pass along something you like about the book by <a target="_blank" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/issues/new?assignees=thurber%2C+crvernon&labels=documentation%2C+triage&template=custom.md&title=Publication+Feedback">opening an issue</a> on GitHub!
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+  <section id="preface">
+<span id="id1"></span><h1>Preface<a class="headerlink" href="#preface" title="Link to this heading">#</a></h1>
+<p>This online book is meant to provide an open science “living” resource on uncertainty characterization methods for the MultiSector Dynamics (MSD) community and other technical communities confronting sustainability, climate, and energy transition challenges. The last decade has seen rapid growth in science efforts seeking to address the interconnected nature of these challenges across scales, sectors, and systems. Accompanying these advances is the growing realization that the deep integration of research from many disciplinary fields is non-trivial and raises important questions. How and why models are developed seems to have an obvious answer (“to gain understanding”). But what does it actually mean to gain understanding? What if a small change in a model or its data fundamentally changes our perceptions of what we thought we understood? What controls the outcomes of our model(s)? How do we understand the implications of model coupling, such as when one model is on the receiving end of several other models that are considered “input data”?</p>
+<p>The often quoted “All models are wrong, but some are useful.” (George Box) is a bit of a conflation trap, often used to excuse known weaknesses in complex models as just an unavoidable outcome of being a modeler. In fact, the quote actually refers to a specific class of small-scale statistical models within an application context that assures a much higher degree of understanding and data quality control than is typical for the coupled human-natural systems applications in the MSD area. Moreover, Box was actually warning readers to avoid overparameterization and emphasizing the need to better understand what underlying factors cause your model to be wrong <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id204" title="G. E. P. Box. Science and Statistics. Journal of the American Statistical Association, 71(356):791–799, 1976. URL: https://www.tandfonline.com/doi/abs/10.1080/01621459.1976.10480949, doi:10.1080/01621459.1976.10480949.">1</a>]</span>.</p>
+<p>So, in short, there is a tension when attaining better performance by means of increasing the complexity of a model or model-based workflow. Box highlights that a modeler requires a clear diagnostic understanding of this performance-complexity tradeoff. If we move from small-scale models simulating readily-observed phenomena to the MSD context, things get quite a bit more complicated. How can we provide robust insights for unseen futures that emerge across a myriad of human and natural systems? Sometimes even asking, “what is a model?” or “what is data?” is complicated (e.g., data assimilated weather products, satellite-based signals translated through retrieval algorithms, demographic changes, resource demands, etc.). This MSD guidance text seeks to help readers navigate these challenges. It is meant to serve as an evolving resource that helps the MSD community learn how to better address uncertainty while working with complex chains of models bridging sectors, scales, and systems. It is not intended to be an exhaustive resource, but instead should be seen as a guided tour through state-of-the-science methods in uncertainty characterization, including global sensitivity analysis and exploratory modeling, to provide insights into complex human-natural systems interactions.</p>
+<p>To aid readers in navigating the text, the key goals for each chapter are summarized below.</p>
+<p><a class="reference internal" href="1_introduction.html#introduction"><span class="std std-numref">Chapter 1</span></a> uses the <a class="reference external" href="https://im3.pnnl.gov">Integrated Multisector Multiscale Modeling</a> project as a living lab to encapsulate the challenges that emerge in bridging disciplines to make consequential model-based insights while acknowledging the tremendous array of uncertainties that shape them.</p>
+<p><a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#diagnostic-modeling"><span class="std std-numref">Chapter 2</span></a> helps the reader to better understand the importance of using diagnostic modeling to interrogate why uncertain model behaviors may emerge. The chapter also aids readers to better understand the diverse disciplinary perspectives that exist on how best to pursue consequential model-based discoveries.</p>
+<p><a class="reference internal" href="3_sensitivity_analysis_the_basics.html#sensitivity-analysis-the-basics"><span class="std std-numref">Chapter 3</span></a> is a technical tools-focused primer for readers on the key elements of uncertainty characterization that includes ensemble-based design of experiments, quantitative methods for computing global sensitivities, and a summary of existing software packages.</p>
+<p><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#sensitivity-analysis"><span class="std std-numref">Chapter 4</span></a> narrates for readers how and why the tools from the previous chapter can be applied in a range of tasks from diagnosing model performance to formal exploratory modeling methods for making consequential model-based discoveries.</p>
+<p>The supplemental appendices provided in the text are also important resources for readers. They provide a glossary to help bridge terminology challenges, a brief summary of uncertainty quantification tools for more advanced readers, and a suite of Jupyter notebook tutorials that provide hands-on training tied to the contents of <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#sensitivity-analysis-the-basics"><span class="std std-numref">Chapter 3</span></a> and <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#sensitivity-analysis"><span class="std std-numref">Chapter 4</span></a>.</p>
+<p>This text was written with a number of different audiences in mind.</p>
+<p>Technical experts in uncertainty may find this to be a helpful and unique resource bridging a number of perspectives that have not been combined in prior books (e.g., formal model diagnostics, global sensitivity analysis, and exploratory modeling under deep uncertainty).</p>
+<p>Readers from different sector-specific and disciplinary-specific backgrounds can use this text to better understand potential differences and opportunities in how to make model-based insights.</p>
+<p>Academic or junior researchers can utilize this freely available text for training and teaching resources that include hands-on coding experiences.</p>
+<p>This text itself represents our strong commitment to open science and will evolve as a living resource as the communities of researchers provide feedback, innovations, and future tools.</p>
+</section>
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"step-5-explore-relationship-between-uncertain-factors-and-performance"], [45, "step-5-explore-relationship-between-uncertain-factors-and-performance"]], "Step 5: Interpret our results": [[40, "step-5-interpret-our-results"], [45, "step-5-interpret-our-results"]], "Suggested Citation": [[49, "suggested-citation"]], "Summary of designs of experiments overviewed in this section. * Depends on the sample size.": [[12, "id32"], [23, "id96"]], "Synthetic Stationary Generator to Better Quantify Natural Variability": [[44, "synthetic-stationary-generator-to-better-quantify-natural-variability"], [45, "synthetic-stationary-generator-to-better-quantify-natural-variability"]], "Synthetic generation of input time series": [[11, "synthetic-generation-of-input-time-series"], [12, "synthetic-generation-of-input-time-series"], [23, "synthetic-generation-of-input-time-series"]], "Testing out a note": [[52, "testing-out-a-note"]], "The Critical Final Step: Predictive Checks": [[36, 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Analysis: Diagnostic &amp; Exploratory Modeling", "<span class=\"section-number\">5. </span>Conclusion", "Glossary", "Introduction", "Parametric Bootstrap", "Pre-Calibration", "Markov Chain Monte Carlo", "Other Methods", "The Critical First Step: How to Choose a Prior Distribution", "The Critical Final Step: Predictive Checks", "Key Take-Home Points", "<span class=\"section-number\">1. </span>Uncertainty Quantification", "Fishery Dynamics Tutorial", "Sobol SA Tutorial", "Logistic Regression Tutorial", "HYMOD Dynamics Tutorial", "Time-evolving scenario discovery for infrastructure pathways", "A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios Tutorial", "<span class=\"section-number\">2. </span>Jupyter Notebook Tutorials", "<span class=\"section-number\">3. </span>Plotting Code Samples", "Bibliography", "Acknowledgements", "Suggested Citation", "Code of Conduct", "Contribution Guide", "Examples", "Addressing Uncertainty in MultiSector Dynamics Research", 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+		<div class="text-md italic text-slate-50 text-shadow pt-1 px-3 text-center lg:text-left leading-tight backdrop-blur-sm svelte-uz1u3c"><div style="">Patrick M. Reed, Antonia Hadjimichael, Keyvan Malek,</div>
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+					<div class="text-md text-slate-900 mx-3 mt-3 leading-tight">Technical applications supporting diagnostic model evaluation and exploration.
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+    <div class="flex flex-row items-stretch justify-center"><div style="box-shadow: inset 0 2px 4px 0 rgb(0 0 0 / 0.2);" class="bg-slate-50 rounded-l-md overflow-auto"><pre class="m-4 text-sm text-slate-900 w-80 md:w-96 text-left leading-tight "></pre></div>
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+<div class="w-full flex flex-col items-center justify-center text-center px-3 pb-12"><div class="max-w-4xl text-sm italic text-slate-400">This e-book was developed by the <a href="https://im3.pnnl.gov" class="font-bold text-slate-500 hover:text-sky-700">Integrated MultiSector, Multiscale Modeling (IM3)</a>
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+		<a href="https://climatemodeling.science.energy.gov/program/multisector-dynamics" class="font-bold text-slate-500 hover:text-sky-700">MultiSector Dynamics</a>, Earth and Environmental System Modeling Program.
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+<span id="id1"></span><h1><span class="section-number">1. </span>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h1>
+<p>This guidance text has been developed in support of the Integrated Multisector Multiscale Modeling (IM3) Science Focus Area’s objective to formally integrate uncertainty into its research tasks. IM3 is focused on innovative modeling to explore how human and natural system landscapes in the United States co-evolve in response to short-term shocks and long-term influences. The project’s challenging scope is to advance our ability to study the interactions between energy, water, land, and urban systems, at scales ranging from local (~1km) to the contiguous United States, while consistently addressing influences such as population change, technology change, heat waves, and drought. Uncertainty and careful model-driven scientific insights are central to the project’s science objectives shown below.</p>
+<p><strong>IM3 Key MSD Science Objectives:</strong></p>
+<p><em>Develop flexible, open-source, and integrated modeling capabilities that capture the structure, dynamic behavior, and emergent properties of the multiscale interactions within and between human and natural systems.</em></p>
+<p><em>Use these capabilities to study the evolution, vulnerability, and resilience of interacting human and natural systems and landscapes from local to continental scales, including their responses to the compounding effects of long-term influences and short-term shocks.</em></p>
+<p><em>Understand the implications of uncertainty in data, observations, models, and model coupling approaches for projections of human-natural system dynamics.</em></p>
+<p>Addressing the objectives above poses a strong transdisciplinary challenge that depends on a diversity of models and, more specifically, a consistent framing for making model-based science inferences. The term transdisciplinary science as used here formally implies a deep integration of disciplines to aid our hypothesis-driven understanding of coupled human-natural systems–bridging differences in theory, hypothesis generation, modeling, and modes of inference <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id2" title="National Research Council and others. Convergence: Facilitating transdisciplinary integration of life sciences, physical sciences, engineering, and beyond. National Academies Press, 2014.">2</a>]</span>. The IM3 MSD research foci and questions require a deep integration across disciplines, where new modes of analysis can emerge that rapidly synthesize and exploit advances for making decision-relevant insights that at minimum acknowledge uncertainty and more ideally promote a rigorous quantitative mapping of its effects on the generality of claimed scientific insights. More broadly, diverse scientific disciplines engaged in the science of coupled human-natural systems, ranging from natural sciences to engineering and economics, employ a diversity of numerical computer models to study and understand their underlying systems of focus. The utility of these computer models hinges on their ability to represent the underlying real systems with sufficient fidelity and enable the inference of novel insights. This is particularly challenging in the case of coupled human-natural systems where there exists a multitude of interdependent human and natural processes taking place that could potentially be represented. These processes usually translate into modeled representations that are highly complex, non-linear, and exhibit strong interactions and threshold behaviors <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id3" title="Sondoss Elsawah, Tatiana Filatova, Anthony J Jakeman, Albert J Kettner, Moira L Zellner, Ioannis N Athanasiadis, Serena H Hamilton, Robert L Axtell, Daniel G Brown, Jonathan M Gilligan, and others. Eight grand challenges in socio-environmental systems modeling. Socio-Environmental Systems Modelling, 2:16226–16226, 2020.">3</a>, <a class="reference internal" href="R.Bibliography.html#id4" title="Yacov Y Haimes. Risk modeling of interdependent complex systems of systems: theory and practice. Risk analysis, 38(1):84–98, 2018.">4</a>, <a class="reference internal" href="R.Bibliography.html#id5" title="Dirk Helbing. Globally networked risks and how to respond. Nature, 497(7447):51–59, 2013.">5</a>]</span>. Model complexity and detail have also been increasing as a result of our improving understanding of these processes, the availability of data, and the rapid growth in computing power <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id6" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">6</a>]</span>. As model complexity grows, modelers need to specify a lot more information than before: additional model inputs and relationships as more processes are represented, higher resolution data as more observations are collected, new coupling relationships and interactions as diverse models are being used in combination to answer multisector questions (e.g., the land-water-energy nexus). Typically, not all of this information is well known, nor is the impact of these many uncertainties on model outputs well understood. It is further especially difficult to distinguish the effects of individual as well as interacting sources of uncertainty when modeling coupled systems with multisector and multiscale dynamics <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id7" title="Daniel Wirtz and Wolfgang Nowak. The rocky road to extended simulation frameworks covering uncertainty, inversion, optimization and control. Environmental Modelling &amp; Software, 93:180–192, 2017.">7</a>]</span>.</p>
+<p>Given the challenge and opportunity posed by the disciplinary diversity of IM3, we utilized an informal team-wide survey to understand how the various disciplines typically address uncertainty, emphasizing key literature examples and domain-specific reviews. The feedback received provided perspectives across diverse areas within the Earth sciences, different engineering fields, as well as economics. Although our synthesis of this survey information highlighted some commonality across areas (e.g., the frequent use of scenario-based modeling), we identified key differences in vocabulary, the frequency with which formal uncertainty analysis appears in the disciplinary literature, and technical approaches. The IM3 team’s responses captured a very broad conceptual continuum of methodological traditions, ranging from deterministic (no uncertainty) modeling to the theoretical case of fully engaging in modeling sources of uncertainty. Overall, error-driven analyses that focus on replicating prior observed conditions were reported to be the most prevalent types of studies for all disciplines. It was generally less common for studies to strongly engage with analyzing uncertainty via more formal ensemble analyses and design of experiments, though some areas did show significantly higher levels of activity. Another notable finding from our survey was the apparent lack of focus on understanding how model coupling relationships shape uncertainty. Although these observations are limited to the scope of feedback attained in the team-wide IM3 survey responses and the bodies of literature reported by respondents, we believe they reflect challenges that are common across the MSD community.</p>
+<p>In the IM3 uncertainty-related research that has occurred since this survey, we have observed that differences in terminology and interpretation of terminology across modeling teams can be confounding. One of the goals of this eBook is to provide a common language for uncertainty analysis within IM3 and, hopefully, for the broader MSD community. While individual scientific disciplines would be expected to retain their own terminology, by providing explicit definitions of terms we can facilitate the translation of concepts across transdisciplinary science teams. To begin, we use the term Uncertainty Analysis (UA) as an umbrella phrase covering all methods in this eBook. Next, we distinguish the key terms of uncertainty quantification (UQ) and uncertainty characterization (UC). UQ refers to the formal focus on the full specification of likelihoods as well as the distributional forms necessary to infer the joint probabilistic response across all modeled factors of interest <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id11" title="Roger Cooke and others. Experts in uncertainty: opinion and subjective probability in science. Oxford University Press on Demand, 1991.">8</a>]</span>. UC refers to exploratory modeling of alternative hypotheses to understand the co-evolutionary dynamics of influences and stressors, as well as path dependent changes in the form and function of modelled systems <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id12" title="Enayat A Moallemi, Jan Kwakkel, Fjalar J de Haan, and Brett A Bryan. Exploratory modeling for analyzing coupled human-natural systems under uncertainty. Global Environmental Change, 65:102186, 2020.">9</a>, <a class="reference internal" href="R.Bibliography.html#id13" title="Warren E Walker, Poul Harremoës, Jan Rotmans, Jeroen P Van Der Sluijs, Marjolein BA Van Asselt, Peter Janssen, and Martin P Krayer von Krauss. Defining uncertainty: a conceptual basis for uncertainty management in model-based decision support. Integrated assessment, 4(1):5–17, 2003.">10</a>]</span>. As discussed in later sections, the choice of UC or UQ depends on the specific goals of studies, the availability of data, the types of uncertainties (e.g., well-characterized or deep), and the complexity of underlying models as well as computational limits. Definitions of key uncertainty analysis terms used in this eBook appear below, and our Glossary (<code class="xref std std-numref docutils literal notranslate"><span class="pre">glossary</span></code>) contains a complete list of terms.</p>
+<ul class="simple">
+<li><p><strong>Exploratory modeling</strong>: Use of large ensembles of uncertain conditions to discover decision-relevant combinations of uncertain factors</p></li>
+<li><p><strong>Factor</strong>: Any model component that can affect model outputs: inputs, resolution levels, coupling relationships, model relationships and parameters. In models with acceptable model fidelity these factors may represent elements of the real-world system under study.</p></li>
+<li><dl class="simple">
+<dt><strong>Sensitivity analysis</strong>: Model evaluation to understand the factors and processes that most (or least) control a model’s outputs</dt><dd><ul>
+<li><p><strong>Local sensitivity analysis</strong>: Varying uncertain factors around specific reference values</p></li>
+<li><p><strong>Global sensitivity analysis</strong>: Varying uncertain factors throughout their entire feasible value space</p></li>
+</ul>
+</dd>
+</dl>
+</li>
+<li><p><strong>Uncertainty characterization</strong>: Model evaluation under alternative factor hypotheses to explore their implications for model output uncertainty</p></li>
+<li><p><strong>Uncertainty quantification</strong>: Representation of model output uncertainty using probability distributions</p></li>
+</ul>
+<p>At present, there is no singular guide for confronting the computational and conceptual challenges of the multi-model, transdisciplinary workflows that characterize ambitious projects such as IM3 <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id10" title="Andrea Saltelli, Philip B Stark, William Becker, and Pawel Stano. Climate models as economic guides scientific challenge or quixotic quest? Issues in Science and Technology, 31(3):79–84, 2015.">11</a>]</span>. The primary aim of this text is to begin to address this gap and provide guidance for facing these challenges. <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#diagnostic-modeling"><span class="std std-numref">Chapter 2</span></a> provides an overview of diagnostic modeling and the different perspectives for how we should evaluate our models, <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#sensitivity-analysis-the-basics"><span class="std std-numref">Chapter 3</span></a> summarizes basic methods and concepts for sensitivity analysis, and <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#sensitivity-analysis"><span class="std std-numref">Chapter 4</span></a> delves into more technical applications of sensitivity analysis to support diagnostic model evaluation and exploratory modeling. Finally, <a class="reference internal" href="5_conclusion.html#conclusion"><span class="std std-numref">Chapter 5</span></a> provides some concluding remarks across the UC and UQ topics covered in this text. The appendices of this text include a glossary of the key concepts, an overview of UQ methods, and coding-based illustrative examples of key UC concepts discussed in earlier chapters.</p>
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+<li class="toctree-l1"><a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html">2. Diagnostic Modeling Overview and Perspectives</a></li>
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+  <section id="overview-of-model-diagnostics">
+<h1>Overview of model diagnostics<a class="headerlink" href="#overview-of-model-diagnostics" title="Link to this heading">#</a></h1>
+<p>Model diagnostics provide a rich basis for hypothesis testing, model innovation, and improved inferences when classifying what is controlling highly consequential results (e.g., vulnerability or resilience in coupled human-natural systems). <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a>, adapted from <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id6" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">6</a>]</span>, presents idealized illustrations of the relationship between UC and global sensitivity analysis for two coupled simulation models. The figure illustrates how UC can be used to address how uncertainties in various modeling decisions (e.g., data inputs, parameters, model structures, coupling relationships) can be sampled and simulated to yield the empirical model output distribution(s) of interest. Monte Carlo frameworks allow us to sample and propagate (or integrate) the ensemble response of the model(s) of focus. The first step of any UC analysis is the specification of the initial input distributions as illustrated in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a>. The second step is to perform the Monte Carlo simulations. The question can then be raised, which of the modeling assumptions in our Monte Carlo experiment are the most responsible for the resulting output uncertainty. We can answer this question using global sensitivity analysis as illustrated in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a>. Global sensitivity analysis can be defined as a formal Monte Carlo sampling and analysis of modeling choices (structures, parameters, inputs) to quantify their influence on direct model outputs (or output-informed metrics). UC experiments by themselves do not explain why a particular uncertain outcome is produced, but produce distributions of model outcomes, as portrayed by the yellow curve. The pie chart shown in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a> is a conceptual representation of the results of using a global sensitivity analysis to identify those factors that are most dominantly influencing results, either individually or interactively <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id48" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global Sensitivity Analysis: The Primer. Wiley-Interscience, Chichester, England ; Hoboken, NJ, 1 edition edition, February 2008. ISBN 978-0-470-05997-5.">14</a>]</span>.</p>
+<figure class="margin-caption align-center" id="id4">
+<span id="figure-2-1"></span><a class="reference internal image-reference" href="_images/figure2_1_idealized_uc.png"><img alt="Figure 2.1" src="_images/figure2_1_idealized_uc.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Idealized uncertainty characterization and global sensitivity analysis for two coupled simulation models. Uncertainty coming from various sources (e.g., inputs, model structures, coupling relationships) is propagated through the coupled model(s) to generate empirical distributions of outputs of interest (uncertainty characterization). This model output uncertainty can be decomposed to its origins, by means of sensitivity analysis. Figure adapted from <span id="id3">Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id6" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">6</a>]</span>.</span><a class="headerlink" href="#id4" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>UC and global sensitivity analysis are not independent modeling analyses. As illustrated here, any global sensitivity analysis requires an initial UC hypothesis in the form of statistical assumptions and representations for the modeling choices of focus (structural, parametric, and data inputs). Information from these two model diagnostic tools can then be used to inform data needs for future model runs, experiments to reduce the uncertainty present, or the simplification or enhancement of the model where necessary. Together UC and global sensitivity analysis provide a foundation for diagnostic exploratory modeling that has a consistent focus on the assumptions, structural model forms, alternative parameterizations, and input data sets that are used to characterize the behavioral space of one or more models.</p>
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+  <section id="perspectives-on-diagnostic-model-evaluation">
+<span id="perspectives"></span><h1>Perspectives on diagnostic model evaluation<a class="headerlink" href="#perspectives-on-diagnostic-model-evaluation" title="Link to this heading">#</a></h1>
+<p>When we judge or diagnose models, the terms “verification” and “validation” are commonly used. However, their appropriateness in the context of numerical models representing complex coupled human-natural systems is questionable <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id54" title="Keith Beven. Towards a coherent philosophy for modelling the environment. Proceedings of the royal society of London. Series A: mathematical, physical and engineering sciences, 458(2026):2465–2484, 2002.">15</a>, <a class="reference internal" href="R.Bibliography.html#id118" title="Naomi Oreskes, Kristin Shrader-Frechette, and Kenneth Belitz. Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences. Science, 263(5147):641–646, February 1994. URL: https://science.sciencemag.org/content/263/5147/641 (visited on 2020-04-15), doi:10.1126/science.263.5147.641.">16</a>]</span>. The core issue relates to the fact that these systems are often not fully known or perfectly implemented when modeled. Rather, they are defined within specific system framings and boundary conditions in an evolving learning process with the goal of making continual progress towards attaining higher levels of fidelity in capturing behaviors or properties of interest. Evaluating the fidelity of a model’s performance can be highly challenging. For example, the observations used to evaluate the fidelity of parameterized processes are often measured at a finer resolution than what is represented in the model, creating the challenge of how to manage their relative scales when performing evaluation. In other cases, numerical models may neglect or simplify system processes because sufficient data is not available or the physical mechanisms are not fully known. If sufficient agreement between prediction and observation is not achieved, it is challenging to know whether these types of modeling choices are the cause, or if other issues, such as deficiencies in the input parameters and/or other modeling assumptions are the true cause of errors. Even if there is high agreement between prediction and observation, the model cannot necessarily be considered validated, as it is always possible that the right values were produced for the wrong reasons. For example, low error can stem from a situation where different errors in underlying assumptions or parameters cancel each other out (“compensatory errors”). Furthermore, coupled human-natural system models are often subject to “equifinality”, a situation where multiple parameterized formulations can produce similar outputs or equally acceptable representations of the observed data. There is therefore no uniquely “true” or validated model, and the common practice of selecting “the best” deterministic calibration set is more of an assumption than a finding <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id49" title="Keith Beven. Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in water resources, 16(1):41–51, 1993.">17</a>, <a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>]</span>. The situation becomes even more tenuous when observational data is limited in its scope and/or quality to be insufficient to distinguish model representations or their performance differences.</p>
+<p>These limitations on model verification undermine any purely positivist treatment of model validity: that a model should correctly and precisely represent reality to be valid. Under this perspective, closely related to empiricism, statistical tests should be used to compare the model’s output with observations and only through empirical verification can a model or theory be deemed credible. A criticism to this viewpoint (besides the aforementioned challenges for model verification) is that it reduces the justification of a model to the single criterion of predictive ability and accuracy <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id119" title="Yaman Barlas and Stanley Carpenter. Philosophical roots of model validation: Two paradigms. System Dynamics Review, 6(2):148–166, 1990. doi:10.1002/sdr.4260060203.">19</a>]</span>. Authors have argued that this ignores the explanatory power held in models and other procedures, which can also advance scientific knowledge <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id120" title="Stephen Toulmin. From form to function: philosophy and history of science in the 1950s and now. Daedalus, pages 143–162, 1977. Publisher: JSTOR.">20</a>]</span>. These views gave rise to relativist perspectives of science, which instead place more value on model utility in terms of fitness for a specific purpose or inquiry, rather than representational accuracy and predictive ability <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id121" title="George B. Kleindorfer, Liam O'Neill, and Ram Ganeshan. Validation in simulation: Various positions in the philosophy of science. Management Science, 44(8):1087–1099, 1998. Publisher: INFORMS.">21</a>]</span>. This viewpoint appears to be most prevalent among practitioners seeking decision-relevant insights (i.e., inspire new views vs. predict future conditions). The relativist perspective argues for the use of models as heuristics that can enhance our understanding and conceptions of system behaviors or possibilities <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id64" title="Sibel Eker, Elena Rovenskaya, Michael Obersteiner, and Simon Langan. Practice and perspectives in the validation of resource management models. Nature communications, 9(1):1–10, 2018.">22</a>]</span>. In contrast, natural sciences favor a positivist perspective, emphasizing similarity between simulation and observation even in application contexts where it is clear that projections are being made for conditions that have never been observed and the system of focus will have evolved structurally beyond the model representation being employed (e.g., decadal to centennial evolution of human-natural systems).</p>
+<p>These differences in prevalent perspectives are mirrored in how model validation is defined by the two camps: From the relativist perspective, validation is seen as a process of incremental “confidence building” in a model as a mechanism for insight <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id122" title="Yaman Barlas. Formal aspects of model validity and validation in system dynamics. System Dynamics Review: The Journal of the System Dynamics Society, 12(3):183–210, 1996. Publisher: Wiley Online Library.">23</a>]</span>, whereas in natural sciences validation is framed as a way to classify a model as having an acceptable representation of physical reality <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id118" title="Naomi Oreskes, Kristin Shrader-Frechette, and Kenneth Belitz. Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences. Science, 263(5147):641–646, February 1994. URL: https://science.sciencemag.org/content/263/5147/641 (visited on 2020-04-15), doi:10.1126/science.263.5147.641.">16</a>]</span>. Even though the relativist viewpoint does not dismiss the importance of representational accuracy, it does place it within a larger process of establishing confidence through a variety of tools. These tools, not necessarily quantitative, include communicating information between practitioners and modelers, interpreting a multitude of model outputs, and contrasting preferences and viewpoints.</p>
+<p>On the technical side of the argument, differing views on the methodology of model validation appear as early as in the 1960’s. <span id="id9">Naylor and Finger [<a class="reference internal" href="R.Bibliography.html#id123" title="Thomas H. Naylor and Joseph Michael Finger. Verification of computer simulation models. Management science, 14(2):B–92, 1967. Publisher: INFORMS.">24</a>]</span> argue that model validation should not be limited to a single metric or test of performance (e.g., a single error metric), but should rather be extended to multiple tests that reflect different aspects of a model’s structure and behavior. This and similar arguments are made in literature to this day <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id46" title="Hoshin V. Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008. Publisher: Wiley Online Library.">12</a>, <a class="reference internal" href="R.Bibliography.html#id52" title="Keith J Beven. On hypothesis testing in hydrology: why falsification of models is still a really good idea. Wiley Interdisciplinary Reviews: Water, 5(3):e1278, 2018.">25</a>, <a class="reference internal" href="R.Bibliography.html#id51" title="Hoshin V Gupta, Martyn P Clark, Jasper A Vrugt, Gab Abramowitz, and Ming Ye. Towards a comprehensive assessment of model structural adequacy. Water Resources Research, 2012.">26</a>, <a class="reference internal" href="R.Bibliography.html#id124" title="Praveen Kumar. Typology of hydrologic predictability. Water Resources Research, 2011. URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2010WR009769 (visited on 2020-04-15), doi:10.1029/2010WR009769.">27</a>, <a class="reference internal" href="R.Bibliography.html#id125" title="Grey S. Nearing, Benjamin L. Ruddell, Andrew R. Bennett, Cristina Prieto, and Hoshin V. Gupta. Does Information Theory Provide a New Paradigm for Earth Science? Hypothesis Testing. Water Resources Research, 56(2):e2019WR024918, 2020. doi:10.1029/2019WR024918.">28</a>]</span> and are primarily founded on two premises. First, that even though modelers widely recognize that their models are abstractions of the truth, they still make truth claims based on traditional performance metrics that measure the divergence of their model from observation <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id125" title="Grey S. Nearing, Benjamin L. Ruddell, Andrew R. Bennett, Cristina Prieto, and Hoshin V. Gupta. Does Information Theory Provide a New Paradigm for Earth Science? Hypothesis Testing. Water Resources Research, 56(2):e2019WR024918, 2020. doi:10.1029/2019WR024918.">28</a>]</span>. Second, that the natural systems mimicked by the models contain many processes that exhibit significant heterogeneity at various temporal and spatial scales. This heterogeneity is lost when a single performance measure is used, as a result of the inherent loss of process information occurring when transitioning from a highly dimensional and interactive system to the dimension of a single metric <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id54" title="Keith Beven. Towards a coherent philosophy for modelling the environment. Proceedings of the royal society of London. Series A: mathematical, physical and engineering sciences, 458(2026):2465–2484, 2002.">15</a>]</span>. These arguments are further elaborated in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#sensitivity-analysis"><span class="std std-numref">Chapter 4</span></a>.</p>
+<p>Multiple authors have proposed that the traditional reliance on single measures of model performance should be replaced by the evaluation of several model signatures (characteristics) to identify model structural errors and achieve a sufficient assessment of model performance <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id46" title="Hoshin V. Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008. Publisher: Wiley Online Library.">12</a>, <a class="reference internal" href="R.Bibliography.html#id126" title="Hoshin Vijai Gupta, Soroosh Sorooshian, and Patrice Ogou Yapo. Toward improved calibration of hydrologic models: Multiple and noncommensurable measures of information. Water Resources Research, 34(4):751–763, 1998. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/97WR03495 (visited on 2020-04-07), doi:10.1029/97WR03495.">29</a>, <a class="reference internal" href="R.Bibliography.html#id168" title="Francesca Pianosi and Thorsten Wagener. Understanding the time-varying importance of different uncertainty sources in hydrological modelling using global sensitivity analysis. Hydrological Processes, pages 3991–4003, November 2017. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/hyp.10968%4010.1111/%28ISSN%291099-1085.Kieth-Beven, doi:10.1002/hyp.10968&#64;10.1111/(ISSN)1099-1085.Kieth-Beven.">30</a>, <a class="reference internal" href="R.Bibliography.html#id167" title="Charles Rougé, Patrick M. Reed, Danielle S. Grogan, Shan Zuidema, Alexander Prusevich, Stanley Glidden, Jonathan R. Lamontagne, and Richard B. Lammers. Coordination and Control: Limits in Standard Representations of Multi-Reservoir Operations in Hydrological Modeling. Hydrology and Earth System Sciences Discussions, pages 1–37, November 2019. URL: https://www.hydrol-earth-syst-sci-discuss.net/hess-2019-589/, doi:https://doi.org/10.5194/hess-2019-589.">31</a>]</span>. There is however a point of departure here, especially when models are used to produce inferences that can inform decisions. When agencies and practitioners use models of their systems for public decisions, those models have already met sufficient conditions for credibility (e.g., acceptable representational fidelity), but may face broader tests on their salience and legitimacy in informing negotiated decisions <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id64" title="Sibel Eker, Elena Rovenskaya, Michael Obersteiner, and Simon Langan. Practice and perspectives in the validation of resource management models. Nature communications, 9(1):1–10, 2018.">22</a>, <a class="reference internal" href="R.Bibliography.html#id127" title="David W. Cash, William C. Clark, Frank Alcock, Nancy M. Dickson, Noelle Eckley, David H. Guston, Jill Jäger, and Ronald B. Mitchell. Knowledge systems for sustainable development. Proceedings of the national academy of sciences, 100(14):8086–8091, 2003. Publisher: National Acad Sciences.">32</a>, <a class="reference internal" href="R.Bibliography.html#id128" title="Dave D. White, Amber Wutich, Kelli L. Larson, Patricia Gober, Timothy Lant, and Clea Senneville. Credibility, salience, and legitimacy of boundary objects: water managers' assessment of a simulation model in an immersive decision theater. Science and Public Policy, 37(3):219–232, April 2010. Publisher: Oxford Academic. URL: https://academic.oup.com/spp/article/37/3/219/1626552 (visited on 2020-05-12), doi:10.3152/030234210X497726.">33</a>]</span>. This presents a new challenge to model validation, that of selecting decision-relevant performance metrics, reflective of the system’s stakeholders’ viewpoints, so that the most consequential uncertainties are identified and addressed <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id129" title="Andrea Saltelli and Silvio Funtowicz. When all models are wrong. Issues in Science and Technology, 30(2):79–85, 2014. Publisher: JSTOR.">34</a>]</span>. For complex multisector models at the intersection of climatic, hydrologic, agricultural, energy, or other processes, the output space is made up of a multitude of states and variables, with very different levels of salience to the system’s stakeholders and to their goals being achieved <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id203" title="Thorsten Wagener and Francesca Pianosi. What has Global Sensitivity Analysis ever done for us? A systematic review to support scientific advancement and to inform policy-making in earth system modelling. Earth-Science Reviews, 194:1–18, July 2019. URL: https://www.sciencedirect.com/science/article/pii/S0012825218300990 (visited on 2021-08-30), doi:10.1016/j.earscirev.2019.04.006.">35</a>]</span>. This is further complicated when such systems are also institutionally and dynamically complex. As a result, a broader set of qualitative and quantitative performance metrics is necessary to evaluate models of such complex systems, one that embraces the plurality of value systems, agencies and perspectives present. For IM3, even though the goal is to develop better projections of future vulnerability and resilience in co-evolving human-natural systems and not to provide decision support per se, it is critical for our multisector, multiscale model evaluation processes to represent stakeholders’ adaptive decision processes credibly.</p>
+<p>As a final point, when a model is used in a projection mode, its results are also subject to additional uncertainty, as there is no guarantee that the model’s functionality and predictive ability will stay the same as the baseline, where the verification and validation tests were conducted. This challenge requires an additional expansion of the scope of model evaluation: a broader set of uncertain conditions needs to be explored, spanning beyond historical observation and exploring a wide range of unprecedented conditions. This perspective on modeling, termed exploratory <span id="id17">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>]</span>, views models as computational experiments that can be used to explore vast ensembles of potential scenarios to identify those with consequential effects. Exploratory modeling literature explicitly orients experiments toward stakeholder consequences and decision-relevant inferences and shifts the focus from predicting future conditions to <em>discovering</em> which conditions lead to undesirable or desirable consequences.</p>
+<p>This evolution in modeling perspectives can be mirrored by the IM3 family of models in a progression from evaluating models relative to observed history to advanced formalized analyses to make inferences on multisector, multiscale vulnerabilities and resilience. Exploratory modeling approaches can help fashion experiments with large numbers of alternative hypotheses on the co-evolutionary dynamics of influences, stressors, as well as path-dependent changes in the form and function of human-natural systems <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id130" title="Christopher P. Weaver, Robert J. Lempert, Casey Brown, John A. Hall, David Revell, and Daniel Sarewitz. Improving the contribution of climate model information to decision making: the value and demands of robust decision frameworks. Wiley Interdisciplinary Reviews: Climate Change, 4(1):39–60, 2013. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/wcc.202 (visited on 2019-10-01), doi:10.1002/wcc.202.">37</a>]</span>. The aim of this text is to therefore guide the reader through the use of sensitivity analysis (SA) methods across these perspectives on diagnostic and exploratory modeling.</p>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#perspectives-on-diagnostic-model-evaluation">2.2. Perspectives on diagnostic model evaluation</a></li>
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+  <section id="diagnostic-modeling-overview-and-perspectives">
+<span id="diagnostic-modeling"></span><h1><span class="section-number">2. </span>Diagnostic Modeling Overview and Perspectives<a class="headerlink" href="#diagnostic-modeling-overview-and-perspectives" title="Link to this heading">#</a></h1>
+<p>This text prescribes a formal model diagnostic approach that is a deliberative and iterative combination of state-of-the-art UC and global sensitivity analysis techniques that progresses from observed history-based fidelity evaluations to forward looking resilience and vulnerability inferences <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id46" title="Hoshin V. Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008. Publisher: Wiley Online Library.">12</a>, <a class="reference internal" href="R.Bibliography.html#id47" title="Antonia Hadjimichael, Julianne Quinn, and Patrick Reed. Advancing Diagnostic Model Evaluation to Better Understand Water Shortage Mechanisms in Institutionally Complex River Basins. Water Resources Research, 56(10):e2020WR028079, 2020. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR028079 (visited on 2020-10-16), doi:10.1029/2020WR028079.">13</a>]</span>.</p>
+<section id="overview-of-model-diagnostics">
+<h2><span class="section-number">2.1. </span>Overview of model diagnostics<a class="headerlink" href="#overview-of-model-diagnostics" title="Link to this heading">#</a></h2>
+<p>Model diagnostics provide a rich basis for hypothesis testing, model innovation, and improved inferences when classifying what is controlling highly consequential results (e.g., vulnerability or resilience in coupled human-natural systems). <a class="reference internal" href="#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a>, adapted from <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id6" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">6</a>]</span>, presents idealized illustrations of the relationship between UC and global sensitivity analysis for two coupled simulation models. The figure illustrates how UC can be used to address how uncertainties in various modeling decisions (e.g., data inputs, parameters, model structures, coupling relationships) can be sampled and simulated to yield the empirical model output distribution(s) of interest. Monte Carlo frameworks allow us to sample and propagate (or integrate) the ensemble response of the model(s) of focus. The first step of any UC analysis is the specification of the initial input distributions as illustrated in <a class="reference internal" href="#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a>. The second step is to perform the Monte Carlo simulations. The question can then be raised, which of the modeling assumptions in our Monte Carlo experiment are the most responsible for the resulting output uncertainty. We can answer this question using global sensitivity analysis as illustrated in <a class="reference internal" href="#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a>. Global sensitivity analysis can be defined as a formal Monte Carlo sampling and analysis of modeling choices (structures, parameters, inputs) to quantify their influence on direct model outputs (or output-informed metrics). UC experiments by themselves do not explain why a particular uncertain outcome is produced, but produce distributions of model outcomes, as portrayed by the yellow curve. The pie chart shown in <a class="reference internal" href="#figure-2-1"><span class="std std-numref">Fig. 2.1</span></a> is a conceptual representation of the results of using a global sensitivity analysis to identify those factors that are most dominantly influencing results, either individually or interactively <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id48" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global Sensitivity Analysis: The Primer. Wiley-Interscience, Chichester, England ; Hoboken, NJ, 1 edition edition, February 2008. ISBN 978-0-470-05997-5.">14</a>]</span>.</p>
+<figure class="margin-caption align-center" id="id23">
+<span id="figure-2-1"></span><a class="reference internal image-reference" href="_images/figure2_1_idealized_uc.png"><img alt="Figure 2.1" src="_images/figure2_1_idealized_uc.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 2.1 </span><span class="caption-text">Idealized uncertainty characterization and global sensitivity analysis for two coupled simulation models. Uncertainty coming from various sources (e.g., inputs, model structures, coupling relationships) is propagated through the coupled model(s) to generate empirical distributions of outputs of interest (uncertainty characterization). This model output uncertainty can be decomposed to its origins, by means of sensitivity analysis. Figure adapted from <span id="id4">Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id6" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">6</a>]</span>.</span><a class="headerlink" href="#id23" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>UC and global sensitivity analysis are not independent modeling analyses. As illustrated here, any global sensitivity analysis requires an initial UC hypothesis in the form of statistical assumptions and representations for the modeling choices of focus (structural, parametric, and data inputs). Information from these two model diagnostic tools can then be used to inform data needs for future model runs, experiments to reduce the uncertainty present, or the simplification or enhancement of the model where necessary. Together UC and global sensitivity analysis provide a foundation for diagnostic exploratory modeling that has a consistent focus on the assumptions, structural model forms, alternative parameterizations, and input data sets that are used to characterize the behavioral space of one or more models.</p>
+</section>
+<section id="perspectives-on-diagnostic-model-evaluation">
+<span id="perspectives"></span><h2><span class="section-number">2.2. </span>Perspectives on diagnostic model evaluation<a class="headerlink" href="#perspectives-on-diagnostic-model-evaluation" title="Link to this heading">#</a></h2>
+<p>When we judge or diagnose models, the terms “verification” and “validation” are commonly used. However, their appropriateness in the context of numerical models representing complex coupled human-natural systems is questionable <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id54" title="Keith Beven. Towards a coherent philosophy for modelling the environment. Proceedings of the royal society of London. Series A: mathematical, physical and engineering sciences, 458(2026):2465–2484, 2002.">15</a>, <a class="reference internal" href="R.Bibliography.html#id118" title="Naomi Oreskes, Kristin Shrader-Frechette, and Kenneth Belitz. Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences. Science, 263(5147):641–646, February 1994. URL: https://science.sciencemag.org/content/263/5147/641 (visited on 2020-04-15), doi:10.1126/science.263.5147.641.">16</a>]</span>. The core issue relates to the fact that these systems are often not fully known or perfectly implemented when modeled. Rather, they are defined within specific system framings and boundary conditions in an evolving learning process with the goal of making continual progress towards attaining higher levels of fidelity in capturing behaviors or properties of interest. Evaluating the fidelity of a model’s performance can be highly challenging. For example, the observations used to evaluate the fidelity of parameterized processes are often measured at a finer resolution than what is represented in the model, creating the challenge of how to manage their relative scales when performing evaluation. In other cases, numerical models may neglect or simplify system processes because sufficient data is not available or the physical mechanisms are not fully known. If sufficient agreement between prediction and observation is not achieved, it is challenging to know whether these types of modeling choices are the cause, or if other issues, such as deficiencies in the input parameters and/or other modeling assumptions are the true cause of errors. Even if there is high agreement between prediction and observation, the model cannot necessarily be considered validated, as it is always possible that the right values were produced for the wrong reasons. For example, low error can stem from a situation where different errors in underlying assumptions or parameters cancel each other out (“compensatory errors”). Furthermore, coupled human-natural system models are often subject to “equifinality”, a situation where multiple parameterized formulations can produce similar outputs or equally acceptable representations of the observed data. There is therefore no uniquely “true” or validated model, and the common practice of selecting “the best” deterministic calibration set is more of an assumption than a finding <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id49" title="Keith Beven. Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in water resources, 16(1):41–51, 1993.">17</a>, <a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>]</span>. The situation becomes even more tenuous when observational data is limited in its scope and/or quality to be insufficient to distinguish model representations or their performance differences.</p>
+<p>These limitations on model verification undermine any purely positivist treatment of model validity: that a model should correctly and precisely represent reality to be valid. Under this perspective, closely related to empiricism, statistical tests should be used to compare the model’s output with observations and only through empirical verification can a model or theory be deemed credible. A criticism to this viewpoint (besides the aforementioned challenges for model verification) is that it reduces the justification of a model to the single criterion of predictive ability and accuracy <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id119" title="Yaman Barlas and Stanley Carpenter. Philosophical roots of model validation: Two paradigms. System Dynamics Review, 6(2):148–166, 1990. doi:10.1002/sdr.4260060203.">19</a>]</span>. Authors have argued that this ignores the explanatory power held in models and other procedures, which can also advance scientific knowledge <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id120" title="Stephen Toulmin. From form to function: philosophy and history of science in the 1950s and now. Daedalus, pages 143–162, 1977. Publisher: JSTOR.">20</a>]</span>. These views gave rise to relativist perspectives of science, which instead place more value on model utility in terms of fitness for a specific purpose or inquiry, rather than representational accuracy and predictive ability <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id121" title="George B. Kleindorfer, Liam O'Neill, and Ram Ganeshan. Validation in simulation: Various positions in the philosophy of science. Management Science, 44(8):1087–1099, 1998. Publisher: INFORMS.">21</a>]</span>. This viewpoint appears to be most prevalent among practitioners seeking decision-relevant insights (i.e., inspire new views vs. predict future conditions). The relativist perspective argues for the use of models as heuristics that can enhance our understanding and conceptions of system behaviors or possibilities <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id64" title="Sibel Eker, Elena Rovenskaya, Michael Obersteiner, and Simon Langan. Practice and perspectives in the validation of resource management models. Nature communications, 9(1):1–10, 2018.">22</a>]</span>. In contrast, natural sciences favor a positivist perspective, emphasizing similarity between simulation and observation even in application contexts where it is clear that projections are being made for conditions that have never been observed and the system of focus will have evolved structurally beyond the model representation being employed (e.g., decadal to centennial evolution of human-natural systems).</p>
+<p>These differences in prevalent perspectives are mirrored in how model validation is defined by the two camps: From the relativist perspective, validation is seen as a process of incremental “confidence building” in a model as a mechanism for insight <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id122" title="Yaman Barlas. Formal aspects of model validity and validation in system dynamics. System Dynamics Review: The Journal of the System Dynamics Society, 12(3):183–210, 1996. Publisher: Wiley Online Library.">23</a>]</span>, whereas in natural sciences validation is framed as a way to classify a model as having an acceptable representation of physical reality <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id118" title="Naomi Oreskes, Kristin Shrader-Frechette, and Kenneth Belitz. Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences. Science, 263(5147):641–646, February 1994. URL: https://science.sciencemag.org/content/263/5147/641 (visited on 2020-04-15), doi:10.1126/science.263.5147.641.">16</a>]</span>. Even though the relativist viewpoint does not dismiss the importance of representational accuracy, it does place it within a larger process of establishing confidence through a variety of tools. These tools, not necessarily quantitative, include communicating information between practitioners and modelers, interpreting a multitude of model outputs, and contrasting preferences and viewpoints.</p>
+<p>On the technical side of the argument, differing views on the methodology of model validation appear as early as in the 1960’s. <span id="id13">Naylor and Finger [<a class="reference internal" href="R.Bibliography.html#id123" title="Thomas H. Naylor and Joseph Michael Finger. Verification of computer simulation models. Management science, 14(2):B–92, 1967. Publisher: INFORMS.">24</a>]</span> argue that model validation should not be limited to a single metric or test of performance (e.g., a single error metric), but should rather be extended to multiple tests that reflect different aspects of a model’s structure and behavior. This and similar arguments are made in literature to this day <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id46" title="Hoshin V. Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008. Publisher: Wiley Online Library.">12</a>, <a class="reference internal" href="R.Bibliography.html#id52" title="Keith J Beven. On hypothesis testing in hydrology: why falsification of models is still a really good idea. Wiley Interdisciplinary Reviews: Water, 5(3):e1278, 2018.">25</a>, <a class="reference internal" href="R.Bibliography.html#id51" title="Hoshin V Gupta, Martyn P Clark, Jasper A Vrugt, Gab Abramowitz, and Ming Ye. Towards a comprehensive assessment of model structural adequacy. Water Resources Research, 2012.">26</a>, <a class="reference internal" href="R.Bibliography.html#id124" title="Praveen Kumar. Typology of hydrologic predictability. Water Resources Research, 2011. URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2010WR009769 (visited on 2020-04-15), doi:10.1029/2010WR009769.">27</a>, <a class="reference internal" href="R.Bibliography.html#id125" title="Grey S. Nearing, Benjamin L. Ruddell, Andrew R. Bennett, Cristina Prieto, and Hoshin V. Gupta. Does Information Theory Provide a New Paradigm for Earth Science? Hypothesis Testing. Water Resources Research, 56(2):e2019WR024918, 2020. doi:10.1029/2019WR024918.">28</a>]</span> and are primarily founded on two premises. First, that even though modelers widely recognize that their models are abstractions of the truth, they still make truth claims based on traditional performance metrics that measure the divergence of their model from observation <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id125" title="Grey S. Nearing, Benjamin L. Ruddell, Andrew R. Bennett, Cristina Prieto, and Hoshin V. Gupta. Does Information Theory Provide a New Paradigm for Earth Science? Hypothesis Testing. Water Resources Research, 56(2):e2019WR024918, 2020. doi:10.1029/2019WR024918.">28</a>]</span>. Second, that the natural systems mimicked by the models contain many processes that exhibit significant heterogeneity at various temporal and spatial scales. This heterogeneity is lost when a single performance measure is used, as a result of the inherent loss of process information occurring when transitioning from a highly dimensional and interactive system to the dimension of a single metric <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id54" title="Keith Beven. Towards a coherent philosophy for modelling the environment. Proceedings of the royal society of London. Series A: mathematical, physical and engineering sciences, 458(2026):2465–2484, 2002.">15</a>]</span>. These arguments are further elaborated in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#sensitivity-analysis"><span class="std std-numref">Chapter 4</span></a>.</p>
+<p>Multiple authors have proposed that the traditional reliance on single measures of model performance should be replaced by the evaluation of several model signatures (characteristics) to identify model structural errors and achieve a sufficient assessment of model performance <span id="id17">[<a class="reference internal" href="R.Bibliography.html#id46" title="Hoshin V. Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008. Publisher: Wiley Online Library.">12</a>, <a class="reference internal" href="R.Bibliography.html#id126" title="Hoshin Vijai Gupta, Soroosh Sorooshian, and Patrice Ogou Yapo. Toward improved calibration of hydrologic models: Multiple and noncommensurable measures of information. Water Resources Research, 34(4):751–763, 1998. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/97WR03495 (visited on 2020-04-07), doi:10.1029/97WR03495.">29</a>, <a class="reference internal" href="R.Bibliography.html#id168" title="Francesca Pianosi and Thorsten Wagener. Understanding the time-varying importance of different uncertainty sources in hydrological modelling using global sensitivity analysis. Hydrological Processes, pages 3991–4003, November 2017. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/hyp.10968%4010.1111/%28ISSN%291099-1085.Kieth-Beven, doi:10.1002/hyp.10968&#64;10.1111/(ISSN)1099-1085.Kieth-Beven.">30</a>, <a class="reference internal" href="R.Bibliography.html#id167" title="Charles Rougé, Patrick M. Reed, Danielle S. Grogan, Shan Zuidema, Alexander Prusevich, Stanley Glidden, Jonathan R. Lamontagne, and Richard B. Lammers. Coordination and Control: Limits in Standard Representations of Multi-Reservoir Operations in Hydrological Modeling. Hydrology and Earth System Sciences Discussions, pages 1–37, November 2019. URL: https://www.hydrol-earth-syst-sci-discuss.net/hess-2019-589/, doi:https://doi.org/10.5194/hess-2019-589.">31</a>]</span>. There is however a point of departure here, especially when models are used to produce inferences that can inform decisions. When agencies and practitioners use models of their systems for public decisions, those models have already met sufficient conditions for credibility (e.g., acceptable representational fidelity), but may face broader tests on their salience and legitimacy in informing negotiated decisions <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id64" title="Sibel Eker, Elena Rovenskaya, Michael Obersteiner, and Simon Langan. Practice and perspectives in the validation of resource management models. Nature communications, 9(1):1–10, 2018.">22</a>, <a class="reference internal" href="R.Bibliography.html#id127" title="David W. Cash, William C. Clark, Frank Alcock, Nancy M. Dickson, Noelle Eckley, David H. Guston, Jill Jäger, and Ronald B. Mitchell. Knowledge systems for sustainable development. Proceedings of the national academy of sciences, 100(14):8086–8091, 2003. Publisher: National Acad Sciences.">32</a>, <a class="reference internal" href="R.Bibliography.html#id128" title="Dave D. White, Amber Wutich, Kelli L. Larson, Patricia Gober, Timothy Lant, and Clea Senneville. Credibility, salience, and legitimacy of boundary objects: water managers' assessment of a simulation model in an immersive decision theater. Science and Public Policy, 37(3):219–232, April 2010. Publisher: Oxford Academic. URL: https://academic.oup.com/spp/article/37/3/219/1626552 (visited on 2020-05-12), doi:10.3152/030234210X497726.">33</a>]</span>. This presents a new challenge to model validation, that of selecting decision-relevant performance metrics, reflective of the system’s stakeholders’ viewpoints, so that the most consequential uncertainties are identified and addressed <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id129" title="Andrea Saltelli and Silvio Funtowicz. When all models are wrong. Issues in Science and Technology, 30(2):79–85, 2014. Publisher: JSTOR.">34</a>]</span>. For complex multisector models at the intersection of climatic, hydrologic, agricultural, energy, or other processes, the output space is made up of a multitude of states and variables, with very different levels of salience to the system’s stakeholders and to their goals being achieved <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id203" title="Thorsten Wagener and Francesca Pianosi. What has Global Sensitivity Analysis ever done for us? A systematic review to support scientific advancement and to inform policy-making in earth system modelling. Earth-Science Reviews, 194:1–18, July 2019. URL: https://www.sciencedirect.com/science/article/pii/S0012825218300990 (visited on 2021-08-30), doi:10.1016/j.earscirev.2019.04.006.">35</a>]</span>. This is further complicated when such systems are also institutionally and dynamically complex. As a result, a broader set of qualitative and quantitative performance metrics is necessary to evaluate models of such complex systems, one that embraces the plurality of value systems, agencies and perspectives present. For IM3, even though the goal is to develop better projections of future vulnerability and resilience in co-evolving human-natural systems and not to provide decision support per se, it is critical for our multisector, multiscale model evaluation processes to represent stakeholders’ adaptive decision processes credibly.</p>
+<p>As a final point, when a model is used in a projection mode, its results are also subject to additional uncertainty, as there is no guarantee that the model’s functionality and predictive ability will stay the same as the baseline, where the verification and validation tests were conducted. This challenge requires an additional expansion of the scope of model evaluation: a broader set of uncertain conditions needs to be explored, spanning beyond historical observation and exploring a wide range of unprecedented conditions. This perspective on modeling, termed exploratory <span id="id21">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>]</span>, views models as computational experiments that can be used to explore vast ensembles of potential scenarios to identify those with consequential effects. Exploratory modeling literature explicitly orients experiments toward stakeholder consequences and decision-relevant inferences and shifts the focus from predicting future conditions to <em>discovering</em> which conditions lead to undesirable or desirable consequences.</p>
+<p>This evolution in modeling perspectives can be mirrored by the IM3 family of models in a progression from evaluating models relative to observed history to advanced formalized analyses to make inferences on multisector, multiscale vulnerabilities and resilience. Exploratory modeling approaches can help fashion experiments with large numbers of alternative hypotheses on the co-evolutionary dynamics of influences, stressors, as well as path-dependent changes in the form and function of human-natural systems <span id="id22">[<a class="reference internal" href="R.Bibliography.html#id130" title="Christopher P. Weaver, Robert J. Lempert, Casey Brown, John A. Hall, David Revell, and Daniel Sarewitz. Improving the contribution of climate model information to decision making: the value and demands of robust decision frameworks. Wiley Interdisciplinary Reviews: Climate Change, 4(1):39–60, 2013. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/wcc.202 (visited on 2019-10-01), doi:10.1002/wcc.202.">37</a>]</span>. The aim of this text is to therefore guide the reader through the use of sensitivity analysis (SA) methods across these perspectives on diagnostic and exploratory modeling.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The following articles are suggested as fundamental reading for the information presented in this section:</p>
+<ul class="simple">
+<li><p>Naomi Oreskes, Kristin Shrader–Frechette, and Kenneth Belitz. Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences. <em>Science</em>, 263 (5147): 641-646, February 1994. URL: <a class="reference external" href="https://science.sciencemag.org/content/263/5147/641">https://science.sciencemag.org/content/263/5147/641</a>. DOI: <a class="reference external" href="https://doi.org/10.1126/science.263.5147.641">https://doi.org/10.1126/science.263.5147.641</a>.</p></li>
+<li><p>Keith Beven. Towards a coherent philosophy for modelling the environment. <em>Proceedings of the Royal Society of London</em>. Series A: mathematical, physical and engineering sciences, 458 (2026): 2465-2484, 2002.</p></li>
+<li><p>Eker, S., Rovenskaya, E., Obersteiner, M., Langan, S., 2018. Practice and perspectives in the validation of resource management models. <em>Nature Communications</em> 9, 1–10. <a class="reference external" href="https://doi.org/10.1038/s41467-018-07811-9">https://doi.org/10.1038/s41467-018-07811-9</a></p></li>
+</ul>
+<p>The following articles can be used as supplemental reading:</p>
+<ul class="simple">
+<li><p>Canham, C.D., Cole, J.J., Lauenroth, W.K. (Eds.), 2003. Models in Ecosystem Science. Princeton University Press. <a class="reference external" href="https://doi.org/10.2307/j.ctv1dwq0tq">https://doi.org/10.2307/j.ctv1dwq0tq</a></p></li>
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+    <h1>Global Versus Local Sensitivity</h1>
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+  <section id="global-versus-local-sensitivity">
+<span id="global-vs-local"></span><h1>Global Versus Local Sensitivity<a class="headerlink" href="#global-versus-local-sensitivity" title="Link to this heading">#</a></h1>
+<p>Out of the several definitions for sensitivity analysis presented in the literature, the most widely used has been proposed by <span id="id1">Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span> as “the study of how uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input”. In other words, sensitivity analysis explores the relationship between the model’s <span class="math notranslate nohighlight">\(N\)</span> input variables, <span class="math notranslate nohighlight">\(x=[x_1,x_2,...,x_N]\)</span>, and <span class="math notranslate nohighlight">\(M\)</span> output variables, <span class="math notranslate nohighlight">\(y=[y_1,y_2,...,y_M]\)</span> with <span class="math notranslate nohighlight">\(y=g(x)\)</span>, where <span class="math notranslate nohighlight">\(g\)</span> is the model that maps the model inputs to the outputs <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>]</span>.</p>
+<p>Historically, there have been two broad categories of sensitivity analysis techniques: local and global. Local sensitivity analysis is performed by varying model parameters around specific reference values, with the goal of exploring how small input perturbations influence model performance. Due to its ease-of-use and limited computational demands, this approach has been widely used in literature, but has important limitations <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id18" title="O Rakovec, Mary C Hill, MP Clark, AH Weerts, AJ Teuling, and R Uijlenhoet. Distributed evaluation of local sensitivity analysis (delsa), with application to hydrologic models. Water Resources Research, 50(1):409–426, 2014.">40</a>, <a class="reference internal" href="R.Bibliography.html#id19" title="Andrea Saltelli and Paola Annoni. How to avoid a perfunctory sensitivity analysis. Environmental Modelling &amp; Software, 25(12):1508–1517, 2010.">41</a>]</span>. If the model is not linear, the results of local sensitivity analysis can be heavily biased, as they are strongly influenced by independence assumptions and a limited exploration of model inputs (e.g., <span id="id4">Tang <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id20" title="Yong Tang, Patrick Reed, Thibaut Wagener, and K van Werkhoven. Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation. Hydrology and Earth System Sciences, 11(2):793–817, 2007.">42</a>]</span>). If the model’s factors interact, local sensitivity analysis will underestimate their importance, as it does not account for those effects (e.g., <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id23" title="Nicholas AS Hamm, Jim W Hall, and MG Anderson. Variance-based sensitivity analysis of the probability of hydrologically induced slope instability. Computers &amp; geosciences, 32(6):803–817, 2006.">43</a>]</span>). In general, as local sensitivity analysis only partially and locally explores a model’s parametric space, it is not considered a valid approach for nonlinear models <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id24" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">44</a>]</span>. This is illustrated in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (a-b), presenting contour plots of a model response (<span class="math notranslate nohighlight">\(y\)</span>) with an additive linear model (a) and with a nonlinear model (b). In a linear model without interactions between the input terms <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span>, local sensitivity analysis (assuming deviations from some reference values) can produce appropriate sensitivity indices (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (a)). If however, factors <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> interact, the local and partial consideration of the space can not properly account for each factor’s effects on the model response (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (b)), as it is only informative at the reference value where it is applied. In contrast, a global sensitivity analysis varies uncertain factors within the entire feasible space of variable model responses (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (c)). This approach reveals the global effects of each parameter on the model output, including any interactive effects. For models that cannot be proven linear, global sensitivity analysis is preferred and this text is primarily discussing global sensitivity analysis methods. In the text that follows, whenever we use the term sensitivity analysis we are referring to its global application.</p>
+<figure class="margin-caption align-center" id="id7">
+<span id="figure-3-1"></span><a class="reference internal image-reference" href="_images/figure3_1_global_versus_local.png"><img alt="Figure 3.1" src="_images/figure3_1_global_versus_local.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Treatment of a two-dimensional space of variability by local (panels a-b) and global (panel c) sensitivity analyses. Panels depict contour plots with the value of a model response (<span class="math notranslate nohighlight">\(y\)</span>) changing with changes in the values of input terms <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span>. Local sensitivity analysis is only an appropriate approach to sensitivity in the case of linear models without interactions between terms, for example in panel (a), where <span class="math notranslate nohighlight">\(y=3x_1+5x_2\)</span>. In the case of more complex models, for example in panels (b-c), where <span class="math notranslate nohighlight">\(y={1 \above 1pt e^{x^2_1+x^2_2}} + {50 \above 1pt e^{(0.1x_1)^2+(0.1x_2)^3}}\)</span>, local sensitivity will miscalculate sensitivity indices as the assessed changes in the value <span class="math notranslate nohighlight">\(y\)</span> depend on the assumed base values chose for <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> (panel (b)). In these cases, global sensitivity methods should be used instead (panel (c)). The points in panel (c) are generated using a uniform random sample of <span class="math notranslate nohighlight">\(n=50\)</span>, but many other methods are available.</span><a class="headerlink" href="#id7" title="Link to this image">#</a></p>
+</figcaption>
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+  <section id="why-perform-sensitivity-analysis">
+<span id="why-sa"></span><h1>Why Perform Sensitivity Analysis<a class="headerlink" href="#why-perform-sensitivity-analysis" title="Link to this heading">#</a></h1>
+<p>It is important to understand the many ways in which a SA might be of use to your modeling effort. Most commonly, one might be motivated to perform sensitivity analysis for the following reasons:</p>
+<p><em>Model evaluation</em>: Sensitivity analysis can be used to gauge model inferences when assumptions about the structure of the model or its parameterization are dubious or have changed. For instance, consider a numerical model that uses a set of calibrated parameter values to produce outputs, which we then use to inform decisions about the real-world system represented. One might like to know if small changes in these parameter values significantly change this model’s output and the decisions it informs or if, instead, our parameter inferences yield stable model behavior regardless of the uncertainty present in the specific parameterized processes or properties. This can either discredit or lend credence to the model at hand, as well as any inferences drawn that are founded on its accurate representation of the system. Sensitivity analysis can identify which uncertain model factors cause this undesirable model behavior.</p>
+<p><em>Model simplification</em>: Sensitivity analysis can also be used to identify factors or components of the model that appear to have limited effects on direct outputs or metrics of interest. Consider a model that has been developed in an organization for the purposes of a specific research question and is later used in the context of a different application. Some processes represented in significant detail might no longer be of the same importance while consuming significant data or computational resources, as different outputs might be pertinent to the new application. Sensitivity analysis can be used to identify unimportant model components and simplify them to nominal values and reduced model forms. Model complexity and computational costs can therefore be reduced.</p>
+<p><em>Model refinement</em>: Alternatively, sensitivity analysis can reveal the factors or processes that are highly influential to the outputs or metrics of interest, by assessing their relative importance. In the context of model evaluation, this can inform which model components warrant additional investigation or measurement so the uncertainty surrounding them and the resulting model outputs or metrics of interest can be reduced.</p>
+<p><em>Exploratory modeling</em>: When sufficient credence has been established in the model, sensitivity analysis can be applied to a host of other inquiries. Inferences about the factors and processes that most (or least) control a model’s outputs of interest can be extrapolated to the real system they represent and be used in a heuristic manner to inform model-based inferences. On this foundation, a model paired with the advanced techniques presented in this text can be used to “discover” decision relevant and highly consequential outcomes (i.e., scenario discovery, discussed in more detail in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a> <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>, <a class="reference internal" href="R.Bibliography.html#id16" title="Benjamin P Bryant and Robert J Lempert. Thinking inside the box: a participatory, computer-assisted approach to scenario discovery. Technological Forecasting and Social Change, 77(1):34–49, 2010.">45</a>]</span>).</p>
+<p>The nature and context of the model shapes the specific objectives of applying a sensitivity analysis, as well as methods and tools most appropriate and defensible for each application setting <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id203" title="Thorsten Wagener and Francesca Pianosi. What has Global Sensitivity Analysis ever done for us? A systematic review to support scientific advancement and to inform policy-making in earth system modelling. Earth-Science Reviews, 194:1–18, July 2019. URL: https://www.sciencedirect.com/science/article/pii/S0012825218300990 (visited on 2021-08-30), doi:10.1016/j.earscirev.2019.04.006.">35</a>, <a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id25" title="Andrea Saltelli and Stefano Tarantola. On the relative importance of input factors in mathematical models: safety assessment for nuclear waste disposal. Journal of the American Statistical Association, 97(459):702–709, 2002.">46</a>]</span>. The three most common sensitivity analysis modes (<em>Factor Prioritization</em>, <em>Factor Fixing</em>, and <em>Factor Mapping</em>) are presented below, but the reader should be aware that other uses have been proposed in the literature (e.g., <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id26" title="Barry Anderson, Emanuele Borgonovo, Marzio Galeotti, and Roberto Roson. Uncertainty in climate change modeling: can global sensitivity analysis be of help? Risk analysis, 34(2):271–293, 2014.">47</a>, <a class="reference internal" href="R.Bibliography.html#id27" title="Emanuele Borgonovo. Sensitivity analysis with finite changes: an application to modified eoq models. European Journal of Operational Research, 200(1):127–138, 2010.">48</a>]</span>).</p>
+<p><em>Factor prioritization</em>: This sensitivity analysis application mode (also referred to as <em>factor ranking</em>) refers to when one would like to identify the uncertain factors that have the greatest impact on the variability of the output, and which, when fixed to their true value (i.e., if there were no uncertainty regarding their value), would lead to the greatest reduction in output variability <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. Information from this type of analysis can be crucial to model improvement as these factors can become the focus of future measurement campaigns or numerical experiments so that uncertainty in the model output can be reduced. The impact of each uncertain input on the variance of the model output is often used as the criterion for factor prioritization. <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (a) shows the effects of three uncertain variables (<span class="math notranslate nohighlight">\(X_1\)</span>, <span class="math notranslate nohighlight">\(X_2\)</span>, and <span class="math notranslate nohighlight">\(X_3\)</span>) on the variance of output <span class="math notranslate nohighlight">\(Y\)</span>. <span class="math notranslate nohighlight">\(V(E(Y|X_i))\)</span> indicates the variance in <span class="math notranslate nohighlight">\(Y\)</span> if factor <span class="math notranslate nohighlight">\(X_i\)</span> is left to vary freely while all other factors remain fixed to nominal values. In this case, factor <span class="math notranslate nohighlight">\(X_2\)</span> makes the largest contribution to the variability of output <span class="math notranslate nohighlight">\(Y\)</span> and it should therefore be prioritized. In the context of risk analysis, factor prioritization can be used to reduce output variance to below a given tolerable threshold (also known as variance cutting).</p>
+<p><em>Factor fixing</em>: This mode of sensitivity analysis (also referred to as <em>factor screening</em>) aims to identify the model components that have a negligible effect or make no significant contributions to the variability of the outputs or metrics of interest (usually referred to as non-influential <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>). In the stylized example of <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (a), <span class="math notranslate nohighlight">\(X_1\)</span> makes the smallest contribution to the variability of output <span class="math notranslate nohighlight">\(Y\)</span> suggesting that the uncertainty in its value could be negligible and the factor itself fixed in subsequent model executions. Eliminating these factors or processes in the model or fixing them to a nominal value can help reduce model complexity as well as the unnecessary computational burden of subsequent model runs, results processing, or other sensitivity analyses (the fewer uncertain factors considered, the fewer runs are necessary to illuminate their effects on the output). Significance of the outcome can be gauged in a variety of manners, depending on the application. For instance, if applying a variance-based method, a minimum threshold value of contribution to the variance could be considered as a significance ‘cutoff’, and factors with indices below that value can be considered non-influential. Conclusions about factor fixing should be made carefully, considering all of the effects a factor has, individually and in interaction with other factors (explained in more detail in the <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#variance-based-methods"><span class="std std-numref">Chapter 3.4.5</span></a>).</p>
+<p><em>Factor mapping</em>: Finally, factor mapping can be used to pinpoint which values of uncertain factors lead to model outputs within a given range of the output space <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. In the context of model diagnostics, it is possible that the model’s output changes in ways considered impossible based on the represented processes, or other observed evidence. In this situation, factor mapping can be used to identify which uncertain model factors cause this undesirable model behavior by ‘filtering’ model runs that are considered ‘non-behavioral’ <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id29" title="Neil R Edwards, David Cameron, and Jonathan Rougier. Precalibrating an intermediate complexity climate model. Climate dynamics, 37(7):1469–1482, 2011.">50</a>, <a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id31" title="RC Spear and GM Hornberger. Eutrophication in peel inlet—ii. identification of critical uncertainties via generalized sensitivity analysis. Water research, 14(1):43–49, 1980.">52</a>]</span>. In <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (b), region <span class="math notranslate nohighlight">\(B\)</span> of the output space <span class="math notranslate nohighlight">\(Y\)</span> denotes the set of behavioral model outcomes and region <span class="math notranslate nohighlight">\(\bar{B}\)</span> denotes the set of non-behavioral outcomes, resulting from the entirety of input space <span class="math notranslate nohighlight">\(X\)</span>. Factor mapping refers to the process of tracing which factor values of input space <span class="math notranslate nohighlight">\(X\)</span> produce the behavioral model outcomes in the output space.</p>
+<figure class="margin-caption align-center" id="id8">
+<span id="figure-3-2"></span><a class="reference internal image-reference" href="_images/figure3_2_factor_mapping.png"><img alt="Figure 3.2" src="_images/figure3_2_factor_mapping.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Factor prioritization, factor fixing and factor mapping settings of sensitivity analysis.</span><a class="headerlink" href="#id8" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The language used above reflects a use of sensitivity analysis for model fidelity evaluation and refinement. However, as previously mentioned, when a model has been established as a sufficiently accurate representation of the system, sensitivity analysis can produce additional inferences (i.e., exploratory modeling and scenario discovery). For instance, under the factor mapping use, the analyst can now focus on undesirable system states and discover which factors are most responsible for them: for instance, “population growth of above 25% would be responsible for unacceptably high energy demands”. Factor prioritization and factor fixing can be used to make equivalent inferences, such as “growing populations and increasing temperatures are the leading factors for changing energy demands” (prioritizing of factors) or “changing dietary needs are inconsequential to increasing energy demands for this region” (a factor that can be fixed in subsequent model runs). All these inferences hinge on the assumption that the real system’s stakeholders consider the model states faithful enough representations of system states. As elaborated in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>, this view on sensitivity analysis is founded on a relativist perspective on modeling, which tends to place more value on model usefulness rather than strict accuracy of representation in terms of error. As such, sensitivity analysis performed with decision-making relevance in mind will focus on model outputs or metrics that are consequential and decision relevant (e.g., energy demand in the examples above).</p>
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+<h1>One-At-a-Time (OAT)<a class="headerlink" href="#one-at-a-time-oat" title="Link to this heading">#</a></h1>
+<p>In this approach, only one model factor is changed at a time while all others are kept fixed across each iteration in a sampling sequence. The OAT method assumes that model factors of focus are linearly independent (i.e., there are no interactions) and can analyze how factors individually influence model outputs or metrics of interest. While popular given its ease of implementation, OAT  is ultimately limited in its exploration of a model’s sensitivities <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. It is primarily used with local sensitivity techniques with similar criticisms: applying this sampling scheme on a system with nonlinear and interactive processes will miss important information on the effect uncertain factors have on the model. OAT samplings can be repeated multiple times in a more sophisticated manner and across different locations of the parameter space to overcome some of these challenges, which would increase computational costs and negate the main reasons for its selection. Given these limitations OAT methods could be used as preliminary, low-cost analyses of the factors’ individual effects, but should ultimately be complemented with more sophisticated methods.</p>
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+    <h1>Full and Fractional Factorial Sampling</h1>
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+  <section id="full-and-fractional-factorial-sampling">
+<h1>Full and Fractional Factorial Sampling<a class="headerlink" href="#full-and-fractional-factorial-sampling" title="Link to this heading">#</a></h1>
+<p>In full factorial sampling each factor is treated as being discrete by considering two or more levels (or intervals) of its values. The sampling process then generates samples within each possible combination of levels, corresponding to each parameter. This scheme produces a more comprehensive sampling of the factors’ variability space, as it accounts for all candidate combinations of factor levels (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a)). If the number of levels is the same across all factors, the number of generated samples is estimated using <span class="math notranslate nohighlight">\(n^k\)</span>, where <span class="math notranslate nohighlight">\(n\)</span> is the number of levels and <span class="math notranslate nohighlight">\(k\)</span> is the number of factors. For example, <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a) presents a full factorial sampling of three uncertain factors <span class="math notranslate nohighlight">\((x_1,\)</span> <span class="math notranslate nohighlight">\(x_2,\)</span> and <span class="math notranslate nohighlight">\(x_3)\)</span>, each considered as having four discrete levels. The total number of samples necessary for such an experiment is <span class="math notranslate nohighlight">\(4^3=64\)</span>. As the number of factors increases, the number of simulations necessary will also grow exponentially, making full factorial sampling computationally burdensome (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (b)). As a result, it is common in the literature to apply full factorial sampling at only two levels per factor, typically the two extremes <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id32" title="Douglas C Montgomery. Design and analysis of experiments. John wiley &amp; sons, 2017.">60</a>]</span>. This significantly reduces computational burden but is only considered appropriate in cases where factors can indeed only assume two discrete values (e.g., when testing the effects of epistemic uncertainty and comparing between model structure A and model structure B). In the case of physical parameters on continuous distributions (e.g., when considering the effects of measurement uncertainty in a temperature sensor), discretizing the range of a factor to only extreme levels can bias its estimated importance.</p>
+<p>Fractional factorial sampling is a widely used alternative to full factorial sampling that allows the analyst to significantly reduce the number of simulations by focusing on the main effects of a factor and seeking to avoid model runs that yield redundant response information <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. In other words, if one can reasonably assume that higher-order interactions are negligible, information about the most significant effects and lower-order interactions (e.g., effects from pairs of factors) can be obtained using a fraction of the full factorial design. Traditionally, fractional factorial design has also been limited to two levels <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id32" title="Douglas C Montgomery. Design and analysis of experiments. John wiley &amp; sons, 2017.">60</a>]</span>, referred to as Fractional Factorial designs 2k-p <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id34" title="George EP Box and J Stuart Hunter. The 2 k—p fractional factorial designs. Technometrics, 3(3):311–351, 1961.">61</a>]</span>. Recently, Generalized Fractional Factorial designs have also been proposed that allow for the structured generation of samples at more than two levels per factor <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id33" title="Izabella Surowiec, Ludvig Vikstrom, Gustaf Hector, Erik Johansson, Conny Vikstrom, and Johan Trygg. Generalized subset designs in analytical chemistry. Analytical chemistry, 89(12):6491–6497, 2017.">62</a>]</span>. Consider a case where the modeling team dealing with the problem in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a) cannot afford to perform 64 simulations of their model. They can afford 32 runs for their experiment and instead decide to fractionally sample the variability space of their factors. A potential design of such a sampling strategy is presented in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (c).</p>
+<figure class="margin-caption align-center" id="id6">
+<span id="figure-3-3"></span><a class="reference internal image-reference" href="_images/figure3_3_alternative_designs.png"><img alt="Figure 3.3" src="_images/figure3_3_alternative_designs.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Alternative designs of experiments and their computational costs for three uncertain factors <span class="math notranslate nohighlight">\((x_1,\)</span> <span class="math notranslate nohighlight">\(x_2,\)</span> and <span class="math notranslate nohighlight">\(x_3)\)</span>. (a) Full factorial design sampling of three factors at four levels, at a total of 64 samples; (b) exponential growth of necessary number of samples when applying full factorial design at four levels; (c) fractional factorial design of three factors at four levels, at a total of 32 samples; and (d) Latin Hypercube sample of three factors with uniform distributions, at a total of 32 samples.</span><a class="headerlink" href="#id6" title="Link to this image">#</a></p>
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+  <section id="latin-hypercube-sampling-lhs">
+<h1>Latin Hypercube Sampling (LHS)<a class="headerlink" href="#latin-hypercube-sampling-lhs" title="Link to this heading">#</a></h1>
+<p>Latin hypercube sampling (LHS) <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id91" title="Michael D McKay, Richard J Beckman, and William J Conover. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21(1):239-2451, 1979.">63</a>]</span> is one of the most common methods in space-filling experimental designs. With this sampling technique, for <span class="math notranslate nohighlight">\(N\)</span> uncertain factors, an <span class="math notranslate nohighlight">\(N\)</span>-dimensional hypercube is generated, with each factor divided into an equal number of levels depending on the total number of samples to be generated. Equal numbers of samples are then randomly generated at each level, across all factors. In this manner, latin hypercube design guarantees sampling from every level of the variability space and without any overlaps. When the number of samples generated is much larger than the number of uncertain factors, LHS can be very effective in examining the effects of each factor <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. LHS is an attractive technique, because it guarantees a diverse coverage of the space, through the use of subintervals, without being constrained to discrete levels for each factor - compare <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (c) with <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (d) for the same number of samples.</p>
+<p>LHS is less effective when the number of samples is not much larger than the number of uncertain factors, and the effects of each factor cannot be appropriately distinguished. The samples between factors can also be highly correlated, biasing any subsequent sensitivity analysis results. To address this, the sampling scheme can be modified to control for the correlation in parameters while maximizing the information derived. An example of such modification is through the use of orthogonal arrays <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id21" title="Boxin Tang. Orthogonal array-based latin hypercubes. Journal of the American statistical association, 88(424):1392–1397, 1993.">64</a>]</span>.</p>
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+  <section id="low-discrepancy-sequences">
+<h1>Low-Discrepancy Sequences<a class="headerlink" href="#low-discrepancy-sequences" title="Link to this heading">#</a></h1>
+<p>Low-discrepancy sequences is another sampling technique that employs a pseudo-random generator for Monte Carlo sampling <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>, <a class="reference internal" href="R.Bibliography.html#id92" title="SK Zaremba. The mathematical basis of monte carlo and quasi-monte carlo methods. SIAM review, 10(3):303–314, 1968.">66</a>]</span>. These quasi-Monte Carlo methods eliminate ‘lumpiness’ across samples (i.e, the presence of gaps and clusters) by minimizing discrepancy across the hypercube samples. Discrepancy can be quantitatively measured using the deviations of sampled points from a uniform distribution <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>, <a class="reference internal" href="R.Bibliography.html#id95" title="Sergei Kucherenko, Daniel Albrecht, and Andrea Saltelli. Exploring multi-dimensional spaces: a comparison of latin hypercube and quasi monte carlo sampling techniques. arXiv preprint arXiv:1505.02350, 2015.">67</a>]</span>. Low-discrepancy sequences ensure that the number of samples in any subspace of the variability hypercube is approximately the same. This is not something guaranteed by Latin Hypercube sampling, and even though its design can be improved through optimization with various criteria, such adjustments are limited to small sample sizes and low dimensions <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id95" title="Sergei Kucherenko, Daniel Albrecht, and Andrea Saltelli. Exploring multi-dimensional spaces: a comparison of latin hypercube and quasi monte carlo sampling techniques. arXiv preprint arXiv:1505.02350, 2015.">67</a>, <a class="reference internal" href="R.Bibliography.html#id93" title="Bertrand Iooss, Loïc Boussouf, Vincent Feuillard, and Amandine Marrel. Numerical studies of the metamodel fitting and validation processes. arXiv preprint arXiv:1001.1049, 2010.">68</a>, <a class="reference internal" href="R.Bibliography.html#id94" title="Ruichen Jin, Wei Chen, and Agus Sudjianto. An efficient algorithm for constructing optimal design of computer experiments. In International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, volume 37009, 545–554. 2003.">69</a>, <a class="reference internal" href="R.Bibliography.html#id96" title="Max D Morris and Toby J Mitchell. Exploratory designs for computational experiments. Journal of statistical planning and inference, 43(3):381–402, 1995.">70</a>, <a class="reference internal" href="R.Bibliography.html#id97" title="Jeong-Soo Park. Optimal latin-hypercube designs for computer experiments. Journal of statistical planning and inference, 39(1):95–111, 1994.">71</a>]</span>. In contrast, the Sobol sequence <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id98" title="Ilya M Sobol. Uniformly distributed sequences with an additional uniform property. USSR Computational Mathematics and Mathematical Physics, 16(5):236–242, 1976.">72</a>, <a class="reference internal" href="R.Bibliography.html#id99" title="Il'ya Meerovich Sobol'. On the distribution of points in a cube and the approximate evaluation of integrals. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 7(4):784–802, 1967.">73</a>]</span>, one of the most widely used sampling techniques, utilizes the low-discrepancy approach to uniformly fill the sampled factor space. A core advantage of this style of sampling is that it takes far fewer samples (i.e., simulations) to attain a much lower level of error in estimating model output statistics (e.g., the mean and variance of outputs).</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial which uses Sobol sequence sampling for the purposes of a Sobol sensitivity analysis:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA using SALib Jupyter Notebook</a></p>
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+  <section id="other-types-of-sampling">
+<h1>Other types of sampling<a class="headerlink" href="#other-types-of-sampling" title="Link to this heading">#</a></h1>
+<p>The sampling techniques mentioned so far are general sampling methods useful for a variety of applications beyond sensitivity analysis. There are however techniques that have been developed for specific sensitivity analysis methods. Examples of these methods include the Morris One-At-a-Time <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id132" title="Max D Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.">74</a>]</span>, Fourier Amplitude Sensitivity Test (FAST; <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id100" title="RI Cukier, CM Fortuin, Kurt E Shuler, AG Petschek, and JH Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. i theory. The Journal of chemical physics, 59(8):3873–3878, 1973.">75</a>]</span>), Extended FAST <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id101" title="Andrea Saltelli, Stefano Tarantola, and KP-S Chan. A quantitative model-independent method for global sensitivity analysis of model output. Technometrics, 41(1):39–56, 1999.">76</a>]</span>, and Extended Sobol methods <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>]</span>. For example, the Morris sampling strategy builds a number of trajectories (usually referred to as repetitions and denoted by <span class="math notranslate nohighlight">\(r\)</span>) in the input space each composed of <span class="math notranslate nohighlight">\(N+1\)</span> factor points, where <span class="math notranslate nohighlight">\(N\)</span> is the number of uncertain factors. The first point of the trajectory is selected randomly and the subsequent <span class="math notranslate nohighlight">\(N\)</span> points are generated by moving one factor at a time by a fixed amount. Each factor is perturbed once along the trajectory, while the starting points of all of the trajectories are randomly and uniformly distributed. Several variations of this strategy also exist in the literature; for more details on each approach and their differences the reader is directed to <span id="id5">Pianosi <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>.</p>
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+<h1>Synthetic generation of input time series<a class="headerlink" href="#synthetic-generation-of-input-time-series" title="Link to this heading">#</a></h1>
+<p>Models often have input time series or processes with strong temporal and/or spatial correlations (e.g., streamflow, energy demand, pricing of commodities, etc.) that, while they might not immediately come to mind as factors to be examined in sensitivity analysis, can be treated as such. Synthetic input time series are used for a variety of reasons, for example, when observations are not available or are limited, or when past observations are not considered sufficiently representative to capture rare or extreme events of interest <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id103" title="Jonathan D Herman, Harrison B Zeff, Jonathan R Lamontagne, Patrick M Reed, and Gregory W Characklis. Synthetic drought scenario generation to support bottom-up water supply vulnerability assessments. Journal of Water Resources Planning and Management, 142(11):04016050, 2016.">78</a>, <a class="reference internal" href="R.Bibliography.html#id104" title="PCD Milly, Julio Betancourt, Malin Falkenmark, Robert M Hirsch, Zbigniew W Kundzewicz, Dennis P Lettenmaier, and Ronald J Stouffer. Stationarity is dead: whither water management? Earth, 4:20, 2008.">79</a>]</span>. Synthetic generation of input time series provides a valuable tool to consider non-stationarity and incorporate potential stressors, such as climate change impacts into input time series <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id105" title="Edoardo Borgomeo, Christopher L Farmer, and Jim W Hall. Numerical rivers: a synthetic streamflow generator for water resources vulnerability assessments. Water Resources Research, 51(7):5382–5405, 2015.">80</a>]</span>. For example, a century of record will be insufficient to capture very high impact rare extreme events (e.g., persistent multi-year droughts). A large body of statistical literature exists focusing on the topics of synthetic weather <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id74" title="Manuel Herrera, Sukumar Natarajan, David A Coley, Tristan Kershaw, Alfonso P Ramallo-González, Matthew Eames, Daniel Fosas, and Michael Wood. A review of current and future weather data for building simulation. Building Services Engineering Research and Technology, 38(5):602–627, 2017.">81</a>, <a class="reference internal" href="R.Bibliography.html#id75" title="Daniel S Wilks and Robert L Wilby. The weather generation game: a review of stochastic weather models. Progress in physical geography, 23(3):329–357, 1999.">82</a>]</span> and streamflow <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id76" title="JR Lamontagne and JR Stedinger. Generating synthetic streamflow forecasts with specified precision. Journal of Water Resources Planning and Management, 144(4):04018007, 2018.">83</a>, <a class="reference internal" href="R.Bibliography.html#id77" title="Sanghamitra Medda and Kalyan Kumar Bhar. Comparison of single-site and multi-site stochastic models for streamflow generation. Applied Water Science, 9(3):67, 2019.">84</a>]</span> generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound (hot, severe drought) extremes. It is beyond the scope of this text to review these methods, but readers are encouraged to explore the studies cited above as well as the following publications for discussions and comparisons of these methods: <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id103" title="Jonathan D Herman, Harrison B Zeff, Jonathan R Lamontagne, Patrick M Reed, and Gregory W Characklis. Synthetic drought scenario generation to support bottom-up water supply vulnerability assessments. Journal of Water Resources Planning and Management, 142(11):04016050, 2016.">78</a>, <a class="reference internal" href="R.Bibliography.html#id105" title="Edoardo Borgomeo, Christopher L Farmer, and Jim W Hall. Numerical rivers: a synthetic streamflow generator for water resources vulnerability assessments. Water Resources Research, 51(7):5382–5405, 2015.">80</a>, <a class="reference internal" href="R.Bibliography.html#id106" title="Brian R Kirsch, Gregory W Characklis, and Harrison B Zeff. Evaluating the impact of alternative hydro-climate scenarios on transfer agreements: practical improvement for generating synthetic streamflows. Journal of Water Resources Planning and Management, 139(4):396–406, 2013.">85</a>, <a class="reference internal" href="R.Bibliography.html#id107" title="Daniel P Loucks and Eelco Van Beek. Water resource systems planning and management: An introduction to methods, models, and applications. Springer, 2017.">86</a>, <a class="reference internal" href="R.Bibliography.html#id108" title="Scott Steinschneider, Sungwook Wi, and Casey Brown. The integrated effects of climate and hydrologic uncertainty on future flood risk assessments. Hydrological Processes, 29(12):2823–2839, 2015.">87</a>, <a class="reference internal" href="R.Bibliography.html#id109" title="Richard M Vogel. Stochastic watershed models for hydrologic risk management. Water Security, 1:28–35, 2017.">88</a>, <a class="reference internal" href="R.Bibliography.html#id110" title="Richard M Vogel and Jery R Stedinger. The value of stochastic streamflow models in overyear reservoir design applications. Water Resources Research, 24(9):1483–1490, 1988.">89</a>]</span>. The use of these methods for the purposes of exploratory modeling, especially in the context of well-characterized versus deep uncertainty, is further discussed in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a>.</p>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#one-at-a-time-oat">One-At-a-Time (OAT)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#full-and-fractional-factorial-sampling">Full and Fractional Factorial Sampling</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#latin-hypercube-sampling-lhs">Latin Hypercube Sampling (LHS)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#low-discrepancy-sequences">Low-Discrepancy Sequences</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-types-of-sampling">Other types of sampling</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-generation-of-input-time-series">Synthetic generation of input time series</a></li>
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+  <section id="design-of-experiments">
+<span id="id1"></span><h1>Design of Experiments<a class="headerlink" href="#design-of-experiments" title="Link to this heading">#</a></h1>
+<p>Before conducting a sensitivity analysis, the first element that needs to be clarified is the uncertainty space of the model <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id111" title="Jon C Helton, Jay Dean Johnson, Cedric J Sallaberry, and Curt B Storlie. Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliability Engineering &amp; System Safety, 91(10-11):1175–1209, 2006.">53</a>]</span>. In other words, how many and which factors making up the mathematical model are considered uncertain and can potentially affect the model output and the inferences drawn from it. Uncertain factors can be model parameters, model structures, inputs, or alternative model resolution levels (scales), all of which can be assessed through the tools presented in this text. Depending on the kind of factor, its variability can be elicited through various means: expert opinion, values reported in the literature, historical observations, its physical meaning (e.g., population values in a city can never be negative), or through the use of more formal UQ methods (<a class="reference internal" href="A1_Uncertainty_Quantification.html#a1-uncertainty-quantification"><span class="std std-numref">Chapter A</span></a>). The model uncertainty space represents the entire space of variability present in each of the uncertain factors of a model. The complexity of most real-world models means that the response function, <span class="math notranslate nohighlight">\(y=g(x)\)</span>, mapping inputs to outputs, is hardly ever available in an analytical form and therefore analytically computing the sensitivity of the output to each uncertain factor becomes impossible. In these cases, sensitivity analysis is only feasible through numerical procedures that employ different strategies to sample the uncertainty space and calculate sensitivity indices.</p>
+<p>A sampling strategy is often referred to as a <em>design of experiments</em> and represents a methodological choice made before conducting any sensitivity analysis. Experimental design was first introduced by <span id="id3">Fisher [<a class="reference internal" href="R.Bibliography.html#id112" title="Ronald Aylmer Fisher. Design of experiments. Br Med J, 1(3923):554–554, 1936.">54</a>]</span> in the context of laboratory or field-based experiments. Its application in sensitivity analysis is similar to setting up a physical experiment in that it is used to discover the behavior of a system under specific conditions. An ideal design of experiments should provide a framework for the extraction of all plausible information about the impact of each factor on the output of the model. The design of experiments is used to set up a simulation platform with the minimum computational cost to answer specific questions that cannot be readily drawn from the data through analytical or common data mining techniques. Models representing coupled human-natural systems usually have a large number of inputs, state variables and parameters, but not all of them exert fundamental control over the numerical process, despite their uncertainty, nor have substantial impacts on the model output, either independently or through their interactions. Each factor influences the model output in different ways that need to be discovered. For example, the influence of a parameter on model output can be linear or non-linear and can be continuous or only be active during specific times or at particular states of the system <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id53" title="JD Herman, PM Reed, and T Wagener. Time-varying sensitivity analysis clarifies the effects of watershed model formulation on model behavior. Water Resources Research, 49(3):1400–1414, 2013.">55</a>, <a class="reference internal" href="R.Bibliography.html#id113" title="Carolina Massmann, Thorsten Wagener, and Hubert Holzmann. A new approach to visualizing time-varying sensitivity indices for environmental model diagnostics across evaluation time-scales. Environmental modelling &amp; software, 51:190–194, 2014.">56</a>]</span>. An effective and efficient design of experiments allows the analyst to explore these complex relationships and evaluate different behaviors of the model for various scientific questions <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id114" title="An Van Schepdael, Aurélie Carlier, and Liesbet Geris. Sensitivity analysis by design of experiments. In Uncertainty in Biology, pages 327–366. Springer, 2016.">57</a>]</span>. The rest of this section overviews some of the most commonly used designs of experiments. Table 1 summarizes the designs discussed.</p>
+<div class="pst-scrollable-table-container"><table class="table" id="id32">
+<caption><span class="caption-text">Summary of designs of experiments overviewed in this section. * Depends on the sample size.</span><a class="headerlink" href="#id32" title="Link to this table">#</a></caption>
+<thead>
+<tr class="row-odd"><th class="head"><p><em>Design of experiments</em></p></th>
+<th class="head"><p><em>Factor interactions considered</em></p></th>
+<th class="head"><p><em>Treatment of factor domains</em></p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>One-At-a-Time (OAT)</p></td>
+<td><p>No - main effects only</p></td>
+<td><p>Continuous (distributions)</p></td>
+</tr>
+<tr class="row-odd"><td><p>Full Factorial Sampling</p></td>
+<td><p>Yes - including total effects</p></td>
+<td><p>Discrete (levels)</p></td>
+</tr>
+<tr class="row-even"><td><p>Fractional Factorial Sampling</p></td>
+<td><p>Yes - only lower-order effects*</p></td>
+<td><p>Discrete (levels)</p></td>
+</tr>
+<tr class="row-odd"><td><p>Latin Hypercube (LH) Sampling</p></td>
+<td><p>Yes - including total effects*</p></td>
+<td><p>Continuous (distributions)</p></td>
+</tr>
+<tr class="row-even"><td><p>Quasi-Random Sampling with Low-Discrepancy Sequences</p></td>
+<td><p>Yes - including total effects*</p></td>
+<td><p>Continuous (distributions)</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>There are a few different approaches to the design of experiments, closely related to the chosen sensitivity analysis approach, which is in turn shaped by the research motivations, scientific questions, and computational constraints at hand (additional discussion of this can be found at the end of <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#sensitivity-analysis-the-basics"><span class="std std-numref">Chapter 3</span></a>). For example, in a sensitivity analysis using perturbation and derivatives methods, the model input parameters vary from their nominal values one at a time, something that the design of experiments needs to reflect. If, instead, one were to perform sensitivity analysis using a multiple-starts perturbation method, the design of experiments needs to consider that multiple points across the factor space are used. The design of experiments specifically defines two key characteristics of samples that are fed to the numerical model: the number of samples and the range of each factor.</p>
+<p>Generally, sampling can be performed randomly or by applying a stratifying approach. In random sampling, such as Monte Carlo <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id115" title="Nicholas Metropolis and Stanislaw Ulam. The monte carlo method. Journal of the American statistical association, 44(247):335–341, 1949.">58</a>]</span>, samples are randomly generated by a pseudo-random number generator with an a-priori assumption about the distribution of parameters and their possible ranges. Random seeds can also be used to ensure consistency and higher control over the random process. However, this method could leave some gaps in the parameter space and cause clustering in some spaces, especially for a large number of parameters <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id116" title="John Norton. An introduction to sensitivity assessment of simulation models. Environmental Modelling &amp; Software, 69:166–174, 2015.">59</a>]</span>. Most sampling strategies use stratified sampling to mitigate these disadvantages. Stratified sampling techniques divide the domain of each factor into subintervals, often of equal lengths. From each subinterval, an equal number of samples is drawn randomly, or based on the specific locations within the subintervals <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>.</p>
+<section id="one-at-a-time-oat">
+<h2>One-At-a-Time (OAT)<a class="headerlink" href="#one-at-a-time-oat" title="Link to this heading">#</a></h2>
+<p>In this approach, only one model factor is changed at a time while all others are kept fixed across each iteration in a sampling sequence. The OAT method assumes that model factors of focus are linearly independent (i.e., there are no interactions) and can analyze how factors individually influence model outputs or metrics of interest. While popular given its ease of implementation, OAT  is ultimately limited in its exploration of a model’s sensitivities <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. It is primarily used with local sensitivity techniques with similar criticisms: applying this sampling scheme on a system with nonlinear and interactive processes will miss important information on the effect uncertain factors have on the model. OAT samplings can be repeated multiple times in a more sophisticated manner and across different locations of the parameter space to overcome some of these challenges, which would increase computational costs and negate the main reasons for its selection. Given these limitations OAT methods could be used as preliminary, low-cost analyses of the factors’ individual effects, but should ultimately be complemented with more sophisticated methods.</p>
+</section>
+<section id="full-and-fractional-factorial-sampling">
+<h2>Full and Fractional Factorial Sampling<a class="headerlink" href="#full-and-fractional-factorial-sampling" title="Link to this heading">#</a></h2>
+<p>In full factorial sampling each factor is treated as being discrete by considering two or more levels (or intervals) of its values. The sampling process then generates samples within each possible combination of levels, corresponding to each parameter. This scheme produces a more comprehensive sampling of the factors’ variability space, as it accounts for all candidate combinations of factor levels (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a)). If the number of levels is the same across all factors, the number of generated samples is estimated using <span class="math notranslate nohighlight">\(n^k\)</span>, where <span class="math notranslate nohighlight">\(n\)</span> is the number of levels and <span class="math notranslate nohighlight">\(k\)</span> is the number of factors. For example, <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a) presents a full factorial sampling of three uncertain factors <span class="math notranslate nohighlight">\((x_1,\)</span> <span class="math notranslate nohighlight">\(x_2,\)</span> and <span class="math notranslate nohighlight">\(x_3)\)</span>, each considered as having four discrete levels. The total number of samples necessary for such an experiment is <span class="math notranslate nohighlight">\(4^3=64\)</span>. As the number of factors increases, the number of simulations necessary will also grow exponentially, making full factorial sampling computationally burdensome (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (b)). As a result, it is common in the literature to apply full factorial sampling at only two levels per factor, typically the two extremes <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id32" title="Douglas C Montgomery. Design and analysis of experiments. John wiley &amp; sons, 2017.">60</a>]</span>. This significantly reduces computational burden but is only considered appropriate in cases where factors can indeed only assume two discrete values (e.g., when testing the effects of epistemic uncertainty and comparing between model structure A and model structure B). In the case of physical parameters on continuous distributions (e.g., when considering the effects of measurement uncertainty in a temperature sensor), discretizing the range of a factor to only extreme levels can bias its estimated importance.</p>
+<p>Fractional factorial sampling is a widely used alternative to full factorial sampling that allows the analyst to significantly reduce the number of simulations by focusing on the main effects of a factor and seeking to avoid model runs that yield redundant response information <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. In other words, if one can reasonably assume that higher-order interactions are negligible, information about the most significant effects and lower-order interactions (e.g., effects from pairs of factors) can be obtained using a fraction of the full factorial design. Traditionally, fractional factorial design has also been limited to two levels <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id32" title="Douglas C Montgomery. Design and analysis of experiments. John wiley &amp; sons, 2017.">60</a>]</span>, referred to as Fractional Factorial designs 2k-p <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id34" title="George EP Box and J Stuart Hunter. The 2 k—p fractional factorial designs. Technometrics, 3(3):311–351, 1961.">61</a>]</span>. Recently, Generalized Fractional Factorial designs have also been proposed that allow for the structured generation of samples at more than two levels per factor <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id33" title="Izabella Surowiec, Ludvig Vikstrom, Gustaf Hector, Erik Johansson, Conny Vikstrom, and Johan Trygg. Generalized subset designs in analytical chemistry. Analytical chemistry, 89(12):6491–6497, 2017.">62</a>]</span>. Consider a case where the modeling team dealing with the problem in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a) cannot afford to perform 64 simulations of their model. They can afford 32 runs for their experiment and instead decide to fractionally sample the variability space of their factors. A potential design of such a sampling strategy is presented in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (c).</p>
+<figure class="margin-caption align-center" id="id33">
+<span id="figure-3-3"></span><a class="reference internal image-reference" href="_images/figure3_3_alternative_designs.png"><img alt="Figure 3.3" src="_images/figure3_3_alternative_designs.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Alternative designs of experiments and their computational costs for three uncertain factors <span class="math notranslate nohighlight">\((x_1,\)</span> <span class="math notranslate nohighlight">\(x_2,\)</span> and <span class="math notranslate nohighlight">\(x_3)\)</span>. (a) Full factorial design sampling of three factors at four levels, at a total of 64 samples; (b) exponential growth of necessary number of samples when applying full factorial design at four levels; (c) fractional factorial design of three factors at four levels, at a total of 32 samples; and (d) Latin Hypercube sample of three factors with uniform distributions, at a total of 32 samples.</span><a class="headerlink" href="#id33" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="latin-hypercube-sampling-lhs">
+<h2>Latin Hypercube Sampling (LHS)<a class="headerlink" href="#latin-hypercube-sampling-lhs" title="Link to this heading">#</a></h2>
+<p>Latin hypercube sampling (LHS) <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id91" title="Michael D McKay, Richard J Beckman, and William J Conover. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21(1):239-2451, 1979.">63</a>]</span> is one of the most common methods in space-filling experimental designs. With this sampling technique, for <span class="math notranslate nohighlight">\(N\)</span> uncertain factors, an <span class="math notranslate nohighlight">\(N\)</span>-dimensional hypercube is generated, with each factor divided into an equal number of levels depending on the total number of samples to be generated. Equal numbers of samples are then randomly generated at each level, across all factors. In this manner, latin hypercube design guarantees sampling from every level of the variability space and without any overlaps. When the number of samples generated is much larger than the number of uncertain factors, LHS can be very effective in examining the effects of each factor <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. LHS is an attractive technique, because it guarantees a diverse coverage of the space, through the use of subintervals, without being constrained to discrete levels for each factor - compare <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (c) with <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (d) for the same number of samples.</p>
+<p>LHS is less effective when the number of samples is not much larger than the number of uncertain factors, and the effects of each factor cannot be appropriately distinguished. The samples between factors can also be highly correlated, biasing any subsequent sensitivity analysis results. To address this, the sampling scheme can be modified to control for the correlation in parameters while maximizing the information derived. An example of such modification is through the use of orthogonal arrays <span id="id17">[<a class="reference internal" href="R.Bibliography.html#id21" title="Boxin Tang. Orthogonal array-based latin hypercubes. Journal of the American statistical association, 88(424):1392–1397, 1993.">64</a>]</span>.</p>
+</section>
+<section id="low-discrepancy-sequences">
+<h2>Low-Discrepancy Sequences<a class="headerlink" href="#low-discrepancy-sequences" title="Link to this heading">#</a></h2>
+<p>Low-discrepancy sequences is another sampling technique that employs a pseudo-random generator for Monte Carlo sampling <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>, <a class="reference internal" href="R.Bibliography.html#id92" title="SK Zaremba. The mathematical basis of monte carlo and quasi-monte carlo methods. SIAM review, 10(3):303–314, 1968.">66</a>]</span>. These quasi-Monte Carlo methods eliminate ‘lumpiness’ across samples (i.e, the presence of gaps and clusters) by minimizing discrepancy across the hypercube samples. Discrepancy can be quantitatively measured using the deviations of sampled points from a uniform distribution <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>, <a class="reference internal" href="R.Bibliography.html#id95" title="Sergei Kucherenko, Daniel Albrecht, and Andrea Saltelli. Exploring multi-dimensional spaces: a comparison of latin hypercube and quasi monte carlo sampling techniques. arXiv preprint arXiv:1505.02350, 2015.">67</a>]</span>. Low-discrepancy sequences ensure that the number of samples in any subspace of the variability hypercube is approximately the same. This is not something guaranteed by Latin Hypercube sampling, and even though its design can be improved through optimization with various criteria, such adjustments are limited to small sample sizes and low dimensions <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id95" title="Sergei Kucherenko, Daniel Albrecht, and Andrea Saltelli. Exploring multi-dimensional spaces: a comparison of latin hypercube and quasi monte carlo sampling techniques. arXiv preprint arXiv:1505.02350, 2015.">67</a>, <a class="reference internal" href="R.Bibliography.html#id93" title="Bertrand Iooss, Loïc Boussouf, Vincent Feuillard, and Amandine Marrel. Numerical studies of the metamodel fitting and validation processes. arXiv preprint arXiv:1001.1049, 2010.">68</a>, <a class="reference internal" href="R.Bibliography.html#id94" title="Ruichen Jin, Wei Chen, and Agus Sudjianto. An efficient algorithm for constructing optimal design of computer experiments. In International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, volume 37009, 545–554. 2003.">69</a>, <a class="reference internal" href="R.Bibliography.html#id96" title="Max D Morris and Toby J Mitchell. Exploratory designs for computational experiments. Journal of statistical planning and inference, 43(3):381–402, 1995.">70</a>, <a class="reference internal" href="R.Bibliography.html#id97" title="Jeong-Soo Park. Optimal latin-hypercube designs for computer experiments. Journal of statistical planning and inference, 39(1):95–111, 1994.">71</a>]</span>. In contrast, the Sobol sequence <span id="id21">[<a class="reference internal" href="R.Bibliography.html#id98" title="Ilya M Sobol. Uniformly distributed sequences with an additional uniform property. USSR Computational Mathematics and Mathematical Physics, 16(5):236–242, 1976.">72</a>, <a class="reference internal" href="R.Bibliography.html#id99" title="Il'ya Meerovich Sobol'. On the distribution of points in a cube and the approximate evaluation of integrals. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 7(4):784–802, 1967.">73</a>]</span>, one of the most widely used sampling techniques, utilizes the low-discrepancy approach to uniformly fill the sampled factor space. A core advantage of this style of sampling is that it takes far fewer samples (i.e., simulations) to attain a much lower level of error in estimating model output statistics (e.g., the mean and variance of outputs).</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial which uses Sobol sequence sampling for the purposes of a Sobol sensitivity analysis:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA using SALib Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="other-types-of-sampling">
+<h2>Other types of sampling<a class="headerlink" href="#other-types-of-sampling" title="Link to this heading">#</a></h2>
+<p>The sampling techniques mentioned so far are general sampling methods useful for a variety of applications beyond sensitivity analysis. There are however techniques that have been developed for specific sensitivity analysis methods. Examples of these methods include the Morris One-At-a-Time <span id="id22">[<a class="reference internal" href="R.Bibliography.html#id132" title="Max D Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.">74</a>]</span>, Fourier Amplitude Sensitivity Test (FAST; <span id="id23">[<a class="reference internal" href="R.Bibliography.html#id100" title="RI Cukier, CM Fortuin, Kurt E Shuler, AG Petschek, and JH Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. i theory. The Journal of chemical physics, 59(8):3873–3878, 1973.">75</a>]</span>), Extended FAST <span id="id24">[<a class="reference internal" href="R.Bibliography.html#id101" title="Andrea Saltelli, Stefano Tarantola, and KP-S Chan. A quantitative model-independent method for global sensitivity analysis of model output. Technometrics, 41(1):39–56, 1999.">76</a>]</span>, and Extended Sobol methods <span id="id25">[<a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>]</span>. For example, the Morris sampling strategy builds a number of trajectories (usually referred to as repetitions and denoted by <span class="math notranslate nohighlight">\(r\)</span>) in the input space each composed of <span class="math notranslate nohighlight">\(N+1\)</span> factor points, where <span class="math notranslate nohighlight">\(N\)</span> is the number of uncertain factors. The first point of the trajectory is selected randomly and the subsequent <span class="math notranslate nohighlight">\(N\)</span> points are generated by moving one factor at a time by a fixed amount. Each factor is perturbed once along the trajectory, while the starting points of all of the trajectories are randomly and uniformly distributed. Several variations of this strategy also exist in the literature; for more details on each approach and their differences the reader is directed to <span id="id26">Pianosi <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>.</p>
+</section>
+<section id="synthetic-generation-of-input-time-series">
+<h2>Synthetic generation of input time series<a class="headerlink" href="#synthetic-generation-of-input-time-series" title="Link to this heading">#</a></h2>
+<p>Models often have input time series or processes with strong temporal and/or spatial correlations (e.g., streamflow, energy demand, pricing of commodities, etc.) that, while they might not immediately come to mind as factors to be examined in sensitivity analysis, can be treated as such. Synthetic input time series are used for a variety of reasons, for example, when observations are not available or are limited, or when past observations are not considered sufficiently representative to capture rare or extreme events of interest <span id="id27">[<a class="reference internal" href="R.Bibliography.html#id103" title="Jonathan D Herman, Harrison B Zeff, Jonathan R Lamontagne, Patrick M Reed, and Gregory W Characklis. Synthetic drought scenario generation to support bottom-up water supply vulnerability assessments. Journal of Water Resources Planning and Management, 142(11):04016050, 2016.">78</a>, <a class="reference internal" href="R.Bibliography.html#id104" title="PCD Milly, Julio Betancourt, Malin Falkenmark, Robert M Hirsch, Zbigniew W Kundzewicz, Dennis P Lettenmaier, and Ronald J Stouffer. Stationarity is dead: whither water management? Earth, 4:20, 2008.">79</a>]</span>. Synthetic generation of input time series provides a valuable tool to consider non-stationarity and incorporate potential stressors, such as climate change impacts into input time series <span id="id28">[<a class="reference internal" href="R.Bibliography.html#id105" title="Edoardo Borgomeo, Christopher L Farmer, and Jim W Hall. Numerical rivers: a synthetic streamflow generator for water resources vulnerability assessments. Water Resources Research, 51(7):5382–5405, 2015.">80</a>]</span>. For example, a century of record will be insufficient to capture very high impact rare extreme events (e.g., persistent multi-year droughts). A large body of statistical literature exists focusing on the topics of synthetic weather <span id="id29">[<a class="reference internal" href="R.Bibliography.html#id74" title="Manuel Herrera, Sukumar Natarajan, David A Coley, Tristan Kershaw, Alfonso P Ramallo-González, Matthew Eames, Daniel Fosas, and Michael Wood. A review of current and future weather data for building simulation. Building Services Engineering Research and Technology, 38(5):602–627, 2017.">81</a>, <a class="reference internal" href="R.Bibliography.html#id75" title="Daniel S Wilks and Robert L Wilby. The weather generation game: a review of stochastic weather models. Progress in physical geography, 23(3):329–357, 1999.">82</a>]</span> and streamflow <span id="id30">[<a class="reference internal" href="R.Bibliography.html#id76" title="JR Lamontagne and JR Stedinger. Generating synthetic streamflow forecasts with specified precision. Journal of Water Resources Planning and Management, 144(4):04018007, 2018.">83</a>, <a class="reference internal" href="R.Bibliography.html#id77" title="Sanghamitra Medda and Kalyan Kumar Bhar. Comparison of single-site and multi-site stochastic models for streamflow generation. Applied Water Science, 9(3):67, 2019.">84</a>]</span> generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound (hot, severe drought) extremes. It is beyond the scope of this text to review these methods, but readers are encouraged to explore the studies cited above as well as the following publications for discussions and comparisons of these methods: <span id="id31">[<a class="reference internal" href="R.Bibliography.html#id103" title="Jonathan D Herman, Harrison B Zeff, Jonathan R Lamontagne, Patrick M Reed, and Gregory W Characklis. Synthetic drought scenario generation to support bottom-up water supply vulnerability assessments. Journal of Water Resources Planning and Management, 142(11):04016050, 2016.">78</a>, <a class="reference internal" href="R.Bibliography.html#id105" title="Edoardo Borgomeo, Christopher L Farmer, and Jim W Hall. Numerical rivers: a synthetic streamflow generator for water resources vulnerability assessments. Water Resources Research, 51(7):5382–5405, 2015.">80</a>, <a class="reference internal" href="R.Bibliography.html#id106" title="Brian R Kirsch, Gregory W Characklis, and Harrison B Zeff. Evaluating the impact of alternative hydro-climate scenarios on transfer agreements: practical improvement for generating synthetic streamflows. Journal of Water Resources Planning and Management, 139(4):396–406, 2013.">85</a>, <a class="reference internal" href="R.Bibliography.html#id107" title="Daniel P Loucks and Eelco Van Beek. Water resource systems planning and management: An introduction to methods, models, and applications. Springer, 2017.">86</a>, <a class="reference internal" href="R.Bibliography.html#id108" title="Scott Steinschneider, Sungwook Wi, and Casey Brown. The integrated effects of climate and hydrologic uncertainty on future flood risk assessments. Hydrological Processes, 29(12):2823–2839, 2015.">87</a>, <a class="reference internal" href="R.Bibliography.html#id109" title="Richard M Vogel. Stochastic watershed models for hydrologic risk management. Water Security, 1:28–35, 2017.">88</a>, <a class="reference internal" href="R.Bibliography.html#id110" title="Richard M Vogel and Jery R Stedinger. The value of stochastic streamflow models in overyear reservoir design applications. Water Resources Research, 24(9):1483–1490, 1988.">89</a>]</span>. The use of these methods for the purposes of exploratory modeling, especially in the context of well-characterized versus deep uncertainty, is further discussed in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a>.</p>
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+  <section id="derivative-based-methods">
+<h1>Derivative-based Methods<a class="headerlink" href="#derivative-based-methods" title="Link to this heading">#</a></h1>
+<p>Derivative-based methods explore how model outputs are affected by perturbations in a single model input around a particular input value. These methods are local and are performed using OAT sampling. For simplicity of mathematical notations, let us assume that the model <span class="math notranslate nohighlight">\(g(X)\)</span> only returns one output. Following <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id117" title="Emanuele Borgonovo. Sensitivity analysis of model output with input constraints: a generalized rationale for local methods. Risk Analysis: An International Journal, 28(3):667–680, 2008.">90</a>]</span> and <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>, the sensitivity index, <span class="math notranslate nohighlight">\(S_i\)</span> , of the model’s <em>i</em>-th input factor, <span class="math notranslate nohighlight">\(x_i\)</span> , can be measured using the partial derivative evaluated at a nominal value, <span class="math notranslate nohighlight">\(\bar{x}\)</span>, of the vector of inputs:</p>
+<div class="math notranslate nohighlight">
+\[S_i (\bar{x}) = \frac{\partial g}{\partial x} |_{\bar{x}{^c{_i}}}\]</div>
+<p>where <em>c</em><sub>i</sub> is the scaling factor. In most applications however, the relationship <span class="math notranslate nohighlight">\(g(X)\)</span> is not fully known in its analytical form, and therefore the above partial derivative is usually approximated:</p>
+<div class="math notranslate nohighlight">
+\[S_i (\bar{x}) = \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i\]</div>
+<p>Using this approximation, the <em>i</em>-th input factor is perturbed by a magnitude of <span class="math notranslate nohighlight">\(\Delta_i\)</span>, and its relative importance is calculated. Derivative-based methods are some of the oldest sensitivity analysis methods as they only require <span class="math notranslate nohighlight">\(N+1\)</span> model evaluations to estimate indices for <span class="math notranslate nohighlight">\(N\)</span> uncertain factors. As described above, being computationally very cheap comes at the cost of not being able to explore the entire input space, but only (local) perturbations to the nominal value. Additionally, as these methods examine the effects of each input factor one at a time, they cannot assess parametric interactions or capture the interacting nature of many real systems and the models that abstract them.</p>
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+    <h1>Elementary Effect Methods</h1>
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+  <section id="elementary-effect-methods">
+<h1>Elementary Effect Methods<a class="headerlink" href="#elementary-effect-methods" title="Link to this heading">#</a></h1>
+<p>Elementary effect (EE) SA methods provide a solution to the local nature of the derivative-based methods by exploring the entire parametric range of each input parameter <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>. However, EE methods still use OAT sampling and do not vary all input parameters simultaneously while exploring the parametric space. The OAT nature of EEs methods therefore prevents them from properly capturing the interactions between uncertain factors. EEs methods are computationally efficient compared to their All-At-a-Time (AAT) counterparts, making them more suitable when computational capacity is a limiting factor, while still allowing for some inferences regarding factor interactions.
+The most popular EE method is the Method of Morris <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id132" title="Max D Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.">74</a>]</span>. Following the notation by <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>, this method calculates global sensitivity using the mean of the EEs (finite differences) of each parameter at different locations:</p>
+<div class="math notranslate nohighlight">
+\[S_i = \mu_i^* = \frac{1}{r}\sum_{j=1}^r EE^j_i = \frac{1}{r}\sum_{j=1}^r \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i\]</div>
+<p>with <span class="math notranslate nohighlight">\(r\)</span> representing the number of sample repetitions (also refered to as trajectories) in the input space, usually set between 4 and 10 <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. Each <span class="math notranslate nohighlight">\(x_j\)</span> represents the points of each trajectory, with <span class="math notranslate nohighlight">\(j=1,…, r\)</span>, selected as described in the sampling strategy for this method, found above. This method also produces the standard deviation of the EEs:</p>
+<div class="math notranslate nohighlight">
+\[\sigma_i = \sqrt{\frac{1}{r}\sum_{j=1}^r(EE_i^j-\frac{1}{r}\sum_{j=1}^r EE^j_i)^2}\]</div>
+<p>which is a measure of parametric interactions. Higher values of <span class="math notranslate nohighlight">\(\sigma_i\)</span> suggest model responses at different levels of factor <span class="math notranslate nohighlight">\(x_i\)</span> are significantly different, which indicates considerable interactions between that and other uncertain factors. The values of <span class="math notranslate nohighlight">\(\mu_i^*\)</span> and <span class="math notranslate nohighlight">\(\sigma_i\)</span> for each factor allow us to draw several different conclusions, illustrated in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-4"><span class="std std-numref">Fig. 3.4</span></a>, following the example by <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>. In this example, factors <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> can be said to have an influence on the model outputs, with <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> having some interactive or non-linear effects. Depending on the orders of magnitude of <span class="math notranslate nohighlight">\(\mu_i^*\)</span> and <span class="math notranslate nohighlight">\(\sigma_i\)</span> one can indirectly deduce whether the factors have strong interactive effects, for example if a factor <span class="math notranslate nohighlight">\(\sigma_i &lt;&lt; \mu_i^*\)</span> then the relationship between that factor and the output can be assumed to be largely linear (note that this is still an OAT method and assumptions on factor interactions should be strongly caveated). Extensions of the Method of Morris have also been developed specifically for the purposes of factor fixing and explorations of parametric interactions (e.g., <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id27" title="Emanuele Borgonovo. Sensitivity analysis with finite changes: an application to modified eoq models. European Journal of Operational Research, 200(1):127–138, 2010.">48</a>, <a class="reference internal" href="R.Bibliography.html#id133" title="Francesca Campolongo and Roger Braddock. The use of graph theory in the sensitivity analysis of the model output: a second order screening method. Reliability Engineering &amp; System Safety, 64(1):1–12, 1999.">92</a>, <a class="reference internal" href="R.Bibliography.html#id134" title="Roger A Cropp and Roger D Braddock. The new morris method: an efficient second-order screening method. Reliability Engineering &amp; System Safety, 78(1):77–83, 2002.">93</a>]</span>).</p>
+<figure class="margin-caption align-center" id="id7">
+<span id="figure-3-4"></span><a class="reference internal image-reference" href="_images/figure3_4_morris_method.png"><img alt="Figure 3.4" src="_images/figure3_4_morris_method.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Illustrative results of the Morris Method. Factors <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> have an influence on the model outputs, with <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> having interactive or non-linear effects. Whether or not a factor should be considered influential to the output depends on the output selected and is specific to the research context and purpose of the analysis, as discussed in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#why-sa"><span class="std std-numref">Chapter 3.2</span></a>.</span><a class="headerlink" href="#id7" title="Link to this image">#</a></p>
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+  <section id="regression-based-methods">
+<h1>Regression-based Methods<a class="headerlink" href="#regression-based-methods" title="Link to this heading">#</a></h1>
+<p>Regression analysis is one of the oldest ways of investigating parametric importance and sensitivity <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. Here, we describe some of the most popular regression-based sensitivity indices. One of the main sensitivity indices of this category is the standardized regression coefficient (SRC). To calculate SRC, a linear regression relationship needs to be fitted between the input vector, <span class="math notranslate nohighlight">\(x\)</span>, and the model output of interest by using a least-square minimizing method:</p>
+<div class="math notranslate nohighlight">
+\[y = b_0 + \sum_{i=1}^N b_ix_i\]</div>
+<p>where <span class="math notranslate nohighlight">\(b_0\)</span> and <span class="math notranslate nohighlight">\(b_i\)</span> (corresponding to the <em>i</em>-th model input) are regression coefficients. The following relationship can then be used to calculate the SRCs for different input values:</p>
+<div class="math notranslate nohighlight">
+\[S_i=SRC_i=b_i\frac{\sigma_i}{\sigma_y}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\sigma_i\)</span> and <span class="math notranslate nohighlight">\(\sigma_y\)</span> are standard deviations of <em>i</em>-th model input and output, respectively.</p>
+<p>Several other regression-based indices explore the correlation between input and output parameters as a proxy to model parametric sensitivity <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>, <a class="reference internal" href="R.Bibliography.html#id135" title="Jon C Helton. Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal. Reliability Engineering &amp; System Safety, 42(2-3):327–367, 1993.">94</a>, <a class="reference internal" href="R.Bibliography.html#id136" title="Gemma Manache and Charles S Melching. Identification of reliable regression-and correlation-based sensitivity measures for importance ranking of water-quality model parameters. Environmental Modelling &amp; Software, 23(5):549–562, 2008.">95</a>]</span>. The Pearson correlation coefficient (PCC) can be used when a linear relationship exists between an uncertain factor, <span class="math notranslate nohighlight">\(x_i\)</span>, and the output <span class="math notranslate nohighlight">\(y\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i=PCC=\frac{cov(x_i,y)}{\sigma_i\sigma_y}\]</div>
+<p>In cases when there are outliers in the data or the relationship between the uncertain factors and the output is not linear, rank-based correlation coefficients are preferred, for example, Spearman’s rank correlation coefficient (SRCC):</p>
+<div class="math notranslate nohighlight">
+\[S_i=SRCC=\frac{cov(rx_i,ryi)}{\sigma_{ri}\sigma_{ry}}\]</div>
+<p>where the raw values of <span class="math notranslate nohighlight">\(x_i\)</span> and <span class="math notranslate nohighlight">\(y\)</span> and converted to ranks <span class="math notranslate nohighlight">\(rx_i\)</span> and <span class="math notranslate nohighlight">\(ry\)</span> respectively, which instead represent a measurement of the strength of the monotonic relationship, rather than linear relationship, between the input and output. Other regression-based metrics include the partial correlations coefficient, the partial rank correlations coefficient, and the Nash-Sutcliffe coefficient, more discussion on which can be found in <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>.</p>
+<p>Tree-based regression techniques have also been used for sensitivity analysis in an effort to address the challenges faced with nonlinear models <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id137" title="F Pappenberger and Keith J Beven. Ignorance is bliss: or seven reasons not to use uncertainty analysis. Water resources research, 2006.">96</a>]</span>. Examples of these methods include the Patient Rule Induction Method (PRIM; <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id79" title="Jerome H Friedman and Nicholas I Fisher. Bump hunting in high-dimensional data. Statistics and Computing, 9(2):123–143, 1999.">97</a>]</span>) and Classification And Regression Trees (CART; <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id80" title="Leo Breiman, Jerome Friedman, Charles J Stone, and Richard A Olshen. Classification and regression trees. CRC press, 1984.">98</a>]</span>). CART-based approaches also include boosting and bagging extensions <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id89" title="Yoav Freund, Robert Schapire, and Naoki Abe. A short introduction to boosting. Journal-Japanese Society For Artificial Intelligence, 14(771-780):1612, 1999.">99</a>, <a class="reference internal" href="R.Bibliography.html#id138" title="Leo Breiman. Bagging predictors. Machine learning, 24(2):123–140, 1996.">100</a>]</span>. These methods are particularly useful when sensitivity analysis is used for factor mapping (i.e., when trying to identify which uncertain model factors produce a certain model behavior). <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a> elaborates on the use of these methods. Regression-based sensitivity analysis methods are global by nature and can explore the entire space of variables. However, the true level of comprehensiveness depends on the design of experiments and the number of simulations providing data to establish the regression relationships. Although they are usually computationally efficient, they do not produce significant information about parametric interactions <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>]</span>.</p>
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+  <section id="regional-sensitivity-analysis">
+<h1>Regional Sensitivity Analysis<a class="headerlink" href="#regional-sensitivity-analysis" title="Link to this heading">#</a></h1>
+<p>Another method primarily applied for basic factor mapping applications is Regional Sensitivity Analysis (RSA; <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id157" title="George M Hornberger and Robert C Spear. Approach to the preliminary analysis of environmental systems. J. Environ. Mgmt., 12(1):7–18, 1981.">101</a>]</span>). RSA is a global sensitivity analysis method that is typically implemented using standard sampling methods such as latin hypercube sampling. It is performed by specifying a condition on the output space (e.g., an upper threshold) and classifying outputs that meet the condition as behavioral and the ones that fail it as non-behavioral (illustrated in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (b)). Note that the specified threshold depends on the nature of the problem, model, and the research question. It can reflect model-performance metrics (such as errors) or consequential decision-relevant metrics (such as unacceptable system outcomes). The behavioral and non-behavioral outputs are then traced back to their originating sampled factors, where differences between the distributions of samples can be used to determine their significance in producing each part of the output. The Kolmogorov-Smirnov divergence is commonly used to quantify the difference between the distribution of behavioral and non-behavioral parameters <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i=|F_{x_i|y_b} (y \in Y_b)-F_{x_i|y_{nb}} (y \in Y_{nb})|\]</div>
+<p>where <span class="math notranslate nohighlight">\(Y_b\)</span> represents the set of behavioral outputs, and <span class="math notranslate nohighlight">\(F_{x_i|y_b}\)</span> is the empirical cumulative distribution function of the values of <span class="math notranslate nohighlight">\(x_i\)</span> associated with values of <span class="math notranslate nohighlight">\(y\)</span> that belong in the behavioral set. The <span class="math notranslate nohighlight">\(nb\)</span> notation indicates the equivalent elements related to the non-behavioral set. Large differences between the two distributions indicate stronger effects by the parameters on the respective part of the output space.</p>
+<p>Used in a factor mapping setting, RSA can be applied for scenario discovery <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id71" title="Robert J Lempert, David G Groves, Steven W Popper, and Steve C Bankes. A general, analytic method for generating robust strategies and narrative scenarios. Management science, 52(4):514–528, 2006.">102</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>, the Generalized Likelihood Uncertainty Estimation method (GLUE; <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>, <a class="reference internal" href="R.Bibliography.html#id158" title="Keith Beven and Andrew Binley. Glue: 20 years on. Hydrological processes, 28(24):5897–5918, 2014.">104</a>, <a class="reference internal" href="R.Bibliography.html#id159" title="Roberta-Serena Blasone, Jasper A Vrugt, Henrik Madsen, Dan Rosbjerg, Bruce A Robinson, and George A Zyvoloski. Generalized likelihood uncertainty estimation (glue) using adaptive markov chain monte carlo sampling. Advances in Water Resources, 31(4):630–648, 2008.">105</a>]</span>) and other hybrid sensitivity analysis methods (e.g., <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id160" title="SA Cryer and PL Havens. Regional sensitivity analysis using a fractional factorial method for the usda model gleams. Environmental modelling &amp; software, 14(6):613–624, 1999.">106</a>, <a class="reference internal" href="R.Bibliography.html#id161" title="Pengfei Wei, Zhenzhou Lu, and Xiukai Yuan. Monte carlo simulation for moment-independent sensitivity analysis. Reliability Engineering &amp; System Safety, 110:60–67, 2013.">107</a>]</span>). The fundamental shortcomings of RSA are that, in some cases, it could be hard to interpret the difference between behavioral and non-behavioral sample sets, and that insights about parametric correlations and interactions cannot always be uncovered <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. For more elaborate discussions and illustrations of the RSA method, readers are directed to <span id="id7">Tang <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id20" title="Yong Tang, Patrick Reed, Thibaut Wagener, and K van Werkhoven. Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation. Hydrology and Earth System Sciences, 11(2):793–817, 2007.">42</a>], Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>], Young [<a class="reference internal" href="R.Bibliography.html#id162" title="Peter Young. Data-based mechanistic modelling, generalised sensitivity and dominant mode analysis. Computer Physics Communications, 117(1-2):113–129, 1999.">108</a>]</span> and references therein.</p>
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+  <section id="variance-based-methods">
+<span id="id1"></span><h1>Variance-based Methods<a class="headerlink" href="#variance-based-methods" title="Link to this heading">#</a></h1>
+<p>Variance-based sensitivity analysis methods hypothesize that various specified model factors contribute differently to the variation of model outputs; therefore, decomposition and analysis of output variance can determine a model’s sensitivity to input parameters <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>]</span>. The most popular variance-based method is the Sobol method, which is a global sensitivity analysis method that takes into account complex and nonlinear factor interaction when calculating sensitivity indices, and employs more sophisticated sampling methods (e.g., the Sobol sampling method). The Sobol method is able to calculate three types of sensitivity indices that provide different types of information about model sensitivities. These indices include first-order, higher-order (e.g., second-, third-, etc. orders), and total-order sensitivities.</p>
+<p>The first-order sensitivity index indicates the percent of model output variance contributed by a factor individually (i.e., the effect of varying <span class="math notranslate nohighlight">\(x_i\)</span> alone) and is obtained using the following <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>, <a class="reference internal" href="R.Bibliography.html#id148" title="Andrea Saltelli. Making best use of model evaluations to compute sensitivity indices. Computer physics communications, 145(2):280–297, 2002.">109</a>]</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i^1=\frac{V_{x_i}[E_{x_{\sim i}}(x_i)]}{V(y)}\]</div>
+<p>with <span class="math notranslate nohighlight">\(E\)</span> and <span class="math notranslate nohighlight">\(V\)</span> denoting the expected value and the variance, respectively. <span class="math notranslate nohighlight">\(x_{\sim i}\)</span> denotes all factors expect for <span class="math notranslate nohighlight">\(x_i\)</span>. The first-order sensitivity index (<span class="math notranslate nohighlight">\(S_i^1\)</span>) can therefore also be thought of as the portion of total output variance (<span class="math notranslate nohighlight">\(V_y\)</span>) that can be reduced if the uncertainty in factor <span class="math notranslate nohighlight">\(x_i\)</span> is eliminated <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id149" title="Andrea Saltelli. Sensitivity analysis for importance assessment. Risk analysis, 22(3):579–590, 2002.">110</a>]</span>. First-order sensitivity indices are usually used to understand the independent effect of a factor and to distinguish its individual versus interactive influence. It would be expected for linearly independent factors that they would only have first order indices (no interactions) that should correspond well with sensitivities obtained from simpler methods using OAT sampling.</p>
+<p>Higher-order sensitivity indices explore the interaction between two or more parameters that contribute to model output variations. For example, a second-order index indicates how interactions between a pair of factors can lead to change in model output variance and is calculated using the following relationship:</p>
+<div class="math notranslate nohighlight">
+\[S_{ij}^2=\frac{V_{x_{i,j}}[E_{x_{\sim i,j}}(x_i,x_j)]}{V(y)}\]</div>
+<p>with <span class="math notranslate nohighlight">\(i \ne j\)</span>. Higher order indices can be calculated by similar extensions (i.e., fixing additional operators together), but it is usually computationally expensive in practice.</p>
+<p>The total sensitivity analysis index represents the entire influence of an input factor on model outputs including all of its interactions with other factors <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id150" title="Toshimitsu Homma and Andrea Saltelli. Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering &amp; System Safety, 52(1):1–17, 1996.">111</a>]</span>. In other words, total-order indices include first-order and all higher-order interactions associated with each factor and can be estimated calculated using the following:</p>
+<div class="math notranslate nohighlight">
+\[S_i^T= \frac{E_{x_{\sim i}}[V_{x_i}(x_{\sim i})]}{V(y)} = 1 - \frac{V_{x_{\sim i}}[E_{x_{i}}(x_{\sim i})]}{V(y)}\]</div>
+<p>This index reveals the expected portion of variance that remains if uncertainty is eliminated in all factors but <span class="math notranslate nohighlight">\(x_i\)</span> <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id149" title="Andrea Saltelli. Sensitivity analysis for importance assessment. Risk analysis, 22(3):579–590, 2002.">110</a>]</span>. The total sensitivity index is the overall best measure of sensitivity as it captures the full individual and interactive effects of model factors.</p>
+<p>Besides the Sobol method, there are some other variance-based sensitivity analysis methods, such as the Fourier amplitude sensitivity test (FAST; <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id100" title="RI Cukier, CM Fortuin, Kurt E Shuler, AG Petschek, and JH Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. i theory. The Journal of chemical physics, 59(8):3873–3878, 1973.">75</a>, <a class="reference internal" href="R.Bibliography.html#id151" title="Gregory J McRae, William R Goodin, and John H Seinfeld. Development of a second-generation mathematical model for urban air pollution—i. model formulation. Atmospheric Environment (1967), 16(4):679–696, 1982.">112</a>]</span>) and extended-FAST <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id152" title="Andrea Saltelli and Ricardo Bolado. An alternative way to compute fourier amplitude sensitivity test (fast). Computational Statistics &amp; Data Analysis, 26(4):445–460, 1998.">113</a>, <a class="reference internal" href="R.Bibliography.html#id153" title="MA Vazquez-Cruz, R Guzman-Cruz, IL Lopez-Cruz, O Cornejo-Perez, I Torres-Pacheco, and RG Guevara-Gonzalez. Global sensitivity analysis by means of efast and sobol’methods and calibration of reduced state-variable tomgro model using genetic algorithms. Computers and Electronics in Agriculture, 100:1–12, 2014.">114</a>]</span>, that have been used by the scientific community. However, Sobol remains by far the most common method of this class. Variance-based techniques have been widely used and have proved to be powerful in a variety of applications. Despite their popularity, some authors have expressed concerns about the methods’ appropriateness in some settings. Specifically, the presence of heavy-tailed distributions or outliers, or when model outputs are multimodal can bias the sensitivity indices produced by these methods <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id154" title="Benjamin Auder and Bertrand Iooss. Global sensitivity analysis based on entropy. In Safety, reliability and risk analysis-Proceedings of the ESREL 2008 Conference, 2107–2115. 2008.">115</a>, <a class="reference internal" href="R.Bibliography.html#id155" title="Farkhondeh Khorashadi Zadeh, Jiri Nossent, Fanny Sarrazin, Francesca Pianosi, Ann van Griensven, Thorsten Wagener, and Willy Bauwens. Comparison of variance-based and moment-independent global sensitivity analysis approaches by application to the swat model. Environmental Modelling &amp; Software, 91:210–222, 2017.">116</a>, <a class="reference internal" href="R.Bibliography.html#id156" title="Francesca Pianosi and Thorsten Wagener. A simple and efficient method for global sensitivity analysis based on cumulative distribution functions. Environmental Modelling &amp; Software, 67:1–11, 2015.">117</a>]</span>. Moment-independent measures, discussed below, attempt to overcome these challenges.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial which demonstrates the application of a Sobol sensitivity analysis:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA using SALib Jupyter Notebook</a></p>
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+  <section id="analysis-of-variance-anova">
+<h1>Analysis of Variance (ANOVA)<a class="headerlink" href="#analysis-of-variance-anova" title="Link to this heading">#</a></h1>
+<p>Analysis of Variance (ANOVA) was first introduced by <span id="id1">Fisher and others [<a class="reference internal" href="R.Bibliography.html#id144" title="Ronald Aylmer Fisher and others. Statistical methods for research workers. Statistical methods for research workers., 1934.">118</a>]</span> and has since become a popular factor analysis method in physical experiments. ANOVA can be used as a sensitivity analysis method in computational experiments with a factorial design of experiment (referred to as factorial ANOVA). Note that Sobol can also be categorized as an ANOVA sensitivity analysis method, and that is why Sobol is sometimes referred to as a functional ANOVA <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id145" title="Loıc Brevault, Mathieu Balesdent, Nicolas Bérend, and Rodolphe Le Riche. Comparison of different global sensitivity analysis methods for aerospace vehicle optimal design. In 10th World Congress on Structural and Multidisciplinary Optimization, WCSMO-10. 2013.">119</a>]</span>. Factorial ANOVA methods are particularly suited for models and problems that have discrete input spaces, significantly reducing the computational time. More information about these methods can be found in <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id145" title="Loıc Brevault, Mathieu Balesdent, Nicolas Bérend, and Rodolphe Le Riche. Comparison of different global sensitivity analysis methods for aerospace vehicle optimal design. In 10th World Congress on Structural and Multidisciplinary Optimization, WCSMO-10. 2013.">119</a>, <a class="reference internal" href="R.Bibliography.html#id146" title="GEB Archer, Andrea Saltelli, and IM Sobol. Sensitivity measures, anova-like techniques and the use of bootstrap. Journal of Statistical Computation and Simulation, 58(2):99–120, 1997.">120</a>, <a class="reference internal" href="R.Bibliography.html#id147" title="Art B Owen. Variance components and generalized sobol'indices. SIAM/ASA Journal on Uncertainty Quantification, 1(1):19–41, 2013.">121</a>]</span>.</p>
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+  <section id="moment-independent-density-based-methods">
+<h1>Moment-Independent (Density-Based) Methods<a class="headerlink" href="#moment-independent-density-based-methods" title="Link to this heading">#</a></h1>
+<p>These methods typically compare the entire distribution (i.e., not just the variance) of input and output parameters in order to determine the sensitivity of the output to a particular input variable. Several moment-independent sensitivity analysis methods have been proposed in recent years. The delta (<span class="math notranslate nohighlight">\(\delta\)</span>) moment-independent method calculates the difference between unconditional and conditional cumulative distribution functions of the output. The method was first introduced by <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id142" title="Emanuele Borgonovo. Measuring uncertainty importance: investigation and comparison of alternative approaches. Risk analysis, 26(5):1349–1361, 2006.">122</a>, <a class="reference internal" href="R.Bibliography.html#id141" title="Emanuele Borgonovo. A new uncertainty importance measure. Reliability Engineering &amp; System Safety, 92(6):771–784, 2007.">123</a>]</span> and has become widely used in various disciplines. The <span class="math notranslate nohighlight">\(\delta\)</span> sensitivity index is defined as follows:</p>
+<div class="math notranslate nohighlight">
+\[S_i=\delta_i=\frac{1}{2}E_{x_i}|f_y(y)-f_{y|x_i}(y)|dy\]</div>
+<p>where <span class="math notranslate nohighlight">\(f_y(y)\)</span> is the probability density function of the entire model output <span class="math notranslate nohighlight">\(y\)</span>, and <span class="math notranslate nohighlight">\(f_{y|x_i}(y)\)</span> is the conditional density of <span class="math notranslate nohighlight">\(y\)</span>, given that factor <span class="math notranslate nohighlight">\(x_i\)</span> assumes a fixed value. The <span class="math notranslate nohighlight">\(\delta_i\)</span> sensitivity indicator therefore represents the normalized expected shift in the distribution of <span class="math notranslate nohighlight">\(y\)</span> provoked by <span class="math notranslate nohighlight">\(x_i\)</span>. Moment-independent methods are advantageous in cases where we are concerned about the entire distribution of events, such as when uncertain factors lead to more extreme events in a system <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id47" title="Antonia Hadjimichael, Julianne Quinn, and Patrick Reed. Advancing Diagnostic Model Evaluation to Better Understand Water Shortage Mechanisms in Institutionally Complex River Basins. Water Resources Research, 56(10):e2020WR028079, 2020. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR028079 (visited on 2020-10-16), doi:10.1029/2020WR028079.">13</a>]</span>. Further, they can be used with a pre-existing sample of data, without requiring a specific sampling scheme, unlike the previously reviewed methods <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id143" title="Elmar Plischke, Emanuele Borgonovo, and Curtis L Smith. Global sensitivity measures from given data. European Journal of Operational Research, 226(3):536–550, 2013.">124</a>]</span>. The <span class="math notranslate nohighlight">\(\delta\)</span> sensitivity index does not include interactions between factors and it is therefore akin to the first order index produced by the Sobol method. Interactions between factors can still be estimated using this method, by conditioning the calculation on more than one uncertain factor being fixed <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id141" title="Emanuele Borgonovo. A new uncertainty importance measure. Reliability Engineering &amp; System Safety, 92(6):771–784, 2007.">123</a>]</span>.</p>
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+    <h1>Sensitivity Analysis Methods</h1>
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+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivative-based-methods">Derivative-based Methods</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#elementary-effect-methods">Elementary Effect Methods</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#regression-based-methods">Regression-based Methods</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#regional-sensitivity-analysis">Regional Sensitivity Analysis</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#variance-based-methods">Variance-based Methods</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysis-of-variance-anova">Analysis of Variance (ANOVA)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#moment-independent-density-based-methods">Moment-Independent (Density-Based) Methods</a></li>
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+  <section id="sensitivity-analysis-methods">
+<span id="id1"></span><h1>Sensitivity Analysis Methods<a class="headerlink" href="#sensitivity-analysis-methods" title="Link to this heading">#</a></h1>
+<p>In this section, we describe some of the most widely applied sensitivity analysis methods along with their mathematical definitions. We also provide a detailed discussion on applying each method, as well as a comparison of and their features and limitations.</p>
+<section id="derivative-based-methods">
+<h2>Derivative-based Methods<a class="headerlink" href="#derivative-based-methods" title="Link to this heading">#</a></h2>
+<p>Derivative-based methods explore how model outputs are affected by perturbations in a single model input around a particular input value. These methods are local and are performed using OAT sampling. For simplicity of mathematical notations, let us assume that the model <span class="math notranslate nohighlight">\(g(X)\)</span> only returns one output. Following <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id117" title="Emanuele Borgonovo. Sensitivity analysis of model output with input constraints: a generalized rationale for local methods. Risk Analysis: An International Journal, 28(3):667–680, 2008.">90</a>]</span> and <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>, the sensitivity index, <span class="math notranslate nohighlight">\(S_i\)</span> , of the model’s <em>i</em>-th input factor, <span class="math notranslate nohighlight">\(x_i\)</span> , can be measured using the partial derivative evaluated at a nominal value, <span class="math notranslate nohighlight">\(\bar{x}\)</span>, of the vector of inputs:</p>
+<div class="math notranslate nohighlight">
+\[S_i (\bar{x}) = \frac{\partial g}{\partial x} |_{\bar{x}{^c{_i}}}\]</div>
+<p>where <em>c</em><sub>i</sub> is the scaling factor. In most applications however, the relationship <span class="math notranslate nohighlight">\(g(X)\)</span> is not fully known in its analytical form, and therefore the above partial derivative is usually approximated:</p>
+<div class="math notranslate nohighlight">
+\[S_i (\bar{x}) = \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i\]</div>
+<p>Using this approximation, the <em>i</em>-th input factor is perturbed by a magnitude of <span class="math notranslate nohighlight">\(\Delta_i\)</span>, and its relative importance is calculated. Derivative-based methods are some of the oldest sensitivity analysis methods as they only require <span class="math notranslate nohighlight">\(N+1\)</span> model evaluations to estimate indices for <span class="math notranslate nohighlight">\(N\)</span> uncertain factors. As described above, being computationally very cheap comes at the cost of not being able to explore the entire input space, but only (local) perturbations to the nominal value. Additionally, as these methods examine the effects of each input factor one at a time, they cannot assess parametric interactions or capture the interacting nature of many real systems and the models that abstract them.</p>
+</section>
+<section id="elementary-effect-methods">
+<h2>Elementary Effect Methods<a class="headerlink" href="#elementary-effect-methods" title="Link to this heading">#</a></h2>
+<p>Elementary effect (EE) SA methods provide a solution to the local nature of the derivative-based methods by exploring the entire parametric range of each input parameter <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>. However, EE methods still use OAT sampling and do not vary all input parameters simultaneously while exploring the parametric space. The OAT nature of EEs methods therefore prevents them from properly capturing the interactions between uncertain factors. EEs methods are computationally efficient compared to their All-At-a-Time (AAT) counterparts, making them more suitable when computational capacity is a limiting factor, while still allowing for some inferences regarding factor interactions.
+The most popular EE method is the Method of Morris <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id132" title="Max D Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.">74</a>]</span>. Following the notation by <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>, this method calculates global sensitivity using the mean of the EEs (finite differences) of each parameter at different locations:</p>
+<div class="math notranslate nohighlight">
+\[S_i = \mu_i^* = \frac{1}{r}\sum_{j=1}^r EE^j_i = \frac{1}{r}\sum_{j=1}^r \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i\]</div>
+<p>with <span class="math notranslate nohighlight">\(r\)</span> representing the number of sample repetitions (also refered to as trajectories) in the input space, usually set between 4 and 10 <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. Each <span class="math notranslate nohighlight">\(x_j\)</span> represents the points of each trajectory, with <span class="math notranslate nohighlight">\(j=1,…, r\)</span>, selected as described in the sampling strategy for this method, found above. This method also produces the standard deviation of the EEs:</p>
+<div class="math notranslate nohighlight">
+\[\sigma_i = \sqrt{\frac{1}{r}\sum_{j=1}^r(EE_i^j-\frac{1}{r}\sum_{j=1}^r EE^j_i)^2}\]</div>
+<p>which is a measure of parametric interactions. Higher values of <span class="math notranslate nohighlight">\(\sigma_i\)</span> suggest model responses at different levels of factor <span class="math notranslate nohighlight">\(x_i\)</span> are significantly different, which indicates considerable interactions between that and other uncertain factors. The values of <span class="math notranslate nohighlight">\(\mu_i^*\)</span> and <span class="math notranslate nohighlight">\(\sigma_i\)</span> for each factor allow us to draw several different conclusions, illustrated in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-4"><span class="std std-numref">Fig. 3.4</span></a>, following the example by <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>. In this example, factors <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> can be said to have an influence on the model outputs, with <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> having some interactive or non-linear effects. Depending on the orders of magnitude of <span class="math notranslate nohighlight">\(\mu_i^*\)</span> and <span class="math notranslate nohighlight">\(\sigma_i\)</span> one can indirectly deduce whether the factors have strong interactive effects, for example if a factor <span class="math notranslate nohighlight">\(\sigma_i &lt;&lt; \mu_i^*\)</span> then the relationship between that factor and the output can be assumed to be largely linear (note that this is still an OAT method and assumptions on factor interactions should be strongly caveated). Extensions of the Method of Morris have also been developed specifically for the purposes of factor fixing and explorations of parametric interactions (e.g., <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id27" title="Emanuele Borgonovo. Sensitivity analysis with finite changes: an application to modified eoq models. European Journal of Operational Research, 200(1):127–138, 2010.">48</a>, <a class="reference internal" href="R.Bibliography.html#id133" title="Francesca Campolongo and Roger Braddock. The use of graph theory in the sensitivity analysis of the model output: a second order screening method. Reliability Engineering &amp; System Safety, 64(1):1–12, 1999.">92</a>, <a class="reference internal" href="R.Bibliography.html#id134" title="Roger A Cropp and Roger D Braddock. The new morris method: an efficient second-order screening method. Reliability Engineering &amp; System Safety, 78(1):77–83, 2002.">93</a>]</span>).</p>
+<figure class="margin-caption align-center" id="id41">
+<span id="figure-3-4"></span><a class="reference internal image-reference" href="_images/figure3_4_morris_method.png"><img alt="Figure 3.4" src="_images/figure3_4_morris_method.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Illustrative results of the Morris Method. Factors <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> have an influence on the model outputs, with <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> having interactive or non-linear effects. Whether or not a factor should be considered influential to the output depends on the output selected and is specific to the research context and purpose of the analysis, as discussed in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#why-sa"><span class="std std-numref">Chapter 3.2</span></a>.</span><a class="headerlink" href="#id41" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="regression-based-methods">
+<h2>Regression-based Methods<a class="headerlink" href="#regression-based-methods" title="Link to this heading">#</a></h2>
+<p>Regression analysis is one of the oldest ways of investigating parametric importance and sensitivity <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. Here, we describe some of the most popular regression-based sensitivity indices. One of the main sensitivity indices of this category is the standardized regression coefficient (SRC). To calculate SRC, a linear regression relationship needs to be fitted between the input vector, <span class="math notranslate nohighlight">\(x\)</span>, and the model output of interest by using a least-square minimizing method:</p>
+<div class="math notranslate nohighlight">
+\[y = b_0 + \sum_{i=1}^N b_ix_i\]</div>
+<p>where <span class="math notranslate nohighlight">\(b_0\)</span> and <span class="math notranslate nohighlight">\(b_i\)</span> (corresponding to the <em>i</em>-th model input) are regression coefficients. The following relationship can then be used to calculate the SRCs for different input values:</p>
+<div class="math notranslate nohighlight">
+\[S_i=SRC_i=b_i\frac{\sigma_i}{\sigma_y}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\sigma_i\)</span> and <span class="math notranslate nohighlight">\(\sigma_y\)</span> are standard deviations of <em>i</em>-th model input and output, respectively.</p>
+<p>Several other regression-based indices explore the correlation between input and output parameters as a proxy to model parametric sensitivity <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>, <a class="reference internal" href="R.Bibliography.html#id135" title="Jon C Helton. Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal. Reliability Engineering &amp; System Safety, 42(2-3):327–367, 1993.">94</a>, <a class="reference internal" href="R.Bibliography.html#id136" title="Gemma Manache and Charles S Melching. Identification of reliable regression-and correlation-based sensitivity measures for importance ranking of water-quality model parameters. Environmental Modelling &amp; Software, 23(5):549–562, 2008.">95</a>]</span>. The Pearson correlation coefficient (PCC) can be used when a linear relationship exists between an uncertain factor, <span class="math notranslate nohighlight">\(x_i\)</span>, and the output <span class="math notranslate nohighlight">\(y\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i=PCC=\frac{cov(x_i,y)}{\sigma_i\sigma_y}\]</div>
+<p>In cases when there are outliers in the data or the relationship between the uncertain factors and the output is not linear, rank-based correlation coefficients are preferred, for example, Spearman’s rank correlation coefficient (SRCC):</p>
+<div class="math notranslate nohighlight">
+\[S_i=SRCC=\frac{cov(rx_i,ryi)}{\sigma_{ri}\sigma_{ry}}\]</div>
+<p>where the raw values of <span class="math notranslate nohighlight">\(x_i\)</span> and <span class="math notranslate nohighlight">\(y\)</span> and converted to ranks <span class="math notranslate nohighlight">\(rx_i\)</span> and <span class="math notranslate nohighlight">\(ry\)</span> respectively, which instead represent a measurement of the strength of the monotonic relationship, rather than linear relationship, between the input and output. Other regression-based metrics include the partial correlations coefficient, the partial rank correlations coefficient, and the Nash-Sutcliffe coefficient, more discussion on which can be found in <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>.</p>
+<p>Tree-based regression techniques have also been used for sensitivity analysis in an effort to address the challenges faced with nonlinear models <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id137" title="F Pappenberger and Keith J Beven. Ignorance is bliss: or seven reasons not to use uncertainty analysis. Water resources research, 2006.">96</a>]</span>. Examples of these methods include the Patient Rule Induction Method (PRIM; <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id79" title="Jerome H Friedman and Nicholas I Fisher. Bump hunting in high-dimensional data. Statistics and Computing, 9(2):123–143, 1999.">97</a>]</span>) and Classification And Regression Trees (CART; <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id80" title="Leo Breiman, Jerome Friedman, Charles J Stone, and Richard A Olshen. Classification and regression trees. CRC press, 1984.">98</a>]</span>). CART-based approaches also include boosting and bagging extensions <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id89" title="Yoav Freund, Robert Schapire, and Naoki Abe. A short introduction to boosting. Journal-Japanese Society For Artificial Intelligence, 14(771-780):1612, 1999.">99</a>, <a class="reference internal" href="R.Bibliography.html#id138" title="Leo Breiman. Bagging predictors. Machine learning, 24(2):123–140, 1996.">100</a>]</span>. These methods are particularly useful when sensitivity analysis is used for factor mapping (i.e., when trying to identify which uncertain model factors produce a certain model behavior). <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a> elaborates on the use of these methods. Regression-based sensitivity analysis methods are global by nature and can explore the entire space of variables. However, the true level of comprehensiveness depends on the design of experiments and the number of simulations providing data to establish the regression relationships. Although they are usually computationally efficient, they do not produce significant information about parametric interactions <span id="id17">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>]</span>.</p>
+</section>
+<section id="regional-sensitivity-analysis">
+<h2>Regional Sensitivity Analysis<a class="headerlink" href="#regional-sensitivity-analysis" title="Link to this heading">#</a></h2>
+<p>Another method primarily applied for basic factor mapping applications is Regional Sensitivity Analysis (RSA; <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id157" title="George M Hornberger and Robert C Spear. Approach to the preliminary analysis of environmental systems. J. Environ. Mgmt., 12(1):7–18, 1981.">101</a>]</span>). RSA is a global sensitivity analysis method that is typically implemented using standard sampling methods such as latin hypercube sampling. It is performed by specifying a condition on the output space (e.g., an upper threshold) and classifying outputs that meet the condition as behavioral and the ones that fail it as non-behavioral (illustrated in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (b)). Note that the specified threshold depends on the nature of the problem, model, and the research question. It can reflect model-performance metrics (such as errors) or consequential decision-relevant metrics (such as unacceptable system outcomes). The behavioral and non-behavioral outputs are then traced back to their originating sampled factors, where differences between the distributions of samples can be used to determine their significance in producing each part of the output. The Kolmogorov-Smirnov divergence is commonly used to quantify the difference between the distribution of behavioral and non-behavioral parameters <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i=|F_{x_i|y_b} (y \in Y_b)-F_{x_i|y_{nb}} (y \in Y_{nb})|\]</div>
+<p>where <span class="math notranslate nohighlight">\(Y_b\)</span> represents the set of behavioral outputs, and <span class="math notranslate nohighlight">\(F_{x_i|y_b}\)</span> is the empirical cumulative distribution function of the values of <span class="math notranslate nohighlight">\(x_i\)</span> associated with values of <span class="math notranslate nohighlight">\(y\)</span> that belong in the behavioral set. The <span class="math notranslate nohighlight">\(nb\)</span> notation indicates the equivalent elements related to the non-behavioral set. Large differences between the two distributions indicate stronger effects by the parameters on the respective part of the output space.</p>
+<p>Used in a factor mapping setting, RSA can be applied for scenario discovery <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id71" title="Robert J Lempert, David G Groves, Steven W Popper, and Steve C Bankes. A general, analytic method for generating robust strategies and narrative scenarios. Management science, 52(4):514–528, 2006.">102</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>, the Generalized Likelihood Uncertainty Estimation method (GLUE; <span id="id21">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>, <a class="reference internal" href="R.Bibliography.html#id158" title="Keith Beven and Andrew Binley. Glue: 20 years on. Hydrological processes, 28(24):5897–5918, 2014.">104</a>, <a class="reference internal" href="R.Bibliography.html#id159" title="Roberta-Serena Blasone, Jasper A Vrugt, Henrik Madsen, Dan Rosbjerg, Bruce A Robinson, and George A Zyvoloski. Generalized likelihood uncertainty estimation (glue) using adaptive markov chain monte carlo sampling. Advances in Water Resources, 31(4):630–648, 2008.">105</a>]</span>) and other hybrid sensitivity analysis methods (e.g., <span id="id22">[<a class="reference internal" href="R.Bibliography.html#id160" title="SA Cryer and PL Havens. Regional sensitivity analysis using a fractional factorial method for the usda model gleams. Environmental modelling &amp; software, 14(6):613–624, 1999.">106</a>, <a class="reference internal" href="R.Bibliography.html#id161" title="Pengfei Wei, Zhenzhou Lu, and Xiukai Yuan. Monte carlo simulation for moment-independent sensitivity analysis. Reliability Engineering &amp; System Safety, 110:60–67, 2013.">107</a>]</span>). The fundamental shortcomings of RSA are that, in some cases, it could be hard to interpret the difference between behavioral and non-behavioral sample sets, and that insights about parametric correlations and interactions cannot always be uncovered <span id="id23">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. For more elaborate discussions and illustrations of the RSA method, readers are directed to <span id="id24">Tang <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id20" title="Yong Tang, Patrick Reed, Thibaut Wagener, and K van Werkhoven. Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation. Hydrology and Earth System Sciences, 11(2):793–817, 2007.">42</a>], Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>], Young [<a class="reference internal" href="R.Bibliography.html#id162" title="Peter Young. Data-based mechanistic modelling, generalised sensitivity and dominant mode analysis. Computer Physics Communications, 117(1-2):113–129, 1999.">108</a>]</span> and references therein.</p>
+</section>
+<section id="variance-based-methods">
+<span id="id25"></span><h2>Variance-based Methods<a class="headerlink" href="#variance-based-methods" title="Link to this heading">#</a></h2>
+<p>Variance-based sensitivity analysis methods hypothesize that various specified model factors contribute differently to the variation of model outputs; therefore, decomposition and analysis of output variance can determine a model’s sensitivity to input parameters <span id="id26">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>]</span>. The most popular variance-based method is the Sobol method, which is a global sensitivity analysis method that takes into account complex and nonlinear factor interaction when calculating sensitivity indices, and employs more sophisticated sampling methods (e.g., the Sobol sampling method). The Sobol method is able to calculate three types of sensitivity indices that provide different types of information about model sensitivities. These indices include first-order, higher-order (e.g., second-, third-, etc. orders), and total-order sensitivities.</p>
+<p>The first-order sensitivity index indicates the percent of model output variance contributed by a factor individually (i.e., the effect of varying <span class="math notranslate nohighlight">\(x_i\)</span> alone) and is obtained using the following <span id="id27">[<a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>, <a class="reference internal" href="R.Bibliography.html#id148" title="Andrea Saltelli. Making best use of model evaluations to compute sensitivity indices. Computer physics communications, 145(2):280–297, 2002.">109</a>]</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i^1=\frac{V_{x_i}[E_{x_{\sim i}}(x_i)]}{V(y)}\]</div>
+<p>with <span class="math notranslate nohighlight">\(E\)</span> and <span class="math notranslate nohighlight">\(V\)</span> denoting the expected value and the variance, respectively. <span class="math notranslate nohighlight">\(x_{\sim i}\)</span> denotes all factors expect for <span class="math notranslate nohighlight">\(x_i\)</span>. The first-order sensitivity index (<span class="math notranslate nohighlight">\(S_i^1\)</span>) can therefore also be thought of as the portion of total output variance (<span class="math notranslate nohighlight">\(V_y\)</span>) that can be reduced if the uncertainty in factor <span class="math notranslate nohighlight">\(x_i\)</span> is eliminated <span id="id28">[<a class="reference internal" href="R.Bibliography.html#id149" title="Andrea Saltelli. Sensitivity analysis for importance assessment. Risk analysis, 22(3):579–590, 2002.">110</a>]</span>. First-order sensitivity indices are usually used to understand the independent effect of a factor and to distinguish its individual versus interactive influence. It would be expected for linearly independent factors that they would only have first order indices (no interactions) that should correspond well with sensitivities obtained from simpler methods using OAT sampling.</p>
+<p>Higher-order sensitivity indices explore the interaction between two or more parameters that contribute to model output variations. For example, a second-order index indicates how interactions between a pair of factors can lead to change in model output variance and is calculated using the following relationship:</p>
+<div class="math notranslate nohighlight">
+\[S_{ij}^2=\frac{V_{x_{i,j}}[E_{x_{\sim i,j}}(x_i,x_j)]}{V(y)}\]</div>
+<p>with <span class="math notranslate nohighlight">\(i \ne j\)</span>. Higher order indices can be calculated by similar extensions (i.e., fixing additional operators together), but it is usually computationally expensive in practice.</p>
+<p>The total sensitivity analysis index represents the entire influence of an input factor on model outputs including all of its interactions with other factors <span id="id29">[<a class="reference internal" href="R.Bibliography.html#id150" title="Toshimitsu Homma and Andrea Saltelli. Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering &amp; System Safety, 52(1):1–17, 1996.">111</a>]</span>. In other words, total-order indices include first-order and all higher-order interactions associated with each factor and can be estimated calculated using the following:</p>
+<div class="math notranslate nohighlight">
+\[S_i^T= \frac{E_{x_{\sim i}}[V_{x_i}(x_{\sim i})]}{V(y)} = 1 - \frac{V_{x_{\sim i}}[E_{x_{i}}(x_{\sim i})]}{V(y)}\]</div>
+<p>This index reveals the expected portion of variance that remains if uncertainty is eliminated in all factors but <span class="math notranslate nohighlight">\(x_i\)</span> <span id="id30">[<a class="reference internal" href="R.Bibliography.html#id149" title="Andrea Saltelli. Sensitivity analysis for importance assessment. Risk analysis, 22(3):579–590, 2002.">110</a>]</span>. The total sensitivity index is the overall best measure of sensitivity as it captures the full individual and interactive effects of model factors.</p>
+<p>Besides the Sobol method, there are some other variance-based sensitivity analysis methods, such as the Fourier amplitude sensitivity test (FAST; <span id="id31">[<a class="reference internal" href="R.Bibliography.html#id100" title="RI Cukier, CM Fortuin, Kurt E Shuler, AG Petschek, and JH Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. i theory. The Journal of chemical physics, 59(8):3873–3878, 1973.">75</a>, <a class="reference internal" href="R.Bibliography.html#id151" title="Gregory J McRae, William R Goodin, and John H Seinfeld. Development of a second-generation mathematical model for urban air pollution—i. model formulation. Atmospheric Environment (1967), 16(4):679–696, 1982.">112</a>]</span>) and extended-FAST <span id="id32">[<a class="reference internal" href="R.Bibliography.html#id152" title="Andrea Saltelli and Ricardo Bolado. An alternative way to compute fourier amplitude sensitivity test (fast). Computational Statistics &amp; Data Analysis, 26(4):445–460, 1998.">113</a>, <a class="reference internal" href="R.Bibliography.html#id153" title="MA Vazquez-Cruz, R Guzman-Cruz, IL Lopez-Cruz, O Cornejo-Perez, I Torres-Pacheco, and RG Guevara-Gonzalez. Global sensitivity analysis by means of efast and sobol’methods and calibration of reduced state-variable tomgro model using genetic algorithms. Computers and Electronics in Agriculture, 100:1–12, 2014.">114</a>]</span>, that have been used by the scientific community. However, Sobol remains by far the most common method of this class. Variance-based techniques have been widely used and have proved to be powerful in a variety of applications. Despite their popularity, some authors have expressed concerns about the methods’ appropriateness in some settings. Specifically, the presence of heavy-tailed distributions or outliers, or when model outputs are multimodal can bias the sensitivity indices produced by these methods <span id="id33">[<a class="reference internal" href="R.Bibliography.html#id154" title="Benjamin Auder and Bertrand Iooss. Global sensitivity analysis based on entropy. In Safety, reliability and risk analysis-Proceedings of the ESREL 2008 Conference, 2107–2115. 2008.">115</a>, <a class="reference internal" href="R.Bibliography.html#id155" title="Farkhondeh Khorashadi Zadeh, Jiri Nossent, Fanny Sarrazin, Francesca Pianosi, Ann van Griensven, Thorsten Wagener, and Willy Bauwens. Comparison of variance-based and moment-independent global sensitivity analysis approaches by application to the swat model. Environmental Modelling &amp; Software, 91:210–222, 2017.">116</a>, <a class="reference internal" href="R.Bibliography.html#id156" title="Francesca Pianosi and Thorsten Wagener. A simple and efficient method for global sensitivity analysis based on cumulative distribution functions. Environmental Modelling &amp; Software, 67:1–11, 2015.">117</a>]</span>. Moment-independent measures, discussed below, attempt to overcome these challenges.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial which demonstrates the application of a Sobol sensitivity analysis:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA using SALib Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="analysis-of-variance-anova">
+<h2>Analysis of Variance (ANOVA)<a class="headerlink" href="#analysis-of-variance-anova" title="Link to this heading">#</a></h2>
+<p>Analysis of Variance (ANOVA) was first introduced by <span id="id34">Fisher and others [<a class="reference internal" href="R.Bibliography.html#id144" title="Ronald Aylmer Fisher and others. Statistical methods for research workers. Statistical methods for research workers., 1934.">118</a>]</span> and has since become a popular factor analysis method in physical experiments. ANOVA can be used as a sensitivity analysis method in computational experiments with a factorial design of experiment (referred to as factorial ANOVA). Note that Sobol can also be categorized as an ANOVA sensitivity analysis method, and that is why Sobol is sometimes referred to as a functional ANOVA <span id="id35">[<a class="reference internal" href="R.Bibliography.html#id145" title="Loıc Brevault, Mathieu Balesdent, Nicolas Bérend, and Rodolphe Le Riche. Comparison of different global sensitivity analysis methods for aerospace vehicle optimal design. In 10th World Congress on Structural and Multidisciplinary Optimization, WCSMO-10. 2013.">119</a>]</span>. Factorial ANOVA methods are particularly suited for models and problems that have discrete input spaces, significantly reducing the computational time. More information about these methods can be found in <span id="id36">[<a class="reference internal" href="R.Bibliography.html#id145" title="Loıc Brevault, Mathieu Balesdent, Nicolas Bérend, and Rodolphe Le Riche. Comparison of different global sensitivity analysis methods for aerospace vehicle optimal design. In 10th World Congress on Structural and Multidisciplinary Optimization, WCSMO-10. 2013.">119</a>, <a class="reference internal" href="R.Bibliography.html#id146" title="GEB Archer, Andrea Saltelli, and IM Sobol. Sensitivity measures, anova-like techniques and the use of bootstrap. Journal of Statistical Computation and Simulation, 58(2):99–120, 1997.">120</a>, <a class="reference internal" href="R.Bibliography.html#id147" title="Art B Owen. Variance components and generalized sobol'indices. SIAM/ASA Journal on Uncertainty Quantification, 1(1):19–41, 2013.">121</a>]</span>.</p>
+</section>
+<section id="moment-independent-density-based-methods">
+<h2>Moment-Independent (Density-Based) Methods<a class="headerlink" href="#moment-independent-density-based-methods" title="Link to this heading">#</a></h2>
+<p>These methods typically compare the entire distribution (i.e., not just the variance) of input and output parameters in order to determine the sensitivity of the output to a particular input variable. Several moment-independent sensitivity analysis methods have been proposed in recent years. The delta (<span class="math notranslate nohighlight">\(\delta\)</span>) moment-independent method calculates the difference between unconditional and conditional cumulative distribution functions of the output. The method was first introduced by <span id="id37">[<a class="reference internal" href="R.Bibliography.html#id142" title="Emanuele Borgonovo. Measuring uncertainty importance: investigation and comparison of alternative approaches. Risk analysis, 26(5):1349–1361, 2006.">122</a>, <a class="reference internal" href="R.Bibliography.html#id141" title="Emanuele Borgonovo. A new uncertainty importance measure. Reliability Engineering &amp; System Safety, 92(6):771–784, 2007.">123</a>]</span> and has become widely used in various disciplines. The <span class="math notranslate nohighlight">\(\delta\)</span> sensitivity index is defined as follows:</p>
+<div class="math notranslate nohighlight">
+\[S_i=\delta_i=\frac{1}{2}E_{x_i}|f_y(y)-f_{y|x_i}(y)|dy\]</div>
+<p>where <span class="math notranslate nohighlight">\(f_y(y)\)</span> is the probability density function of the entire model output <span class="math notranslate nohighlight">\(y\)</span>, and <span class="math notranslate nohighlight">\(f_{y|x_i}(y)\)</span> is the conditional density of <span class="math notranslate nohighlight">\(y\)</span>, given that factor <span class="math notranslate nohighlight">\(x_i\)</span> assumes a fixed value. The <span class="math notranslate nohighlight">\(\delta_i\)</span> sensitivity indicator therefore represents the normalized expected shift in the distribution of <span class="math notranslate nohighlight">\(y\)</span> provoked by <span class="math notranslate nohighlight">\(x_i\)</span>. Moment-independent methods are advantageous in cases where we are concerned about the entire distribution of events, such as when uncertain factors lead to more extreme events in a system <span id="id38">[<a class="reference internal" href="R.Bibliography.html#id47" title="Antonia Hadjimichael, Julianne Quinn, and Patrick Reed. Advancing Diagnostic Model Evaluation to Better Understand Water Shortage Mechanisms in Institutionally Complex River Basins. Water Resources Research, 56(10):e2020WR028079, 2020. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR028079 (visited on 2020-10-16), doi:10.1029/2020WR028079.">13</a>]</span>. Further, they can be used with a pre-existing sample of data, without requiring a specific sampling scheme, unlike the previously reviewed methods <span id="id39">[<a class="reference internal" href="R.Bibliography.html#id143" title="Elmar Plischke, Emanuele Borgonovo, and Curtis L Smith. Global sensitivity measures from given data. European Journal of Operational Research, 226(3):536–550, 2013.">124</a>]</span>. The <span class="math notranslate nohighlight">\(\delta\)</span> sensitivity index does not include interactions between factors and it is therefore akin to the first order index produced by the Sobol method. Interactions between factors can still be estimated using this method, by conditioning the calculation on more than one uncertain factor being fixed <span id="id40">[<a class="reference internal" href="R.Bibliography.html#id141" title="Emanuele Borgonovo. A new uncertainty importance measure. Reliability Engineering &amp; System Safety, 92(6):771–784, 2007.">123</a>]</span>.</p>
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+  <section id="how-to-choose-a-sensitivity-analysis-method-model-traits-and-dimensionality">
+<h1>How To Choose A Sensitivity Analysis Method: Model Traits And Dimensionality<a class="headerlink" href="#how-to-choose-a-sensitivity-analysis-method-model-traits-and-dimensionality" title="Link to this heading">#</a></h1>
+<p><a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-5"><span class="std std-numref">Fig. 3.5</span></a>, synthesized from variants found in <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>, presents a graphical synthesis of the methods overviewed in this section, with regards to their appropriateness of application based on the complexity of the model at hand and the computational limits on the number of model evaluations afforded. The bars below each method also indicate the sensitivity analysis purposes they are most appropriate to address, which are in turn a reflection of the motivations and research questions the sensitivity analysis is called to address. Computational intensity is measured as a multiple of the number of model factors that are considered uncertain (<span class="math notranslate nohighlight">\(d\)</span>). Increasing model complexity mandates that more advanced sensitivity analysis methods are applied to address potential nonlinearities, factor interactions, and discontinuities. Such methods can only be performed at increasing computational expense. For example, computationally cheap linear regression should not be used to assess factors’ importance if the model cannot be proven linear and the factors independent, because important relationships will invariably be missed (recall the example in <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-5"><span class="std std-numref">Fig. 3.5</span></a>). When computational limits do constrain applications to make simplified assumptions and sensitivity techniques, any conclusions in such cases should be delivered with clear statements of the appropriate caveats.</p>
+<figure class="margin-caption align-center" id="id7">
+<span id="figure-3-5"></span><a class="reference internal image-reference" href="_images/figure3_5classificationofmethods.png"><img alt="Figure 3_5" src="_images/figure3_5classificationofmethods.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Classification of the sensitivity analysis methods overviewed in this section, with regards to their computational cost (horizontal axis), their appropriateness to model complexity (vertical axis), and the purpose they can be used for (colored bars). d: number of uncertain factors considered; ANOVA: Analysis of Variance; FAST: Fourier Amplitude Sensitivity Test; PRIM: Patient Rule Induction Method; CART: Classification and Regression Trees; SRCC: Spearman’s rank correlation coefficient: NSE: Nash–Sutcliffe efficiency; SRC: standardized regression coefficient; PCC: Pearson correlation coefficient. This figure is synthesized from variants found in <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>.</span><a class="headerlink" href="#id7" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The reader should also be aware that the estimates of computational intensity that are given here are indicative of magnitude and would vary depending on the sampling technique, model complexity and the level of information being asked. For example, a Sobol sensitivity analysis typically requires a sample of size <span class="math notranslate nohighlight">\(n * d+2\)</span> to produce first- and total-order indices, where <span class="math notranslate nohighlight">\(d\)</span> is the number of uncertain factors and <span class="math notranslate nohighlight">\(n\)</span> is a scaling factor, selected ad hoc, depending on model complexity <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id25" title="Andrea Saltelli and Stefano Tarantola. On the relative importance of input factors in mathematical models: safety assessment for nuclear waste disposal. Journal of the American Statistical Association, 97(459):702–709, 2002.">46</a>]</span>. The scaling factor <span class="math notranslate nohighlight">\(n\)</span> is typically set to at least 1000, but it should most appropriately be set on the basis of index convergence. In other words, a prudent analyst would perform the analysis several times with increasing <span class="math notranslate nohighlight">\(n\)</span> and observe at what level the indices converge to stable values <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id140" title="Jiri Nossent, Pieter Elsen, and Willy Bauwens. Sobol’ sensitivity analysis of a complex environmental model. Environmental Modelling &amp; Software, 26(12):1515-1525, 2011. URL: https://www.sciencedirect.com/science/article/pii/S1364815211001939, doi:https://doi.org/10.1016/j.envsoft.2011.08.010.">125</a>]</span>. The level should be the minimum sample size used in subsequent sensitivity analyses of the same system. Furthermore, if the analyst would like to better understand the degrees of interaction between factors, requiring second-order indices, the sample size would have to increase to <span class="math notranslate nohighlight">\(n * 2d+2\)</span> <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id25" title="Andrea Saltelli and Stefano Tarantola. On the relative importance of input factors in mathematical models: safety assessment for nuclear waste disposal. Journal of the American Statistical Association, 97(459):702–709, 2002.">46</a>]</span>.</p>
+<p>Another important consideration is that methods that do not require specific sampling schemes can be performed in conjunction with others without requiring additional model evaluations. None of the regression-based methods, for example, require samples of specific structures or sizes, and can be combined with other methods for complementary purposes. For instance, one could complement a Sobol analysis with an application of CART, using the same data, but to address questions relating to factor mapping (e.g., we know factor <span class="math notranslate nohighlight">\(x_i\)</span>  is important for a model output, but we would like to also know which of its values specifically push the output to undesirable states). Lastly, comparing results from different methods performed together can be especially useful in model diagnostic settings. For example, <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id47" title="Antonia Hadjimichael, Julianne Quinn, and Patrick Reed. Advancing Diagnostic Model Evaluation to Better Understand Water Shortage Mechanisms in Institutionally Complex River Basins. Water Resources Research, 56(10):e2020WR028079, 2020. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR028079 (visited on 2020-10-16), doi:10.1029/2020WR028079.">13</a>]</span> used <span class="math notranslate nohighlight">\(\delta\)</span> indices, first-order Sobol indices, and <span class="math notranslate nohighlight">\(R^2\)</span>  values from linear regression, all performed on the same factors, to derive insights about the effects on factors on different moments of the output distribution and about the linearity of their relationship.</p>
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+  <section id="software-toolkits">
+<span id="id1"></span><h1>Software Toolkits<a class="headerlink" href="#software-toolkits" title="Link to this heading">#</a></h1>
+<p>This section presents available open source sensitivity analysis software tools, based on the programming language they use and the methods they support <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-6"><span class="std std-numref">Fig. 3.6</span></a>. Our review covers five widely used programming languages: R, MATLAB, Julia, Python, and C++, as well as one tool that provides a graphical user interface (GUI). Each available SA tool was assessed on the number of SA methods and design of experiments methods it supports. For example, the <em>sensobol</em> package in R only supports the variance-based Sobol method. However, it is the only package we came across that calculates third-order interactions among parameters. On the other side of the spectrum, there are SA software packages that contain several popular SA methods. For example, <em>SALib</em> in Python <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id139" title="Jon Herman and Will Usher. Salib: an open-source python library for sensitivity analysis. Journal of Open Source Software, 2(9):97, 2017.">126</a>]</span> supports seven different SA methods. The <em>DifferentialEquations</em> package is a comprehensive package developed for Julia, and <em>GlobalSensitivityAnalysis</em> is another Julia package that has mostly adapted SALib methods. <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-6"><span class="std std-numref">Fig. 3.6</span></a> also identifies the SA packages that have been updated since 2018, indicating active support and development.</p>
+<figure class="margin-caption align-center" id="id3">
+<span id="figure-3-6"></span><a class="reference internal image-reference" href="_images/figure3_6_softwaretoolkits.png"><img alt="Figure 3_6" src="_images/figure3_6_softwaretoolkits.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Sensitivity analysis packages available in different programming language platforms (R, Python, Julia, MATLAB, and C++), with the number of methods they support. Packages supporting more than five methods are indicated in pink. Packages updated since 2018 are indicated with asterisks.</span><a class="headerlink" href="#id3" title="Link to this image">#</a></p>
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+    <h1>Sensitivity Analysis: The Basics</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#global-versus-local-sensitivity">3.1. Global Versus Local Sensitivity</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#why-perform-sensitivity-analysis">3.2. Why Perform Sensitivity Analysis</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#design-of-experiments">3.3. Design of Experiments</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#one-at-a-time-oat">3.3.1. One-At-a-Time (OAT)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#full-and-fractional-factorial-sampling">3.3.2. Full and Fractional Factorial Sampling</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#latin-hypercube-sampling-lhs">3.3.3. Latin Hypercube Sampling (LHS)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#low-discrepancy-sequences">3.3.4. Low-Discrepancy Sequences</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#other-types-of-sampling">3.3.5. Other types of sampling</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-generation-of-input-time-series">3.3.6. Synthetic generation of input time series</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sensitivity-analysis-methods">3.4. Sensitivity Analysis Methods</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#derivative-based-methods">3.4.1. Derivative-based Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#elementary-effect-methods">3.4.2. Elementary Effect Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#regression-based-methods">3.4.3. Regression-based Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#regional-sensitivity-analysis">3.4.4. Regional Sensitivity Analysis</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#variance-based-methods">3.4.5. Variance-based Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#analysis-of-variance-anova">3.4.6. Analysis of Variance (ANOVA)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#moment-independent-density-based-methods">3.4.7. Moment-Independent (Density-Based) Methods</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#how-to-choose-a-sensitivity-analysis-method-model-traits-and-dimensionality">3.5. How To Choose A Sensitivity Analysis Method: Model Traits And Dimensionality</a></li>
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+                <article class="bd-article">
+                  
+  <section id="sensitivity-analysis-the-basics">
+<span id="id1"></span><h1><span class="section-number">3. </span>Sensitivity Analysis: The Basics<a class="headerlink" href="#sensitivity-analysis-the-basics" title="Link to this heading">#</a></h1>
+<section id="global-versus-local-sensitivity">
+<span id="global-vs-local"></span><h2><span class="section-number">3.1. </span>Global Versus Local Sensitivity<a class="headerlink" href="#global-versus-local-sensitivity" title="Link to this heading">#</a></h2>
+<p>Out of the several definitions for sensitivity analysis presented in the literature, the most widely used has been proposed by <span id="id2">Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span> as “the study of how uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input”. In other words, sensitivity analysis explores the relationship between the model’s <span class="math notranslate nohighlight">\(N\)</span> input variables, <span class="math notranslate nohighlight">\(x=[x_1,x_2,...,x_N]\)</span>, and <span class="math notranslate nohighlight">\(M\)</span> output variables, <span class="math notranslate nohighlight">\(y=[y_1,y_2,...,y_M]\)</span> with <span class="math notranslate nohighlight">\(y=g(x)\)</span>, where <span class="math notranslate nohighlight">\(g\)</span> is the model that maps the model inputs to the outputs <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>]</span>.</p>
+<p>Historically, there have been two broad categories of sensitivity analysis techniques: local and global. Local sensitivity analysis is performed by varying model parameters around specific reference values, with the goal of exploring how small input perturbations influence model performance. Due to its ease-of-use and limited computational demands, this approach has been widely used in literature, but has important limitations <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id18" title="O Rakovec, Mary C Hill, MP Clark, AH Weerts, AJ Teuling, and R Uijlenhoet. Distributed evaluation of local sensitivity analysis (delsa), with application to hydrologic models. Water Resources Research, 50(1):409–426, 2014.">40</a>, <a class="reference internal" href="R.Bibliography.html#id19" title="Andrea Saltelli and Paola Annoni. How to avoid a perfunctory sensitivity analysis. Environmental Modelling &amp; Software, 25(12):1508–1517, 2010.">41</a>]</span>. If the model is not linear, the results of local sensitivity analysis can be heavily biased, as they are strongly influenced by independence assumptions and a limited exploration of model inputs (e.g., <span id="id5">Tang <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id20" title="Yong Tang, Patrick Reed, Thibaut Wagener, and K van Werkhoven. Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation. Hydrology and Earth System Sciences, 11(2):793–817, 2007.">42</a>]</span>). If the model’s factors interact, local sensitivity analysis will underestimate their importance, as it does not account for those effects (e.g., <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id23" title="Nicholas AS Hamm, Jim W Hall, and MG Anderson. Variance-based sensitivity analysis of the probability of hydrologically induced slope instability. Computers &amp; geosciences, 32(6):803–817, 2006.">43</a>]</span>). In general, as local sensitivity analysis only partially and locally explores a model’s parametric space, it is not considered a valid approach for nonlinear models <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id24" title="Andrea Saltelli, Ksenia Aleksankina, William Becker, Pamela Fennell, Federico Ferretti, Niels Holst, Sushan Li, and Qiongli Wu. Why so many published sensitivity analyses are false: a systematic review of sensitivity analysis practices. Environmental modelling &amp; software, 114:29–39, 2019.">44</a>]</span>. This is illustrated in <a class="reference internal" href="#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (a-b), presenting contour plots of a model response (<span class="math notranslate nohighlight">\(y\)</span>) with an additive linear model (a) and with a nonlinear model (b). In a linear model without interactions between the input terms <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span>, local sensitivity analysis (assuming deviations from some reference values) can produce appropriate sensitivity indices (<a class="reference internal" href="#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (a)). If however, factors <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> interact, the local and partial consideration of the space can not properly account for each factor’s effects on the model response (<a class="reference internal" href="#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (b)), as it is only informative at the reference value where it is applied. In contrast, a global sensitivity analysis varies uncertain factors within the entire feasible space of variable model responses (<a class="reference internal" href="#figure-3-1"><span class="std std-numref">Fig. 3.1</span></a> (c)). This approach reveals the global effects of each parameter on the model output, including any interactive effects. For models that cannot be proven linear, global sensitivity analysis is preferred and this text is primarily discussing global sensitivity analysis methods. In the text that follows, whenever we use the term sensitivity analysis we are referring to its global application.</p>
+<figure class="margin-caption align-center" id="id94">
+<span id="figure-3-1"></span><a class="reference internal image-reference" href="_images/figure3_1_global_versus_local.png"><img alt="Figure 3.1" src="_images/figure3_1_global_versus_local.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 3.1 </span><span class="caption-text">Treatment of a two-dimensional space of variability by local (panels a-b) and global (panel c) sensitivity analyses. Panels depict contour plots with the value of a model response (<span class="math notranslate nohighlight">\(y\)</span>) changing with changes in the values of input terms <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span>. Local sensitivity analysis is only an appropriate approach to sensitivity in the case of linear models without interactions between terms, for example in panel (a), where <span class="math notranslate nohighlight">\(y=3x_1+5x_2\)</span>. In the case of more complex models, for example in panels (b-c), where <span class="math notranslate nohighlight">\(y={1 \above 1pt e^{x^2_1+x^2_2}} + {50 \above 1pt e^{(0.1x_1)^2+(0.1x_2)^3}}\)</span>, local sensitivity will miscalculate sensitivity indices as the assessed changes in the value <span class="math notranslate nohighlight">\(y\)</span> depend on the assumed base values chose for <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> (panel (b)). In these cases, global sensitivity methods should be used instead (panel (c)). The points in panel (c) are generated using a uniform random sample of <span class="math notranslate nohighlight">\(n=50\)</span>, but many other methods are available.</span><a class="headerlink" href="#id94" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="why-perform-sensitivity-analysis">
+<span id="why-sa"></span><h2><span class="section-number">3.2. </span>Why Perform Sensitivity Analysis<a class="headerlink" href="#why-perform-sensitivity-analysis" title="Link to this heading">#</a></h2>
+<p>It is important to understand the many ways in which a SA might be of use to your modeling effort. Most commonly, one might be motivated to perform sensitivity analysis for the following reasons:</p>
+<p><em>Model evaluation</em>: Sensitivity analysis can be used to gauge model inferences when assumptions about the structure of the model or its parameterization are dubious or have changed. For instance, consider a numerical model that uses a set of calibrated parameter values to produce outputs, which we then use to inform decisions about the real-world system represented. One might like to know if small changes in these parameter values significantly change this model’s output and the decisions it informs or if, instead, our parameter inferences yield stable model behavior regardless of the uncertainty present in the specific parameterized processes or properties. This can either discredit or lend credence to the model at hand, as well as any inferences drawn that are founded on its accurate representation of the system. Sensitivity analysis can identify which uncertain model factors cause this undesirable model behavior.</p>
+<p><em>Model simplification</em>: Sensitivity analysis can also be used to identify factors or components of the model that appear to have limited effects on direct outputs or metrics of interest. Consider a model that has been developed in an organization for the purposes of a specific research question and is later used in the context of a different application. Some processes represented in significant detail might no longer be of the same importance while consuming significant data or computational resources, as different outputs might be pertinent to the new application. Sensitivity analysis can be used to identify unimportant model components and simplify them to nominal values and reduced model forms. Model complexity and computational costs can therefore be reduced.</p>
+<p><em>Model refinement</em>: Alternatively, sensitivity analysis can reveal the factors or processes that are highly influential to the outputs or metrics of interest, by assessing their relative importance. In the context of model evaluation, this can inform which model components warrant additional investigation or measurement so the uncertainty surrounding them and the resulting model outputs or metrics of interest can be reduced.</p>
+<p><em>Exploratory modeling</em>: When sufficient credence has been established in the model, sensitivity analysis can be applied to a host of other inquiries. Inferences about the factors and processes that most (or least) control a model’s outputs of interest can be extrapolated to the real system they represent and be used in a heuristic manner to inform model-based inferences. On this foundation, a model paired with the advanced techniques presented in this text can be used to “discover” decision relevant and highly consequential outcomes (i.e., scenario discovery, discussed in more detail in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a> <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>, <a class="reference internal" href="R.Bibliography.html#id16" title="Benjamin P Bryant and Robert J Lempert. Thinking inside the box: a participatory, computer-assisted approach to scenario discovery. Technological Forecasting and Social Change, 77(1):34–49, 2010.">45</a>]</span>).</p>
+<p>The nature and context of the model shapes the specific objectives of applying a sensitivity analysis, as well as methods and tools most appropriate and defensible for each application setting <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id203" title="Thorsten Wagener and Francesca Pianosi. What has Global Sensitivity Analysis ever done for us? A systematic review to support scientific advancement and to inform policy-making in earth system modelling. Earth-Science Reviews, 194:1–18, July 2019. URL: https://www.sciencedirect.com/science/article/pii/S0012825218300990 (visited on 2021-08-30), doi:10.1016/j.earscirev.2019.04.006.">35</a>, <a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id25" title="Andrea Saltelli and Stefano Tarantola. On the relative importance of input factors in mathematical models: safety assessment for nuclear waste disposal. Journal of the American Statistical Association, 97(459):702–709, 2002.">46</a>]</span>. The three most common sensitivity analysis modes (<em>Factor Prioritization</em>, <em>Factor Fixing</em>, and <em>Factor Mapping</em>) are presented below, but the reader should be aware that other uses have been proposed in the literature (e.g., <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id26" title="Barry Anderson, Emanuele Borgonovo, Marzio Galeotti, and Roberto Roson. Uncertainty in climate change modeling: can global sensitivity analysis be of help? Risk analysis, 34(2):271–293, 2014.">47</a>, <a class="reference internal" href="R.Bibliography.html#id27" title="Emanuele Borgonovo. Sensitivity analysis with finite changes: an application to modified eoq models. European Journal of Operational Research, 200(1):127–138, 2010.">48</a>]</span>).</p>
+<p><em>Factor prioritization</em>: This sensitivity analysis application mode (also referred to as <em>factor ranking</em>) refers to when one would like to identify the uncertain factors that have the greatest impact on the variability of the output, and which, when fixed to their true value (i.e., if there were no uncertainty regarding their value), would lead to the greatest reduction in output variability <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. Information from this type of analysis can be crucial to model improvement as these factors can become the focus of future measurement campaigns or numerical experiments so that uncertainty in the model output can be reduced. The impact of each uncertain input on the variance of the model output is often used as the criterion for factor prioritization. <a class="reference internal" href="#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (a) shows the effects of three uncertain variables (<span class="math notranslate nohighlight">\(X_1\)</span>, <span class="math notranslate nohighlight">\(X_2\)</span>, and <span class="math notranslate nohighlight">\(X_3\)</span>) on the variance of output <span class="math notranslate nohighlight">\(Y\)</span>. <span class="math notranslate nohighlight">\(V(E(Y|X_i))\)</span> indicates the variance in <span class="math notranslate nohighlight">\(Y\)</span> if factor <span class="math notranslate nohighlight">\(X_i\)</span> is left to vary freely while all other factors remain fixed to nominal values. In this case, factor <span class="math notranslate nohighlight">\(X_2\)</span> makes the largest contribution to the variability of output <span class="math notranslate nohighlight">\(Y\)</span> and it should therefore be prioritized. In the context of risk analysis, factor prioritization can be used to reduce output variance to below a given tolerable threshold (also known as variance cutting).</p>
+<p><em>Factor fixing</em>: This mode of sensitivity analysis (also referred to as <em>factor screening</em>) aims to identify the model components that have a negligible effect or make no significant contributions to the variability of the outputs or metrics of interest (usually referred to as non-influential <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>). In the stylized example of <a class="reference internal" href="#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (a), <span class="math notranslate nohighlight">\(X_1\)</span> makes the smallest contribution to the variability of output <span class="math notranslate nohighlight">\(Y\)</span> suggesting that the uncertainty in its value could be negligible and the factor itself fixed in subsequent model executions. Eliminating these factors or processes in the model or fixing them to a nominal value can help reduce model complexity as well as the unnecessary computational burden of subsequent model runs, results processing, or other sensitivity analyses (the fewer uncertain factors considered, the fewer runs are necessary to illuminate their effects on the output). Significance of the outcome can be gauged in a variety of manners, depending on the application. For instance, if applying a variance-based method, a minimum threshold value of contribution to the variance could be considered as a significance ‘cutoff’, and factors with indices below that value can be considered non-influential. Conclusions about factor fixing should be made carefully, considering all of the effects a factor has, individually and in interaction with other factors (explained in more detail in the <a class="reference internal" href="#variance-based-methods"><span class="std std-numref">Chapter 3.4.5</span></a>).</p>
+<p><em>Factor mapping</em>: Finally, factor mapping can be used to pinpoint which values of uncertain factors lead to model outputs within a given range of the output space <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. In the context of model diagnostics, it is possible that the model’s output changes in ways considered impossible based on the represented processes, or other observed evidence. In this situation, factor mapping can be used to identify which uncertain model factors cause this undesirable model behavior by ‘filtering’ model runs that are considered ‘non-behavioral’ <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id29" title="Neil R Edwards, David Cameron, and Jonathan Rougier. Precalibrating an intermediate complexity climate model. Climate dynamics, 37(7):1469–1482, 2011.">50</a>, <a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id31" title="RC Spear and GM Hornberger. Eutrophication in peel inlet—ii. identification of critical uncertainties via generalized sensitivity analysis. Water research, 14(1):43–49, 1980.">52</a>]</span>. In <a class="reference internal" href="#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (b), region <span class="math notranslate nohighlight">\(B\)</span> of the output space <span class="math notranslate nohighlight">\(Y\)</span> denotes the set of behavioral model outcomes and region <span class="math notranslate nohighlight">\(\bar{B}\)</span> denotes the set of non-behavioral outcomes, resulting from the entirety of input space <span class="math notranslate nohighlight">\(X\)</span>. Factor mapping refers to the process of tracing which factor values of input space <span class="math notranslate nohighlight">\(X\)</span> produce the behavioral model outcomes in the output space.</p>
+<figure class="margin-caption align-center" id="id95">
+<span id="figure-3-2"></span><a class="reference internal image-reference" href="_images/figure3_2_factor_mapping.png"><img alt="Figure 3.2" src="_images/figure3_2_factor_mapping.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 3.2 </span><span class="caption-text">Factor prioritization, factor fixing and factor mapping settings of sensitivity analysis.</span><a class="headerlink" href="#id95" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The language used above reflects a use of sensitivity analysis for model fidelity evaluation and refinement. However, as previously mentioned, when a model has been established as a sufficiently accurate representation of the system, sensitivity analysis can produce additional inferences (i.e., exploratory modeling and scenario discovery). For instance, under the factor mapping use, the analyst can now focus on undesirable system states and discover which factors are most responsible for them: for instance, “population growth of above 25% would be responsible for unacceptably high energy demands”. Factor prioritization and factor fixing can be used to make equivalent inferences, such as “growing populations and increasing temperatures are the leading factors for changing energy demands” (prioritizing of factors) or “changing dietary needs are inconsequential to increasing energy demands for this region” (a factor that can be fixed in subsequent model runs). All these inferences hinge on the assumption that the real system’s stakeholders consider the model states faithful enough representations of system states. As elaborated in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>, this view on sensitivity analysis is founded on a relativist perspective on modeling, which tends to place more value on model usefulness rather than strict accuracy of representation in terms of error. As such, sensitivity analysis performed with decision-making relevance in mind will focus on model outputs or metrics that are consequential and decision relevant (e.g., energy demand in the examples above).</p>
+</section>
+<section id="design-of-experiments">
+<span id="id15"></span><h2><span class="section-number">3.3. </span>Design of Experiments<a class="headerlink" href="#design-of-experiments" title="Link to this heading">#</a></h2>
+<p>Before conducting a sensitivity analysis, the first element that needs to be clarified is the uncertainty space of the model <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id111" title="Jon C Helton, Jay Dean Johnson, Cedric J Sallaberry, and Curt B Storlie. Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliability Engineering &amp; System Safety, 91(10-11):1175–1209, 2006.">53</a>]</span>. In other words, how many and which factors making up the mathematical model are considered uncertain and can potentially affect the model output and the inferences drawn from it. Uncertain factors can be model parameters, model structures, inputs, or alternative model resolution levels (scales), all of which can be assessed through the tools presented in this text. Depending on the kind of factor, its variability can be elicited through various means: expert opinion, values reported in the literature, historical observations, its physical meaning (e.g., population values in a city can never be negative), or through the use of more formal UQ methods (<a class="reference internal" href="A1_Uncertainty_Quantification.html#a1-uncertainty-quantification"><span class="std std-numref">Chapter A</span></a>). The model uncertainty space represents the entire space of variability present in each of the uncertain factors of a model. The complexity of most real-world models means that the response function, <span class="math notranslate nohighlight">\(y=g(x)\)</span>, mapping inputs to outputs, is hardly ever available in an analytical form and therefore analytically computing the sensitivity of the output to each uncertain factor becomes impossible. In these cases, sensitivity analysis is only feasible through numerical procedures that employ different strategies to sample the uncertainty space and calculate sensitivity indices.</p>
+<p>A sampling strategy is often referred to as a <em>design of experiments</em> and represents a methodological choice made before conducting any sensitivity analysis. Experimental design was first introduced by <span id="id17">Fisher [<a class="reference internal" href="R.Bibliography.html#id112" title="Ronald Aylmer Fisher. Design of experiments. Br Med J, 1(3923):554–554, 1936.">54</a>]</span> in the context of laboratory or field-based experiments. Its application in sensitivity analysis is similar to setting up a physical experiment in that it is used to discover the behavior of a system under specific conditions. An ideal design of experiments should provide a framework for the extraction of all plausible information about the impact of each factor on the output of the model. The design of experiments is used to set up a simulation platform with the minimum computational cost to answer specific questions that cannot be readily drawn from the data through analytical or common data mining techniques. Models representing coupled human-natural systems usually have a large number of inputs, state variables and parameters, but not all of them exert fundamental control over the numerical process, despite their uncertainty, nor have substantial impacts on the model output, either independently or through their interactions. Each factor influences the model output in different ways that need to be discovered. For example, the influence of a parameter on model output can be linear or non-linear and can be continuous or only be active during specific times or at particular states of the system <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id53" title="JD Herman, PM Reed, and T Wagener. Time-varying sensitivity analysis clarifies the effects of watershed model formulation on model behavior. Water Resources Research, 49(3):1400–1414, 2013.">55</a>, <a class="reference internal" href="R.Bibliography.html#id113" title="Carolina Massmann, Thorsten Wagener, and Hubert Holzmann. A new approach to visualizing time-varying sensitivity indices for environmental model diagnostics across evaluation time-scales. Environmental modelling &amp; software, 51:190–194, 2014.">56</a>]</span>. An effective and efficient design of experiments allows the analyst to explore these complex relationships and evaluate different behaviors of the model for various scientific questions <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id114" title="An Van Schepdael, Aurélie Carlier, and Liesbet Geris. Sensitivity analysis by design of experiments. In Uncertainty in Biology, pages 327–366. Springer, 2016.">57</a>]</span>. The rest of this section overviews some of the most commonly used designs of experiments. Table 1 summarizes the designs discussed.</p>
+<div class="pst-scrollable-table-container"><table class="table" id="id96">
+<caption><span class="caption-number">Table 3.1 </span><span class="caption-text">Summary of designs of experiments overviewed in this section. * Depends on the sample size.</span><a class="headerlink" href="#id96" title="Link to this table">#</a></caption>
+<thead>
+<tr class="row-odd"><th class="head"><p><em>Design of experiments</em></p></th>
+<th class="head"><p><em>Factor interactions considered</em></p></th>
+<th class="head"><p><em>Treatment of factor domains</em></p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>One-At-a-Time (OAT)</p></td>
+<td><p>No - main effects only</p></td>
+<td><p>Continuous (distributions)</p></td>
+</tr>
+<tr class="row-odd"><td><p>Full Factorial Sampling</p></td>
+<td><p>Yes - including total effects</p></td>
+<td><p>Discrete (levels)</p></td>
+</tr>
+<tr class="row-even"><td><p>Fractional Factorial Sampling</p></td>
+<td><p>Yes - only lower-order effects*</p></td>
+<td><p>Discrete (levels)</p></td>
+</tr>
+<tr class="row-odd"><td><p>Latin Hypercube (LH) Sampling</p></td>
+<td><p>Yes - including total effects*</p></td>
+<td><p>Continuous (distributions)</p></td>
+</tr>
+<tr class="row-even"><td><p>Quasi-Random Sampling with Low-Discrepancy Sequences</p></td>
+<td><p>Yes - including total effects*</p></td>
+<td><p>Continuous (distributions)</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>There are a few different approaches to the design of experiments, closely related to the chosen sensitivity analysis approach, which is in turn shaped by the research motivations, scientific questions, and computational constraints at hand (additional discussion of this can be found at the end of <a class="reference internal" href="#sensitivity-analysis-the-basics"><span class="std std-numref">Chapter 3</span></a>). For example, in a sensitivity analysis using perturbation and derivatives methods, the model input parameters vary from their nominal values one at a time, something that the design of experiments needs to reflect. If, instead, one were to perform sensitivity analysis using a multiple-starts perturbation method, the design of experiments needs to consider that multiple points across the factor space are used. The design of experiments specifically defines two key characteristics of samples that are fed to the numerical model: the number of samples and the range of each factor.</p>
+<p>Generally, sampling can be performed randomly or by applying a stratifying approach. In random sampling, such as Monte Carlo <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id115" title="Nicholas Metropolis and Stanislaw Ulam. The monte carlo method. Journal of the American statistical association, 44(247):335–341, 1949.">58</a>]</span>, samples are randomly generated by a pseudo-random number generator with an a-priori assumption about the distribution of parameters and their possible ranges. Random seeds can also be used to ensure consistency and higher control over the random process. However, this method could leave some gaps in the parameter space and cause clustering in some spaces, especially for a large number of parameters <span id="id21">[<a class="reference internal" href="R.Bibliography.html#id116" title="John Norton. An introduction to sensitivity assessment of simulation models. Environmental Modelling &amp; Software, 69:166–174, 2015.">59</a>]</span>. Most sampling strategies use stratified sampling to mitigate these disadvantages. Stratified sampling techniques divide the domain of each factor into subintervals, often of equal lengths. From each subinterval, an equal number of samples is drawn randomly, or based on the specific locations within the subintervals <span id="id22">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>.</p>
+<section id="one-at-a-time-oat">
+<h3><span class="section-number">3.3.1. </span>One-At-a-Time (OAT)<a class="headerlink" href="#one-at-a-time-oat" title="Link to this heading">#</a></h3>
+<p>In this approach, only one model factor is changed at a time while all others are kept fixed across each iteration in a sampling sequence. The OAT method assumes that model factors of focus are linearly independent (i.e., there are no interactions) and can analyze how factors individually influence model outputs or metrics of interest. While popular given its ease of implementation, OAT  is ultimately limited in its exploration of a model’s sensitivities <span id="id23">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. It is primarily used with local sensitivity techniques with similar criticisms: applying this sampling scheme on a system with nonlinear and interactive processes will miss important information on the effect uncertain factors have on the model. OAT samplings can be repeated multiple times in a more sophisticated manner and across different locations of the parameter space to overcome some of these challenges, which would increase computational costs and negate the main reasons for its selection. Given these limitations OAT methods could be used as preliminary, low-cost analyses of the factors’ individual effects, but should ultimately be complemented with more sophisticated methods.</p>
+</section>
+<section id="full-and-fractional-factorial-sampling">
+<h3><span class="section-number">3.3.2. </span>Full and Fractional Factorial Sampling<a class="headerlink" href="#full-and-fractional-factorial-sampling" title="Link to this heading">#</a></h3>
+<p>In full factorial sampling each factor is treated as being discrete by considering two or more levels (or intervals) of its values. The sampling process then generates samples within each possible combination of levels, corresponding to each parameter. This scheme produces a more comprehensive sampling of the factors’ variability space, as it accounts for all candidate combinations of factor levels (<a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a)). If the number of levels is the same across all factors, the number of generated samples is estimated using <span class="math notranslate nohighlight">\(n^k\)</span>, where <span class="math notranslate nohighlight">\(n\)</span> is the number of levels and <span class="math notranslate nohighlight">\(k\)</span> is the number of factors. For example, <a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a) presents a full factorial sampling of three uncertain factors <span class="math notranslate nohighlight">\((x_1,\)</span> <span class="math notranslate nohighlight">\(x_2,\)</span> and <span class="math notranslate nohighlight">\(x_3)\)</span>, each considered as having four discrete levels. The total number of samples necessary for such an experiment is <span class="math notranslate nohighlight">\(4^3=64\)</span>. As the number of factors increases, the number of simulations necessary will also grow exponentially, making full factorial sampling computationally burdensome (<a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (b)). As a result, it is common in the literature to apply full factorial sampling at only two levels per factor, typically the two extremes <span id="id24">[<a class="reference internal" href="R.Bibliography.html#id32" title="Douglas C Montgomery. Design and analysis of experiments. John wiley &amp; sons, 2017.">60</a>]</span>. This significantly reduces computational burden but is only considered appropriate in cases where factors can indeed only assume two discrete values (e.g., when testing the effects of epistemic uncertainty and comparing between model structure A and model structure B). In the case of physical parameters on continuous distributions (e.g., when considering the effects of measurement uncertainty in a temperature sensor), discretizing the range of a factor to only extreme levels can bias its estimated importance.</p>
+<p>Fractional factorial sampling is a widely used alternative to full factorial sampling that allows the analyst to significantly reduce the number of simulations by focusing on the main effects of a factor and seeking to avoid model runs that yield redundant response information <span id="id25">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. In other words, if one can reasonably assume that higher-order interactions are negligible, information about the most significant effects and lower-order interactions (e.g., effects from pairs of factors) can be obtained using a fraction of the full factorial design. Traditionally, fractional factorial design has also been limited to two levels <span id="id26">[<a class="reference internal" href="R.Bibliography.html#id32" title="Douglas C Montgomery. Design and analysis of experiments. John wiley &amp; sons, 2017.">60</a>]</span>, referred to as Fractional Factorial designs 2k-p <span id="id27">[<a class="reference internal" href="R.Bibliography.html#id34" title="George EP Box and J Stuart Hunter. The 2 k—p fractional factorial designs. Technometrics, 3(3):311–351, 1961.">61</a>]</span>. Recently, Generalized Fractional Factorial designs have also been proposed that allow for the structured generation of samples at more than two levels per factor <span id="id28">[<a class="reference internal" href="R.Bibliography.html#id33" title="Izabella Surowiec, Ludvig Vikstrom, Gustaf Hector, Erik Johansson, Conny Vikstrom, and Johan Trygg. Generalized subset designs in analytical chemistry. Analytical chemistry, 89(12):6491–6497, 2017.">62</a>]</span>. Consider a case where the modeling team dealing with the problem in <a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (a) cannot afford to perform 64 simulations of their model. They can afford 32 runs for their experiment and instead decide to fractionally sample the variability space of their factors. A potential design of such a sampling strategy is presented in <a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (c).</p>
+<figure class="margin-caption align-center" id="id97">
+<span id="figure-3-3"></span><a class="reference internal image-reference" href="_images/figure3_3_alternative_designs.png"><img alt="Figure 3.3" src="_images/figure3_3_alternative_designs.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 3.3 </span><span class="caption-text">Alternative designs of experiments and their computational costs for three uncertain factors <span class="math notranslate nohighlight">\((x_1,\)</span> <span class="math notranslate nohighlight">\(x_2,\)</span> and <span class="math notranslate nohighlight">\(x_3)\)</span>. (a) Full factorial design sampling of three factors at four levels, at a total of 64 samples; (b) exponential growth of necessary number of samples when applying full factorial design at four levels; (c) fractional factorial design of three factors at four levels, at a total of 32 samples; and (d) Latin Hypercube sample of three factors with uniform distributions, at a total of 32 samples.</span><a class="headerlink" href="#id97" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="latin-hypercube-sampling-lhs">
+<h3><span class="section-number">3.3.3. </span>Latin Hypercube Sampling (LHS)<a class="headerlink" href="#latin-hypercube-sampling-lhs" title="Link to this heading">#</a></h3>
+<p>Latin hypercube sampling (LHS) <span id="id29">[<a class="reference internal" href="R.Bibliography.html#id91" title="Michael D McKay, Richard J Beckman, and William J Conover. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21(1):239-2451, 1979.">63</a>]</span> is one of the most common methods in space-filling experimental designs. With this sampling technique, for <span class="math notranslate nohighlight">\(N\)</span> uncertain factors, an <span class="math notranslate nohighlight">\(N\)</span>-dimensional hypercube is generated, with each factor divided into an equal number of levels depending on the total number of samples to be generated. Equal numbers of samples are then randomly generated at each level, across all factors. In this manner, latin hypercube design guarantees sampling from every level of the variability space and without any overlaps. When the number of samples generated is much larger than the number of uncertain factors, LHS can be very effective in examining the effects of each factor <span id="id30">[<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>]</span>. LHS is an attractive technique, because it guarantees a diverse coverage of the space, through the use of subintervals, without being constrained to discrete levels for each factor - compare <a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (c) with <a class="reference internal" href="#figure-3-3"><span class="std std-numref">Fig. 3.3</span></a> (d) for the same number of samples.</p>
+<p>LHS is less effective when the number of samples is not much larger than the number of uncertain factors, and the effects of each factor cannot be appropriately distinguished. The samples between factors can also be highly correlated, biasing any subsequent sensitivity analysis results. To address this, the sampling scheme can be modified to control for the correlation in parameters while maximizing the information derived. An example of such modification is through the use of orthogonal arrays <span id="id31">[<a class="reference internal" href="R.Bibliography.html#id21" title="Boxin Tang. Orthogonal array-based latin hypercubes. Journal of the American statistical association, 88(424):1392–1397, 1993.">64</a>]</span>.</p>
+</section>
+<section id="low-discrepancy-sequences">
+<h3><span class="section-number">3.3.4. </span>Low-Discrepancy Sequences<a class="headerlink" href="#low-discrepancy-sequences" title="Link to this heading">#</a></h3>
+<p>Low-discrepancy sequences is another sampling technique that employs a pseudo-random generator for Monte Carlo sampling <span id="id32">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>, <a class="reference internal" href="R.Bibliography.html#id92" title="SK Zaremba. The mathematical basis of monte carlo and quasi-monte carlo methods. SIAM review, 10(3):303–314, 1968.">66</a>]</span>. These quasi-Monte Carlo methods eliminate ‘lumpiness’ across samples (i.e, the presence of gaps and clusters) by minimizing discrepancy across the hypercube samples. Discrepancy can be quantitatively measured using the deviations of sampled points from a uniform distribution <span id="id33">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>, <a class="reference internal" href="R.Bibliography.html#id95" title="Sergei Kucherenko, Daniel Albrecht, and Andrea Saltelli. Exploring multi-dimensional spaces: a comparison of latin hypercube and quasi monte carlo sampling techniques. arXiv preprint arXiv:1505.02350, 2015.">67</a>]</span>. Low-discrepancy sequences ensure that the number of samples in any subspace of the variability hypercube is approximately the same. This is not something guaranteed by Latin Hypercube sampling, and even though its design can be improved through optimization with various criteria, such adjustments are limited to small sample sizes and low dimensions <span id="id34">[<a class="reference internal" href="R.Bibliography.html#id95" title="Sergei Kucherenko, Daniel Albrecht, and Andrea Saltelli. Exploring multi-dimensional spaces: a comparison of latin hypercube and quasi monte carlo sampling techniques. arXiv preprint arXiv:1505.02350, 2015.">67</a>, <a class="reference internal" href="R.Bibliography.html#id93" title="Bertrand Iooss, Loïc Boussouf, Vincent Feuillard, and Amandine Marrel. Numerical studies of the metamodel fitting and validation processes. arXiv preprint arXiv:1001.1049, 2010.">68</a>, <a class="reference internal" href="R.Bibliography.html#id94" title="Ruichen Jin, Wei Chen, and Agus Sudjianto. An efficient algorithm for constructing optimal design of computer experiments. In International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, volume 37009, 545–554. 2003.">69</a>, <a class="reference internal" href="R.Bibliography.html#id96" title="Max D Morris and Toby J Mitchell. Exploratory designs for computational experiments. Journal of statistical planning and inference, 43(3):381–402, 1995.">70</a>, <a class="reference internal" href="R.Bibliography.html#id97" title="Jeong-Soo Park. Optimal latin-hypercube designs for computer experiments. Journal of statistical planning and inference, 39(1):95–111, 1994.">71</a>]</span>. In contrast, the Sobol sequence <span id="id35">[<a class="reference internal" href="R.Bibliography.html#id98" title="Ilya M Sobol. Uniformly distributed sequences with an additional uniform property. USSR Computational Mathematics and Mathematical Physics, 16(5):236–242, 1976.">72</a>, <a class="reference internal" href="R.Bibliography.html#id99" title="Il'ya Meerovich Sobol'. On the distribution of points in a cube and the approximate evaluation of integrals. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 7(4):784–802, 1967.">73</a>]</span>, one of the most widely used sampling techniques, utilizes the low-discrepancy approach to uniformly fill the sampled factor space. A core advantage of this style of sampling is that it takes far fewer samples (i.e., simulations) to attain a much lower level of error in estimating model output statistics (e.g., the mean and variance of outputs).</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial which uses Sobol sequence sampling for the purposes of a Sobol sensitivity analysis:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA using SALib Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="other-types-of-sampling">
+<h3><span class="section-number">3.3.5. </span>Other types of sampling<a class="headerlink" href="#other-types-of-sampling" title="Link to this heading">#</a></h3>
+<p>The sampling techniques mentioned so far are general sampling methods useful for a variety of applications beyond sensitivity analysis. There are however techniques that have been developed for specific sensitivity analysis methods. Examples of these methods include the Morris One-At-a-Time <span id="id36">[<a class="reference internal" href="R.Bibliography.html#id132" title="Max D Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.">74</a>]</span>, Fourier Amplitude Sensitivity Test (FAST; <span id="id37">[<a class="reference internal" href="R.Bibliography.html#id100" title="RI Cukier, CM Fortuin, Kurt E Shuler, AG Petschek, and JH Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. i theory. The Journal of chemical physics, 59(8):3873–3878, 1973.">75</a>]</span>), Extended FAST <span id="id38">[<a class="reference internal" href="R.Bibliography.html#id101" title="Andrea Saltelli, Stefano Tarantola, and KP-S Chan. A quantitative model-independent method for global sensitivity analysis of model output. Technometrics, 41(1):39–56, 1999.">76</a>]</span>, and Extended Sobol methods <span id="id39">[<a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>]</span>. For example, the Morris sampling strategy builds a number of trajectories (usually referred to as repetitions and denoted by <span class="math notranslate nohighlight">\(r\)</span>) in the input space each composed of <span class="math notranslate nohighlight">\(N+1\)</span> factor points, where <span class="math notranslate nohighlight">\(N\)</span> is the number of uncertain factors. The first point of the trajectory is selected randomly and the subsequent <span class="math notranslate nohighlight">\(N\)</span> points are generated by moving one factor at a time by a fixed amount. Each factor is perturbed once along the trajectory, while the starting points of all of the trajectories are randomly and uniformly distributed. Several variations of this strategy also exist in the literature; for more details on each approach and their differences the reader is directed to <span id="id40">Pianosi <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>.</p>
+</section>
+<section id="synthetic-generation-of-input-time-series">
+<h3><span class="section-number">3.3.6. </span>Synthetic generation of input time series<a class="headerlink" href="#synthetic-generation-of-input-time-series" title="Link to this heading">#</a></h3>
+<p>Models often have input time series or processes with strong temporal and/or spatial correlations (e.g., streamflow, energy demand, pricing of commodities, etc.) that, while they might not immediately come to mind as factors to be examined in sensitivity analysis, can be treated as such. Synthetic input time series are used for a variety of reasons, for example, when observations are not available or are limited, or when past observations are not considered sufficiently representative to capture rare or extreme events of interest <span id="id41">[<a class="reference internal" href="R.Bibliography.html#id103" title="Jonathan D Herman, Harrison B Zeff, Jonathan R Lamontagne, Patrick M Reed, and Gregory W Characklis. Synthetic drought scenario generation to support bottom-up water supply vulnerability assessments. Journal of Water Resources Planning and Management, 142(11):04016050, 2016.">78</a>, <a class="reference internal" href="R.Bibliography.html#id104" title="PCD Milly, Julio Betancourt, Malin Falkenmark, Robert M Hirsch, Zbigniew W Kundzewicz, Dennis P Lettenmaier, and Ronald J Stouffer. Stationarity is dead: whither water management? Earth, 4:20, 2008.">79</a>]</span>. Synthetic generation of input time series provides a valuable tool to consider non-stationarity and incorporate potential stressors, such as climate change impacts into input time series <span id="id42">[<a class="reference internal" href="R.Bibliography.html#id105" title="Edoardo Borgomeo, Christopher L Farmer, and Jim W Hall. Numerical rivers: a synthetic streamflow generator for water resources vulnerability assessments. Water Resources Research, 51(7):5382–5405, 2015.">80</a>]</span>. For example, a century of record will be insufficient to capture very high impact rare extreme events (e.g., persistent multi-year droughts). A large body of statistical literature exists focusing on the topics of synthetic weather <span id="id43">[<a class="reference internal" href="R.Bibliography.html#id74" title="Manuel Herrera, Sukumar Natarajan, David A Coley, Tristan Kershaw, Alfonso P Ramallo-González, Matthew Eames, Daniel Fosas, and Michael Wood. A review of current and future weather data for building simulation. Building Services Engineering Research and Technology, 38(5):602–627, 2017.">81</a>, <a class="reference internal" href="R.Bibliography.html#id75" title="Daniel S Wilks and Robert L Wilby. The weather generation game: a review of stochastic weather models. Progress in physical geography, 23(3):329–357, 1999.">82</a>]</span> and streamflow <span id="id44">[<a class="reference internal" href="R.Bibliography.html#id76" title="JR Lamontagne and JR Stedinger. Generating synthetic streamflow forecasts with specified precision. Journal of Water Resources Planning and Management, 144(4):04018007, 2018.">83</a>, <a class="reference internal" href="R.Bibliography.html#id77" title="Sanghamitra Medda and Kalyan Kumar Bhar. Comparison of single-site and multi-site stochastic models for streamflow generation. Applied Water Science, 9(3):67, 2019.">84</a>]</span> generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound (hot, severe drought) extremes. It is beyond the scope of this text to review these methods, but readers are encouraged to explore the studies cited above as well as the following publications for discussions and comparisons of these methods: <span id="id45">[<a class="reference internal" href="R.Bibliography.html#id103" title="Jonathan D Herman, Harrison B Zeff, Jonathan R Lamontagne, Patrick M Reed, and Gregory W Characklis. Synthetic drought scenario generation to support bottom-up water supply vulnerability assessments. Journal of Water Resources Planning and Management, 142(11):04016050, 2016.">78</a>, <a class="reference internal" href="R.Bibliography.html#id105" title="Edoardo Borgomeo, Christopher L Farmer, and Jim W Hall. Numerical rivers: a synthetic streamflow generator for water resources vulnerability assessments. Water Resources Research, 51(7):5382–5405, 2015.">80</a>, <a class="reference internal" href="R.Bibliography.html#id106" title="Brian R Kirsch, Gregory W Characklis, and Harrison B Zeff. Evaluating the impact of alternative hydro-climate scenarios on transfer agreements: practical improvement for generating synthetic streamflows. Journal of Water Resources Planning and Management, 139(4):396–406, 2013.">85</a>, <a class="reference internal" href="R.Bibliography.html#id107" title="Daniel P Loucks and Eelco Van Beek. Water resource systems planning and management: An introduction to methods, models, and applications. Springer, 2017.">86</a>, <a class="reference internal" href="R.Bibliography.html#id108" title="Scott Steinschneider, Sungwook Wi, and Casey Brown. The integrated effects of climate and hydrologic uncertainty on future flood risk assessments. Hydrological Processes, 29(12):2823–2839, 2015.">87</a>, <a class="reference internal" href="R.Bibliography.html#id109" title="Richard M Vogel. Stochastic watershed models for hydrologic risk management. Water Security, 1:28–35, 2017.">88</a>, <a class="reference internal" href="R.Bibliography.html#id110" title="Richard M Vogel and Jery R Stedinger. The value of stochastic streamflow models in overyear reservoir design applications. Water Resources Research, 24(9):1483–1490, 1988.">89</a>]</span>. The use of these methods for the purposes of exploratory modeling, especially in the context of well-characterized versus deep uncertainty, is further discussed in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a>.</p>
+</section>
+</section>
+<section id="sensitivity-analysis-methods">
+<span id="id46"></span><h2><span class="section-number">3.4. </span>Sensitivity Analysis Methods<a class="headerlink" href="#sensitivity-analysis-methods" title="Link to this heading">#</a></h2>
+<p>In this section, we describe some of the most widely applied sensitivity analysis methods along with their mathematical definitions. We also provide a detailed discussion on applying each method, as well as a comparison of and their features and limitations.</p>
+<section id="derivative-based-methods">
+<h3><span class="section-number">3.4.1. </span>Derivative-based Methods<a class="headerlink" href="#derivative-based-methods" title="Link to this heading">#</a></h3>
+<p>Derivative-based methods explore how model outputs are affected by perturbations in a single model input around a particular input value. These methods are local and are performed using OAT sampling. For simplicity of mathematical notations, let us assume that the model <span class="math notranslate nohighlight">\(g(X)\)</span> only returns one output. Following <span id="id47">[<a class="reference internal" href="R.Bibliography.html#id117" title="Emanuele Borgonovo. Sensitivity analysis of model output with input constraints: a generalized rationale for local methods. Risk Analysis: An International Journal, 28(3):667–680, 2008.">90</a>]</span> and <span id="id48">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>, the sensitivity index, <span class="math notranslate nohighlight">\(S_i\)</span> , of the model’s <em>i</em>-th input factor, <span class="math notranslate nohighlight">\(x_i\)</span> , can be measured using the partial derivative evaluated at a nominal value, <span class="math notranslate nohighlight">\(\bar{x}\)</span>, of the vector of inputs:</p>
+<div class="math notranslate nohighlight">
+\[S_i (\bar{x}) = \frac{\partial g}{\partial x} |_{\bar{x}{^c{_i}}}\]</div>
+<p>where <em>c</em><sub>i</sub> is the scaling factor. In most applications however, the relationship <span class="math notranslate nohighlight">\(g(X)\)</span> is not fully known in its analytical form, and therefore the above partial derivative is usually approximated:</p>
+<div class="math notranslate nohighlight">
+\[S_i (\bar{x}) = \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i\]</div>
+<p>Using this approximation, the <em>i</em>-th input factor is perturbed by a magnitude of <span class="math notranslate nohighlight">\(\Delta_i\)</span>, and its relative importance is calculated. Derivative-based methods are some of the oldest sensitivity analysis methods as they only require <span class="math notranslate nohighlight">\(N+1\)</span> model evaluations to estimate indices for <span class="math notranslate nohighlight">\(N\)</span> uncertain factors. As described above, being computationally very cheap comes at the cost of not being able to explore the entire input space, but only (local) perturbations to the nominal value. Additionally, as these methods examine the effects of each input factor one at a time, they cannot assess parametric interactions or capture the interacting nature of many real systems and the models that abstract them.</p>
+</section>
+<section id="elementary-effect-methods">
+<h3><span class="section-number">3.4.2. </span>Elementary Effect Methods<a class="headerlink" href="#elementary-effect-methods" title="Link to this heading">#</a></h3>
+<p>Elementary effect (EE) SA methods provide a solution to the local nature of the derivative-based methods by exploring the entire parametric range of each input parameter <span id="id49">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>. However, EE methods still use OAT sampling and do not vary all input parameters simultaneously while exploring the parametric space. The OAT nature of EEs methods therefore prevents them from properly capturing the interactions between uncertain factors. EEs methods are computationally efficient compared to their All-At-a-Time (AAT) counterparts, making them more suitable when computational capacity is a limiting factor, while still allowing for some inferences regarding factor interactions.
+The most popular EE method is the Method of Morris <span id="id50">[<a class="reference internal" href="R.Bibliography.html#id132" title="Max D Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.">74</a>]</span>. Following the notation by <span id="id51">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>, this method calculates global sensitivity using the mean of the EEs (finite differences) of each parameter at different locations:</p>
+<div class="math notranslate nohighlight">
+\[S_i = \mu_i^* = \frac{1}{r}\sum_{j=1}^r EE^j_i = \frac{1}{r}\sum_{j=1}^r \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i\]</div>
+<p>with <span class="math notranslate nohighlight">\(r\)</span> representing the number of sample repetitions (also refered to as trajectories) in the input space, usually set between 4 and 10 <span id="id52">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. Each <span class="math notranslate nohighlight">\(x_j\)</span> represents the points of each trajectory, with <span class="math notranslate nohighlight">\(j=1,…, r\)</span>, selected as described in the sampling strategy for this method, found above. This method also produces the standard deviation of the EEs:</p>
+<div class="math notranslate nohighlight">
+\[\sigma_i = \sqrt{\frac{1}{r}\sum_{j=1}^r(EE_i^j-\frac{1}{r}\sum_{j=1}^r EE^j_i)^2}\]</div>
+<p>which is a measure of parametric interactions. Higher values of <span class="math notranslate nohighlight">\(\sigma_i\)</span> suggest model responses at different levels of factor <span class="math notranslate nohighlight">\(x_i\)</span> are significantly different, which indicates considerable interactions between that and other uncertain factors. The values of <span class="math notranslate nohighlight">\(\mu_i^*\)</span> and <span class="math notranslate nohighlight">\(\sigma_i\)</span> for each factor allow us to draw several different conclusions, illustrated in <a class="reference internal" href="#figure-3-4"><span class="std std-numref">Fig. 3.4</span></a>, following the example by <span id="id53">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>. In this example, factors <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> can be said to have an influence on the model outputs, with <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> having some interactive or non-linear effects. Depending on the orders of magnitude of <span class="math notranslate nohighlight">\(\mu_i^*\)</span> and <span class="math notranslate nohighlight">\(\sigma_i\)</span> one can indirectly deduce whether the factors have strong interactive effects, for example if a factor <span class="math notranslate nohighlight">\(\sigma_i &lt;&lt; \mu_i^*\)</span> then the relationship between that factor and the output can be assumed to be largely linear (note that this is still an OAT method and assumptions on factor interactions should be strongly caveated). Extensions of the Method of Morris have also been developed specifically for the purposes of factor fixing and explorations of parametric interactions (e.g., <span id="id54">[<a class="reference internal" href="R.Bibliography.html#id27" title="Emanuele Borgonovo. Sensitivity analysis with finite changes: an application to modified eoq models. European Journal of Operational Research, 200(1):127–138, 2010.">48</a>, <a class="reference internal" href="R.Bibliography.html#id133" title="Francesca Campolongo and Roger Braddock. The use of graph theory in the sensitivity analysis of the model output: a second order screening method. Reliability Engineering &amp; System Safety, 64(1):1–12, 1999.">92</a>, <a class="reference internal" href="R.Bibliography.html#id134" title="Roger A Cropp and Roger D Braddock. The new morris method: an efficient second-order screening method. Reliability Engineering &amp; System Safety, 78(1):77–83, 2002.">93</a>]</span>).</p>
+<figure class="margin-caption align-center" id="id98">
+<span id="figure-3-4"></span><a class="reference internal image-reference" href="_images/figure3_4_morris_method.png"><img alt="Figure 3.4" src="_images/figure3_4_morris_method.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 3.4 </span><span class="caption-text">Illustrative results of the Morris Method. Factors <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> have an influence on the model outputs, with <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_4\)</span>, and <span class="math notranslate nohighlight">\(x_5\)</span> having interactive or non-linear effects. Whether or not a factor should be considered influential to the output depends on the output selected and is specific to the research context and purpose of the analysis, as discussed in <a class="reference internal" href="#why-sa"><span class="std std-numref">Chapter 3.2</span></a>.</span><a class="headerlink" href="#id98" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="regression-based-methods">
+<h3><span class="section-number">3.4.3. </span>Regression-based Methods<a class="headerlink" href="#regression-based-methods" title="Link to this heading">#</a></h3>
+<p>Regression analysis is one of the oldest ways of investigating parametric importance and sensitivity <span id="id55">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. Here, we describe some of the most popular regression-based sensitivity indices. One of the main sensitivity indices of this category is the standardized regression coefficient (SRC). To calculate SRC, a linear regression relationship needs to be fitted between the input vector, <span class="math notranslate nohighlight">\(x\)</span>, and the model output of interest by using a least-square minimizing method:</p>
+<div class="math notranslate nohighlight">
+\[y = b_0 + \sum_{i=1}^N b_ix_i\]</div>
+<p>where <span class="math notranslate nohighlight">\(b_0\)</span> and <span class="math notranslate nohighlight">\(b_i\)</span> (corresponding to the <em>i</em>-th model input) are regression coefficients. The following relationship can then be used to calculate the SRCs for different input values:</p>
+<div class="math notranslate nohighlight">
+\[S_i=SRC_i=b_i\frac{\sigma_i}{\sigma_y}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\sigma_i\)</span> and <span class="math notranslate nohighlight">\(\sigma_y\)</span> are standard deviations of <em>i</em>-th model input and output, respectively.</p>
+<p>Several other regression-based indices explore the correlation between input and output parameters as a proxy to model parametric sensitivity <span id="id56">[<a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>, <a class="reference internal" href="R.Bibliography.html#id135" title="Jon C Helton. Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal. Reliability Engineering &amp; System Safety, 42(2-3):327–367, 1993.">94</a>, <a class="reference internal" href="R.Bibliography.html#id136" title="Gemma Manache and Charles S Melching. Identification of reliable regression-and correlation-based sensitivity measures for importance ranking of water-quality model parameters. Environmental Modelling &amp; Software, 23(5):549–562, 2008.">95</a>]</span>. The Pearson correlation coefficient (PCC) can be used when a linear relationship exists between an uncertain factor, <span class="math notranslate nohighlight">\(x_i\)</span>, and the output <span class="math notranslate nohighlight">\(y\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i=PCC=\frac{cov(x_i,y)}{\sigma_i\sigma_y}\]</div>
+<p>In cases when there are outliers in the data or the relationship between the uncertain factors and the output is not linear, rank-based correlation coefficients are preferred, for example, Spearman’s rank correlation coefficient (SRCC):</p>
+<div class="math notranslate nohighlight">
+\[S_i=SRCC=\frac{cov(rx_i,ryi)}{\sigma_{ri}\sigma_{ry}}\]</div>
+<p>where the raw values of <span class="math notranslate nohighlight">\(x_i\)</span> and <span class="math notranslate nohighlight">\(y\)</span> and converted to ranks <span class="math notranslate nohighlight">\(rx_i\)</span> and <span class="math notranslate nohighlight">\(ry\)</span> respectively, which instead represent a measurement of the strength of the monotonic relationship, rather than linear relationship, between the input and output. Other regression-based metrics include the partial correlations coefficient, the partial rank correlations coefficient, and the Nash-Sutcliffe coefficient, more discussion on which can be found in <span id="id57">[<a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>.</p>
+<p>Tree-based regression techniques have also been used for sensitivity analysis in an effort to address the challenges faced with nonlinear models <span id="id58">[<a class="reference internal" href="R.Bibliography.html#id137" title="F Pappenberger and Keith J Beven. Ignorance is bliss: or seven reasons not to use uncertainty analysis. Water resources research, 2006.">96</a>]</span>. Examples of these methods include the Patient Rule Induction Method (PRIM; <span id="id59">[<a class="reference internal" href="R.Bibliography.html#id79" title="Jerome H Friedman and Nicholas I Fisher. Bump hunting in high-dimensional data. Statistics and Computing, 9(2):123–143, 1999.">97</a>]</span>) and Classification And Regression Trees (CART; <span id="id60">[<a class="reference internal" href="R.Bibliography.html#id80" title="Leo Breiman, Jerome Friedman, Charles J Stone, and Richard A Olshen. Classification and regression trees. CRC press, 1984.">98</a>]</span>). CART-based approaches also include boosting and bagging extensions <span id="id61">[<a class="reference internal" href="R.Bibliography.html#id89" title="Yoav Freund, Robert Schapire, and Naoki Abe. A short introduction to boosting. Journal-Japanese Society For Artificial Intelligence, 14(771-780):1612, 1999.">99</a>, <a class="reference internal" href="R.Bibliography.html#id138" title="Leo Breiman. Bagging predictors. Machine learning, 24(2):123–140, 1996.">100</a>]</span>. These methods are particularly useful when sensitivity analysis is used for factor mapping (i.e., when trying to identify which uncertain model factors produce a certain model behavior). <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-scenarios"><span class="std std-numref">Chapter 4.3</span></a> elaborates on the use of these methods. Regression-based sensitivity analysis methods are global by nature and can explore the entire space of variables. However, the true level of comprehensiveness depends on the design of experiments and the number of simulations providing data to establish the regression relationships. Although they are usually computationally efficient, they do not produce significant information about parametric interactions <span id="id62">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id22" title="Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: a review of recent advances. European Journal of Operational Research, 248(3):869–887, 2016.">39</a>]</span>.</p>
+</section>
+<section id="regional-sensitivity-analysis">
+<h3><span class="section-number">3.4.4. </span>Regional Sensitivity Analysis<a class="headerlink" href="#regional-sensitivity-analysis" title="Link to this heading">#</a></h3>
+<p>Another method primarily applied for basic factor mapping applications is Regional Sensitivity Analysis (RSA; <span id="id63">[<a class="reference internal" href="R.Bibliography.html#id157" title="George M Hornberger and Robert C Spear. Approach to the preliminary analysis of environmental systems. J. Environ. Mgmt., 12(1):7–18, 1981.">101</a>]</span>). RSA is a global sensitivity analysis method that is typically implemented using standard sampling methods such as latin hypercube sampling. It is performed by specifying a condition on the output space (e.g., an upper threshold) and classifying outputs that meet the condition as behavioral and the ones that fail it as non-behavioral (illustrated in <a class="reference internal" href="#figure-3-2"><span class="std std-numref">Fig. 3.2</span></a> (b)). Note that the specified threshold depends on the nature of the problem, model, and the research question. It can reflect model-performance metrics (such as errors) or consequential decision-relevant metrics (such as unacceptable system outcomes). The behavioral and non-behavioral outputs are then traced back to their originating sampled factors, where differences between the distributions of samples can be used to determine their significance in producing each part of the output. The Kolmogorov-Smirnov divergence is commonly used to quantify the difference between the distribution of behavioral and non-behavioral parameters <span id="id64">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>]</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i=|F_{x_i|y_b} (y \in Y_b)-F_{x_i|y_{nb}} (y \in Y_{nb})|\]</div>
+<p>where <span class="math notranslate nohighlight">\(Y_b\)</span> represents the set of behavioral outputs, and <span class="math notranslate nohighlight">\(F_{x_i|y_b}\)</span> is the empirical cumulative distribution function of the values of <span class="math notranslate nohighlight">\(x_i\)</span> associated with values of <span class="math notranslate nohighlight">\(y\)</span> that belong in the behavioral set. The <span class="math notranslate nohighlight">\(nb\)</span> notation indicates the equivalent elements related to the non-behavioral set. Large differences between the two distributions indicate stronger effects by the parameters on the respective part of the output space.</p>
+<p>Used in a factor mapping setting, RSA can be applied for scenario discovery <span id="id65">[<a class="reference internal" href="R.Bibliography.html#id71" title="Robert J Lempert, David G Groves, Steven W Popper, and Steve C Bankes. A general, analytic method for generating robust strategies and narrative scenarios. Management science, 52(4):514–528, 2006.">102</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>, the Generalized Likelihood Uncertainty Estimation method (GLUE; <span id="id66">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>, <a class="reference internal" href="R.Bibliography.html#id158" title="Keith Beven and Andrew Binley. Glue: 20 years on. Hydrological processes, 28(24):5897–5918, 2014.">104</a>, <a class="reference internal" href="R.Bibliography.html#id159" title="Roberta-Serena Blasone, Jasper A Vrugt, Henrik Madsen, Dan Rosbjerg, Bruce A Robinson, and George A Zyvoloski. Generalized likelihood uncertainty estimation (glue) using adaptive markov chain monte carlo sampling. Advances in Water Resources, 31(4):630–648, 2008.">105</a>]</span>) and other hybrid sensitivity analysis methods (e.g., <span id="id67">[<a class="reference internal" href="R.Bibliography.html#id160" title="SA Cryer and PL Havens. Regional sensitivity analysis using a fractional factorial method for the usda model gleams. Environmental modelling &amp; software, 14(6):613–624, 1999.">106</a>, <a class="reference internal" href="R.Bibliography.html#id161" title="Pengfei Wei, Zhenzhou Lu, and Xiukai Yuan. Monte carlo simulation for moment-independent sensitivity analysis. Reliability Engineering &amp; System Safety, 110:60–67, 2013.">107</a>]</span>). The fundamental shortcomings of RSA are that, in some cases, it could be hard to interpret the difference between behavioral and non-behavioral sample sets, and that insights about parametric correlations and interactions cannot always be uncovered <span id="id68">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>]</span>. For more elaborate discussions and illustrations of the RSA method, readers are directed to <span id="id69">Tang <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id20" title="Yong Tang, Patrick Reed, Thibaut Wagener, and K van Werkhoven. Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation. Hydrology and Earth System Sciences, 11(2):793–817, 2007.">42</a>], Saltelli <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id28" title="Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley &amp; Sons, 2008.">49</a>], Young [<a class="reference internal" href="R.Bibliography.html#id162" title="Peter Young. Data-based mechanistic modelling, generalised sensitivity and dominant mode analysis. Computer Physics Communications, 117(1-2):113–129, 1999.">108</a>]</span> and references therein.</p>
+</section>
+<section id="variance-based-methods">
+<span id="id70"></span><h3><span class="section-number">3.4.5. </span>Variance-based Methods<a class="headerlink" href="#variance-based-methods" title="Link to this heading">#</a></h3>
+<p>Variance-based sensitivity analysis methods hypothesize that various specified model factors contribute differently to the variation of model outputs; therefore, decomposition and analysis of output variance can determine a model’s sensitivity to input parameters <span id="id71">[<a class="reference internal" href="R.Bibliography.html#id17" title="Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity analysis in practice: a guide to assessing scientific models. Volume 1. Wiley Online Library, 2004.">38</a>, <a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>]</span>. The most popular variance-based method is the Sobol method, which is a global sensitivity analysis method that takes into account complex and nonlinear factor interaction when calculating sensitivity indices, and employs more sophisticated sampling methods (e.g., the Sobol sampling method). The Sobol method is able to calculate three types of sensitivity indices that provide different types of information about model sensitivities. These indices include first-order, higher-order (e.g., second-, third-, etc. orders), and total-order sensitivities.</p>
+<p>The first-order sensitivity index indicates the percent of model output variance contributed by a factor individually (i.e., the effect of varying <span class="math notranslate nohighlight">\(x_i\)</span> alone) and is obtained using the following <span id="id72">[<a class="reference internal" href="R.Bibliography.html#id102" title="Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001.">77</a>, <a class="reference internal" href="R.Bibliography.html#id148" title="Andrea Saltelli. Making best use of model evaluations to compute sensitivity indices. Computer physics communications, 145(2):280–297, 2002.">109</a>]</span>:</p>
+<div class="math notranslate nohighlight">
+\[S_i^1=\frac{V_{x_i}[E_{x_{\sim i}}(x_i)]}{V(y)}\]</div>
+<p>with <span class="math notranslate nohighlight">\(E\)</span> and <span class="math notranslate nohighlight">\(V\)</span> denoting the expected value and the variance, respectively. <span class="math notranslate nohighlight">\(x_{\sim i}\)</span> denotes all factors expect for <span class="math notranslate nohighlight">\(x_i\)</span>. The first-order sensitivity index (<span class="math notranslate nohighlight">\(S_i^1\)</span>) can therefore also be thought of as the portion of total output variance (<span class="math notranslate nohighlight">\(V_y\)</span>) that can be reduced if the uncertainty in factor <span class="math notranslate nohighlight">\(x_i\)</span> is eliminated <span id="id73">[<a class="reference internal" href="R.Bibliography.html#id149" title="Andrea Saltelli. Sensitivity analysis for importance assessment. Risk analysis, 22(3):579–590, 2002.">110</a>]</span>. First-order sensitivity indices are usually used to understand the independent effect of a factor and to distinguish its individual versus interactive influence. It would be expected for linearly independent factors that they would only have first order indices (no interactions) that should correspond well with sensitivities obtained from simpler methods using OAT sampling.</p>
+<p>Higher-order sensitivity indices explore the interaction between two or more parameters that contribute to model output variations. For example, a second-order index indicates how interactions between a pair of factors can lead to change in model output variance and is calculated using the following relationship:</p>
+<div class="math notranslate nohighlight">
+\[S_{ij}^2=\frac{V_{x_{i,j}}[E_{x_{\sim i,j}}(x_i,x_j)]}{V(y)}\]</div>
+<p>with <span class="math notranslate nohighlight">\(i \ne j\)</span>. Higher order indices can be calculated by similar extensions (i.e., fixing additional operators together), but it is usually computationally expensive in practice.</p>
+<p>The total sensitivity analysis index represents the entire influence of an input factor on model outputs including all of its interactions with other factors <span id="id74">[<a class="reference internal" href="R.Bibliography.html#id150" title="Toshimitsu Homma and Andrea Saltelli. Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering &amp; System Safety, 52(1):1–17, 1996.">111</a>]</span>. In other words, total-order indices include first-order and all higher-order interactions associated with each factor and can be estimated calculated using the following:</p>
+<div class="math notranslate nohighlight">
+\[S_i^T= \frac{E_{x_{\sim i}}[V_{x_i}(x_{\sim i})]}{V(y)} = 1 - \frac{V_{x_{\sim i}}[E_{x_{i}}(x_{\sim i})]}{V(y)}\]</div>
+<p>This index reveals the expected portion of variance that remains if uncertainty is eliminated in all factors but <span class="math notranslate nohighlight">\(x_i\)</span> <span id="id75">[<a class="reference internal" href="R.Bibliography.html#id149" title="Andrea Saltelli. Sensitivity analysis for importance assessment. Risk analysis, 22(3):579–590, 2002.">110</a>]</span>. The total sensitivity index is the overall best measure of sensitivity as it captures the full individual and interactive effects of model factors.</p>
+<p>Besides the Sobol method, there are some other variance-based sensitivity analysis methods, such as the Fourier amplitude sensitivity test (FAST; <span id="id76">[<a class="reference internal" href="R.Bibliography.html#id100" title="RI Cukier, CM Fortuin, Kurt E Shuler, AG Petschek, and JH Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. i theory. The Journal of chemical physics, 59(8):3873–3878, 1973.">75</a>, <a class="reference internal" href="R.Bibliography.html#id151" title="Gregory J McRae, William R Goodin, and John H Seinfeld. Development of a second-generation mathematical model for urban air pollution—i. model formulation. Atmospheric Environment (1967), 16(4):679–696, 1982.">112</a>]</span>) and extended-FAST <span id="id77">[<a class="reference internal" href="R.Bibliography.html#id152" title="Andrea Saltelli and Ricardo Bolado. An alternative way to compute fourier amplitude sensitivity test (fast). Computational Statistics &amp; Data Analysis, 26(4):445–460, 1998.">113</a>, <a class="reference internal" href="R.Bibliography.html#id153" title="MA Vazquez-Cruz, R Guzman-Cruz, IL Lopez-Cruz, O Cornejo-Perez, I Torres-Pacheco, and RG Guevara-Gonzalez. Global sensitivity analysis by means of efast and sobol’methods and calibration of reduced state-variable tomgro model using genetic algorithms. Computers and Electronics in Agriculture, 100:1–12, 2014.">114</a>]</span>, that have been used by the scientific community. However, Sobol remains by far the most common method of this class. Variance-based techniques have been widely used and have proved to be powerful in a variety of applications. Despite their popularity, some authors have expressed concerns about the methods’ appropriateness in some settings. Specifically, the presence of heavy-tailed distributions or outliers, or when model outputs are multimodal can bias the sensitivity indices produced by these methods <span id="id78">[<a class="reference internal" href="R.Bibliography.html#id154" title="Benjamin Auder and Bertrand Iooss. Global sensitivity analysis based on entropy. In Safety, reliability and risk analysis-Proceedings of the ESREL 2008 Conference, 2107–2115. 2008.">115</a>, <a class="reference internal" href="R.Bibliography.html#id155" title="Farkhondeh Khorashadi Zadeh, Jiri Nossent, Fanny Sarrazin, Francesca Pianosi, Ann van Griensven, Thorsten Wagener, and Willy Bauwens. Comparison of variance-based and moment-independent global sensitivity analysis approaches by application to the swat model. Environmental Modelling &amp; Software, 91:210–222, 2017.">116</a>, <a class="reference internal" href="R.Bibliography.html#id156" title="Francesca Pianosi and Thorsten Wagener. A simple and efficient method for global sensitivity analysis based on cumulative distribution functions. Environmental Modelling &amp; Software, 67:1–11, 2015.">117</a>]</span>. Moment-independent measures, discussed below, attempt to overcome these challenges.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial which demonstrates the application of a Sobol sensitivity analysis:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA using SALib Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="analysis-of-variance-anova">
+<h3><span class="section-number">3.4.6. </span>Analysis of Variance (ANOVA)<a class="headerlink" href="#analysis-of-variance-anova" title="Link to this heading">#</a></h3>
+<p>Analysis of Variance (ANOVA) was first introduced by <span id="id79">Fisher and others [<a class="reference internal" href="R.Bibliography.html#id144" title="Ronald Aylmer Fisher and others. Statistical methods for research workers. Statistical methods for research workers., 1934.">118</a>]</span> and has since become a popular factor analysis method in physical experiments. ANOVA can be used as a sensitivity analysis method in computational experiments with a factorial design of experiment (referred to as factorial ANOVA). Note that Sobol can also be categorized as an ANOVA sensitivity analysis method, and that is why Sobol is sometimes referred to as a functional ANOVA <span id="id80">[<a class="reference internal" href="R.Bibliography.html#id145" title="Loıc Brevault, Mathieu Balesdent, Nicolas Bérend, and Rodolphe Le Riche. Comparison of different global sensitivity analysis methods for aerospace vehicle optimal design. In 10th World Congress on Structural and Multidisciplinary Optimization, WCSMO-10. 2013.">119</a>]</span>. Factorial ANOVA methods are particularly suited for models and problems that have discrete input spaces, significantly reducing the computational time. More information about these methods can be found in <span id="id81">[<a class="reference internal" href="R.Bibliography.html#id145" title="Loıc Brevault, Mathieu Balesdent, Nicolas Bérend, and Rodolphe Le Riche. Comparison of different global sensitivity analysis methods for aerospace vehicle optimal design. In 10th World Congress on Structural and Multidisciplinary Optimization, WCSMO-10. 2013.">119</a>, <a class="reference internal" href="R.Bibliography.html#id146" title="GEB Archer, Andrea Saltelli, and IM Sobol. Sensitivity measures, anova-like techniques and the use of bootstrap. Journal of Statistical Computation and Simulation, 58(2):99–120, 1997.">120</a>, <a class="reference internal" href="R.Bibliography.html#id147" title="Art B Owen. Variance components and generalized sobol'indices. SIAM/ASA Journal on Uncertainty Quantification, 1(1):19–41, 2013.">121</a>]</span>.</p>
+</section>
+<section id="moment-independent-density-based-methods">
+<h3><span class="section-number">3.4.7. </span>Moment-Independent (Density-Based) Methods<a class="headerlink" href="#moment-independent-density-based-methods" title="Link to this heading">#</a></h3>
+<p>These methods typically compare the entire distribution (i.e., not just the variance) of input and output parameters in order to determine the sensitivity of the output to a particular input variable. Several moment-independent sensitivity analysis methods have been proposed in recent years. The delta (<span class="math notranslate nohighlight">\(\delta\)</span>) moment-independent method calculates the difference between unconditional and conditional cumulative distribution functions of the output. The method was first introduced by <span id="id82">[<a class="reference internal" href="R.Bibliography.html#id142" title="Emanuele Borgonovo. Measuring uncertainty importance: investigation and comparison of alternative approaches. Risk analysis, 26(5):1349–1361, 2006.">122</a>, <a class="reference internal" href="R.Bibliography.html#id141" title="Emanuele Borgonovo. A new uncertainty importance measure. Reliability Engineering &amp; System Safety, 92(6):771–784, 2007.">123</a>]</span> and has become widely used in various disciplines. The <span class="math notranslate nohighlight">\(\delta\)</span> sensitivity index is defined as follows:</p>
+<div class="math notranslate nohighlight">
+\[S_i=\delta_i=\frac{1}{2}E_{x_i}|f_y(y)-f_{y|x_i}(y)|dy\]</div>
+<p>where <span class="math notranslate nohighlight">\(f_y(y)\)</span> is the probability density function of the entire model output <span class="math notranslate nohighlight">\(y\)</span>, and <span class="math notranslate nohighlight">\(f_{y|x_i}(y)\)</span> is the conditional density of <span class="math notranslate nohighlight">\(y\)</span>, given that factor <span class="math notranslate nohighlight">\(x_i\)</span> assumes a fixed value. The <span class="math notranslate nohighlight">\(\delta_i\)</span> sensitivity indicator therefore represents the normalized expected shift in the distribution of <span class="math notranslate nohighlight">\(y\)</span> provoked by <span class="math notranslate nohighlight">\(x_i\)</span>. Moment-independent methods are advantageous in cases where we are concerned about the entire distribution of events, such as when uncertain factors lead to more extreme events in a system <span id="id83">[<a class="reference internal" href="R.Bibliography.html#id47" title="Antonia Hadjimichael, Julianne Quinn, and Patrick Reed. Advancing Diagnostic Model Evaluation to Better Understand Water Shortage Mechanisms in Institutionally Complex River Basins. Water Resources Research, 56(10):e2020WR028079, 2020. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR028079 (visited on 2020-10-16), doi:10.1029/2020WR028079.">13</a>]</span>. Further, they can be used with a pre-existing sample of data, without requiring a specific sampling scheme, unlike the previously reviewed methods <span id="id84">[<a class="reference internal" href="R.Bibliography.html#id143" title="Elmar Plischke, Emanuele Borgonovo, and Curtis L Smith. Global sensitivity measures from given data. European Journal of Operational Research, 226(3):536–550, 2013.">124</a>]</span>. The <span class="math notranslate nohighlight">\(\delta\)</span> sensitivity index does not include interactions between factors and it is therefore akin to the first order index produced by the Sobol method. Interactions between factors can still be estimated using this method, by conditioning the calculation on more than one uncertain factor being fixed <span id="id85">[<a class="reference internal" href="R.Bibliography.html#id141" title="Emanuele Borgonovo. A new uncertainty importance measure. Reliability Engineering &amp; System Safety, 92(6):771–784, 2007.">123</a>]</span>.</p>
+</section>
+</section>
+<section id="how-to-choose-a-sensitivity-analysis-method-model-traits-and-dimensionality">
+<h2><span class="section-number">3.5. </span>How To Choose A Sensitivity Analysis Method: Model Traits And Dimensionality<a class="headerlink" href="#how-to-choose-a-sensitivity-analysis-method-model-traits-and-dimensionality" title="Link to this heading">#</a></h2>
+<p><a class="reference internal" href="#figure-3-5"><span class="std std-numref">Fig. 3.5</span></a>, synthesized from variants found in <span id="id86">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>, presents a graphical synthesis of the methods overviewed in this section, with regards to their appropriateness of application based on the complexity of the model at hand and the computational limits on the number of model evaluations afforded. The bars below each method also indicate the sensitivity analysis purposes they are most appropriate to address, which are in turn a reflection of the motivations and research questions the sensitivity analysis is called to address. Computational intensity is measured as a multiple of the number of model factors that are considered uncertain (<span class="math notranslate nohighlight">\(d\)</span>). Increasing model complexity mandates that more advanced sensitivity analysis methods are applied to address potential nonlinearities, factor interactions, and discontinuities. Such methods can only be performed at increasing computational expense. For example, computationally cheap linear regression should not be used to assess factors’ importance if the model cannot be proven linear and the factors independent, because important relationships will invariably be missed (recall the example in <a class="reference internal" href="#figure-3-5"><span class="std std-numref">Fig. 3.5</span></a>). When computational limits do constrain applications to make simplified assumptions and sensitivity techniques, any conclusions in such cases should be delivered with clear statements of the appropriate caveats.</p>
+<figure class="margin-caption align-center" id="id99">
+<span id="figure-3-5"></span><a class="reference internal image-reference" href="_images/figure3_5classificationofmethods.png"><img alt="Figure 3_5" src="_images/figure3_5classificationofmethods.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 3.5 </span><span class="caption-text">Classification of the sensitivity analysis methods overviewed in this section, with regards to their computational cost (horizontal axis), their appropriateness to model complexity (vertical axis), and the purpose they can be used for (colored bars). d: number of uncertain factors considered; ANOVA: Analysis of Variance; FAST: Fourier Amplitude Sensitivity Test; PRIM: Patient Rule Induction Method; CART: Classification and Regression Trees; SRCC: Spearman’s rank correlation coefficient: NSE: Nash–Sutcliffe efficiency; SRC: standardized regression coefficient; PCC: Pearson correlation coefficient. This figure is synthesized from variants found in <span id="id87">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id131" title="Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. In Uncertainty management in simulation-optimization of complex systems, pages 101–122. Springer, 2015.">91</a>]</span>.</span><a class="headerlink" href="#id99" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The reader should also be aware that the estimates of computational intensity that are given here are indicative of magnitude and would vary depending on the sampling technique, model complexity and the level of information being asked. For example, a Sobol sensitivity analysis typically requires a sample of size <span class="math notranslate nohighlight">\(n * d+2\)</span> to produce first- and total-order indices, where <span class="math notranslate nohighlight">\(d\)</span> is the number of uncertain factors and <span class="math notranslate nohighlight">\(n\)</span> is a scaling factor, selected ad hoc, depending on model complexity <span id="id88">[<a class="reference internal" href="R.Bibliography.html#id25" title="Andrea Saltelli and Stefano Tarantola. On the relative importance of input factors in mathematical models: safety assessment for nuclear waste disposal. Journal of the American Statistical Association, 97(459):702–709, 2002.">46</a>]</span>. The scaling factor <span class="math notranslate nohighlight">\(n\)</span> is typically set to at least 1000, but it should most appropriately be set on the basis of index convergence. In other words, a prudent analyst would perform the analysis several times with increasing <span class="math notranslate nohighlight">\(n\)</span> and observe at what level the indices converge to stable values <span id="id89">[<a class="reference internal" href="R.Bibliography.html#id140" title="Jiri Nossent, Pieter Elsen, and Willy Bauwens. Sobol’ sensitivity analysis of a complex environmental model. Environmental Modelling &amp; Software, 26(12):1515-1525, 2011. URL: https://www.sciencedirect.com/science/article/pii/S1364815211001939, doi:https://doi.org/10.1016/j.envsoft.2011.08.010.">125</a>]</span>. The level should be the minimum sample size used in subsequent sensitivity analyses of the same system. Furthermore, if the analyst would like to better understand the degrees of interaction between factors, requiring second-order indices, the sample size would have to increase to <span class="math notranslate nohighlight">\(n * 2d+2\)</span> <span id="id90">[<a class="reference internal" href="R.Bibliography.html#id25" title="Andrea Saltelli and Stefano Tarantola. On the relative importance of input factors in mathematical models: safety assessment for nuclear waste disposal. Journal of the American Statistical Association, 97(459):702–709, 2002.">46</a>]</span>.</p>
+<p>Another important consideration is that methods that do not require specific sampling schemes can be performed in conjunction with others without requiring additional model evaluations. None of the regression-based methods, for example, require samples of specific structures or sizes, and can be combined with other methods for complementary purposes. For instance, one could complement a Sobol analysis with an application of CART, using the same data, but to address questions relating to factor mapping (e.g., we know factor <span class="math notranslate nohighlight">\(x_i\)</span>  is important for a model output, but we would like to also know which of its values specifically push the output to undesirable states). Lastly, comparing results from different methods performed together can be especially useful in model diagnostic settings. For example, <span id="id91">[<a class="reference internal" href="R.Bibliography.html#id47" title="Antonia Hadjimichael, Julianne Quinn, and Patrick Reed. Advancing Diagnostic Model Evaluation to Better Understand Water Shortage Mechanisms in Institutionally Complex River Basins. Water Resources Research, 56(10):e2020WR028079, 2020. URL: http://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR028079 (visited on 2020-10-16), doi:10.1029/2020WR028079.">13</a>]</span> used <span class="math notranslate nohighlight">\(\delta\)</span> indices, first-order Sobol indices, and <span class="math notranslate nohighlight">\(R^2\)</span>  values from linear regression, all performed on the same factors, to derive insights about the effects on factors on different moments of the output distribution and about the linearity of their relationship.</p>
+</section>
+<section id="software-toolkits">
+<span id="id92"></span><h2><span class="section-number">3.6. </span>Software Toolkits<a class="headerlink" href="#software-toolkits" title="Link to this heading">#</a></h2>
+<p>This section presents available open source sensitivity analysis software tools, based on the programming language they use and the methods they support <a class="reference internal" href="#figure-3-6"><span class="std std-numref">Fig. 3.6</span></a>. Our review covers five widely used programming languages: R, MATLAB, Julia, Python, and C++, as well as one tool that provides a graphical user interface (GUI). Each available SA tool was assessed on the number of SA methods and design of experiments methods it supports. For example, the <em>sensobol</em> package in R only supports the variance-based Sobol method. However, it is the only package we came across that calculates third-order interactions among parameters. On the other side of the spectrum, there are SA software packages that contain several popular SA methods. For example, <em>SALib</em> in Python <span id="id93">[<a class="reference internal" href="R.Bibliography.html#id139" title="Jon Herman and Will Usher. Salib: an open-source python library for sensitivity analysis. Journal of Open Source Software, 2(9):97, 2017.">126</a>]</span> supports seven different SA methods. The <em>DifferentialEquations</em> package is a comprehensive package developed for Julia, and <em>GlobalSensitivityAnalysis</em> is another Julia package that has mostly adapted SALib methods. <a class="reference internal" href="#figure-3-6"><span class="std std-numref">Fig. 3.6</span></a> also identifies the SA packages that have been updated since 2018, indicating active support and development.</p>
+<figure class="margin-caption align-center" id="id100">
+<span id="figure-3-6"></span><a class="reference internal image-reference" href="_images/figure3_6_softwaretoolkits.png"><img alt="Figure 3_6" src="_images/figure3_6_softwaretoolkits.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 3.6 </span><span class="caption-text">Sensitivity analysis packages available in different programming language platforms (R, Python, Julia, MATLAB, and C++), with the number of methods they support. Packages supporting more than five methods are indicated in pink. Packages updated since 2018 are indicated with asterisks.</span><a class="headerlink" href="#id100" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The following articles are suggested as fundamental reading for the information presented in this section:</p>
+<ul class="simple">
+<li><p>Wagener, T., Pianosi, F., 2019. What has Global Sensitivity Analysis ever done for us? A systematic review to support scientific advancement and to inform policy-making in earth system modelling. <em>Earth-Science Reviews</em> 194, 1–18. <a class="reference external" href="https://doi.org/10.1016/j.earscirev.2019.04.006">https://doi.org/10.1016/j.earscirev.2019.04.006</a></p></li>
+<li><p>Pianosi, F., Beven, K., Freer, J., Hall, J.W., Rougier, J., Stephenson, D.B., Wagener, T., 2016. Sensitivity analysis of environmental models: A systematic review with practical workflow. Environmental Modelling &amp; Software 79, 214–232. <a class="reference external" href="https://doi.org/10.1016/j.envsoft.2016.02.008">https://doi.org/10.1016/j.envsoft.2016.02.008</a></p></li>
+</ul>
+<p>The following articles can be used as supplemental reading:</p>
+<ul class="simple">
+<li><p>Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S., 2008. Global Sensitivity Analysis: The Primer, 1st edition. ed. Wiley-Interscience, Chichester, England ; Hoboken, NJ.</p></li>
+<li><p>Montgomery, D.C., 2017. Design and analysis of experiments. John Wiley &amp; Sons.</p></li>
+<li><p>Iooss, B., Lemaître, P., 2015. A Review on Global Sensitivity Analysis Methods, in: Dellino, G., Meloni, C. (Eds.), Uncertainty Management in Simulation-Optimization of Complex Systems: Algorithms and Applications, Operations Research/Computer Science Interfaces Series. Springer US, Boston, MA, pp. 101–122. <a class="reference external" href="https://doi.org/10.1007/978-1-4899-7547-8_5">https://doi.org/10.1007/978-1-4899-7547-8_5</a></p></li>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#global-versus-local-sensitivity">3.1. Global Versus Local Sensitivity</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#why-perform-sensitivity-analysis">3.2. Why Perform Sensitivity Analysis</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#design-of-experiments">3.3. Design of Experiments</a><ul class="nav section-nav flex-column">
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-generation-of-input-time-series">3.3.6. Synthetic generation of input time series</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sensitivity-analysis-methods">3.4. Sensitivity Analysis Methods</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#derivative-based-methods">3.4.1. Derivative-based Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#elementary-effect-methods">3.4.2. Elementary Effect Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#regression-based-methods">3.4.3. Regression-based Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#regional-sensitivity-analysis">3.4.4. Regional Sensitivity Analysis</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#variance-based-methods">3.4.5. Variance-based Methods</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#analysis-of-variance-anova">3.4.6. Analysis of Variance (ANOVA)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#moment-independent-density-based-methods">3.4.7. Moment-Independent (Density-Based) Methods</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#how-to-choose-a-sensitivity-analysis-method-model-traits-and-dimensionality">3.5. How To Choose A Sensitivity Analysis Method: Model Traits And Dimensionality</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#software-toolkits">3.6. Software Toolkits</a></li>
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+    <h1>Understanding Errors: What Is Controlling Model Performance?</h1>
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+  <section id="understanding-errors-what-is-controlling-model-performance">
+<span id="understanding-errors"></span><h1>Understanding Errors: What Is Controlling Model Performance?<a class="headerlink" href="#understanding-errors-what-is-controlling-model-performance" title="Link to this heading">#</a></h1>
+<p>Sensitivity analysis is a diagnostic tool when reconciling model outputs with observed data. It is helpful for clarifying how and under what conditions modeling choices (structure, parameterization, data inputs, etc.) propagate through model components and manifest in their effects on model outputs. This exploration is performed through carefully designed sampling of multiple combinations of input parameters and subsequent evaluation of the model structures that are emerging as controlling factors. Model structure and parameterization are two of the most commonly explored aspects of models that have been a central focus when evaluating their performance relative to available observations <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id49" title="Keith Beven. Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in water resources, 16(1):41–51, 1993.">17</a>]</span>. Addressing these issues plays an important role in establishing credibility in model predictions, particularly in the positivist natural sciences literature. Traditional model evaluations compare the model with observed data, and then rely on expert judgements of its acceptability based on the closeness between simulation and observation with one or a small number of selected metrics. This approach can be myopic, as it is often impossible to use one metric to attribute a certain error and to link that with different parts of the model and its parameters <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id50" title="Hoshin V Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008.">127</a>]</span>. This means that, even when the error or fitting measure between the model estimates and observations is very small, it is not guaranteed that all the components in the model accurately represent the conceptual reality that the model is abstracting: propagated errors in different parts of the model might cancel each other out, or multiple parameterized implementations of the model can yield similar performance (i.e., equifinality <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id49" title="Keith Beven. Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in water resources, 16(1):41–51, 1993.">17</a>]</span>).</p>
+<p>The inherent complexity of a system hinders accepting or rejecting a model based on one performance measure and different types of measures can aid in evaluating the various components of a model as essentially a multiobjective problem <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id52" title="Keith J Beven. On hypothesis testing in hydrology: why falsification of models is still a really good idea. Wiley Interdisciplinary Reviews: Water, 5(3):e1278, 2018.">25</a>, <a class="reference internal" href="R.Bibliography.html#id51" title="Hoshin V Gupta, Martyn P Clark, Jasper A Vrugt, Gab Abramowitz, and Ming Ye. Towards a comprehensive assessment of model structural adequacy. Water Resources Research, 2012.">26</a>]</span>. In addition, natural systems mimicked by the models contain various interacting components that might act differently across spatial and temporal domains <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id53" title="JD Herman, PM Reed, and T Wagener. Time-varying sensitivity analysis clarifies the effects of watershed model formulation on model behavior. Water Resources Research, 49(3):1400–1414, 2013.">55</a>]</span>. This heterogeneity is lost when a single performance measure is used, as a highly dimensional and interactive system becomes aggregated through the averaging of spatial or temporal output errors <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id54" title="Keith Beven. Towards a coherent philosophy for modelling the environment. Proceedings of the royal society of London. Series A: mathematical, physical and engineering sciences, 458(2026):2465–2484, 2002.">15</a>]</span>. Therefore, diagnostic error analyses should consider multiple error signatures across different scales and states of concern when seeking to understand how model performance relates to observed data (<a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-1"><span class="std std-numref">Fig. 4.1</span></a>). Diverse error signatures can be used to measure the consistency of underlying processes and behaviors of the model and to evaluate the dynamics of model controls under changing temporal and spatial conditions <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id56" title="Keith Beven and Jim Freer. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the glue methodology. Journal of hydrology, 249(1-4):11–29, 2001.">128</a>]</span>. Within this framework, even minimal information extracted from the data can be beneficial as it helps us unearth structural inadequacies in the model. In this context, proper selection of measures of model performance and the number of measures could play consequential roles in our understanding of the model and its predictions <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id55" title="Cameron McPhail, HR Maier, JH Kwakkel, M Giuliani, A Castelletti, and S Westra. Robustness metrics: how are they calculated, when should they be used and why do they give different results? Earth's Future, 6(2):169–191, 2018.">129</a>]</span>.</p>
+<p>As discussed earlier, instead of the traditional focus on using deterministic prediction that results in a single error measure, many plausible states and spaces could be searched for making different inferences and quantifying uncertainties. This process also requires estimates of prior probability distributions of all the important parameters and quantification of model behavior across input space. One strategy to reduce the search space is filtering of some model alternatives that are not consistent with observations and known system behaviors. Those implausible parts of the search space can be referred to as non-physical or non-behavioral alternatives <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id29" title="Neil R Edwards, David Cameron, and Jonathan Rougier. Precalibrating an intermediate complexity climate model. Climate dynamics, 37(7):1469–1482, 2011.">50</a>, <a class="reference internal" href="R.Bibliography.html#id158" title="Keith Beven and Andrew Binley. Glue: 20 years on. Hydrological processes, 28(24):5897–5918, 2014.">104</a>]</span>. This step is conducted before the Bayesian calibration exercise (see <a class="reference internal" href="A1_Uncertainty_Quantification.html#a1-uncertainty-quantification"><span class="std std-numref">Chapter A</span></a>).</p>
+<p>A comprehensive model diagnostic workflow typically entails the components demonstrated in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-1"><span class="std std-numref">Fig. 4.1</span></a>. The workflow begins with the selection of model input parameters and their plausible ranges. After the parameter selection, we need to specify the design of experiment (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#design-of-experiments"><span class="std std-numref">Chapter 3.3</span></a>) and the sensitivity analysis method (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#sensitivity-analysis-methods"><span class="std std-numref">Chapter 3.4</span></a>) to be used. As previously discussed, these methods require different numbers of model simulations, and each method provides a different insights into the direct effects and interactions of the uncertain factors. In addition, the simulation time of the model and the available computational resources are two of the primary considerations that influence these decisions. After identifying the appropriate methods, we generate a matrix of input parameters, where each set of input parameters will be used to conduct a model simulation. The model can include one or more output variables that fluctuate in time and space. The next step is to analyze model performance by comparing model outputs with observations. As discussed earlier, the positivist model evaluation paradigm focuses on a single model performance metric (error), leading to a loss of information about model parameters and the suitability of the model’s structure. However, a thorough investigation of the temporal and spatial signatures of model outputs using various performance metrics or time- and space-varying sensitivity analyses can shed more light on the fitness of each parameter set and the model’s internal structure. This analysis provides diagnostic feedback on the importance and range of model parameters and can guide further improvement of the model algorithm.</p>
+<figure class="margin-caption align-center" id="id10">
+<span id="figure-4-1"></span><a class="reference internal image-reference" href="_images/figure4_1_diagnostic_workflow.png"><img alt="Figure 4.1" src="_images/figure4_1_diagnostic_workflow.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Diagnostic evaluation of model fidelity using sensitivity analysis methods.</span><a class="headerlink" href="#id10" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following badge to try out an interactive tutorial on implementing a time-varying sensitivity analysis of HYMOD model parameters:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">HYMOD Jupyter Notebook</a></p>
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+    <h1>Consequential Dynamics: What is Controlling Model Behaviors of Interest?</h1>
+    <!-- Table of contents -->
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+  <section id="consequential-dynamics-what-is-controlling-model-behaviors-of-interest">
+<h1>Consequential Dynamics: What is Controlling Model Behaviors of Interest?<a class="headerlink" href="#consequential-dynamics-what-is-controlling-model-behaviors-of-interest" title="Link to this heading">#</a></h1>
+<p>Consequential changes in dynamic systems can take many forms, but most dynamic behavior can be categorized in a few basic patterns. Feedback structures inherent in a system, be they positive or negative, generate these patterns which can be grouped into three groups for the simplest of systems: exponential growth, goal seeking, and oscillation (<a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-2"><span class="std std-numref">Fig. 4.2</span></a>). A positive (or self-reinforcing) feedback gives rise to exponential growth, a negative (or self-correcting) feedback gives rise to a goal seeking mode, and negative feedbacks with time delays give rise to oscillatory behavior. Nonlinear interactions between the system’s feedback structures can give rise to more complex dynamic behavior modes, examples of which are also shown in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-2"><span class="std std-numref">Fig. 4.2</span></a>, adapted from <span id="id1">Sterman [<a class="reference internal" href="R.Bibliography.html#id57" title="John D Sterman. System dynamics modeling: tools for learning in a complex world. California management review, 43(4):8–25, 2001.">130</a>]</span>.</p>
+<figure class="margin-caption align-center" id="id16">
+<span id="figure-4-2"></span><a class="reference internal image-reference" href="_images/figure4_2_behavior_modes.png"><img alt="Figure 4.2" src="_images/figure4_2_behavior_modes.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Common modes of behavior in dynamic systems, occurring based on the presence of positive and negative feedback relationships, and linear and non-linear interactions. Adapted from <span id="id2">Sterman [<a class="reference internal" href="R.Bibliography.html#id57" title="John D Sterman. System dynamics modeling: tools for learning in a complex world. California management review, 43(4):8–25, 2001.">130</a>]</span>.</span><a class="headerlink" href="#id16" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The nature of feedback processes in a dynamic system shapes its fundamental behavior: positive feedbacks generate their own growth, negative feedbacks self-limit, seeking balance and equilibrium. In this manner, feedback processes give rise to different regimes, multiples of which could be present in each mode of behavior. Consider a population of mammals growing exponentially until it reaches the carrying capacity of its environment (referred to as S-shaped growth). When the population is exponentially growing, the regime is dominated by positive feedback relationships that reinforce its growth. As the population approaches its carrying capacity limit, negative feedback structures begin to dominate, counteracting the growth and establishing a stable equilibrium. Shifts between regimes can be thought of as tipping points, mathematically defined as unstable equilibria, where the presence of positive feedbacks amplifies disturbances and moves the system to a new equilibrium point. In the case of stable equilibria, the presence of negative feedbacks dampens any small disturbance and maintains the system at a stable state. As different feedback relationships govern each regime, different factors (those making up each feedback mechanism) are activated and shape the states the system is found in, as well as define the points of equilibria.</p>
+<p>For simple stylized models with a small number of states, system dynamics analysis can analytically derive these equilibria, the conditions for their stability and the factors determining them. The ability for this to be performed is, however, significantly challenged when it comes to systems that attempt to more closely resemble real complex systems. We argue this is the case for several reasons. First, besides generally exhibiting complex nonlinear dynamics, real world systems are also made up from larger numbers of interacting elements, which often makes the analytic derivation of system characteristics intractable <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id59" title="John M Anderies, Jean-Denis Mathias, and Marco A Janssen. Knowledge infrastructure and safe operating spaces in social–ecological systems. Proceedings of the National Academy of Sciences, 116(12):5277–5284, 2019.">131</a>, <a class="reference internal" href="R.Bibliography.html#id58" title="Rachata Muneepeerakul and John M Anderies. The emergence and resilience of self-organized governance in coupled infrastructure systems. Proceedings of the National Academy of Sciences, 117(9):4617–4622, 2020.">132</a>]</span>. Second, human-natural systems temporally evolve and transform when human state-aware action is present. Consider, for instance, humans recreationally hunting the aforementioned population of mammals. Humans act based on the mammal population levels by enforcing hunting quotas or establishing protected territories or eliminating other predators. The mammal population reacts in a response, giving birth to ever changing state-action-consequence feedbacks, the path dependencies of which become difficult to diagnose and understand (e.g., <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id60" title="Antonia Hadjimichael, Patrick M Reed, and Julianne D Quinn. Navigating deeply uncertain tradeoffs in harvested predator-prey systems. Complexity, 2020.">133</a>]</span>). Trying to simulate the combination of these two challenges (large numbers of state-aware agents interacting with a natural resource and with each other) produces intractable models that require advanced heuristics to analyze their properties and establish useful inferences.</p>
+<p>Sensitivity analysis paired with exploratory modeling methods offers a promising set of tools to address these challenges. We present a simple demonstrative application based on <span id="id5">Quinn <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id61" title="Julianne D Quinn, Patrick M Reed, and Klaus Keller. Direct policy search for robust multi-objective management of deeply uncertain socio-ecological tipping points. Environmental modelling &amp; software, 92:125–141, 2017.">134</a>]</span>. This stylistic example was first developed by <span id="id6">Carpenter <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id163" title="S. R. Carpenter, D. Ludwig, and W. A. Brock. Management of Eutrophication for Lakes Subject to Potentially Irreversible Change. Ecological Applications, 9(3):751–771, August 1999. URL: http://onlinelibrary.wiley.com/doi/10.1890/1051-0761(1999)009[0751:MOEFLS]2.0.CO;2/abstract, doi:10.1890/1051-0761(1999)009[0751:MOEFLS]2.0.CO;2.">135</a>]</span> and represents a town that must balance its agricultural and industrial productivity with the pollution it creates in a downstream lake. Increased productivity allows for increased profits, which the town aims to maximize, but it also produces more pollution for the lake. Too much phosphorus pollution can cause irreversible eutrophication, a process known as “tipping” the lake. The model of phosphorus in the lake <span class="math notranslate nohighlight">\(X_t\)</span> at time <span class="math notranslate nohighlight">\(t\)</span> is governed by:</p>
+<div class="math notranslate nohighlight">
+\[X_{t+1}= X_{t}+a_{t}+\frac{X_{t}^q} {1+X_{t}^q}-bX_t+\varepsilon\]</div>
+<p>where <span class="math notranslate nohighlight">\(a_t \in [0,0.1]\)</span> is the town’s pollution release at each timestep, <span class="math notranslate nohighlight">\(b\)</span> is the natural decay rate of phosphorus in the lake, <span class="math notranslate nohighlight">\(q\)</span> defines the lake’s recycling rate (primarily through sediments), and <span class="math notranslate nohighlight">\(\varepsilon\)</span> represents uncontrollable natural inflows of pollution modeled as a log-normal distribution with a given mean, <span class="math notranslate nohighlight">\(\mu\)</span>, and standard deviation <span class="math notranslate nohighlight">\(\sigma\)</span>.</p>
+<p>Panels (a-c) in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> plot the fluxes of phosphorus into the lake versus the mass accumulation of phosphorus in the lake. The red line corresponds to the phosphorus sinks in the lake (natural decay), given by <span class="math notranslate nohighlight">\(bX_t\)</span>. The grey shaded area represents the lake’s phosphorus recycling flux, given by <span class="math notranslate nohighlight">\(\frac{X_{t}^q} {1+X_{t}^q}\)</span>. The points of intersection indicate the system’s equilibria, two of which are stable, and one is unstable (also known as the tipping point). The stable equilibrium in the bottom left of the figure reflects an oligotrophic lake, whereas the stable equilibrium in the top right represents a eutrophic lake. With increasing phosphorus values, the tipping point can be crossed, and the lake will experience irreversible eutrophication, as the recycling rate would exceed the removal rate even if the town’s pollution became zero. In the absence of anthropogenic and natural inflows of pollution in the lake (<span class="math notranslate nohighlight">\(a_t\)</span> and <span class="math notranslate nohighlight">\(\varepsilon\)</span> respectively), the area between the bottom-left black point and the white point in the middle can be considered as the safe operating space, before emission levels cross the tipping point.</p>
+<figure class="margin-caption align-center" id="id17">
+<span id="figure-4-3"></span><a class="reference internal image-reference" href="_images/figure4_3_lake_problem_fluxes.png"><img alt="Figure 4.3" src="_images/figure4_3_lake_problem_fluxes.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Fluxes of phosphorus with regards to mass of phosphorus in the lake and sensitivity analysis results, assuming <span class="math notranslate nohighlight">\(b=0.42\)</span> and <span class="math notranslate nohighlight">\(q=2\)</span>. (a) Fluxes of phosphorus assuming no emmisions policy and no natural inflows. (b-c) Fluxes phosphorus when applying two different emissions policies. The “Best economic policy” and the “Most reliable policy” have been identified by <span id="id7">Quinn <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id61" title="Julianne D Quinn, Patrick M Reed, and Klaus Keller. Direct policy search for robust multi-objective management of deeply uncertain socio-ecological tipping points. Environmental modelling &amp; software, 92:125–141, 2017.">134</a>]</span> and can be found at <span id="id8">Quinn [<a class="reference internal" href="R.Bibliography.html#id164" title="Julianne Quinn. Julianneq/Lake_problem_dps. December 2017. original-date: 2017-02-06T18:33:54Z. URL: https://github.com/julianneq/Lake_Problem_DPS (visited on 2021-06-14).">136</a>]</span>. (d) Results of a sensitivity analysis on the parameters of the model most consequential to the reliability of the “Most reliable policy”. The code to replicate the sensitivity analysis can be found at <span id="id9">Hadka [<a class="reference internal" href="R.Bibliography.html#id165" title="David Hadka. Project-Platypus/Rhodium. 2017. original-date: 2015-10-29T18:08:43Z. URL: https://github.com/Project-Platypus/Rhodium (visited on 2021-06-14).">137</a>]</span>. Panels (a-c) are used courtesy of Julianne Quinn, University of Virginia.</span><a class="headerlink" href="#id17" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The town has identified two potential policies that can be used to manage this lake, one that maximizes its economic profits (“best economic policy”) and one that maximizes the time below the tipping point (“most reliable policy”). Panels (b-c) in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> add the emissions from these policies to the recycling flux and show how the equilibria points shift as a result. In both cases the stable oligotrophic equilibrium increases and the tipping point decreases, narrowing the safe operating space <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id59" title="John M Anderies, Jean-Denis Mathias, and Marco A Janssen. Knowledge infrastructure and safe operating spaces in social–ecological systems. Proceedings of the National Academy of Sciences, 116(12):5277–5284, 2019.">131</a>, <a class="reference internal" href="R.Bibliography.html#id166" title="Stephen R. Carpenter, William A. Brock, Carl Folke, Egbert H. van Nes, and Marten Scheffer. Allowing variance may enlarge the safe operating space for exploited ecosystems. Proceedings of the National Academy of Sciences, 112(46):14384–14389, November 2015. URL: http://www.pnas.org/content/112/46/14384 (visited on 2017-08-18), doi:10.1073/pnas.1511804112.">138</a>]</span>. The best economic policy results in a much narrower space of action, with the tipping point very close to the oligotrophic equilibrium. The performance of both policies depends significantly on the system parameters. For example, a higher value of <span class="math notranslate nohighlight">\(b\)</span>, the natural decay rate, would shift the red line upward, moving the equilibria points and widening the safe operating space. Inversely, a higher value of <span class="math notranslate nohighlight">\(q\)</span>, the lake’s recycling rate, would shift the recycling line upward, moving the tipping point lower and decreasing the safe operating space. The assumptions under which these policies were identified are therefore critical to their performance and any potential uncertainty in the parameter values could be detrimental to the system’s objectives being met.</p>
+<p>Sensitivity analysis can be used to clarify the role these parameters play on policy performance. <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> (d) shows the results of a Sobol sensitivity analysis on the reliability of the “most reliable” policy in a radial convergence diagram. The significance of each parameter is indicated by the size of circles corresponding to it. The size of the interior dark circle indicates the parameter’s first-order effects and the size of the exterior circle indicates the parameter’s total-order effects. The thickness of the lines between two parameters indicated the extent of their interaction (second-order effects). In this case, parameters <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(q\)</span> appear to have the most significant importance on the system, followed by the mean, <span class="math notranslate nohighlight">\(\mu\)</span>, of the natural inflows. All these parameters function in a manner that shifts the location of the three equilibria and therefore policies that are identified ignoring this parametric uncertainty might fail to meet their intended goals.</p>
+<p>It is worth mentioning that current sensitivity analysis methods are somewhat challenged in addressing several system dynamics analysis questions. The fundamental reason is that sensitivity analysis methods and tools have been developed to gauge numerical sensitivity of model output to changes in factor values. This is natural, as most simulation studies (e.g., all aforementioned examples) have been traditionally concerned with this type of sensitivity. In system dynamics modeling, however, a more important and pertinent concern is changes between regimes or between behavior modes (also known as bifurcations) as a result of changes in model factors <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id57" title="John D Sterman. System dynamics modeling: tools for learning in a complex world. California management review, 43(4):8–25, 2001.">130</a>, <a class="reference internal" href="R.Bibliography.html#id63" title="Mustafa Hekimoğlu and Yaman Barlas. Sensitivity analysis for models with multiple behavior modes: a method based on behavior pattern measures. System Dynamics Review, 32(3-4):332–362, 2016.">139</a>]</span>. This poses two new challenges. First, identifying a change in regime depends on several characteristics besides a change in output value, like the rate and direction of change. Second, behavior mode changes are qualitative and discontinuous, as equilibria change in stability but also move in and out of existence.</p>
+<p>Despite these challenges, recent advanced sensitivity analysis methods can help illuminate which factors in a system are most important in shaping boundary conditions (tipping points) between different regimes and determining changes in behavior modes. Reviewing such methods is outside the scope of this text, but the reader is directed to the examples of <span id="id12">Eker <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id64" title="Sibel Eker, Elena Rovenskaya, Michael Obersteiner, and Simon Langan. Practice and perspectives in the validation of resource management models. Nature communications, 9(1):1–10, 2018.">22</a>]</span> and <span id="id13">Hadjimichael <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id60" title="Antonia Hadjimichael, Patrick M Reed, and Julianne D Quinn. Navigating deeply uncertain tradeoffs in harvested predator-prey systems. Complexity, 2020.">133</a>]</span>, who apply parameterised perturbation on the functional relationships of a system to study the effects of model structural uncertainty on model outputs and bifurcations, and <span id="id14">Hekimoğlu and Barlas [<a class="reference internal" href="R.Bibliography.html#id63" title="Mustafa Hekimoğlu and Yaman Barlas. Sensitivity analysis for models with multiple behavior modes: a method based on behavior pattern measures. System Dynamics Review, 32(3-4):332–362, 2016.">139</a>]</span> and <span id="id15">Steinmann <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id65" title="Patrick Steinmann, Willem L Auping, and Jan H Kwakkel. Behavior-based scenario discovery using time series clustering. Technological Forecasting and Social Change, 156:120052, 2020.">140</a>]</span> who, following wide sampling of uncertain inputs, cluster the resulting time series in modes of behavior and identify most important factors for each.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following badge to try out an interactive tutorial on performing a sensitivity analysis to discover consequential dynamics:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/fishery_dynamics.ipynb">Factor Discovery Jupyter Notebook</a></p>
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+  <section id="consequential-scenarios-what-is-controlling-consequential-outcomes">
+<span id="consequential-scenarios"></span><h1>Consequential Scenarios: What is Controlling Consequential Outcomes?<a class="headerlink" href="#consequential-scenarios-what-is-controlling-consequential-outcomes" title="Link to this heading">#</a></h1>
+<p>As overviewed in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>, most models are abstractions of systems in the real world. When sufficient confidence has been established in a model, it can then act as a surrogate for the actual system, in that the consequences of potential stressors, proposed actions or other changes can be evaluated by computer model simulations <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>]</span>. A model simulation then represents a computational experiment, which can be used to assess how the modeled system would behave should the various changes come to be. Steven Bankes coined the term exploratory modeling to describe the use of large sets of such computational experiments to investigate their implications on the system. <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a> presents a typical workflow of an exploratory modeling application. Exploratory modeling approaches typically use sampling designs to generate large ensembles of states that represent combinations of changes happening together, spanning the entire range of potential values a factor might take (indicated in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a> by numbers 2-5). This perspective on modeling is particularly relevant to studies making long term projections into the future.</p>
+<figure class="margin-caption align-center" id="id26">
+<span id="figure-4-4"></span><a class="reference internal image-reference" href="_images/figure4_4_exploratory_workflow.png"><img alt="Figure 4_4" src="_images/figure4_4_exploratory_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">A typical exploratory modeling workflow</span><a class="headerlink" href="#id26" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>In the long-term policy analysis literature, exploratory modeling has prominently placed itself as an alternative to traditional narrative scenario or assumptions-based planning approaches, in what can be summarized in the following two-pronged critique <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>, <a class="reference internal" href="R.Bibliography.html#id67" title="Steven C Bankes, Robert J Lempert, and Steven W Popper. Computer-assisted reasoning. Computing in Science &amp; Engineering, 3(2):71–77, 2001.">141</a>, <a class="reference internal" href="R.Bibliography.html#id66" title="Robert J. Lempert, Steven W. Popper, and Steven C. Bankes. Shaping the Next One Hundred Years. RAND Corporation, 2003. URL: https://www.rand.org/pubs/monograph_reports/MR1626.html (visited on 2017-09-14).">142</a>]</span>. The most prevalent criticism sees that the future and how it might evolve is both highly complex and deeply uncertain. Despite its benefits for interpretation and intuitive appeal, a small number of scenarios invariably misses many other potential futures that did not get selected as sufficiently representative of the future. This is especially the case for aggregate, narrative scenarios that describe simultaneous changes in multiple sectors together (e.g., “increased energy demand, combined with high agricultural land use and large economic growth”), such as the emission scenarios produced by the Intergovernmental Panel on Climate Change <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id68" title="Jonathan R Lamontagne, Patrick M Reed, Robert Link, Katherine V Calvin, Leon E Clarke, and James A Edmonds. Large ensemble analytic framework for consequence-driven discovery of climate change scenarios. Earth's Future, 6(3):488–504, 2018.">143</a>]</span>. The bias introduced by this reduced set of potential changes can skew inferences drawn from the model, particularly when the original narrative scenarios are focused on a single or narrow set of measures of system behavior.</p>
+<p>The second main criticism of traditional narrative scenario-based planning methods is that they provide no systematic way to distinguish which of the constituent factors lead to the undesirable consequences produced by a scenario. Narrative scenarios (e.g., the scenario matrix framework of RCPs-SSPs-SPAs; <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id69" title="Brian C O’Neill, Elmar Kriegler, Keywan Riahi, Kristie L Ebi, Stephane Hallegatte, Timothy R Carter, Ritu Mathur, and Detlef P van Vuuren. A new scenario framework for climate change research: the concept of shared socioeconomic pathways. Climatic change, 122(3):387–400, 2014.">144</a>]</span>) encompass multiple changes happening together selected to span the range of potential changes but are not typically generated in a systematic factorial manner that considers the multiple ways the factors can be combined. This has two critical limitations. It obfuscates the role each component factor plays in the system, both in isolation and in combination with others (e.g., “is it the increased energy demand or the high agricultural land use that cause unbearable water stress?”). It also renders the delineation of how much change in a factor is critical near impossible. Consider, for example, narrative scenario A with a 5% increase in energy demand, and scenario B with a 30% increase in energy demand, which would have dire consequences. At which point between 5% and 30% do the dire consequences actually begin to occur? Such questions cannot be answered without a wide exploration of the space of potential changes. It should be noted that for some levels of model complexity and computational demands (e.g., global-scale models) there is little feasible recourse beyond the use of narrative scenarios.</p>
+<p>Exploratory modeling is typically paired with scenario discovery methods (indicated by number 9 in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>) that identify which of the scenarios (also known as states of the world) generated indeed have consequences of interest for stakeholders and policy makers, in an approach referred to as ensemble-based scenario-discovery <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id16" title="Benjamin P Bryant and Robert J Lempert. Thinking inside the box: a participatory, computer-assisted approach to scenario discovery. Technological Forecasting and Social Change, 77(1):34–49, 2010.">45</a>, <a class="reference internal" href="R.Bibliography.html#id71" title="Robert J Lempert, David G Groves, Steven W Popper, and Steve C Bankes. A general, analytic method for generating robust strategies and narrative scenarios. Management science, 52(4):514–528, 2006.">102</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>. This approach therefore flips the planning analysis from one that attempts to predict future system conditions to one that attempts to discover the (un)desirable future conditions. Ensemble-based scenario discovery can thus inform what modeling choices yield the most consequential behavioral changes or outcomes, especially when considering deeply uncertain, scenario-informed projections <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id12" title="Enayat A Moallemi, Jan Kwakkel, Fjalar J de Haan, and Brett A Bryan. Exploratory modeling for analyzing coupled human-natural systems under uncertainty. Global Environmental Change, 65:102186, 2020.">9</a>, <a class="reference internal" href="R.Bibliography.html#id72" title="Warren E Walker, Marjolijn Haasnoot, and Jan H Kwakkel. Adapt or perish: a review of planning approaches for adaptation under deep uncertainty. Sustainability, 5(3):955–979, 2013.">145</a>]</span>. The relative likelihoods and relevance of the discovered scenarios can be subsequently evaluated by the practitioners a posteriori, within a richer context of knowing the wider set of potential consequences <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id73" title="Suraje Dessai, Mike Hulme, Robert Lempert, and Roger Pielke Jr. Climate prediction: a limit to adaptation. Adapting to climate change: thresholds, values, governance, 64:78, 2009.">146</a>]</span>. This can include changing how an analysis is framed (number 10 in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>). For instance, one could initially focus on ensemble modeling of vulnerability using a single uncertain factor that is assumed to be well characterized by historical observations (e.g., streamflow; this step is represented by numbers 2-3 in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>). The analysis can then shift to include projections of more factors treated as deeply uncertain (e.g., urbanization, population demands, temperature, and snow-melt) to yield a far wider space of challenging projected futures. UC experiments contrasting these two framings can be highly valuable for tracing how vulnerability inferences change as the modeled space of futures expands from the historical baseline <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id61" title="Julianne D Quinn, Patrick M Reed, and Klaus Keller. Direct policy search for robust multi-objective management of deeply uncertain socio-ecological tipping points. Environmental modelling &amp; software, 92:125–141, 2017.">134</a>]</span>.</p>
+<p>An important nuance to be clarified here is that the focus or purpose of a modeling exercise plays a major role in whether a given factor of interest is considered well-characterized or deeply uncertain. Take the example context of characterizing temperature or streamflow extremes, where for each state variable of interest for a given location of focus there is a century of historical observations. Clearly, the observation technologies will have evolved over time uniquely for temperature and streamflow measurements and they likely lack replicate experiments (data uncertainty). A century of record will be insufficient to capture very high impact and rare extreme events (i.e., increasingly poor structural/parametric inference for the distributions of specific extreme single or compound events). The mechanistic processes as well as their evolving variability will be interdependent but uniquely different for each of these state variables. A large body of statistical literature exists focusing on the topics of synthetic weather <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id74" title="Manuel Herrera, Sukumar Natarajan, David A Coley, Tristan Kershaw, Alfonso P Ramallo-González, Matthew Eames, Daniel Fosas, and Michael Wood. A review of current and future weather data for building simulation. Building Services Engineering Research and Technology, 38(5):602–627, 2017.">81</a>, <a class="reference internal" href="R.Bibliography.html#id75" title="Daniel S Wilks and Robert L Wilby. The weather generation game: a review of stochastic weather models. Progress in physical geography, 23(3):329–357, 1999.">82</a>]</span> or streamflow <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id76" title="JR Lamontagne and JR Stedinger. Generating synthetic streamflow forecasts with specified precision. Journal of Water Resources Planning and Management, 144(4):04018007, 2018.">83</a>, <a class="reference internal" href="R.Bibliography.html#id77" title="Sanghamitra Medda and Kalyan Kumar Bhar. Comparison of single-site and multi-site stochastic models for streamflow generation. Applied Water Science, 9(3):67, 2019.">84</a>]</span> generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound extremes. These history-focused approaches can be viewed as providing well-characterized quantifications of streamflow or temperature distributions; however, they do not capture how coupled natural-human processes can fundamentally change their dynamics when transitioning to projections of longer-term futures (e.g., streamflow and temperature in 2055). Consequently, changing the focus of the modeling to making long term projections of future streamflow or temperature now makes these processes deeply uncertain.</p>
+<p>Scenario discovery methods (number 9 in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>) can be qualitative or quantitative and they generally attempt to distinguish futures in which a system or proposed policies to manage the system meet or miss their goals <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>. The emphasis placed by exploratory modeling on model outputs that have decision relevant consequences represents a shift toward a broader class of metrics that are reflective of the stakeholders’ concerns, agency and preferences (also discussed in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>). As a result, sensitivity analysis and scenario discovery methods in this context are therefore applied to performance metrics that go beyond model error but are rather focused on broader metrics such as the resilience of a sector, the reliability of a process, or the vulnerability of a population in the face of uncertainty. In exploratory modeling literature, this metric is most typically—but not always—a measure of robustness (number 8 in Fig. 13). Robustness is a property of a system or a design choice capturing its insensitivity to uncertainty and can be measured via a variety of means, most recently reviewed by <span id="id12">Herman <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id78" title="Jonathan D Herman, Patrick M Reed, Harrison B Zeff, and Gregory W Characklis. How should robustness be defined for water systems planning under change? Journal of Water Resources Planning and Management, 141(10):04015012, 2015.">147</a>]</span> and <span id="id13">McPhail <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id55" title="Cameron McPhail, HR Maier, JH Kwakkel, M Giuliani, A Castelletti, and S Westra. Robustness metrics: how are they calculated, when should they be used and why do they give different results? Earth's Future, 6(2):169–191, 2018.">129</a>]</span>.</p>
+<p>Scenario discovery is typically performed through the use of algorithms applied on large databases of model runs, generated through exploratory modeling, with each model run representing the performance of the system in one potential state of the world. The algorithms seek to identify the combinations of factor values (e.g., future conditions) that best distinguish the cases in which the system does or does not meet its objectives. The most widely known classification algorithms are the Patient Rule Induction Method (PRIM; <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id79" title="Jerome H Friedman and Nicholas I Fisher. Bump hunting in high-dimensional data. Statistics and Computing, 9(2):123–143, 1999.">97</a>]</span>) and Classification and Regression Trees (CART; <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id80" title="Leo Breiman, Jerome Friedman, Charles J Stone, and Richard A Olshen. Classification and regression trees. CRC press, 1984.">98</a>]</span>). These factor mapping algorithms create orthogonal boundaries (multi-dimensional hypercubes) between states of the world that are successful or unsuccessful in meeting the system’s goals <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>]</span>. The algorithms attempt to strike a balance between simplicity of classification (and as a result, interpretability) and accuracy <span id="id17">[<a class="reference internal" href="R.Bibliography.html#id16" title="Benjamin P Bryant and Robert J Lempert. Thinking inside the box: a participatory, computer-assisted approach to scenario discovery. Technological Forecasting and Social Change, 77(1):34–49, 2010.">45</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>, <a class="reference internal" href="R.Bibliography.html#id82" title="Robert J Lempert. Robust decision making (rdm). In Decision making under deep uncertainty, pages 23–51. Springer, Cham, 2019.">148</a>]</span>.</p>
+<p>Even though these approaches have been shown to yield interpretable and relevant scenarios <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id83" title="Jan H Kwakkel and Marjolijn Haasnoot. Supporting dmdu: a taxonomy of approaches and tools. In Decision Making under Deep Uncertainty, pages 355–374. Springer, Cham, 2019.">149</a>]</span>, several authors have pointed out the limitations of these methods with regards to their division of space in orthogonal behavioral and non-behavioral regions <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id84" title="BC Trindade, PM Reed, and GW Characklis. Deeply uncertain pathways: integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134:103442, 2019.">150</a>]</span>. Due to their reliance on boundaries orthogonal to the uncertainty axes, PRIM and CART cannot capture interactions between the various uncertain factors considered, which can often be significant <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id85" title="Julianne D Quinn, Patrick M Reed, Matteo Giuliani, Andrea Castelletti, Jared W Oyler, and Robert E Nicholas. Exploring how changing monsoonal dynamics and human pressures challenge multireservoir management for flood protection, hydropower production, and agricultural water supply. Water Resources Research, 54(7):4638–4662, 2018.">151</a>]</span>. More advanced methods have been proposed to address this drawback, with logistic regression being perhaps the most prominent <span id="id21">[<a class="reference internal" href="R.Bibliography.html#id85" title="Julianne D Quinn, Patrick M Reed, Matteo Giuliani, Andrea Castelletti, Jared W Oyler, and Robert E Nicholas. Exploring how changing monsoonal dynamics and human pressures challenge multireservoir management for flood protection, hydropower production, and agricultural water supply. Water Resources Research, 54(7):4638–4662, 2018.">151</a>, <a class="reference internal" href="R.Bibliography.html#id86" title="DF Gold, PM Reed, BC Trindade, and GW Characklis. Identifying actionable compromises: navigating multi-city robustness conflicts to discover cooperative safe operating spaces for regional water supply portfolios. Water Resources Research, 55(11):9024–9050, 2019.">152</a>, <a class="reference internal" href="R.Bibliography.html#id87" title="JR Lamontagne, PM Reed, G Marangoni, K Keller, and GG Garner. Robust abatement pathways to tolerable climate futures require immediate global action. Nature Climate Change, 9(4):290–294, 2019.">153</a>]</span>. Logistic regression can produce boundaries that are not necessarily orthogonal to each uncertainty axis, nor necessarily linear, if interactive terms between two parameters are used to build the regression model. It also describes the probability that a state of the world belongs to the scenarios that lead to failure. This feature allows users to define regions of success based on a gradient of estimated probability of success in those worlds, unlike PRIM which only classifies states of the world in two regions <span id="id22">[<a class="reference internal" href="R.Bibliography.html#id85" title="Julianne D Quinn, Patrick M Reed, Matteo Giuliani, Andrea Castelletti, Jared W Oyler, and Robert E Nicholas. Exploring how changing monsoonal dynamics and human pressures challenge multireservoir management for flood protection, hydropower production, and agricultural water supply. Water Resources Research, 54(7):4638–4662, 2018.">151</a>, <a class="reference internal" href="R.Bibliography.html#id202" title="Antonia Hadjimichael, Julianne Quinn, Erin Wilson, Patrick Reed, Leon Basdekas, David Yates, and Michelle Garrison. Defining Robustness, Vulnerabilities, and Consequential Scenarios for Diverse Stakeholder Interests in Institutionally Complex River Basins. Earth's Future, 8(7):e2020EF001503, 2020. URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020EF001503 (visited on 2020-07-13), doi:10.1029/2020EF001503.">154</a>]</span>.</p>
+<p>Another more advanced factor mapping method is boosted trees <span id="id23">[<a class="reference internal" href="R.Bibliography.html#id89" title="Yoav Freund, Robert Schapire, and Naoki Abe. A short introduction to boosting. Journal-Japanese Society For Artificial Intelligence, 14(771-780):1612, 1999.">99</a>, <a class="reference internal" href="R.Bibliography.html#id88" title="Harris Drucker and Corinna Cortes. Boosting decision trees. Advances in neural information processing systems, pages 479–485, 1996.">155</a>]</span>. Boosted trees can avoid two limitations inherent to the application of logistic regression: i) to build a nonlinear classification model the interactive term between two uncertainties needs to be pre-specified and cannot be discovered (e.g., we need to know a-priori whether factor <span class="math notranslate nohighlight">\(x_1\)</span> interacts with <span class="math notranslate nohighlight">\(x_2\)</span> in a relationship that looks like <span class="math notranslate nohighlight">\(x_1\)</span>·<span class="math notranslate nohighlight">\(x_2\)</span> or <span class="math notranslate nohighlight">\(x_1^{x_2}\)</span>); and ii) the subspaces defined are always convex. The application of such a factor mapping algorithm is limited in the presence of threshold-based rules with discrete actions in a modeled system (e.g., “if network capacity is low, build new infrastructure”), which results in failure regions that are nonlinear and non-convex <span id="id24">[<a class="reference internal" href="R.Bibliography.html#id84" title="BC Trindade, PM Reed, and GW Characklis. Deeply uncertain pathways: integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134:103442, 2019.">150</a>]</span>. Boosting works by creating an ensemble of classifiers and forcing some of them to focus on the hard-to-learn parts of the problem, and others to focus on the easy-to-learn parts. Boosting applied to CART trees can avoid the aforementioned challenges faced by other scenario discovery methods, while resisting overfitting <span id="id25">[<a class="reference internal" href="R.Bibliography.html#id90" title="Kevin P Murphy. Machine learning: a probabilistic perspective. MIT press, 2012.">156</a>]</span>, assuring the identified success and failure regions are still easy to interpret.</p>
+<p>Below we provide an example application of two scenario discovery methods, PRIM and logistic regression, using the lake problem introduced in the previous section. From the sensitivity analysis results presented in <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> (d), we can already infer that parameters <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(q\)</span> have important effects on model outputs (i.e., we have performed factor prioritization). Scenario discovery (i.e., factor mapping) complements this analysis by further identifying the specific values of b and q that can lead to consequential and undesirable outcomes. For the purposes of demonstration, we can assume the undesirable outcome in this case is defined as the management policy failing to achieve 90% reliability in a state of the world.</p>
+<figure class="margin-caption align-center" id="id27">
+<span id="figure-4-5"></span><a class="reference internal image-reference" href="_images/figure14lake_problem_SD.png"><img alt="Figure 14." src="_images/figure14lake_problem_SD.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Scenario discovery for the lake problem, using (a) PRIM and (b) logistic regression.</span><a class="headerlink" href="#id27" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#figure-4-5"><span class="std std-numref">Fig. 4.5</span></a> shows the results of scenario discovery, performed through (a) PRIM and (b) logistic regression. Each point in the two panels indicates a potential state of the world, generated through Latin Hypercube Sampling. Each point is colored by whether the policy meets the above performance criterion, with blue indicating success and red indicating failure. PRIM identifies several orthogonal areas of interest, one of which is shown in panel (a). As discussed above, this necessary orthogonality limits how PRIM identifies areas of success (the area within the box). As factors  <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(q\)</span> interact in this system, the transition boundary between the regions of success and failure is not orthogonal to any of the axes. As a result, a large number of points in the bottom right and the top left of the figure are left outside of the identified region. Logistic regression can overcome this limitation by identifying a diagonal boundary between the two regions, seen in panel (b). This method also produces a gradient of estimated probability of success across these regions.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial on performing factor mapping using logistic regression:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/basin_users_logistic_regression.ipynb">Logistic Regression Jupyter Notebook</a></p>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#understanding-errors-what-is-controlling-model-performance">4.1. Understanding Errors: What Is Controlling Model Performance?</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#consequential-dynamics-what-is-controlling-model-behaviors-of-interest">4.2. Consequential Dynamics: What is Controlling Model Behaviors of Interest?</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#consequential-scenarios-what-is-controlling-consequential-outcomes">4.3. Consequential Scenarios: What is Controlling Consequential Outcomes?</a></li>
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+  <section id="sensitivity-analysis-diagnostic-exploratory-modeling">
+<span id="sensitivity-analysis"></span><h1><span class="section-number">4. </span>Sensitivity Analysis: Diagnostic &amp; Exploratory Modeling<a class="headerlink" href="#sensitivity-analysis-diagnostic-exploratory-modeling" title="Link to this heading">#</a></h1>
+<section id="understanding-errors-what-is-controlling-model-performance">
+<span id="understanding-errors"></span><h2><span class="section-number">4.1. </span>Understanding Errors: What Is Controlling Model Performance?<a class="headerlink" href="#understanding-errors-what-is-controlling-model-performance" title="Link to this heading">#</a></h2>
+<p>Sensitivity analysis is a diagnostic tool when reconciling model outputs with observed data. It is helpful for clarifying how and under what conditions modeling choices (structure, parameterization, data inputs, etc.) propagate through model components and manifest in their effects on model outputs. This exploration is performed through carefully designed sampling of multiple combinations of input parameters and subsequent evaluation of the model structures that are emerging as controlling factors. Model structure and parameterization are two of the most commonly explored aspects of models that have been a central focus when evaluating their performance relative to available observations <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id49" title="Keith Beven. Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in water resources, 16(1):41–51, 1993.">17</a>]</span>. Addressing these issues plays an important role in establishing credibility in model predictions, particularly in the positivist natural sciences literature. Traditional model evaluations compare the model with observed data, and then rely on expert judgements of its acceptability based on the closeness between simulation and observation with one or a small number of selected metrics. This approach can be myopic, as it is often impossible to use one metric to attribute a certain error and to link that with different parts of the model and its parameters <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id50" title="Hoshin V Gupta, Thorsten Wagener, and Yuqiong Liu. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. Hydrological Processes: An International Journal, 22(18):3802–3813, 2008.">127</a>]</span>. This means that, even when the error or fitting measure between the model estimates and observations is very small, it is not guaranteed that all the components in the model accurately represent the conceptual reality that the model is abstracting: propagated errors in different parts of the model might cancel each other out, or multiple parameterized implementations of the model can yield similar performance (i.e., equifinality <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id49" title="Keith Beven. Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in water resources, 16(1):41–51, 1993.">17</a>]</span>).</p>
+<p>The inherent complexity of a system hinders accepting or rejecting a model based on one performance measure and different types of measures can aid in evaluating the various components of a model as essentially a multiobjective problem <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id52" title="Keith J Beven. On hypothesis testing in hydrology: why falsification of models is still a really good idea. Wiley Interdisciplinary Reviews: Water, 5(3):e1278, 2018.">25</a>, <a class="reference internal" href="R.Bibliography.html#id51" title="Hoshin V Gupta, Martyn P Clark, Jasper A Vrugt, Gab Abramowitz, and Ming Ye. Towards a comprehensive assessment of model structural adequacy. Water Resources Research, 2012.">26</a>]</span>. In addition, natural systems mimicked by the models contain various interacting components that might act differently across spatial and temporal domains <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id30" title="Francesca Pianosi, Keith Beven, Jim Freer, Jim W Hall, Jonathan Rougier, David B Stephenson, and Thorsten Wagener. Sensitivity analysis of environmental models: a systematic review with practical workflow. Environmental Modelling &amp; Software, 79:214–232, 2016.">51</a>, <a class="reference internal" href="R.Bibliography.html#id53" title="JD Herman, PM Reed, and T Wagener. Time-varying sensitivity analysis clarifies the effects of watershed model formulation on model behavior. Water Resources Research, 49(3):1400–1414, 2013.">55</a>]</span>. This heterogeneity is lost when a single performance measure is used, as a highly dimensional and interactive system becomes aggregated through the averaging of spatial or temporal output errors <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id54" title="Keith Beven. Towards a coherent philosophy for modelling the environment. Proceedings of the royal society of London. Series A: mathematical, physical and engineering sciences, 458(2026):2465–2484, 2002.">15</a>]</span>. Therefore, diagnostic error analyses should consider multiple error signatures across different scales and states of concern when seeking to understand how model performance relates to observed data (<a class="reference internal" href="#figure-4-1"><span class="std std-numref">Fig. 4.1</span></a>). Diverse error signatures can be used to measure the consistency of underlying processes and behaviors of the model and to evaluate the dynamics of model controls under changing temporal and spatial conditions <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id56" title="Keith Beven and Jim Freer. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the glue methodology. Journal of hydrology, 249(1-4):11–29, 2001.">128</a>]</span>. Within this framework, even minimal information extracted from the data can be beneficial as it helps us unearth structural inadequacies in the model. In this context, proper selection of measures of model performance and the number of measures could play consequential roles in our understanding of the model and its predictions <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id55" title="Cameron McPhail, HR Maier, JH Kwakkel, M Giuliani, A Castelletti, and S Westra. Robustness metrics: how are they calculated, when should they be used and why do they give different results? Earth's Future, 6(2):169–191, 2018.">129</a>]</span>.</p>
+<p>As discussed earlier, instead of the traditional focus on using deterministic prediction that results in a single error measure, many plausible states and spaces could be searched for making different inferences and quantifying uncertainties. This process also requires estimates of prior probability distributions of all the important parameters and quantification of model behavior across input space. One strategy to reduce the search space is filtering of some model alternatives that are not consistent with observations and known system behaviors. Those implausible parts of the search space can be referred to as non-physical or non-behavioral alternatives <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id29" title="Neil R Edwards, David Cameron, and Jonathan Rougier. Precalibrating an intermediate complexity climate model. Climate dynamics, 37(7):1469–1482, 2011.">50</a>, <a class="reference internal" href="R.Bibliography.html#id158" title="Keith Beven and Andrew Binley. Glue: 20 years on. Hydrological processes, 28(24):5897–5918, 2014.">104</a>]</span>. This step is conducted before the Bayesian calibration exercise (see <a class="reference internal" href="A1_Uncertainty_Quantification.html#a1-uncertainty-quantification"><span class="std std-numref">Chapter A</span></a>).</p>
+<p>A comprehensive model diagnostic workflow typically entails the components demonstrated in <a class="reference internal" href="#figure-4-1"><span class="std std-numref">Fig. 4.1</span></a>. The workflow begins with the selection of model input parameters and their plausible ranges. After the parameter selection, we need to specify the design of experiment (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#design-of-experiments"><span class="std std-numref">Chapter 3.3</span></a>) and the sensitivity analysis method (<a class="reference internal" href="3_sensitivity_analysis_the_basics.html#sensitivity-analysis-methods"><span class="std std-numref">Chapter 3.4</span></a>) to be used. As previously discussed, these methods require different numbers of model simulations, and each method provides a different insights into the direct effects and interactions of the uncertain factors. In addition, the simulation time of the model and the available computational resources are two of the primary considerations that influence these decisions. After identifying the appropriate methods, we generate a matrix of input parameters, where each set of input parameters will be used to conduct a model simulation. The model can include one or more output variables that fluctuate in time and space. The next step is to analyze model performance by comparing model outputs with observations. As discussed earlier, the positivist model evaluation paradigm focuses on a single model performance metric (error), leading to a loss of information about model parameters and the suitability of the model’s structure. However, a thorough investigation of the temporal and spatial signatures of model outputs using various performance metrics or time- and space-varying sensitivity analyses can shed more light on the fitness of each parameter set and the model’s internal structure. This analysis provides diagnostic feedback on the importance and range of model parameters and can guide further improvement of the model algorithm.</p>
+<figure class="margin-caption align-center" id="id50">
+<span id="figure-4-1"></span><a class="reference internal image-reference" href="_images/figure4_1_diagnostic_workflow.png"><img alt="Figure 4.1" src="_images/figure4_1_diagnostic_workflow.png" style="width: 500px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 4.1 </span><span class="caption-text">Diagnostic evaluation of model fidelity using sensitivity analysis methods.</span><a class="headerlink" href="#id50" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following badge to try out an interactive tutorial on implementing a time-varying sensitivity analysis of HYMOD model parameters:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">HYMOD Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="consequential-dynamics-what-is-controlling-model-behaviors-of-interest">
+<h2><span class="section-number">4.2. </span>Consequential Dynamics: What is Controlling Model Behaviors of Interest?<a class="headerlink" href="#consequential-dynamics-what-is-controlling-model-behaviors-of-interest" title="Link to this heading">#</a></h2>
+<p>Consequential changes in dynamic systems can take many forms, but most dynamic behavior can be categorized in a few basic patterns. Feedback structures inherent in a system, be they positive or negative, generate these patterns which can be grouped into three groups for the simplest of systems: exponential growth, goal seeking, and oscillation (<a class="reference internal" href="#figure-4-2"><span class="std std-numref">Fig. 4.2</span></a>). A positive (or self-reinforcing) feedback gives rise to exponential growth, a negative (or self-correcting) feedback gives rise to a goal seeking mode, and negative feedbacks with time delays give rise to oscillatory behavior. Nonlinear interactions between the system’s feedback structures can give rise to more complex dynamic behavior modes, examples of which are also shown in <a class="reference internal" href="#figure-4-2"><span class="std std-numref">Fig. 4.2</span></a>, adapted from <span id="id10">Sterman [<a class="reference internal" href="R.Bibliography.html#id57" title="John D Sterman. System dynamics modeling: tools for learning in a complex world. California management review, 43(4):8–25, 2001.">130</a>]</span>.</p>
+<figure class="margin-caption align-center" id="id51">
+<span id="figure-4-2"></span><a class="reference internal image-reference" href="_images/figure4_2_behavior_modes.png"><img alt="Figure 4.2" src="_images/figure4_2_behavior_modes.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 4.2 </span><span class="caption-text">Common modes of behavior in dynamic systems, occurring based on the presence of positive and negative feedback relationships, and linear and non-linear interactions. Adapted from <span id="id11">Sterman [<a class="reference internal" href="R.Bibliography.html#id57" title="John D Sterman. System dynamics modeling: tools for learning in a complex world. California management review, 43(4):8–25, 2001.">130</a>]</span>.</span><a class="headerlink" href="#id51" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The nature of feedback processes in a dynamic system shapes its fundamental behavior: positive feedbacks generate their own growth, negative feedbacks self-limit, seeking balance and equilibrium. In this manner, feedback processes give rise to different regimes, multiples of which could be present in each mode of behavior. Consider a population of mammals growing exponentially until it reaches the carrying capacity of its environment (referred to as S-shaped growth). When the population is exponentially growing, the regime is dominated by positive feedback relationships that reinforce its growth. As the population approaches its carrying capacity limit, negative feedback structures begin to dominate, counteracting the growth and establishing a stable equilibrium. Shifts between regimes can be thought of as tipping points, mathematically defined as unstable equilibria, where the presence of positive feedbacks amplifies disturbances and moves the system to a new equilibrium point. In the case of stable equilibria, the presence of negative feedbacks dampens any small disturbance and maintains the system at a stable state. As different feedback relationships govern each regime, different factors (those making up each feedback mechanism) are activated and shape the states the system is found in, as well as define the points of equilibria.</p>
+<p>For simple stylized models with a small number of states, system dynamics analysis can analytically derive these equilibria, the conditions for their stability and the factors determining them. The ability for this to be performed is, however, significantly challenged when it comes to systems that attempt to more closely resemble real complex systems. We argue this is the case for several reasons. First, besides generally exhibiting complex nonlinear dynamics, real world systems are also made up from larger numbers of interacting elements, which often makes the analytic derivation of system characteristics intractable <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id59" title="John M Anderies, Jean-Denis Mathias, and Marco A Janssen. Knowledge infrastructure and safe operating spaces in social–ecological systems. Proceedings of the National Academy of Sciences, 116(12):5277–5284, 2019.">131</a>, <a class="reference internal" href="R.Bibliography.html#id58" title="Rachata Muneepeerakul and John M Anderies. The emergence and resilience of self-organized governance in coupled infrastructure systems. Proceedings of the National Academy of Sciences, 117(9):4617–4622, 2020.">132</a>]</span>. Second, human-natural systems temporally evolve and transform when human state-aware action is present. Consider, for instance, humans recreationally hunting the aforementioned population of mammals. Humans act based on the mammal population levels by enforcing hunting quotas or establishing protected territories or eliminating other predators. The mammal population reacts in a response, giving birth to ever changing state-action-consequence feedbacks, the path dependencies of which become difficult to diagnose and understand (e.g., <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id60" title="Antonia Hadjimichael, Patrick M Reed, and Julianne D Quinn. Navigating deeply uncertain tradeoffs in harvested predator-prey systems. Complexity, 2020.">133</a>]</span>). Trying to simulate the combination of these two challenges (large numbers of state-aware agents interacting with a natural resource and with each other) produces intractable models that require advanced heuristics to analyze their properties and establish useful inferences.</p>
+<p>Sensitivity analysis paired with exploratory modeling methods offers a promising set of tools to address these challenges. We present a simple demonstrative application based on <span id="id14">Quinn <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id61" title="Julianne D Quinn, Patrick M Reed, and Klaus Keller. Direct policy search for robust multi-objective management of deeply uncertain socio-ecological tipping points. Environmental modelling &amp; software, 92:125–141, 2017.">134</a>]</span>. This stylistic example was first developed by <span id="id15">Carpenter <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id163" title="S. R. Carpenter, D. Ludwig, and W. A. Brock. Management of Eutrophication for Lakes Subject to Potentially Irreversible Change. Ecological Applications, 9(3):751–771, August 1999. URL: http://onlinelibrary.wiley.com/doi/10.1890/1051-0761(1999)009[0751:MOEFLS]2.0.CO;2/abstract, doi:10.1890/1051-0761(1999)009[0751:MOEFLS]2.0.CO;2.">135</a>]</span> and represents a town that must balance its agricultural and industrial productivity with the pollution it creates in a downstream lake. Increased productivity allows for increased profits, which the town aims to maximize, but it also produces more pollution for the lake. Too much phosphorus pollution can cause irreversible eutrophication, a process known as “tipping” the lake. The model of phosphorus in the lake <span class="math notranslate nohighlight">\(X_t\)</span> at time <span class="math notranslate nohighlight">\(t\)</span> is governed by:</p>
+<div class="math notranslate nohighlight">
+\[X_{t+1}= X_{t}+a_{t}+\frac{X_{t}^q} {1+X_{t}^q}-bX_t+\varepsilon\]</div>
+<p>where <span class="math notranslate nohighlight">\(a_t \in [0,0.1]\)</span> is the town’s pollution release at each timestep, <span class="math notranslate nohighlight">\(b\)</span> is the natural decay rate of phosphorus in the lake, <span class="math notranslate nohighlight">\(q\)</span> defines the lake’s recycling rate (primarily through sediments), and <span class="math notranslate nohighlight">\(\varepsilon\)</span> represents uncontrollable natural inflows of pollution modeled as a log-normal distribution with a given mean, <span class="math notranslate nohighlight">\(\mu\)</span>, and standard deviation <span class="math notranslate nohighlight">\(\sigma\)</span>.</p>
+<p>Panels (a-c) in <a class="reference internal" href="#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> plot the fluxes of phosphorus into the lake versus the mass accumulation of phosphorus in the lake. The red line corresponds to the phosphorus sinks in the lake (natural decay), given by <span class="math notranslate nohighlight">\(bX_t\)</span>. The grey shaded area represents the lake’s phosphorus recycling flux, given by <span class="math notranslate nohighlight">\(\frac{X_{t}^q} {1+X_{t}^q}\)</span>. The points of intersection indicate the system’s equilibria, two of which are stable, and one is unstable (also known as the tipping point). The stable equilibrium in the bottom left of the figure reflects an oligotrophic lake, whereas the stable equilibrium in the top right represents a eutrophic lake. With increasing phosphorus values, the tipping point can be crossed, and the lake will experience irreversible eutrophication, as the recycling rate would exceed the removal rate even if the town’s pollution became zero. In the absence of anthropogenic and natural inflows of pollution in the lake (<span class="math notranslate nohighlight">\(a_t\)</span> and <span class="math notranslate nohighlight">\(\varepsilon\)</span> respectively), the area between the bottom-left black point and the white point in the middle can be considered as the safe operating space, before emission levels cross the tipping point.</p>
+<figure class="margin-caption align-center" id="id52">
+<span id="figure-4-3"></span><a class="reference internal image-reference" href="_images/figure4_3_lake_problem_fluxes.png"><img alt="Figure 4.3" src="_images/figure4_3_lake_problem_fluxes.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 4.3 </span><span class="caption-text">Fluxes of phosphorus with regards to mass of phosphorus in the lake and sensitivity analysis results, assuming <span class="math notranslate nohighlight">\(b=0.42\)</span> and <span class="math notranslate nohighlight">\(q=2\)</span>. (a) Fluxes of phosphorus assuming no emmisions policy and no natural inflows. (b-c) Fluxes phosphorus when applying two different emissions policies. The “Best economic policy” and the “Most reliable policy” have been identified by <span id="id16">Quinn <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id61" title="Julianne D Quinn, Patrick M Reed, and Klaus Keller. Direct policy search for robust multi-objective management of deeply uncertain socio-ecological tipping points. Environmental modelling &amp; software, 92:125–141, 2017.">134</a>]</span> and can be found at <span id="id17">Quinn [<a class="reference internal" href="R.Bibliography.html#id164" title="Julianne Quinn. Julianneq/Lake_problem_dps. December 2017. original-date: 2017-02-06T18:33:54Z. URL: https://github.com/julianneq/Lake_Problem_DPS (visited on 2021-06-14).">136</a>]</span>. (d) Results of a sensitivity analysis on the parameters of the model most consequential to the reliability of the “Most reliable policy”. The code to replicate the sensitivity analysis can be found at <span id="id18">Hadka [<a class="reference internal" href="R.Bibliography.html#id165" title="David Hadka. Project-Platypus/Rhodium. 2017. original-date: 2015-10-29T18:08:43Z. URL: https://github.com/Project-Platypus/Rhodium (visited on 2021-06-14).">137</a>]</span>. Panels (a-c) are used courtesy of Julianne Quinn, University of Virginia.</span><a class="headerlink" href="#id52" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The town has identified two potential policies that can be used to manage this lake, one that maximizes its economic profits (“best economic policy”) and one that maximizes the time below the tipping point (“most reliable policy”). Panels (b-c) in <a class="reference internal" href="#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> add the emissions from these policies to the recycling flux and show how the equilibria points shift as a result. In both cases the stable oligotrophic equilibrium increases and the tipping point decreases, narrowing the safe operating space <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id59" title="John M Anderies, Jean-Denis Mathias, and Marco A Janssen. Knowledge infrastructure and safe operating spaces in social–ecological systems. Proceedings of the National Academy of Sciences, 116(12):5277–5284, 2019.">131</a>, <a class="reference internal" href="R.Bibliography.html#id166" title="Stephen R. Carpenter, William A. Brock, Carl Folke, Egbert H. van Nes, and Marten Scheffer. Allowing variance may enlarge the safe operating space for exploited ecosystems. Proceedings of the National Academy of Sciences, 112(46):14384–14389, November 2015. URL: http://www.pnas.org/content/112/46/14384 (visited on 2017-08-18), doi:10.1073/pnas.1511804112.">138</a>]</span>. The best economic policy results in a much narrower space of action, with the tipping point very close to the oligotrophic equilibrium. The performance of both policies depends significantly on the system parameters. For example, a higher value of <span class="math notranslate nohighlight">\(b\)</span>, the natural decay rate, would shift the red line upward, moving the equilibria points and widening the safe operating space. Inversely, a higher value of <span class="math notranslate nohighlight">\(q\)</span>, the lake’s recycling rate, would shift the recycling line upward, moving the tipping point lower and decreasing the safe operating space. The assumptions under which these policies were identified are therefore critical to their performance and any potential uncertainty in the parameter values could be detrimental to the system’s objectives being met.</p>
+<p>Sensitivity analysis can be used to clarify the role these parameters play on policy performance. <a class="reference internal" href="#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> (d) shows the results of a Sobol sensitivity analysis on the reliability of the “most reliable” policy in a radial convergence diagram. The significance of each parameter is indicated by the size of circles corresponding to it. The size of the interior dark circle indicates the parameter’s first-order effects and the size of the exterior circle indicates the parameter’s total-order effects. The thickness of the lines between two parameters indicated the extent of their interaction (second-order effects). In this case, parameters <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(q\)</span> appear to have the most significant importance on the system, followed by the mean, <span class="math notranslate nohighlight">\(\mu\)</span>, of the natural inflows. All these parameters function in a manner that shifts the location of the three equilibria and therefore policies that are identified ignoring this parametric uncertainty might fail to meet their intended goals.</p>
+<p>It is worth mentioning that current sensitivity analysis methods are somewhat challenged in addressing several system dynamics analysis questions. The fundamental reason is that sensitivity analysis methods and tools have been developed to gauge numerical sensitivity of model output to changes in factor values. This is natural, as most simulation studies (e.g., all aforementioned examples) have been traditionally concerned with this type of sensitivity. In system dynamics modeling, however, a more important and pertinent concern is changes between regimes or between behavior modes (also known as bifurcations) as a result of changes in model factors <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id57" title="John D Sterman. System dynamics modeling: tools for learning in a complex world. California management review, 43(4):8–25, 2001.">130</a>, <a class="reference internal" href="R.Bibliography.html#id63" title="Mustafa Hekimoğlu and Yaman Barlas. Sensitivity analysis for models with multiple behavior modes: a method based on behavior pattern measures. System Dynamics Review, 32(3-4):332–362, 2016.">139</a>]</span>. This poses two new challenges. First, identifying a change in regime depends on several characteristics besides a change in output value, like the rate and direction of change. Second, behavior mode changes are qualitative and discontinuous, as equilibria change in stability but also move in and out of existence.</p>
+<p>Despite these challenges, recent advanced sensitivity analysis methods can help illuminate which factors in a system are most important in shaping boundary conditions (tipping points) between different regimes and determining changes in behavior modes. Reviewing such methods is outside the scope of this text, but the reader is directed to the examples of <span id="id21">Eker <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id64" title="Sibel Eker, Elena Rovenskaya, Michael Obersteiner, and Simon Langan. Practice and perspectives in the validation of resource management models. Nature communications, 9(1):1–10, 2018.">22</a>]</span> and <span id="id22">Hadjimichael <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id60" title="Antonia Hadjimichael, Patrick M Reed, and Julianne D Quinn. Navigating deeply uncertain tradeoffs in harvested predator-prey systems. Complexity, 2020.">133</a>]</span>, who apply parameterised perturbation on the functional relationships of a system to study the effects of model structural uncertainty on model outputs and bifurcations, and <span id="id23">Hekimoğlu and Barlas [<a class="reference internal" href="R.Bibliography.html#id63" title="Mustafa Hekimoğlu and Yaman Barlas. Sensitivity analysis for models with multiple behavior modes: a method based on behavior pattern measures. System Dynamics Review, 32(3-4):332–362, 2016.">139</a>]</span> and <span id="id24">Steinmann <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id65" title="Patrick Steinmann, Willem L Auping, and Jan H Kwakkel. Behavior-based scenario discovery using time series clustering. Technological Forecasting and Social Change, 156:120052, 2020.">140</a>]</span> who, following wide sampling of uncertain inputs, cluster the resulting time series in modes of behavior and identify most important factors for each.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following badge to try out an interactive tutorial on performing a sensitivity analysis to discover consequential dynamics:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/fishery_dynamics.ipynb">Factor Discovery Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="consequential-scenarios-what-is-controlling-consequential-outcomes">
+<span id="consequential-scenarios"></span><h2><span class="section-number">4.3. </span>Consequential Scenarios: What is Controlling Consequential Outcomes?<a class="headerlink" href="#consequential-scenarios-what-is-controlling-consequential-outcomes" title="Link to this heading">#</a></h2>
+<p>As overviewed in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>, most models are abstractions of systems in the real world. When sufficient confidence has been established in a model, it can then act as a surrogate for the actual system, in that the consequences of potential stressors, proposed actions or other changes can be evaluated by computer model simulations <span id="id25">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>]</span>. A model simulation then represents a computational experiment, which can be used to assess how the modeled system would behave should the various changes come to be. Steven Bankes coined the term exploratory modeling to describe the use of large sets of such computational experiments to investigate their implications on the system. <a class="reference internal" href="#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a> presents a typical workflow of an exploratory modeling application. Exploratory modeling approaches typically use sampling designs to generate large ensembles of states that represent combinations of changes happening together, spanning the entire range of potential values a factor might take (indicated in <a class="reference internal" href="#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a> by numbers 2-5). This perspective on modeling is particularly relevant to studies making long term projections into the future.</p>
+<figure class="margin-caption align-center" id="id53">
+<span id="figure-4-4"></span><a class="reference internal image-reference" href="_images/figure4_4_exploratory_workflow.png"><img alt="Figure 4_4" src="_images/figure4_4_exploratory_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 4.4 </span><span class="caption-text">A typical exploratory modeling workflow</span><a class="headerlink" href="#id53" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>In the long-term policy analysis literature, exploratory modeling has prominently placed itself as an alternative to traditional narrative scenario or assumptions-based planning approaches, in what can be summarized in the following two-pronged critique <span id="id26">[<a class="reference internal" href="R.Bibliography.html#id15" title="Steve Bankes. Exploratory Modeling for Policy Analysis. Operations Research, 41(3):435–449, June 1993. URL: https://pubsonline.informs.org/doi/abs/10.1287/opre.41.3.435 (visited on 2018-09-11), doi:10.1287/opre.41.3.435.">36</a>, <a class="reference internal" href="R.Bibliography.html#id67" title="Steven C Bankes, Robert J Lempert, and Steven W Popper. Computer-assisted reasoning. Computing in Science &amp; Engineering, 3(2):71–77, 2001.">141</a>, <a class="reference internal" href="R.Bibliography.html#id66" title="Robert J. Lempert, Steven W. Popper, and Steven C. Bankes. Shaping the Next One Hundred Years. RAND Corporation, 2003. URL: https://www.rand.org/pubs/monograph_reports/MR1626.html (visited on 2017-09-14).">142</a>]</span>. The most prevalent criticism sees that the future and how it might evolve is both highly complex and deeply uncertain. Despite its benefits for interpretation and intuitive appeal, a small number of scenarios invariably misses many other potential futures that did not get selected as sufficiently representative of the future. This is especially the case for aggregate, narrative scenarios that describe simultaneous changes in multiple sectors together (e.g., “increased energy demand, combined with high agricultural land use and large economic growth”), such as the emission scenarios produced by the Intergovernmental Panel on Climate Change <span id="id27">[<a class="reference internal" href="R.Bibliography.html#id68" title="Jonathan R Lamontagne, Patrick M Reed, Robert Link, Katherine V Calvin, Leon E Clarke, and James A Edmonds. Large ensemble analytic framework for consequence-driven discovery of climate change scenarios. Earth's Future, 6(3):488–504, 2018.">143</a>]</span>. The bias introduced by this reduced set of potential changes can skew inferences drawn from the model, particularly when the original narrative scenarios are focused on a single or narrow set of measures of system behavior.</p>
+<p>The second main criticism of traditional narrative scenario-based planning methods is that they provide no systematic way to distinguish which of the constituent factors lead to the undesirable consequences produced by a scenario. Narrative scenarios (e.g., the scenario matrix framework of RCPs-SSPs-SPAs; <span id="id28">[<a class="reference internal" href="R.Bibliography.html#id69" title="Brian C O’Neill, Elmar Kriegler, Keywan Riahi, Kristie L Ebi, Stephane Hallegatte, Timothy R Carter, Ritu Mathur, and Detlef P van Vuuren. A new scenario framework for climate change research: the concept of shared socioeconomic pathways. Climatic change, 122(3):387–400, 2014.">144</a>]</span>) encompass multiple changes happening together selected to span the range of potential changes but are not typically generated in a systematic factorial manner that considers the multiple ways the factors can be combined. This has two critical limitations. It obfuscates the role each component factor plays in the system, both in isolation and in combination with others (e.g., “is it the increased energy demand or the high agricultural land use that cause unbearable water stress?”). It also renders the delineation of how much change in a factor is critical near impossible. Consider, for example, narrative scenario A with a 5% increase in energy demand, and scenario B with a 30% increase in energy demand, which would have dire consequences. At which point between 5% and 30% do the dire consequences actually begin to occur? Such questions cannot be answered without a wide exploration of the space of potential changes. It should be noted that for some levels of model complexity and computational demands (e.g., global-scale models) there is little feasible recourse beyond the use of narrative scenarios.</p>
+<p>Exploratory modeling is typically paired with scenario discovery methods (indicated by number 9 in <a class="reference internal" href="#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>) that identify which of the scenarios (also known as states of the world) generated indeed have consequences of interest for stakeholders and policy makers, in an approach referred to as ensemble-based scenario-discovery <span id="id29">[<a class="reference internal" href="R.Bibliography.html#id16" title="Benjamin P Bryant and Robert J Lempert. Thinking inside the box: a participatory, computer-assisted approach to scenario discovery. Technological Forecasting and Social Change, 77(1):34–49, 2010.">45</a>, <a class="reference internal" href="R.Bibliography.html#id71" title="Robert J Lempert, David G Groves, Steven W Popper, and Steve C Bankes. A general, analytic method for generating robust strategies and narrative scenarios. Management science, 52(4):514–528, 2006.">102</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>. This approach therefore flips the planning analysis from one that attempts to predict future system conditions to one that attempts to discover the (un)desirable future conditions. Ensemble-based scenario discovery can thus inform what modeling choices yield the most consequential behavioral changes or outcomes, especially when considering deeply uncertain, scenario-informed projections <span id="id30">[<a class="reference internal" href="R.Bibliography.html#id12" title="Enayat A Moallemi, Jan Kwakkel, Fjalar J de Haan, and Brett A Bryan. Exploratory modeling for analyzing coupled human-natural systems under uncertainty. Global Environmental Change, 65:102186, 2020.">9</a>, <a class="reference internal" href="R.Bibliography.html#id72" title="Warren E Walker, Marjolijn Haasnoot, and Jan H Kwakkel. Adapt or perish: a review of planning approaches for adaptation under deep uncertainty. Sustainability, 5(3):955–979, 2013.">145</a>]</span>. The relative likelihoods and relevance of the discovered scenarios can be subsequently evaluated by the practitioners a posteriori, within a richer context of knowing the wider set of potential consequences <span id="id31">[<a class="reference internal" href="R.Bibliography.html#id73" title="Suraje Dessai, Mike Hulme, Robert Lempert, and Roger Pielke Jr. Climate prediction: a limit to adaptation. Adapting to climate change: thresholds, values, governance, 64:78, 2009.">146</a>]</span>. This can include changing how an analysis is framed (number 10 in <a class="reference internal" href="#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>). For instance, one could initially focus on ensemble modeling of vulnerability using a single uncertain factor that is assumed to be well characterized by historical observations (e.g., streamflow; this step is represented by numbers 2-3 in <a class="reference internal" href="#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>). The analysis can then shift to include projections of more factors treated as deeply uncertain (e.g., urbanization, population demands, temperature, and snow-melt) to yield a far wider space of challenging projected futures. UC experiments contrasting these two framings can be highly valuable for tracing how vulnerability inferences change as the modeled space of futures expands from the historical baseline <span id="id32">[<a class="reference internal" href="R.Bibliography.html#id61" title="Julianne D Quinn, Patrick M Reed, and Klaus Keller. Direct policy search for robust multi-objective management of deeply uncertain socio-ecological tipping points. Environmental modelling &amp; software, 92:125–141, 2017.">134</a>]</span>.</p>
+<p>An important nuance to be clarified here is that the focus or purpose of a modeling exercise plays a major role in whether a given factor of interest is considered well-characterized or deeply uncertain. Take the example context of characterizing temperature or streamflow extremes, where for each state variable of interest for a given location of focus there is a century of historical observations. Clearly, the observation technologies will have evolved over time uniquely for temperature and streamflow measurements and they likely lack replicate experiments (data uncertainty). A century of record will be insufficient to capture very high impact and rare extreme events (i.e., increasingly poor structural/parametric inference for the distributions of specific extreme single or compound events). The mechanistic processes as well as their evolving variability will be interdependent but uniquely different for each of these state variables. A large body of statistical literature exists focusing on the topics of synthetic weather <span id="id33">[<a class="reference internal" href="R.Bibliography.html#id74" title="Manuel Herrera, Sukumar Natarajan, David A Coley, Tristan Kershaw, Alfonso P Ramallo-González, Matthew Eames, Daniel Fosas, and Michael Wood. A review of current and future weather data for building simulation. Building Services Engineering Research and Technology, 38(5):602–627, 2017.">81</a>, <a class="reference internal" href="R.Bibliography.html#id75" title="Daniel S Wilks and Robert L Wilby. The weather generation game: a review of stochastic weather models. Progress in physical geography, 23(3):329–357, 1999.">82</a>]</span> or streamflow <span id="id34">[<a class="reference internal" href="R.Bibliography.html#id76" title="JR Lamontagne and JR Stedinger. Generating synthetic streamflow forecasts with specified precision. Journal of Water Resources Planning and Management, 144(4):04018007, 2018.">83</a>, <a class="reference internal" href="R.Bibliography.html#id77" title="Sanghamitra Medda and Kalyan Kumar Bhar. Comparison of single-site and multi-site stochastic models for streamflow generation. Applied Water Science, 9(3):67, 2019.">84</a>]</span> generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound extremes. These history-focused approaches can be viewed as providing well-characterized quantifications of streamflow or temperature distributions; however, they do not capture how coupled natural-human processes can fundamentally change their dynamics when transitioning to projections of longer-term futures (e.g., streamflow and temperature in 2055). Consequently, changing the focus of the modeling to making long term projections of future streamflow or temperature now makes these processes deeply uncertain.</p>
+<p>Scenario discovery methods (number 9 in <a class="reference internal" href="#figure-4-4"><span class="std std-numref">Fig. 4.4</span></a>) can be qualitative or quantitative and they generally attempt to distinguish futures in which a system or proposed policies to manage the system meet or miss their goals <span id="id35">[<a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>]</span>. The emphasis placed by exploratory modeling on model outputs that have decision relevant consequences represents a shift toward a broader class of metrics that are reflective of the stakeholders’ concerns, agency and preferences (also discussed in <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>). As a result, sensitivity analysis and scenario discovery methods in this context are therefore applied to performance metrics that go beyond model error but are rather focused on broader metrics such as the resilience of a sector, the reliability of a process, or the vulnerability of a population in the face of uncertainty. In exploratory modeling literature, this metric is most typically—but not always—a measure of robustness (number 8 in Fig. 13). Robustness is a property of a system or a design choice capturing its insensitivity to uncertainty and can be measured via a variety of means, most recently reviewed by <span id="id36">Herman <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id78" title="Jonathan D Herman, Patrick M Reed, Harrison B Zeff, and Gregory W Characklis. How should robustness be defined for water systems planning under change? Journal of Water Resources Planning and Management, 141(10):04015012, 2015.">147</a>]</span> and <span id="id37">McPhail <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id55" title="Cameron McPhail, HR Maier, JH Kwakkel, M Giuliani, A Castelletti, and S Westra. Robustness metrics: how are they calculated, when should they be used and why do they give different results? Earth's Future, 6(2):169–191, 2018.">129</a>]</span>.</p>
+<p>Scenario discovery is typically performed through the use of algorithms applied on large databases of model runs, generated through exploratory modeling, with each model run representing the performance of the system in one potential state of the world. The algorithms seek to identify the combinations of factor values (e.g., future conditions) that best distinguish the cases in which the system does or does not meet its objectives. The most widely known classification algorithms are the Patient Rule Induction Method (PRIM; <span id="id38">[<a class="reference internal" href="R.Bibliography.html#id79" title="Jerome H Friedman and Nicholas I Fisher. Bump hunting in high-dimensional data. Statistics and Computing, 9(2):123–143, 1999.">97</a>]</span>) and Classification and Regression Trees (CART; <span id="id39">[<a class="reference internal" href="R.Bibliography.html#id80" title="Leo Breiman, Jerome Friedman, Charles J Stone, and Richard A Olshen. Classification and regression trees. CRC press, 1984.">98</a>]</span>). These factor mapping algorithms create orthogonal boundaries (multi-dimensional hypercubes) between states of the world that are successful or unsuccessful in meeting the system’s goals <span id="id40">[<a class="reference internal" href="R.Bibliography.html#id81" title="Ishaan L Dalal, Deian Stefan, and Jared Harwayne-Gidansky. Low discrepancy sequences for monte carlo simulations on reconfigurable platforms. In 2008 International Conference on Application-Specific Systems, Architectures and Processors, 108–113. IEEE, 2008.">65</a>]</span>. The algorithms attempt to strike a balance between simplicity of classification (and as a result, interpretability) and accuracy <span id="id41">[<a class="reference internal" href="R.Bibliography.html#id16" title="Benjamin P Bryant and Robert J Lempert. Thinking inside the box: a participatory, computer-assisted approach to scenario discovery. Technological Forecasting and Social Change, 77(1):34–49, 2010.">45</a>, <a class="reference internal" href="R.Bibliography.html#id70" title="David G Groves and Robert J Lempert. A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1):73–85, 2007.">103</a>, <a class="reference internal" href="R.Bibliography.html#id82" title="Robert J Lempert. Robust decision making (rdm). In Decision making under deep uncertainty, pages 23–51. Springer, Cham, 2019.">148</a>]</span>.</p>
+<p>Even though these approaches have been shown to yield interpretable and relevant scenarios <span id="id42">[<a class="reference internal" href="R.Bibliography.html#id83" title="Jan H Kwakkel and Marjolijn Haasnoot. Supporting dmdu: a taxonomy of approaches and tools. In Decision Making under Deep Uncertainty, pages 355–374. Springer, Cham, 2019.">149</a>]</span>, several authors have pointed out the limitations of these methods with regards to their division of space in orthogonal behavioral and non-behavioral regions <span id="id43">[<a class="reference internal" href="R.Bibliography.html#id84" title="BC Trindade, PM Reed, and GW Characklis. Deeply uncertain pathways: integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134:103442, 2019.">150</a>]</span>. Due to their reliance on boundaries orthogonal to the uncertainty axes, PRIM and CART cannot capture interactions between the various uncertain factors considered, which can often be significant <span id="id44">[<a class="reference internal" href="R.Bibliography.html#id85" title="Julianne D Quinn, Patrick M Reed, Matteo Giuliani, Andrea Castelletti, Jared W Oyler, and Robert E Nicholas. Exploring how changing monsoonal dynamics and human pressures challenge multireservoir management for flood protection, hydropower production, and agricultural water supply. Water Resources Research, 54(7):4638–4662, 2018.">151</a>]</span>. More advanced methods have been proposed to address this drawback, with logistic regression being perhaps the most prominent <span id="id45">[<a class="reference internal" href="R.Bibliography.html#id85" title="Julianne D Quinn, Patrick M Reed, Matteo Giuliani, Andrea Castelletti, Jared W Oyler, and Robert E Nicholas. Exploring how changing monsoonal dynamics and human pressures challenge multireservoir management for flood protection, hydropower production, and agricultural water supply. Water Resources Research, 54(7):4638–4662, 2018.">151</a>, <a class="reference internal" href="R.Bibliography.html#id86" title="DF Gold, PM Reed, BC Trindade, and GW Characklis. Identifying actionable compromises: navigating multi-city robustness conflicts to discover cooperative safe operating spaces for regional water supply portfolios. Water Resources Research, 55(11):9024–9050, 2019.">152</a>, <a class="reference internal" href="R.Bibliography.html#id87" title="JR Lamontagne, PM Reed, G Marangoni, K Keller, and GG Garner. Robust abatement pathways to tolerable climate futures require immediate global action. Nature Climate Change, 9(4):290–294, 2019.">153</a>]</span>. Logistic regression can produce boundaries that are not necessarily orthogonal to each uncertainty axis, nor necessarily linear, if interactive terms between two parameters are used to build the regression model. It also describes the probability that a state of the world belongs to the scenarios that lead to failure. This feature allows users to define regions of success based on a gradient of estimated probability of success in those worlds, unlike PRIM which only classifies states of the world in two regions <span id="id46">[<a class="reference internal" href="R.Bibliography.html#id85" title="Julianne D Quinn, Patrick M Reed, Matteo Giuliani, Andrea Castelletti, Jared W Oyler, and Robert E Nicholas. Exploring how changing monsoonal dynamics and human pressures challenge multireservoir management for flood protection, hydropower production, and agricultural water supply. Water Resources Research, 54(7):4638–4662, 2018.">151</a>, <a class="reference internal" href="R.Bibliography.html#id202" title="Antonia Hadjimichael, Julianne Quinn, Erin Wilson, Patrick Reed, Leon Basdekas, David Yates, and Michelle Garrison. Defining Robustness, Vulnerabilities, and Consequential Scenarios for Diverse Stakeholder Interests in Institutionally Complex River Basins. Earth's Future, 8(7):e2020EF001503, 2020. URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020EF001503 (visited on 2020-07-13), doi:10.1029/2020EF001503.">154</a>]</span>.</p>
+<p>Another more advanced factor mapping method is boosted trees <span id="id47">[<a class="reference internal" href="R.Bibliography.html#id89" title="Yoav Freund, Robert Schapire, and Naoki Abe. A short introduction to boosting. Journal-Japanese Society For Artificial Intelligence, 14(771-780):1612, 1999.">99</a>, <a class="reference internal" href="R.Bibliography.html#id88" title="Harris Drucker and Corinna Cortes. Boosting decision trees. Advances in neural information processing systems, pages 479–485, 1996.">155</a>]</span>. Boosted trees can avoid two limitations inherent to the application of logistic regression: i) to build a nonlinear classification model the interactive term between two uncertainties needs to be pre-specified and cannot be discovered (e.g., we need to know a-priori whether factor <span class="math notranslate nohighlight">\(x_1\)</span> interacts with <span class="math notranslate nohighlight">\(x_2\)</span> in a relationship that looks like <span class="math notranslate nohighlight">\(x_1\)</span>·<span class="math notranslate nohighlight">\(x_2\)</span> or <span class="math notranslate nohighlight">\(x_1^{x_2}\)</span>); and ii) the subspaces defined are always convex. The application of such a factor mapping algorithm is limited in the presence of threshold-based rules with discrete actions in a modeled system (e.g., “if network capacity is low, build new infrastructure”), which results in failure regions that are nonlinear and non-convex <span id="id48">[<a class="reference internal" href="R.Bibliography.html#id84" title="BC Trindade, PM Reed, and GW Characklis. Deeply uncertain pathways: integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134:103442, 2019.">150</a>]</span>. Boosting works by creating an ensemble of classifiers and forcing some of them to focus on the hard-to-learn parts of the problem, and others to focus on the easy-to-learn parts. Boosting applied to CART trees can avoid the aforementioned challenges faced by other scenario discovery methods, while resisting overfitting <span id="id49">[<a class="reference internal" href="R.Bibliography.html#id90" title="Kevin P Murphy. Machine learning: a probabilistic perspective. MIT press, 2012.">156</a>]</span>, assuring the identified success and failure regions are still easy to interpret.</p>
+<p>Below we provide an example application of two scenario discovery methods, PRIM and logistic regression, using the lake problem introduced in the previous section. From the sensitivity analysis results presented in <a class="reference internal" href="#figure-4-3"><span class="std std-numref">Fig. 4.3</span></a> (d), we can already infer that parameters <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(q\)</span> have important effects on model outputs (i.e., we have performed factor prioritization). Scenario discovery (i.e., factor mapping) complements this analysis by further identifying the specific values of b and q that can lead to consequential and undesirable outcomes. For the purposes of demonstration, we can assume the undesirable outcome in this case is defined as the management policy failing to achieve 90% reliability in a state of the world.</p>
+<figure class="margin-caption align-center" id="id54">
+<span id="figure-4-5"></span><a class="reference internal image-reference" href="_images/figure14lake_problem_SD.png"><img alt="Figure 14." src="_images/figure14lake_problem_SD.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. 4.5 </span><span class="caption-text">Scenario discovery for the lake problem, using (a) PRIM and (b) logistic regression.</span><a class="headerlink" href="#id54" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p><a class="reference internal" href="#figure-4-5"><span class="std std-numref">Fig. 4.5</span></a> shows the results of scenario discovery, performed through (a) PRIM and (b) logistic regression. Each point in the two panels indicates a potential state of the world, generated through Latin Hypercube Sampling. Each point is colored by whether the policy meets the above performance criterion, with blue indicating success and red indicating failure. PRIM identifies several orthogonal areas of interest, one of which is shown in panel (a). As discussed above, this necessary orthogonality limits how PRIM identifies areas of success (the area within the box). As factors  <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(q\)</span> interact in this system, the transition boundary between the regions of success and failure is not orthogonal to any of the axes. As a result, a large number of points in the bottom right and the top left of the figure are left outside of the identified region. Logistic regression can overcome this limitation by identifying a diagonal boundary between the two regions, seen in panel (b). This method also produces a gradient of estimated probability of success across these regions.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following link to try out an interactive tutorial on performing factor mapping using logistic regression:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/basin_users_logistic_regression.ipynb">Logistic Regression Jupyter Notebook</a></p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The following articles are suggested as fundamental reading for the information presented in this section:</p>
+<ul class="simple">
+<li><p>Gupta, H.V., Wagener, T., Liu, Y., 2008. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. <em>Hydrological Processes: An International Journal</em> 22, 3802–3813.</p></li>
+<li><p>Bankes, S., 1993. Exploratory Modeling for Policy Analysis. <em>Operations Research</em> 41, 435–449. <a class="reference external" href="https://doi.org/10.1287/opre.41.3.435">https://doi.org/10.1287/opre.41.3.435</a></p></li>
+<li><p>Groves, D.G., Lempert, R.J., 2007. A new analytic method for finding policy-relevant scenarios. <em>Global Environmental Change</em> 17, 73–85. <a class="reference external" href="https://doi.org/10.1016/j.gloenvcha.2006.11.006">https://doi.org/10.1016/j.gloenvcha.2006.11.006</a></p></li>
+</ul>
+<p>The following articles can be used as supplemental reading:</p>
+<ul class="simple">
+<li><p>Marchau, V.A.W.J., Walker, W.E., Bloemen, P.J.T.M., Popper, S.W. (Eds.), 2019. Decision Making under Deep Uncertainty: From Theory to Practice. Springer International Publishing. <a class="reference external" href="https://doi.org/10.1007/978-3-030-05252-2">https://doi.org/10.1007/978-3-030-05252-2</a></p></li>
+<li><p>Sterman, J.D., 1994. Learning in and about complex systems. System Dynamics Review 10, 291–330. <a class="reference external" href="https://doi.org/10.1002/sdr.4260100214">https://doi.org/10.1002/sdr.4260100214</a></p></li>
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+<span id="id1"></span><h1><span class="section-number">5. </span>Conclusion<a class="headerlink" href="#conclusion" title="Link to this heading">#</a></h1>
+<p>As noted in the Introduction (<a class="reference internal" href="1_introduction.html#introduction"><span class="std std-numref">Chapter 1</span></a>), the computational and conceptual challenges of the multi-model, transdisciplinary workflows that characterize ambitious projects such as IM3 have limited UC and UQ analyses. Moreover, the very nature and purpose of modeling and diagnostic model evaluation can have very diverse philosophical framings depending on the disciplines involved (see <code class="xref std std-numref docutils literal notranslate"><span class="pre">Figure_1_1</span></code> and <a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#perspectives"><span class="std std-numref">Chapter 2.2</span></a>). The guidance provided in this text can be used to frame consistent and rigorous experimental designs for better understanding the consequences and insights from our modeling choices when seeking to capture complex human-natural systems. The progression of sections of this text provide a thorough introduction of the concepts and definitions of diagnostic model evaluation, sensitivity analysis and UC. In addition, we comprehensively discuss how specific modeling objectives and applications should guide the selection of appropriate techniques; broadly, these can include model diagnostics, in-depth analysis of the behavior of the abstracted system, and projections under conditions of deep uncertainty. This text also contains a detailed presentation of the main sensitivity analysis methods and a discussion of their features and main limitations. Readers are also provided with an overview of computer tools and platforms that have been developed and could be considered in addressing IM3 scientific questions. The appendices of this text include an overview of UQ methods, a terminology glossary of the key concepts as well as example test cases and scripts to showcase various UC related capabilities.</p>
+<p>Although we distinguish the UC and UQ model diagnostics, the reader should note that we suggest an overall consistent approach to both in this text by emphasizing “exploratory modeling” (see review by <span id="id2">Moallemi <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id12" title="Enayat A Moallemi, Jan Kwakkel, Fjalar J de Haan, and Brett A Bryan. Exploratory modeling for analyzing coupled human-natural systems under uncertainty. Global Environmental Change, 65:102186, 2020.">9</a>]</span>). Although data support, model complexity, and computational limits strongly distinguish the feasibility and appropriateness of various UC diagnostic tools (e.g., see <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#figure-3-5"><span class="std std-numref">Fig. 3.5</span></a>), we overall recommend that modelers view their work through the lens of cycles of learning. Iterative and deliberative exploration of model-based hypotheses and inferences for transdisciplinary teams is non-trivial and ultimately critical for mapping where innovations or insights are most consequential. Overall, we recommend approaching modeling with an openness to the diverse disciplinary perspectives such as those mirrored by the IM3 family of models in a progression from evaluating models relative to observed history to advanced formalized analyses to make inferences on multi-sector, multi-scale vulnerabilities and resilience. Exploratory modeling approaches can help fashion experiments with large numbers of alternative hypotheses on the co-evolutionary dynamics of influences, stressors, as well as path-dependent changes in the form and function of coupled human-natural systems <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id130" title="Christopher P. Weaver, Robert J. Lempert, Casey Brown, John A. Hall, David Revell, and Daniel Sarewitz. Improving the contribution of climate model information to decision making: the value and demands of robust decision frameworks. Wiley Interdisciplinary Reviews: Climate Change, 4(1):39–60, 2013. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/wcc.202 (visited on 2019-10-01), doi:10.1002/wcc.202.">37</a>]</span>. This text guides the reader through the use of sensitivity analysis and uncertainty methods across the diverse perspectives that have shaped modern diagnostic and exploratory modeling.</p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="A1_Uncertainty_Quantification.html">A. Uncertainty Quantification</a></li>
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+<span id="id1"></span><h1>Glossary<a class="headerlink" href="#glossary" title="Link to this heading">#</a></h1>
+<p><strong>Design of experiment</strong>: Provides a framework for the extraction of all plausible information about the impact of each factor on the output of the numerical model</p>
+<p><strong>Exploratory modeling</strong>: Use of large ensembles of uncertain conditions to discover decision-relevant combinations of uncertain factors</p>
+<p><strong>Factor</strong>: Any model component that can affect model outputs: inputs, resolution levels, coupling relationships, model relationships and parameters. In models with acceptable model fidelity these factors may represent elements of the real-world system under study.</p>
+<p><strong>Factor mapping</strong>: A technique to identify which uncertain model factors lead to certain model behavior</p>
+<p><strong>Factor prioritization</strong>: A technique to identify the uncertain factors which, when fixed to their true value, would lead to the greatest reduction in output variability</p>
+<p><strong>Factor screening</strong>: A technique to identify model components that have a negligible effect or make no significant contributions to the variability of the outputs or metrics of interest</p>
+<p><strong>First-, second-, total-order effects</strong>: First-order effects indicate the percent of model output variance contributed by a factor individually. Second-order effects capture how interactions between a pair of parameter input variables can lead to change in model output. Total-order effects consider all the effects a factor has, individually and in interaction with other factors.</p>
+<p><strong>Hindcasting</strong>: A type of predictive check that uses the model to estimate output for past events to see how well the output matches the known results.</p>
+<p><strong>Pre-calibration</strong>: A hybrid uncertainty assessment method that involves identifying a plausible set of parameters using some prespecified screening criterion, such as the distance from the model results to the observations.</p>
+<p><strong>Prior</strong>: The best assessment of the probability of an event based on existing knowledge before a new experiment is conducted</p>
+<p><strong>Posterior</strong>: The revised or updated probability of an event after taking into account new information</p>
+<p><strong>Probabilistic inversion</strong>: Uses additional information, for instance, a probabilistic expert assessment or survey result, to update an existing prior distribution</p>
+<p><strong>Return level</strong>: A value that is expected to be equaled or exceeded on average once every interval of time (T) (with a probability of 1/T)</p>
+<p><strong>Return period</strong>: The estimated time interval between events of a similar size or intensity/</p>
+<p><strong>Sampling</strong>: The process of selecting model parameters or inputs that characterize the model uncertainty space.</p>
+<p><strong>Scenario discovery</strong>: Use of large ensembles of uncertain conditions to discover decision-relevant combinations of uncertain factors</p>
+<p><strong>Sensitivity analysis</strong>: Conducted to understand the factors and processes that most (or least) control a model’s outputs</p>
+<blockquote>
+<div><p><em>Local sensitivity analysis</em>: Model evaluation performed by varying uncertain factors around specific reference values</p>
+<p><em>Global sensitivity analysis</em>: Model evaluation performed by varying uncertain factors throughout their entire feasible value space</p>
+</div></blockquote>
+<p><strong>Uncertainty</strong></p>
+<p><em>Deep uncertainty</em>: Refers to situations where expert opinions consulted on a decision do not know or cannot agree on system boundaries, or the outcomes of interest and their relative importance, or the prior probability Distribution for the various uncertain factors present</p>
+<p><em>Epistemic uncertainty</em>: Systematic uncertainty that comes about due to the lack of knowledge or data to choose the best model</p>
+<p><em>Ontological uncertainty</em>: Uncertainties due to processes, interactions, or futures, that are not contained within current conceptual models</p>
+<p><em>Aleatory uncertainty</em>: Uncertainty due to natural randomness in processes</p>
+<p><em>Uncertainty characterization</em>: Model evaluation under alternative factor hypotheses to explore their implications for model output uncertainty</p>
+<p><em>Uncertainty quantification</em>: Representation of model output uncertainty using probability distributions</p>
+<p><strong>Variance decomposition</strong>: A technique to partition how much of the variability in a model’s output is due to different explanatory variables.</p>
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+<h1>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h1>
+<p>As defined in <a class="reference internal" href="1_introduction.html#introduction"><span class="std std-numref">Chapter 1</span></a>, uncertainty quantification (UQ) refers to the formal focus on the full specification of likelihoods as well as distributional forms necessary to infer the joint  probabilistic response across all modeled factors of interest <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id11" title="Roger Cooke and others. Experts in uncertainty: opinion and subjective probability in science. Oxford University Press on Demand, 1991.">8</a>]</span>. This is in contrast to UC (the primary focus of the main document of this book), which is instead aimed at identifying which modeling choices yield the most consequential changes or outcomes and exploring alternative hypotheses related to the form and function of modeled systems <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id12" title="Enayat A Moallemi, Jan Kwakkel, Fjalar J de Haan, and Brett A Bryan. Exploratory modeling for analyzing coupled human-natural systems under uncertainty. Global Environmental Change, 65:102186, 2020.">9</a>, <a class="reference internal" href="R.Bibliography.html#id13" title="Warren E Walker, Poul Harremoës, Jan Rotmans, Jeroen P Van Der Sluijs, Marjolein BA Van Asselt, Peter Janssen, and Martin P Krayer von Krauss. Defining uncertainty: a conceptual basis for uncertainty management in model-based decision support. Integrated assessment, 4(1):5–17, 2003.">10</a>]</span>.</p>
+<p>UQ is important for quantifying the relative merits of hypotheses for at least three main reasons. First, identifying model parameters that are consistent with observations is an important part of model development. Due to several effects, including correlations between parameters, simplified or incomplete model structures (relative to the full real-world dynamics), and uncertainty in the observations, many different combinations of parameter values can be consistent with the model structure and the observations to varying extents. Accounting for this uncertainty is conceptually preferable to selecting a single “best fit” parameter vector, particularly as consistency with historical or present observations does not necessarily  guarantee skillful future projections.</p>
+<p>The act of quantification requires specific assumptions about distributional forms and likelihoods, which may be more or less justified depending on prior information about the system or model behavior. As a result, UQ is well-suited for studies accounting for or addressing hypotheses related to systems with a relatively large amount of available data and models which are computationally inexpensive, particularly when the emphasis is on prediction. As shown in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-1"><span class="std std-numref">Fig. A.1</span></a>, there is a fundamental tradeoff between the available number of model evaluations (for a fixed computational budget) and the number of parameters treated as uncertain. Sensitivity analyses are therefore part of a typical UQ workflow to identify which factors can be fixed and which ought to be prioritized in the UQ.</p>
+<figure class="margin-caption align-center" id="id4">
+<span id="figure-a1-1"></span><a class="reference internal image-reference" href="_images/figureA1_1_UQ_approaches.png"><img alt="Figure A1.1" src="_images/figureA1_1_UQ_approaches.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Overview of selected existing approaches for uncertainty quantification and their appropriateness given the number of uncertain model parameters and the number of available model simulations. Green shading denotes regions suitable for uncertainty quantification and red shading indicates regions more appropriate for uncertainty characterization.</span><a class="headerlink" href="#id4" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The choice of a particular UQ method depends on both the desired level of quantification and the ability to navigate the tradeoff between computational expense and the number of uncertain parameters (<a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-1"><span class="std std-numref">Fig. A.1</span></a>). For example, Markov chain Monte Carlo with a full system model can provide an improved representation of uncertainty compared to the coarser pre-calibration approach <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id37" title="Kelsey L. Ruckert, Gary Shaffer, David Pollard, Yawen Guan, Tony E. Wong, Chris E. Forest, and Klaus Keller. Assessing the Impact of Retreat Mechanisms in a Simple Antarctic Ice Sheet Model Using Bayesian Calibration. PLOS ONE, 12(1):e0170052, January 2017. doi:10.1371/journal.pone.0170052.">157</a>]</span>, but requires many more model evaluations. The use of a surrogate model to approximate the full system model can reduce the number of needed model evaluations by several orders of magnitude, but the uncertainty quantification can only accommodate a limited number of parameters.</p>
+<p>The remainder of this appendix will focus on introducing workflows for particular UQ methods, including a brief discussion of advantages and limitations.</p>
+</section>
+
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+  <section id="parametric-bootstrap">
+<h1>Parametric Bootstrap<a class="headerlink" href="#parametric-bootstrap" title="Link to this heading">#</a></h1>
+<p>The parametric bootstrap <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id14" title="B. Efron and R. Tibshirani. Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy. Statistical Science, 1(1):54–75, February 1986. doi:10.1214/ss/1177013815.">158</a>]</span> refers to a process of model recalibration to alternate realizations of the data. The bootstrap was originally developed to estimate standard errors and confidence intervals without ascertaining key assumptions that might not hold given the available data. In a setting where observations can be viewed as independent realizations of an underlying stochastic process, a sufficiently rich dataset can be treated as a population representing the data distribution. New datasets are then generated by resampling from the data with replacement, and the model can be refit to each new dataset using maximum-likelihood estimation. The resulting distribution of estimates can then be viewed as a representation of parametric uncertainty.</p>
+<p>A typical workflow for the parametric bootstrap is shown in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-2"><span class="std std-numref">Fig. A.2</span></a>. After identifying outputs of interest and preparing the data, the parametric model is fit by some procedure such as minimizing root-mean-square-error or maximizing the likelihood. Alternate datasets are constructed by resampling from the population or by generating new samples from the fitted data-generating process. It is important at this step that the resampled quantities are independent of one another. For example, in the context of temporally- or spatially-correlated data, such as time series, the raw observations cannot be treated as independent realizations. However, the residuals resulting from fitting the model to the data could be (depending on their structure). For example, if the residuals are treated as independent, they can then be resampled with replacement, and these residuals added to the original model fit to create new realizations. If the residuals are assumed to be the result of an autoregressive process, this process could be fit to the original residual series and new residuals be created using this model <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id35" title="Ryan L. Sriver, Robert J. Lempert, Per Wikman-Svahn, and Klaus Keller. Characterizing uncertain sea-level rise projections to support investment decisions. PLOS ONE, 13(2):e0190641, February 2018. URL: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0190641 (visited on 2021-06-09), doi:10.1371/journal.pone.0190641.">159</a>]</span>. The model is then refit to each new realization.</p>
+<figure class="margin-caption align-center" id="id4">
+<span id="figure-a1-2"></span><a class="reference internal image-reference" href="_images/figureA1_2_bootstrap_workflow.png"><img alt="Figure A1.2" src="_images/figureA1_2_bootstrap_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Workflow for the parametric bootstrap.</span><a class="headerlink" href="#id4" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The bootstrap is computationally convenient, particularly as the process of fitting the model to each realization can be easily parallelized. This approach also requires minimal prior assumptions. However, due to the assumption that the available data are representative of the underlying data distribution, the bootstrap can neglect key uncertainties which might influence the results. For example, when using an autoregressive process to generate new residuals, uncertainty in the autocorrelation parameter and innovation variance is neglected, which may bias estimates of, for example, low-probability but high-impact events <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id36" title="Kelsey L. Ruckert, Yawen Guan, Alexander M. R. Bakker, Chris E. Forest, and Klaus Keller. The effects of time-varying observation errors on semi-empirical sea-level projections. Climatic Change, 140(3):349–360, February 2017. URL: https://doi.org/10.1007/s10584-016-1858-z (visited on 2021-06-09), doi:10.1007/s10584-016-1858-z.">160</a>]</span>.</p>
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+    <h1>Pre-Calibration</h1>
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+  <section id="pre-calibration">
+<h1>Pre-Calibration<a class="headerlink" href="#pre-calibration" title="Link to this heading">#</a></h1>
+<p>Pre-calibration <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>, <a class="reference internal" href="R.Bibliography.html#id39" title="Neil R Edwards, David Cameron, and Jonathan Rougier. Precalibrating an intermediate complexity climate model. Clim. Dyn., 37(7-8):1469–1482, 2011. URL: http://dx.doi.org/10.1007/s00382-010-0921-0, doi:10.1007/s00382-010-0921-0.">161</a>, <a class="reference internal" href="R.Bibliography.html#id40" title="Alexis Boukouvalas, Pete Sykes, Dan Cornford, and Hugo Maruri-Aguilar. Bayesian Precalibration of a Large Stochastic Microsimulation Model. IEEE Transactions on Intelligent Transportation Systems, 15(3):1337–1347, June 2014. doi:10.1109/TITS.2014.2304394.">162</a>]</span> involves the identification of a plausible set of parameters using some prespecified screening criterion, such as the distance from the model results to the observations (based on an appropriate metric for the desired matching features, such as root-mean-squared error). A typical workflow is shown in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-3"><span class="std std-numref">Fig. A.3</span></a>. Parameter values are obtained by systematically sampling the input space (see <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#design-of-experiments"><span class="std std-numref">Chapter 3.3</span></a>). After the model is evaluated at the samples, only those passing the distance criterion are retained. This selects a subset of the parameter space as “plausible” based on the screening criterion, though there is no assignment of probabilities within this plausible region.</p>
+<figure class="margin-caption align-center" id="id7">
+<span id="figure-a1-3"></span><a class="reference internal image-reference" href="_images/figureA1_3_precal_workflow.png"><img alt="Figure A1.3" src="_images/figureA1_3_precal_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Workflow for pre-calibration.</span><a class="headerlink" href="#id7" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>Pre-calibration can be useful for models which are inexpensive enough that a reasonable
+number of samples can be used to represent the parameter space, but which are too expensive to facilitate full uncertainty quantification. High-dimensional parameter spaces, which can be problematic for the uncertainty quantification methods below, may also be explored using pre-calibration. One key prerequisite to using this method is the ability to place a meaningful distance metric on the output space.</p>
+<p>However, pre-calibration results in a very coarse characterization of uncertainty, especially when considering a large number of parameters, as more samples are needed to fully characterize the parameter space. Due to the inability to evaluate the relative probability of regions of the parameter space beyond the binary plausible-and-implausible characterization, pre-calibration can also result in degraded hindcast and projection skills and parameter estimates <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id37" title="Kelsey L. Ruckert, Gary Shaffer, David Pollard, Yawen Guan, Tony E. Wong, Chris E. Forest, and Klaus Keller. Assessing the Impact of Retreat Mechanisms in a Simple Antarctic Ice Sheet Model Using Bayesian Calibration. PLOS ONE, 12(1):e0170052, January 2017. doi:10.1371/journal.pone.0170052.">157</a>, <a class="reference internal" href="R.Bibliography.html#id41" title="David Makowski, Daniel Wallach, and Marie Tremblay. Using a Bayesian approach to parameter estimation; comparison of the GLUE and MCMC methods. Agronomie, 22(2):191–203, 2002. Publisher: EDP Sciences.">163</a>, <a class="reference internal" href="R.Bibliography.html#id42" title="Mahyar Shafii, Bryan Tolson, and Loren Shawn Matott. Uncertainty-based multi-criteria calibration of rainfall-runoff models: a comparative study. Stochastic Environmental Research and Risk Assessment, 28(6):1493–1510, August 2014. doi:10.1007/s00477-014-0855-x.">164</a>]</span>.</p>
+<p>A related method, widely used in hydrological studies, is generalized likelihood uncertainty estimation, or GLUE <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>]</span>. Unlike pre-calibration, the underlying argument for GLUE relies on the concept of equifinality <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id43" title="Keith Beven and Jim Freer. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology. Journal of hydrology, 249(1-4):11–29, 2001. Publisher: Elsevier.">165</a>]</span>, which posits that it is impossible to find a uniquely well-performing parameter vector for models of abstract environmental systems <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id43" title="Keith Beven and Jim Freer. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology. Journal of hydrology, 249(1-4):11–29, 2001. Publisher: Elsevier.">165</a>, <a class="reference internal" href="R.Bibliography.html#id44" title="Jasper A. Vrugt and Keith J. Beven. Embracing equifinality with efficiency: Limits of Acceptability sampling using the DREAM(LOA) algorithm. Journal of Hydrology, 559:954–971, April 2018. doi:10.1016/j.jhydrol.2018.02.026.">166</a>]</span>. In other words, there exist multiple parameter vectors which perform equally or similarly well. As with pre-calibration, GLUE uses a goodness-of-fit measure (though this is called a “likelihood” in the GLUE literature, as opposed to a statistical likelihood function <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id45" title="Jery R. Stedinger, Richard M. Vogel, Seung Uk Lee, and Rebecca Batchelder. Appraisal of the generalized likelihood uncertainty estimation (GLUE) method. Water Resources Research, 2008. doi:10.1029/2008WR006822.">167</a>]</span>) to evaluate samples. After setting a threshold of acceptable performance with respect to that measure, samples are evaluated and classified into “behavioral” or “non-behavioral” according to the threshold.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following badge to try out an interactive tutorial on utilizing Pre-Calibration and GLUE for HYMOD model calibration:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">Pre-Calibration Jupyter Notebook</a></p>
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+    <h1>Markov Chain Monte Carlo</h1>
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+  <section id="markov-chain-monte-carlo">
+<h1>Markov Chain Monte Carlo<a class="headerlink" href="#markov-chain-monte-carlo" title="Link to this heading">#</a></h1>
+<p>Markov chain Monte Carlo (MCMC) is a “gold standard” approach to full uncertainty quantification. MCMC refers to a category of algorithms which systematically sample from a target distribution (in this case, the posterior distribution) by constructing a Markov chain. A Markov chain is a probabilistic structure consisting of a state space, an initial probability distribution over the states, and a transition distribution between states. If a Markov chain satisfies certain properties <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id170" title="Christian Robert and George Casella. Monte Carlo Statistical Methods. Springer Science &amp; Business Media, March 2013. ISBN 978-1-4757-3071-5.">168</a>, <a class="reference internal" href="R.Bibliography.html#id169" title="Christian P. Robert. The Metropolis–Hastings Algorithm. In Wiley StatsRef: Statistics Reference Online, pages 1–15. American Cancer Society, 2015. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat07834 (visited on 2021-06-14), doi:10.1002/9781118445112.stat07834.">169</a>]</span>, the probability of being in each state will eventually converge to a stable, or stationary, distribution, regardless of the initial probabilities.</p>
+<p>MCMC algorithms construct a Markov chain of samples from a parameter space (the combination of model and statistical parameters). This Markov chain is constructed so that the stationary distribution is a target distribution, in this case the (Bayesian) posterior distribution. As a result, after the transient period, the resulting samples can be viewed as a set of dependent samples from the posterior (the dependence is due to the autocorrelation between samples resulting from the Markov chain transitions). Expected values can be computed from these samples (for example, using batch-means estimators <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id171" title="James M. Flegal, Murali Haran, and Galin L. Jones. Markov Chain Monte Carlo: Can We Trust the Third Significant Figure? Statistical Science, 23(2):250–260, May 2008. Publisher: Institute of Mathematical Statistics. URL: https://projecteuclid.org/journals/statistical-science/volume-23/issue-2/Markov-Chain-Monte-Carlo--Can-We-Trust-the-Third/10.1214/08-STS257.full (visited on 2021-06-14), doi:10.1214/08-STS257.">170</a>]</span>), or the chain can be sub-sampled or thinned and the resulting samples used as independent Monte Carlo samples due to the reduced or eliminated autocorrelation.</p>
+<p>A general workflow for MCMC is shown in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-4"><span class="std std-numref">Fig. A.4</span></a>. The first decision is whether to use the full model or a surrogate model (or emulator). Typical surrogates include Gaussian process emulation <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id172" title="Carla Currin, Toby Mitchell, Max Morris, and Don Ylvisaker. Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments. Journal of the American Statistical Association, 86(416):953–963, December 1991. Publisher: Taylor &amp; Francis _eprint: https://www.tandfonline.com/doi/pdf/10.1080/01621459.1991.10475138. URL: https://www.tandfonline.com/doi/abs/10.1080/01621459.1991.10475138 (visited on 2021-06-14), doi:10.1080/01621459.1991.10475138.">171</a>, <a class="reference internal" href="R.Bibliography.html#id173" title="Jerome Sacks, William J. Welch, Toby J. Mitchell, and Henry P. Wynn. Design and Analysis of Computer Experiments. Statistical Science, 4(4):409–423, 1989. Publisher: Institute of Mathematical Statistics. URL: https://www.jstor.org/stable/2245858 (visited on 2021-06-14).">172</a>]</span>, polynomial chaos expansions <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id174" title="Roger G. Ghanem and Pol D. Spanos. Spectral Stochastic Finite‐Element Formulation for Reliability Analysis. Journal of Engineering Mechanics, 117(10):2351–2372, October 1991. Publisher: American Society of Civil Engineers. URL: https://ascelibrary.org/doi/abs/10.1061/%28ASCE%290733-9399%281991%29117%3A10%282351%29 (visited on 2021-06-14), doi:10.1061/(ASCE)0733-9399(1991)117:10(2351).">173</a>, <a class="reference internal" href="R.Bibliography.html#id175" title="Dongbin Xiu and George Em Karniadakis. The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations. SIAM Journal on Scientific Computing, 24(2):619–644, January 2002. Publisher: Society for Industrial and Applied Mathematics. URL: https://epubs.siam.org/doi/abs/10.1137/S1064827501387826 (visited on 2021-06-14), doi:10.1137/S1064827501387826.">174</a>]</span>, support vector machines <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id176" title="Angelo Ciccazzo, Gianni Di Pillo, and Vittorio Latorre. A SVM Surrogate Model-Based Method for Parametric Yield Optimization. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 35(7):1224–1228, July 2016. Conference Name: IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. doi:10.1109/TCAD.2015.2501307.">175</a>, <a class="reference internal" href="R.Bibliography.html#id177" title="W. Andrew Pruett and Robert L. Hester. The Creation of Surrogate Models for Fast Estimation of Complex Model Outcomes. PLOS ONE, 11(6):e0156574, June 2016. Publisher: Public Library of Science. URL: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0156574 (visited on 2021-06-14), doi:10.1371/journal.pone.0156574.">176</a>]</span>, and neural networks <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id178" title="John Eason and Selen Cremaschi. Adaptive sequential sampling for surrogate model generation with artificial neural networks. Computers &amp; Chemical Engineering, 68:220–232, September 2014. URL: https://www.sciencedirect.com/science/article/pii/S0098135414001719 (visited on 2021-06-14), doi:10.1016/j.compchemeng.2014.05.021.">177</a>, <a class="reference internal" href="R.Bibliography.html#id179" title="Dirk Gorissen, Luciano De Tommasi, Karel Crombecq, and Tom Dhaene. Sequential modeling of a low noise amplifier with neural networks and active learning. Neural Computing and Applications, 18(5):485–494, June 2009. URL: https://doi.org/10.1007/s00521-008-0223-1 (visited on 2021-06-14), doi:10.1007/s00521-008-0223-1.">178</a>]</span>. Surrogate modeling can be faster, but requires a sufficient number of model evaluations for the surrogate to accurately represent the model’s response surface, and this typically limits the number of parameters which can be included in the analysis.</p>
+<figure class="margin-caption align-center" id="id16">
+<span id="figure-a1-4"></span><a class="reference internal image-reference" href="_images/figureA1_4_mcmc_workflow.png"><img alt="Figure A1.4" src="_images/figureA1_4_mcmc_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Workflow for Markov chain Monte Carlo.</span><a class="headerlink" href="#id16" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>After selecting the variables which will be treated as uncertain, the next step is to specify the likelihood based on the selected surrogate model or the structure of the data-model residuals. For example, it may not always be appropriate to treat the residuals as independent and identically distributed (as is commonly done in linear regression). A mis-specification of the residual structure can result in biases and over- or under-confident inferences and projections <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id180" title="Jenný Brynjarsdóttir and Anthony O'Hagan. Learning about physical parameters: the importance of model discrepancy. Inverse Problems, 30(11):114007, October 2014. Publisher: IOP Publishing. URL: https://doi.org/10.1088/0266-5611/30/11/114007 (visited on 2021-06-14), doi:10.1088/0266-5611/30/11/114007.">179</a>]</span>.</p>
+<p>After specifying the prior distributions (see <a class="reference internal" href="A1_Uncertainty_Quantification.html#critical-first-step"><span class="std std-numref">Chapter A.6</span></a>), the selected MCMC algorithm should be used to draw samples from the posterior distribution. There are many MCMC algorithms, all of which have advantages and disadvantages for a particular problem. These include the Metropolis-Hastings algorithm <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id169" title="Christian P. Robert. The Metropolis–Hastings Algorithm. In Wiley StatsRef: Statistics Reference Online, pages 1–15. American Cancer Society, 2015. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat07834 (visited on 2021-06-14), doi:10.1002/9781118445112.stat07834.">169</a>]</span> and Hamiltonian Monte Carlo <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id181" title="Michael Betancourt. A conceptual introduction to Hamiltonian Monte Carlo. arXiv preprint arXiv:1701.02434, 2017.">180</a>, <a class="reference internal" href="R.Bibliography.html#id182" title="Radford M. Neal. MCMC using Hamiltonian dynamics. Handbook of markov chain monte carlo, 2(11):2, 2011.">181</a>]</span>. Software packages typically implement one MCMC method, sometimes designed for a particular problem setting or likelihood specification. For example, R’s <em>adaptMCMC</em> implements an adaptive Metropolis-Hastings algorithm <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id183" title="Matti Vihola. Robust adaptive Metropolis algorithm with coerced acceptance rate. Statistics and Computing, 22(5):997–1008, September 2012. URL: https://doi.org/10.1007/s11222-011-9269-5 (visited on 2021-06-14), doi:10.1007/s11222-011-9269-5.">182</a>]</span>, while <em>NIMBLE</em> <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id184" title="Perry de Valpine, Daniel Turek, Christopher J. Paciorek, Clifford Anderson-Bergman, Duncan Temple Lang, and Rastislav Bodik. Programming With Models: Writing Statistical Algorithms for General Model Structures With NIMBLE. Journal of Computational and Graphical Statistics, 26(2):403–413, April 2017. Publisher: Taylor &amp; Francis _eprint: https://doi.org/10.1080/10618600.2016.1172487. URL: https://doi.org/10.1080/10618600.2016.1172487 (visited on 2021-06-14), doi:10.1080/10618600.2016.1172487.">183</a>, <a class="reference internal" href="R.Bibliography.html#id185" title="NIMBLE Development Team. NIMBLE: MCMC, Particle Filtering, and Programmable Hierarchical Modeling. May 2021. URL: https://zenodo.org/record/4829693 (visited on 2021-06-14), doi:10.5281/zenodo.4829693.">184</a>]</span> uses a user-customizable Metropolis-Hastings implementation, as well as functionality for Gibbs sampling (which is a special case of Metropolis-Hastings where the prior distribution has a convenient mathematical form). Some recent implementations, such as <em>Stan</em> <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id186" title="Stan Development Team. Stan Modeling Language Users Guide and Reference Manual. 2021. URL: https://mc-stan.org/docs/2_27/stan-users-guide/index.html (visited on 2021-06-14).">185</a>]</span>, <em>pyMC3</em> <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id187" title="John Salvatier, Thomas V. Wiecki, and Christopher Fonnesbeck. Probabilistic programming in Python using PyMC3. PeerJ Computer Science, 2:e55, April 2016. Publisher: PeerJ Inc. URL: https://peerj.com/articles/cs-55 (visited on 2021-06-14), doi:10.7717/peerj-cs.55.">186</a>]</span>, and <em>Turing</em> <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id188" title="Hong Ge, Kai Xu, and Zoubin Ghahramani. Turing: A Language for Flexible Probabilistic Inference. In International Conference on Artificial Intelligence and Statistics, 1682–1690. PMLR, March 2018. ISSN: 2640-3498. URL: http://proceedings.mlr.press/v84/ge18b.html (visited on 2021-06-14).">187</a>]</span> allow different algorithms to be used.</p>
+<p>A main consideration when using MCMC algorithms is testing for convergence to the target distribution. As convergence is guaranteed only for a sufficiently large number of transitions, it is impossible to conclude for certain that a chain has converged for a fixed number of iterations. However, several heuristics have been developed <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id171" title="James M. Flegal, Murali Haran, and Galin L. Jones. Markov Chain Monte Carlo: Can We Trust the Third Significant Figure? Statistical Science, 23(2):250–260, May 2008. Publisher: Institute of Mathematical Statistics. URL: https://projecteuclid.org/journals/statistical-science/volume-23/issue-2/Markov-Chain-Monte-Carlo--Can-We-Trust-the-Third/10.1214/08-STS257.full (visited on 2021-06-14), doi:10.1214/08-STS257.">170</a>, <a class="reference internal" href="R.Bibliography.html#id189" title="Andrew Gelman and Donald B. Rubin. Inference from Iterative Simulation Using Multiple Sequences. Statistical Science, 7(4):457–472, November 1992. Publisher: Institute of Mathematical Statistics. URL: https://projecteuclid.org/journals/statistical-science/volume-7/issue-4/Inference-from-Iterative-Simulation-Using-Multiple-Sequences/10.1214/ss/1177011136.full (visited on 2021-06-14), doi:10.1214/ss/1177011136.">188</a>]</span> to increase evidence that convergence has occurred.</p>
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+<li class="toctree-l1"><a class="reference internal" href="3_sensitivity_analysis_the_basics.html">3. Sensitivity Analysis: The Basics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html">4. Sensitivity Analysis: Diagnostic &amp; Exploratory Modeling</a></li>
+<li class="toctree-l1"><a class="reference internal" href="5_conclusion.html">5. Conclusion</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
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+<h1>Other Methods<a class="headerlink" href="#other-methods" title="Link to this heading">#</a></h1>
+<p>Other common methods for UQ exist. These include sequential Monte Carlo, otherwise known as particle filtering <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id190" title="Pierre Del Moral, Arnaud Doucet, and Ajay Jasra. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(3):411–436, 2006. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9868.2006.00553.x. URL: http://rss.onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9868.2006.00553.x (visited on 2021-06-14), doi:10.1111/j.1467-9868.2006.00553.x.">189</a>, <a class="reference internal" href="R.Bibliography.html#id191" title="Arnaud Doucet, Simon Godsill, and Christophe Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 10(3):197–208, July 2000. URL: https://doi.org/10.1023/A:1008935410038 (visited on 2021-06-14), doi:10.1023/A:1008935410038.">190</a>, <a class="reference internal" href="R.Bibliography.html#id192" title="Jane Liu and Mike West. Combined Parameter and State Estimation in Simulation-Based Filtering. In Arnaud Doucet, Nando de Freitas, and Neil Gordon, editors, Sequential Monte Carlo Methods in Practice, Statistics for Engineering and Information Science, pages 197–223. Springer, New York, NY, 2001. URL: https://doi.org/10.1007/978-1-4757-3437-9_10 (visited on 2021-06-14), doi:10.1007/978-1-4757-3437-9_10.">191</a>]</span>, where a number of particles are used to evaluate samples. An advantage of sequential Monte Carlo is that the vast majority of the computation can be parallelized, unlike with standard MCMC. A major weakness is the potential for degeneracy <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id191" title="Arnaud Doucet, Simon Godsill, and Christophe Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 10(3):197–208, July 2000. URL: https://doi.org/10.1023/A:1008935410038 (visited on 2021-06-14), doi:10.1023/A:1008935410038.">190</a>]</span>, where many particles have extremely small weights, resulting in the effective use of only a few samples.</p>
+<p>Another method is approximate Bayesian computation (ABC) <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id193" title="Stefano Cabras, Maria Eugenia Castellanos Nueda, and Erlis Ruli. Approximate Bayesian Computation by Modelling Summary Statistics in a Quasi-likelihood Framework. Bayesian Analysis, 10(2):411–439, June 2015. Publisher: International Society for Bayesian Analysis. URL: https://projecteuclid.org/journals/bayesian-analysis/volume-10/issue-2/Approximate-Bayesian-Computation-by-Modelling-Summary-Statistics-in-a-Quasi/10.1214/14-BA921.full (visited on 2021-06-14), doi:10.1214/14-BA921.">192</a>, <a class="reference internal" href="R.Bibliography.html#id194" title="Jarno Lintusaari, Michael U. Gutmann, Ritabrata Dutta, Samuel Kaski, and Jukka Corander. Fundamentals and Recent Developments in Approximate Bayesian Computation. Systematic Biology, 66(1):e66–e82, January 2017. URL: https://doi.org/10.1093/sysbio/syw077 (visited on 2021-06-14), doi:10.1093/sysbio/syw077.">193</a>, <a class="reference internal" href="R.Bibliography.html#id195" title="Mikael Sunnåker, Alberto Giovanni Busetto, Elina Numminen, Jukka Corander, Matthieu Foll, and Christophe Dessimoz. Approximate Bayesian Computation. PLOS Computational Biology, 9(1):e1002803, January 2013. Publisher: Public Library of Science. URL: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002803 (visited on 2021-06-14), doi:10.1371/journal.pcbi.1002803.">194</a>]</span>. ABC is a likelihood-free approach that compares model output to a set of summary statistics. ABC is therefore well-suited for models and residual structures which do not lend themselves to a computationally-tractable likelihood, but the resulting inferences are known to be biased if the set of summary statistics is not sufficient, which can be difficult to know a-priori.</p>
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+    <h1>The Critical First Step: How to Choose a Prior Distribution</h1>
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+<span id="critical-first-step"></span><h1>The Critical First Step: How to Choose a Prior Distribution<a class="headerlink" href="#the-critical-first-step-how-to-choose-a-prior-distribution" title="Link to this heading">#</a></h1>
+<p>Prior distributions play an important role in Bayesian uncertainty quantification, particularly when data is limited relative to the dimension of the model. Bayesian updating can be thought of as an information filter, where each additional datum is added to the information contained in the prior; eventually, the prior makes relatively little impact. In real world problems, it can be extremely difficult to assess how much data is required for the choice of prior to become less relevant. The choice of prior can also be influential when conducting SA prior to or without UQ. This is because a prior distribution for a sensitivity analysis for a given parameter which is much wider than the region where the model response surface is sensitive to the parameter value might cause the sensitivity calculation to underestimate the response in that potentially critical region. Similarly, a prior which is too narrow may miss regions where the model responds to the parameter altogether.</p>
+<p>Ideally, prior distributions are constructed independently of any analysis of the new data considered. This is because using data to inform the prior as well as to compute the likelihood reuses information in a potentially inappropriate way, which can lead to overconfident inferences. Following <span id="id1">Jaynes [<a class="reference internal" href="R.Bibliography.html#id196" title="Edwin T. Jaynes. Probability theory: the logic of science. Washington University St. Louis, MO, 1996.">195</a>], Gelman <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id197" title="Andrew Gelman, Daniel Simpson, and Michael Betancourt. The Prior Can Often Only Be Understood in the Context of the Likelihood. Entropy, 19(10):555, October 2017. Number: 10 Publisher: Multidisciplinary Digital Publishing Institute. URL: https://www.mdpi.com/1099-4300/19/10/555 (visited on 2021-06-14), doi:10.3390/e19100555.">196</a>]</span> refers to the ideal prior as one which encodes all available information about the model. For practical reasons (difficulty of construction or computational inconvenience), most priors fail to achieve this ideal. These compromises mean that priors should be transparently articulated and justified, so that the impact of the choice of prior can be fully understood. When there is ambiguity about an appropriate prior, such as how fat the tails should be, an analyst should examine how sensitive the UQ results are to the choice of prior.</p>
+<p>Priors can also be classified in terms of the information encoded by them, demonstrated in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a>. Non-informative priors (illustrated in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a> (a)) allegedly correspond to (and are frequently justified by) a position of ignorance. A classic example is the use of a uniform distribution. A uniform prior can, however, be problematic, as it can lead to improper inferences by giving extremely large values the same prior probability as values which may seem more likely <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id197" title="Andrew Gelman, Daniel Simpson, and Michael Betancourt. The Prior Can Often Only Be Understood in the Context of the Likelihood. Entropy, 19(10):555, October 2017. Number: 10 Publisher: Multidisciplinary Digital Publishing Institute. URL: https://www.mdpi.com/1099-4300/19/10/555 (visited on 2021-06-14), doi:10.3390/e19100555.">196</a>, <a class="reference internal" href="R.Bibliography.html#id198" title="Christian Robert. The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer Science &amp; Business Media, 2007.">197</a>]</span>, and therefore does not really reflect a state of complete ignorance. In the extreme case of a uniform prior over the entire real line, every particular region has effectively a prior weight of zero, even though not all regions are a priori unlikely <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id198" title="Christian Robert. The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer Science &amp; Business Media, 2007.">197</a>]</span>. Moreover, a uniform prior which excludes possible parameter values is not actually noninformative, as it assigns zero probability to those values while jumping to a nonzero probability as soon as the boundary is crossed. While a uniform prior can be problematic for the task of uncertainty quantification, it may be useful for an initial sensitivity analysis to identify the boundary of any regions where the model is sensitive to the parameter.</p>
+<figure class="margin-caption align-center" id="id5">
+<span id="figure-a1-5"></span><a class="reference internal image-reference" href="_images/figureA1_5_priors_posteriors.png"><img alt="Figure A1.5" src="_images/figureA1_5_priors_posteriors.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Impact of priors on posterior inferences. These plots show the results of inference for a linear regression model with 15 data points. The true value of the parameter is equal to -3. All priors have mean 0. In panel (a), a non-informative prior allows the tails of the posterior to extend freely, which may result in unreasonably large parameter values. In panel (b), a weakly informative prior constrains the tails more, but allows them to extend without too much restriction. In panel (c), an informative prior strongly constrains the tails of the posterior and biases the inference closer towards the prior mean (the posterior mean is -0.89 in this case, and closer to -3 in the other two cases).</span><a class="headerlink" href="#id5" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>Informative priors strongly bound the range of probable values (illustrated in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a> (c)). One example is a Gaussian distribution with a relatively small standard deviation, so that large values are assigned a close to null prior probability. Another example is the jump from zero to non-zero probability occurring at the truncation point of a truncated Gaussian, which could be justified based on information that the parameter cannot take on values beyond this point. Without this type of justification, however, priors may be too informative, failing to allow the information contained in the available data to update them.</p>
+<p>Finally, weakly informative priors (illustrated in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a> (b)) fall in between <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id197" title="Andrew Gelman, Daniel Simpson, and Michael Betancourt. The Prior Can Often Only Be Understood in the Context of the Likelihood. Entropy, 19(10):555, October 2017. Number: 10 Publisher: Multidisciplinary Digital Publishing Institute. URL: https://www.mdpi.com/1099-4300/19/10/555 (visited on 2021-06-14), doi:10.3390/e19100555.">196</a>]</span>. They regularize better than non-informative priors, but allow for more inference flexibility than fully informative priors. An example might be a Gaussian distribution with a moderate standard deviation, which still assigns negligible probability for values far away from the mean, but is less constrained than a narrow Gaussian for a reasonably large area. A key note is that it is not necessarily better to be more informative if this cannot be justified by the available information.</p>
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+  <section id="the-critical-final-step-predictive-checks">
+<h1>The Critical Final Step: Predictive Checks<a class="headerlink" href="#the-critical-final-step-predictive-checks" title="Link to this heading">#</a></h1>
+<p>Every UQ workflow requires a number of choices, potentially including selecting prior distributions, the likelihood specification, and any used numerical models. Checking the appropriateness of these choices is an essential step for sound inferences, as misspecification can produce biased results <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id180" title="Jenný Brynjarsdóttir and Anthony O'Hagan. Learning about physical parameters: the importance of model discrepancy. Inverse Problems, 30(11):114007, October 2014. Publisher: IOP Publishing. URL: https://doi.org/10.1088/0266-5611/30/11/114007 (visited on 2021-06-14), doi:10.1088/0266-5611/30/11/114007.">179</a>]</span>. Model checking in this fashion is part of an iterative UQ process, as the results can reveal adjustments to the statistical model or the need to select a different numerical model <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id200" title="Andrew Gelman, Xiao-Li Meng, and Hal Stern. Posterior Predictive Assessment of Model Fitness via Realized Discrepancies. Statistica Sinica, 6(4):733–760, 1996. Publisher: Institute of Statistical Science, Academia Sinica. URL: https://www.jstor.org/stable/24306036 (visited on 2021-06-14).">198</a>, <a class="reference internal" href="R.Bibliography.html#id199" title="Andrew Gelman, Aki Vehtari, Daniel Simpson, Charles C. Margossian, Bob Carpenter, Yuling Yao, Lauren Kennedy, Jonah Gabry, Paul-Christian Bürkner, and Martin Modrák. Bayesian Workflow. arXiv:2011.01808 [stat], November 2020. arXiv: 2011.01808. URL: http://arxiv.org/abs/2011.01808 (visited on 2021-06-14).">199</a>, <a class="reference internal" href="R.Bibliography.html#id201" title="Andrew Gelman and Cosma Rohilla Shalizi. Philosophy and the practice of Bayesian statistics. British Journal of Mathematical and Statistical Psychology, 66(1):8–38, 2013. _eprint: https://bpspsychub.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2044-8317.2011.02037.x. URL: https://bpspsychub.onlinelibrary.wiley.com/doi/abs/10.1111/j.2044-8317.2011.02037.x (visited on 2021-06-14), doi:10.1111/j.2044-8317.2011.02037.x.">200</a>]</span>.</p>
+<p>A classic example is the need to check the structure of residuals for correlations. Many standard statistical models, such as linear regression, assume that the residuals are independent and identically distributed from the error distribution. The presence of correlations, including temporal autocorrelations and spatial correlations, indicates a structural mismatch between the likelihood and the data. In these cases, the likelihood should be adjusted to account for these correlations.</p>
+<p>Checking residuals in this fashion is one example of a predictive check (or a posterior predictive check in the Bayesian setting). One way to view UQ is as a means to recover data-generating processes (associated with each parameter vector) consistent with the observations. Predictive checks compare the inferred data-generating process to the observations to determine whether the model is capable of appropriately capturing uncertainty. After conducting the UQ analysis, alternatively realized datasets are simulated from sampled parameters. These alternative datasets, or their summary statistics, can be tested against the observations to determine adequacy of the fit. Predictive checks are therefore a way of probing various model components to identify shortcomings that might result in biased inferences or poor projections, depending on the goal of the analysis.</p>
+<p>One example of a graphical predictive check for time series models is hindcasting, where predictive intervals are constructed from the alternative datasets and plotted along with the data. Hindcasts demonstrate how well the model is capable of capturing the broader dynamics of the data, as well as whether the parameter distributions produce appropriate levels of output uncertainty. A related quantitative check is the surprise index, which calculates the percentage of data points located within a fixed predictive interval. For example, the 90% predictive interval should contain approximately 90% of the data. More uncertainty than this reflects underconfidence, while less uncertainty reflects overconfidence. This could be the result of priors that are not appropriately informative, or a likelihood that does not account for correlations between data points appropriately. It could also be the result of a numerical model that isn’t sufficiently sensitive to the parameters that are treated as uncertain.</p>
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+<h1>Key Take-Home Points<a class="headerlink" href="#key-take-home-points" title="Link to this heading">#</a></h1>
+<p>When appropriate, UQ is an important component of the exploratory modeling workflow. While a number of parameter sets could be consistent with observations, they may result in divergent model outputs when exposed to different future conditions. This can result in identifying risks which are not visible when selecting a “best fit” parameterization. Quantifying uncertainties also allows us to quantify the support for hypotheses, which is an essential part of the scientific process.</p>
+<p>Due to the scale and complexity of the experiments taking place in IM3, UQ has not been extensively used. The tradeoff between the available number of function evaluations and the number of uncertain parameters illustrated in <a class="reference internal" href="A1_Uncertainty_Quantification.html#figure-a1-1"><span class="std std-numref">Fig. A.1</span></a> is particularly challenging due to the increasing complexity of state-of-the-art models and the movement towards coupled, multisector models. This tradeoff can be addressed somewhat through the use of emulators and parallelizable methods. In particular, when attempting to navigate this tradeoff by limiting the number of uncertain parameters, it is important to carefully iterate with sensitivity analyses to ensure that critical parameters are identified.</p>
+<p>Specifying prior distributions and likelihoods is an ongoing challenge. Prior distributions, in particular, should be treated as deeply uncertain when appropriate. One key advantage of the methods described in this chapter is that they have the potential for increased transparency. When it is not possible to conduct a sensitivity analysis on a number of critical priors due to limited computational budgets, fully specifying and providing a justification for the utilized distributions allows other researchers to identify key assumptions and build on existing work. The same is true for the specification of likelihoods—while likelihood-free methods avoid the need to specify a likelihood function, they require other assumptions or choices, which should be described and justified as transparently as possible.</p>
+<p>We conclude this appendix with some key recommendations:
+1. UQ analysis does not require full confidence in priors and likelihoods. Rather, UQ should be treated as part of an exploratory modeling workflow, where hypotheses related to model structures, prior distributions, and likelihoods can be tested.
+2. For complex multisectoral models, UQ will typically require the use of a reduced set of parameters, either through emulation or by fixing the others to their best-fit values. These parameters should be selected through a careful sensitivity analysis.
+3. Avoid the use of supposedly “non-informative” priors, such as uniform priors, whenever possible. In the absence of strong information about parameter values, the use of weakly informative priors, such as diffuse normals, is preferable.
+4. Be cognizant of the limitations of conclusions that can be drawn by using each method. The bootstrap, for example, may result in overconfidence if the dataset is limited and is not truly representative of the underlying stochastic process.
+5. When using MCMC, Markov chains can not be shown to have converged to the target distribution, but rather evidence can be collected to demonstrate that it is likely that they have.
+6. Conduct predictive checks based on the assumptions underlying the choices made in the analysis, and iteratively update those choices if the assumptions prove to be ill-suited for the problem at hand.</p>
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+    <h1>Uncertainty Quantification</h1>
+    <!-- Table of contents -->
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+                <h2> Contents </h2>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">A.1. Introduction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#parametric-bootstrap">A.2. Parametric Bootstrap</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pre-calibration">A.3. Pre-Calibration</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#markov-chain-monte-carlo">A.4. Markov Chain Monte Carlo</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-methods">A.5. Other Methods</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-critical-first-step-how-to-choose-a-prior-distribution">A.6. The Critical First Step: How to Choose a Prior Distribution</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-critical-final-step-predictive-checks">A.7. The Critical Final Step: Predictive Checks</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#key-take-home-points">A.8. Key Take-Home Points</a></li>
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+  <section id="uncertainty-quantification">
+<span id="a1-uncertainty-quantification"></span><h1><span class="section-number">A. </span>Uncertainty Quantification<a class="headerlink" href="#uncertainty-quantification" title="Link to this heading">#</a></h1>
+<section id="introduction">
+<h2><span class="section-number">A.1. </span>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h2>
+<p>As defined in <a class="reference internal" href="1_introduction.html#introduction"><span class="std std-numref">Chapter 1</span></a>, uncertainty quantification (UQ) refers to the formal focus on the full specification of likelihoods as well as distributional forms necessary to infer the joint  probabilistic response across all modeled factors of interest <span id="id1">[<a class="reference internal" href="R.Bibliography.html#id11" title="Roger Cooke and others. Experts in uncertainty: opinion and subjective probability in science. Oxford University Press on Demand, 1991.">8</a>]</span>. This is in contrast to UC (the primary focus of the main document of this book), which is instead aimed at identifying which modeling choices yield the most consequential changes or outcomes and exploring alternative hypotheses related to the form and function of modeled systems <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id12" title="Enayat A Moallemi, Jan Kwakkel, Fjalar J de Haan, and Brett A Bryan. Exploratory modeling for analyzing coupled human-natural systems under uncertainty. Global Environmental Change, 65:102186, 2020.">9</a>, <a class="reference internal" href="R.Bibliography.html#id13" title="Warren E Walker, Poul Harremoës, Jan Rotmans, Jeroen P Van Der Sluijs, Marjolein BA Van Asselt, Peter Janssen, and Martin P Krayer von Krauss. Defining uncertainty: a conceptual basis for uncertainty management in model-based decision support. Integrated assessment, 4(1):5–17, 2003.">10</a>]</span>.</p>
+<p>UQ is important for quantifying the relative merits of hypotheses for at least three main reasons. First, identifying model parameters that are consistent with observations is an important part of model development. Due to several effects, including correlations between parameters, simplified or incomplete model structures (relative to the full real-world dynamics), and uncertainty in the observations, many different combinations of parameter values can be consistent with the model structure and the observations to varying extents. Accounting for this uncertainty is conceptually preferable to selecting a single “best fit” parameter vector, particularly as consistency with historical or present observations does not necessarily  guarantee skillful future projections.</p>
+<p>The act of quantification requires specific assumptions about distributional forms and likelihoods, which may be more or less justified depending on prior information about the system or model behavior. As a result, UQ is well-suited for studies accounting for or addressing hypotheses related to systems with a relatively large amount of available data and models which are computationally inexpensive, particularly when the emphasis is on prediction. As shown in <a class="reference internal" href="#figure-a1-1"><span class="std std-numref">Fig. A.1</span></a>, there is a fundamental tradeoff between the available number of model evaluations (for a fixed computational budget) and the number of parameters treated as uncertain. Sensitivity analyses are therefore part of a typical UQ workflow to identify which factors can be fixed and which ought to be prioritized in the UQ.</p>
+<figure class="margin-caption align-center" id="id37">
+<span id="figure-a1-1"></span><a class="reference internal image-reference" href="_images/figureA1_1_UQ_approaches.png"><img alt="Figure A1.1" src="_images/figureA1_1_UQ_approaches.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. A.1 </span><span class="caption-text">Overview of selected existing approaches for uncertainty quantification and their appropriateness given the number of uncertain model parameters and the number of available model simulations. Green shading denotes regions suitable for uncertainty quantification and red shading indicates regions more appropriate for uncertainty characterization.</span><a class="headerlink" href="#id37" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The choice of a particular UQ method depends on both the desired level of quantification and the ability to navigate the tradeoff between computational expense and the number of uncertain parameters (<a class="reference internal" href="#figure-a1-1"><span class="std std-numref">Fig. A.1</span></a>). For example, Markov chain Monte Carlo with a full system model can provide an improved representation of uncertainty compared to the coarser pre-calibration approach <span id="id3">[<a class="reference internal" href="R.Bibliography.html#id37" title="Kelsey L. Ruckert, Gary Shaffer, David Pollard, Yawen Guan, Tony E. Wong, Chris E. Forest, and Klaus Keller. Assessing the Impact of Retreat Mechanisms in a Simple Antarctic Ice Sheet Model Using Bayesian Calibration. PLOS ONE, 12(1):e0170052, January 2017. doi:10.1371/journal.pone.0170052.">157</a>]</span>, but requires many more model evaluations. The use of a surrogate model to approximate the full system model can reduce the number of needed model evaluations by several orders of magnitude, but the uncertainty quantification can only accommodate a limited number of parameters.</p>
+<p>The remainder of this appendix will focus on introducing workflows for particular UQ methods, including a brief discussion of advantages and limitations.</p>
+</section>
+<section id="parametric-bootstrap">
+<h2><span class="section-number">A.2. </span>Parametric Bootstrap<a class="headerlink" href="#parametric-bootstrap" title="Link to this heading">#</a></h2>
+<p>The parametric bootstrap <span id="id4">[<a class="reference internal" href="R.Bibliography.html#id14" title="B. Efron and R. Tibshirani. Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy. Statistical Science, 1(1):54–75, February 1986. doi:10.1214/ss/1177013815.">158</a>]</span> refers to a process of model recalibration to alternate realizations of the data. The bootstrap was originally developed to estimate standard errors and confidence intervals without ascertaining key assumptions that might not hold given the available data. In a setting where observations can be viewed as independent realizations of an underlying stochastic process, a sufficiently rich dataset can be treated as a population representing the data distribution. New datasets are then generated by resampling from the data with replacement, and the model can be refit to each new dataset using maximum-likelihood estimation. The resulting distribution of estimates can then be viewed as a representation of parametric uncertainty.</p>
+<p>A typical workflow for the parametric bootstrap is shown in <a class="reference internal" href="#figure-a1-2"><span class="std std-numref">Fig. A.2</span></a>. After identifying outputs of interest and preparing the data, the parametric model is fit by some procedure such as minimizing root-mean-square-error or maximizing the likelihood. Alternate datasets are constructed by resampling from the population or by generating new samples from the fitted data-generating process. It is important at this step that the resampled quantities are independent of one another. For example, in the context of temporally- or spatially-correlated data, such as time series, the raw observations cannot be treated as independent realizations. However, the residuals resulting from fitting the model to the data could be (depending on their structure). For example, if the residuals are treated as independent, they can then be resampled with replacement, and these residuals added to the original model fit to create new realizations. If the residuals are assumed to be the result of an autoregressive process, this process could be fit to the original residual series and new residuals be created using this model <span id="id5">[<a class="reference internal" href="R.Bibliography.html#id35" title="Ryan L. Sriver, Robert J. Lempert, Per Wikman-Svahn, and Klaus Keller. Characterizing uncertain sea-level rise projections to support investment decisions. PLOS ONE, 13(2):e0190641, February 2018. URL: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0190641 (visited on 2021-06-09), doi:10.1371/journal.pone.0190641.">159</a>]</span>. The model is then refit to each new realization.</p>
+<figure class="margin-caption align-center" id="id38">
+<span id="figure-a1-2"></span><a class="reference internal image-reference" href="_images/figureA1_2_bootstrap_workflow.png"><img alt="Figure A1.2" src="_images/figureA1_2_bootstrap_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. A.2 </span><span class="caption-text">Workflow for the parametric bootstrap.</span><a class="headerlink" href="#id38" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The bootstrap is computationally convenient, particularly as the process of fitting the model to each realization can be easily parallelized. This approach also requires minimal prior assumptions. However, due to the assumption that the available data are representative of the underlying data distribution, the bootstrap can neglect key uncertainties which might influence the results. For example, when using an autoregressive process to generate new residuals, uncertainty in the autocorrelation parameter and innovation variance is neglected, which may bias estimates of, for example, low-probability but high-impact events <span id="id6">[<a class="reference internal" href="R.Bibliography.html#id36" title="Kelsey L. Ruckert, Yawen Guan, Alexander M. R. Bakker, Chris E. Forest, and Klaus Keller. The effects of time-varying observation errors on semi-empirical sea-level projections. Climatic Change, 140(3):349–360, February 2017. URL: https://doi.org/10.1007/s10584-016-1858-z (visited on 2021-06-09), doi:10.1007/s10584-016-1858-z.">160</a>]</span>.</p>
+</section>
+<section id="pre-calibration">
+<h2><span class="section-number">A.3. </span>Pre-Calibration<a class="headerlink" href="#pre-calibration" title="Link to this heading">#</a></h2>
+<p>Pre-calibration <span id="id7">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>, <a class="reference internal" href="R.Bibliography.html#id39" title="Neil R Edwards, David Cameron, and Jonathan Rougier. Precalibrating an intermediate complexity climate model. Clim. Dyn., 37(7-8):1469–1482, 2011. URL: http://dx.doi.org/10.1007/s00382-010-0921-0, doi:10.1007/s00382-010-0921-0.">161</a>, <a class="reference internal" href="R.Bibliography.html#id40" title="Alexis Boukouvalas, Pete Sykes, Dan Cornford, and Hugo Maruri-Aguilar. Bayesian Precalibration of a Large Stochastic Microsimulation Model. IEEE Transactions on Intelligent Transportation Systems, 15(3):1337–1347, June 2014. doi:10.1109/TITS.2014.2304394.">162</a>]</span> involves the identification of a plausible set of parameters using some prespecified screening criterion, such as the distance from the model results to the observations (based on an appropriate metric for the desired matching features, such as root-mean-squared error). A typical workflow is shown in <a class="reference internal" href="#figure-a1-3"><span class="std std-numref">Fig. A.3</span></a>. Parameter values are obtained by systematically sampling the input space (see <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#design-of-experiments"><span class="std std-numref">Chapter 3.3</span></a>). After the model is evaluated at the samples, only those passing the distance criterion are retained. This selects a subset of the parameter space as “plausible” based on the screening criterion, though there is no assignment of probabilities within this plausible region.</p>
+<figure class="margin-caption align-center" id="id39">
+<span id="figure-a1-3"></span><a class="reference internal image-reference" href="_images/figureA1_3_precal_workflow.png"><img alt="Figure A1.3" src="_images/figureA1_3_precal_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. A.3 </span><span class="caption-text">Workflow for pre-calibration.</span><a class="headerlink" href="#id39" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>Pre-calibration can be useful for models which are inexpensive enough that a reasonable
+number of samples can be used to represent the parameter space, but which are too expensive to facilitate full uncertainty quantification. High-dimensional parameter spaces, which can be problematic for the uncertainty quantification methods below, may also be explored using pre-calibration. One key prerequisite to using this method is the ability to place a meaningful distance metric on the output space.</p>
+<p>However, pre-calibration results in a very coarse characterization of uncertainty, especially when considering a large number of parameters, as more samples are needed to fully characterize the parameter space. Due to the inability to evaluate the relative probability of regions of the parameter space beyond the binary plausible-and-implausible characterization, pre-calibration can also result in degraded hindcast and projection skills and parameter estimates <span id="id8">[<a class="reference internal" href="R.Bibliography.html#id37" title="Kelsey L. Ruckert, Gary Shaffer, David Pollard, Yawen Guan, Tony E. Wong, Chris E. Forest, and Klaus Keller. Assessing the Impact of Retreat Mechanisms in a Simple Antarctic Ice Sheet Model Using Bayesian Calibration. PLOS ONE, 12(1):e0170052, January 2017. doi:10.1371/journal.pone.0170052.">157</a>, <a class="reference internal" href="R.Bibliography.html#id41" title="David Makowski, Daniel Wallach, and Marie Tremblay. Using a Bayesian approach to parameter estimation; comparison of the GLUE and MCMC methods. Agronomie, 22(2):191–203, 2002. Publisher: EDP Sciences.">163</a>, <a class="reference internal" href="R.Bibliography.html#id42" title="Mahyar Shafii, Bryan Tolson, and Loren Shawn Matott. Uncertainty-based multi-criteria calibration of rainfall-runoff models: a comparative study. Stochastic Environmental Research and Risk Assessment, 28(6):1493–1510, August 2014. doi:10.1007/s00477-014-0855-x.">164</a>]</span>.</p>
+<p>A related method, widely used in hydrological studies, is generalized likelihood uncertainty estimation, or GLUE <span id="id9">[<a class="reference internal" href="R.Bibliography.html#id38" title="Keith Beven and Andrew Binley. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3):279–298, 1992. doi:10.1002/hyp.3360060305.">18</a>]</span>. Unlike pre-calibration, the underlying argument for GLUE relies on the concept of equifinality <span id="id10">[<a class="reference internal" href="R.Bibliography.html#id43" title="Keith Beven and Jim Freer. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology. Journal of hydrology, 249(1-4):11–29, 2001. Publisher: Elsevier.">165</a>]</span>, which posits that it is impossible to find a uniquely well-performing parameter vector for models of abstract environmental systems <span id="id11">[<a class="reference internal" href="R.Bibliography.html#id43" title="Keith Beven and Jim Freer. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology. Journal of hydrology, 249(1-4):11–29, 2001. Publisher: Elsevier.">165</a>, <a class="reference internal" href="R.Bibliography.html#id44" title="Jasper A. Vrugt and Keith J. Beven. Embracing equifinality with efficiency: Limits of Acceptability sampling using the DREAM(LOA) algorithm. Journal of Hydrology, 559:954–971, April 2018. doi:10.1016/j.jhydrol.2018.02.026.">166</a>]</span>. In other words, there exist multiple parameter vectors which perform equally or similarly well. As with pre-calibration, GLUE uses a goodness-of-fit measure (though this is called a “likelihood” in the GLUE literature, as opposed to a statistical likelihood function <span id="id12">[<a class="reference internal" href="R.Bibliography.html#id45" title="Jery R. Stedinger, Richard M. Vogel, Seung Uk Lee, and Rebecca Batchelder. Appraisal of the generalized likelihood uncertainty estimation (GLUE) method. Water Resources Research, 2008. doi:10.1029/2008WR006822.">167</a>]</span>) to evaluate samples. After setting a threshold of acceptable performance with respect to that measure, samples are evaluated and classified into “behavioral” or “non-behavioral” according to the threshold.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Put this into practice! Click the following badge to try out an interactive tutorial on utilizing Pre-Calibration and GLUE for HYMOD model calibration:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">Pre-Calibration Jupyter Notebook</a></p>
+</div>
+</section>
+<section id="markov-chain-monte-carlo">
+<h2><span class="section-number">A.4. </span>Markov Chain Monte Carlo<a class="headerlink" href="#markov-chain-monte-carlo" title="Link to this heading">#</a></h2>
+<p>Markov chain Monte Carlo (MCMC) is a “gold standard” approach to full uncertainty quantification. MCMC refers to a category of algorithms which systematically sample from a target distribution (in this case, the posterior distribution) by constructing a Markov chain. A Markov chain is a probabilistic structure consisting of a state space, an initial probability distribution over the states, and a transition distribution between states. If a Markov chain satisfies certain properties <span id="id13">[<a class="reference internal" href="R.Bibliography.html#id170" title="Christian Robert and George Casella. Monte Carlo Statistical Methods. Springer Science &amp; Business Media, March 2013. ISBN 978-1-4757-3071-5.">168</a>, <a class="reference internal" href="R.Bibliography.html#id169" title="Christian P. Robert. The Metropolis–Hastings Algorithm. In Wiley StatsRef: Statistics Reference Online, pages 1–15. American Cancer Society, 2015. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat07834 (visited on 2021-06-14), doi:10.1002/9781118445112.stat07834.">169</a>]</span>, the probability of being in each state will eventually converge to a stable, or stationary, distribution, regardless of the initial probabilities.</p>
+<p>MCMC algorithms construct a Markov chain of samples from a parameter space (the combination of model and statistical parameters). This Markov chain is constructed so that the stationary distribution is a target distribution, in this case the (Bayesian) posterior distribution. As a result, after the transient period, the resulting samples can be viewed as a set of dependent samples from the posterior (the dependence is due to the autocorrelation between samples resulting from the Markov chain transitions). Expected values can be computed from these samples (for example, using batch-means estimators <span id="id14">[<a class="reference internal" href="R.Bibliography.html#id171" title="James M. Flegal, Murali Haran, and Galin L. Jones. Markov Chain Monte Carlo: Can We Trust the Third Significant Figure? Statistical Science, 23(2):250–260, May 2008. Publisher: Institute of Mathematical Statistics. URL: https://projecteuclid.org/journals/statistical-science/volume-23/issue-2/Markov-Chain-Monte-Carlo--Can-We-Trust-the-Third/10.1214/08-STS257.full (visited on 2021-06-14), doi:10.1214/08-STS257.">170</a>]</span>), or the chain can be sub-sampled or thinned and the resulting samples used as independent Monte Carlo samples due to the reduced or eliminated autocorrelation.</p>
+<p>A general workflow for MCMC is shown in <a class="reference internal" href="#figure-a1-4"><span class="std std-numref">Fig. A.4</span></a>. The first decision is whether to use the full model or a surrogate model (or emulator). Typical surrogates include Gaussian process emulation <span id="id15">[<a class="reference internal" href="R.Bibliography.html#id172" title="Carla Currin, Toby Mitchell, Max Morris, and Don Ylvisaker. Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments. Journal of the American Statistical Association, 86(416):953–963, December 1991. Publisher: Taylor &amp; Francis _eprint: https://www.tandfonline.com/doi/pdf/10.1080/01621459.1991.10475138. URL: https://www.tandfonline.com/doi/abs/10.1080/01621459.1991.10475138 (visited on 2021-06-14), doi:10.1080/01621459.1991.10475138.">171</a>, <a class="reference internal" href="R.Bibliography.html#id173" title="Jerome Sacks, William J. Welch, Toby J. Mitchell, and Henry P. Wynn. Design and Analysis of Computer Experiments. Statistical Science, 4(4):409–423, 1989. Publisher: Institute of Mathematical Statistics. URL: https://www.jstor.org/stable/2245858 (visited on 2021-06-14).">172</a>]</span>, polynomial chaos expansions <span id="id16">[<a class="reference internal" href="R.Bibliography.html#id174" title="Roger G. Ghanem and Pol D. Spanos. Spectral Stochastic Finite‐Element Formulation for Reliability Analysis. Journal of Engineering Mechanics, 117(10):2351–2372, October 1991. Publisher: American Society of Civil Engineers. URL: https://ascelibrary.org/doi/abs/10.1061/%28ASCE%290733-9399%281991%29117%3A10%282351%29 (visited on 2021-06-14), doi:10.1061/(ASCE)0733-9399(1991)117:10(2351).">173</a>, <a class="reference internal" href="R.Bibliography.html#id175" title="Dongbin Xiu and George Em Karniadakis. The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations. SIAM Journal on Scientific Computing, 24(2):619–644, January 2002. Publisher: Society for Industrial and Applied Mathematics. URL: https://epubs.siam.org/doi/abs/10.1137/S1064827501387826 (visited on 2021-06-14), doi:10.1137/S1064827501387826.">174</a>]</span>, support vector machines <span id="id17">[<a class="reference internal" href="R.Bibliography.html#id176" title="Angelo Ciccazzo, Gianni Di Pillo, and Vittorio Latorre. A SVM Surrogate Model-Based Method for Parametric Yield Optimization. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 35(7):1224–1228, July 2016. Conference Name: IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. doi:10.1109/TCAD.2015.2501307.">175</a>, <a class="reference internal" href="R.Bibliography.html#id177" title="W. Andrew Pruett and Robert L. Hester. The Creation of Surrogate Models for Fast Estimation of Complex Model Outcomes. PLOS ONE, 11(6):e0156574, June 2016. Publisher: Public Library of Science. URL: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0156574 (visited on 2021-06-14), doi:10.1371/journal.pone.0156574.">176</a>]</span>, and neural networks <span id="id18">[<a class="reference internal" href="R.Bibliography.html#id178" title="John Eason and Selen Cremaschi. Adaptive sequential sampling for surrogate model generation with artificial neural networks. Computers &amp; Chemical Engineering, 68:220–232, September 2014. URL: https://www.sciencedirect.com/science/article/pii/S0098135414001719 (visited on 2021-06-14), doi:10.1016/j.compchemeng.2014.05.021.">177</a>, <a class="reference internal" href="R.Bibliography.html#id179" title="Dirk Gorissen, Luciano De Tommasi, Karel Crombecq, and Tom Dhaene. Sequential modeling of a low noise amplifier with neural networks and active learning. Neural Computing and Applications, 18(5):485–494, June 2009. URL: https://doi.org/10.1007/s00521-008-0223-1 (visited on 2021-06-14), doi:10.1007/s00521-008-0223-1.">178</a>]</span>. Surrogate modeling can be faster, but requires a sufficient number of model evaluations for the surrogate to accurately represent the model’s response surface, and this typically limits the number of parameters which can be included in the analysis.</p>
+<figure class="margin-caption align-center" id="id40">
+<span id="figure-a1-4"></span><a class="reference internal image-reference" href="_images/figureA1_4_mcmc_workflow.png"><img alt="Figure A1.4" src="_images/figureA1_4_mcmc_workflow.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. A.4 </span><span class="caption-text">Workflow for Markov chain Monte Carlo.</span><a class="headerlink" href="#id40" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>After selecting the variables which will be treated as uncertain, the next step is to specify the likelihood based on the selected surrogate model or the structure of the data-model residuals. For example, it may not always be appropriate to treat the residuals as independent and identically distributed (as is commonly done in linear regression). A mis-specification of the residual structure can result in biases and over- or under-confident inferences and projections <span id="id19">[<a class="reference internal" href="R.Bibliography.html#id180" title="Jenný Brynjarsdóttir and Anthony O'Hagan. Learning about physical parameters: the importance of model discrepancy. Inverse Problems, 30(11):114007, October 2014. Publisher: IOP Publishing. URL: https://doi.org/10.1088/0266-5611/30/11/114007 (visited on 2021-06-14), doi:10.1088/0266-5611/30/11/114007.">179</a>]</span>.</p>
+<p>After specifying the prior distributions (see <a class="reference internal" href="#critical-first-step"><span class="std std-numref">Chapter A.6</span></a>), the selected MCMC algorithm should be used to draw samples from the posterior distribution. There are many MCMC algorithms, all of which have advantages and disadvantages for a particular problem. These include the Metropolis-Hastings algorithm <span id="id20">[<a class="reference internal" href="R.Bibliography.html#id169" title="Christian P. Robert. The Metropolis–Hastings Algorithm. In Wiley StatsRef: Statistics Reference Online, pages 1–15. American Cancer Society, 2015. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat07834 (visited on 2021-06-14), doi:10.1002/9781118445112.stat07834.">169</a>]</span> and Hamiltonian Monte Carlo <span id="id21">[<a class="reference internal" href="R.Bibliography.html#id181" title="Michael Betancourt. A conceptual introduction to Hamiltonian Monte Carlo. arXiv preprint arXiv:1701.02434, 2017.">180</a>, <a class="reference internal" href="R.Bibliography.html#id182" title="Radford M. Neal. MCMC using Hamiltonian dynamics. Handbook of markov chain monte carlo, 2(11):2, 2011.">181</a>]</span>. Software packages typically implement one MCMC method, sometimes designed for a particular problem setting or likelihood specification. For example, R’s <em>adaptMCMC</em> implements an adaptive Metropolis-Hastings algorithm <span id="id22">[<a class="reference internal" href="R.Bibliography.html#id183" title="Matti Vihola. Robust adaptive Metropolis algorithm with coerced acceptance rate. Statistics and Computing, 22(5):997–1008, September 2012. URL: https://doi.org/10.1007/s11222-011-9269-5 (visited on 2021-06-14), doi:10.1007/s11222-011-9269-5.">182</a>]</span>, while <em>NIMBLE</em> <span id="id23">[<a class="reference internal" href="R.Bibliography.html#id184" title="Perry de Valpine, Daniel Turek, Christopher J. Paciorek, Clifford Anderson-Bergman, Duncan Temple Lang, and Rastislav Bodik. Programming With Models: Writing Statistical Algorithms for General Model Structures With NIMBLE. Journal of Computational and Graphical Statistics, 26(2):403–413, April 2017. Publisher: Taylor &amp; Francis _eprint: https://doi.org/10.1080/10618600.2016.1172487. URL: https://doi.org/10.1080/10618600.2016.1172487 (visited on 2021-06-14), doi:10.1080/10618600.2016.1172487.">183</a>, <a class="reference internal" href="R.Bibliography.html#id185" title="NIMBLE Development Team. NIMBLE: MCMC, Particle Filtering, and Programmable Hierarchical Modeling. May 2021. URL: https://zenodo.org/record/4829693 (visited on 2021-06-14), doi:10.5281/zenodo.4829693.">184</a>]</span> uses a user-customizable Metropolis-Hastings implementation, as well as functionality for Gibbs sampling (which is a special case of Metropolis-Hastings where the prior distribution has a convenient mathematical form). Some recent implementations, such as <em>Stan</em> <span id="id24">[<a class="reference internal" href="R.Bibliography.html#id186" title="Stan Development Team. Stan Modeling Language Users Guide and Reference Manual. 2021. URL: https://mc-stan.org/docs/2_27/stan-users-guide/index.html (visited on 2021-06-14).">185</a>]</span>, <em>pyMC3</em> <span id="id25">[<a class="reference internal" href="R.Bibliography.html#id187" title="John Salvatier, Thomas V. Wiecki, and Christopher Fonnesbeck. Probabilistic programming in Python using PyMC3. PeerJ Computer Science, 2:e55, April 2016. Publisher: PeerJ Inc. URL: https://peerj.com/articles/cs-55 (visited on 2021-06-14), doi:10.7717/peerj-cs.55.">186</a>]</span>, and <em>Turing</em> <span id="id26">[<a class="reference internal" href="R.Bibliography.html#id188" title="Hong Ge, Kai Xu, and Zoubin Ghahramani. Turing: A Language for Flexible Probabilistic Inference. In International Conference on Artificial Intelligence and Statistics, 1682–1690. PMLR, March 2018. ISSN: 2640-3498. URL: http://proceedings.mlr.press/v84/ge18b.html (visited on 2021-06-14).">187</a>]</span> allow different algorithms to be used.</p>
+<p>A main consideration when using MCMC algorithms is testing for convergence to the target distribution. As convergence is guaranteed only for a sufficiently large number of transitions, it is impossible to conclude for certain that a chain has converged for a fixed number of iterations. However, several heuristics have been developed <span id="id27">[<a class="reference internal" href="R.Bibliography.html#id171" title="James M. Flegal, Murali Haran, and Galin L. Jones. Markov Chain Monte Carlo: Can We Trust the Third Significant Figure? Statistical Science, 23(2):250–260, May 2008. Publisher: Institute of Mathematical Statistics. URL: https://projecteuclid.org/journals/statistical-science/volume-23/issue-2/Markov-Chain-Monte-Carlo--Can-We-Trust-the-Third/10.1214/08-STS257.full (visited on 2021-06-14), doi:10.1214/08-STS257.">170</a>, <a class="reference internal" href="R.Bibliography.html#id189" title="Andrew Gelman and Donald B. Rubin. Inference from Iterative Simulation Using Multiple Sequences. Statistical Science, 7(4):457–472, November 1992. Publisher: Institute of Mathematical Statistics. URL: https://projecteuclid.org/journals/statistical-science/volume-7/issue-4/Inference-from-Iterative-Simulation-Using-Multiple-Sequences/10.1214/ss/1177011136.full (visited on 2021-06-14), doi:10.1214/ss/1177011136.">188</a>]</span> to increase evidence that convergence has occurred.</p>
+</section>
+<section id="other-methods">
+<h2><span class="section-number">A.5. </span>Other Methods<a class="headerlink" href="#other-methods" title="Link to this heading">#</a></h2>
+<p>Other common methods for UQ exist. These include sequential Monte Carlo, otherwise known as particle filtering <span id="id28">[<a class="reference internal" href="R.Bibliography.html#id190" title="Pierre Del Moral, Arnaud Doucet, and Ajay Jasra. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(3):411–436, 2006. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9868.2006.00553.x. URL: http://rss.onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9868.2006.00553.x (visited on 2021-06-14), doi:10.1111/j.1467-9868.2006.00553.x.">189</a>, <a class="reference internal" href="R.Bibliography.html#id191" title="Arnaud Doucet, Simon Godsill, and Christophe Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 10(3):197–208, July 2000. URL: https://doi.org/10.1023/A:1008935410038 (visited on 2021-06-14), doi:10.1023/A:1008935410038.">190</a>, <a class="reference internal" href="R.Bibliography.html#id192" title="Jane Liu and Mike West. Combined Parameter and State Estimation in Simulation-Based Filtering. In Arnaud Doucet, Nando de Freitas, and Neil Gordon, editors, Sequential Monte Carlo Methods in Practice, Statistics for Engineering and Information Science, pages 197–223. Springer, New York, NY, 2001. URL: https://doi.org/10.1007/978-1-4757-3437-9_10 (visited on 2021-06-14), doi:10.1007/978-1-4757-3437-9_10.">191</a>]</span>, where a number of particles are used to evaluate samples. An advantage of sequential Monte Carlo is that the vast majority of the computation can be parallelized, unlike with standard MCMC. A major weakness is the potential for degeneracy <span id="id29">[<a class="reference internal" href="R.Bibliography.html#id191" title="Arnaud Doucet, Simon Godsill, and Christophe Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 10(3):197–208, July 2000. URL: https://doi.org/10.1023/A:1008935410038 (visited on 2021-06-14), doi:10.1023/A:1008935410038.">190</a>]</span>, where many particles have extremely small weights, resulting in the effective use of only a few samples.</p>
+<p>Another method is approximate Bayesian computation (ABC) <span id="id30">[<a class="reference internal" href="R.Bibliography.html#id193" title="Stefano Cabras, Maria Eugenia Castellanos Nueda, and Erlis Ruli. Approximate Bayesian Computation by Modelling Summary Statistics in a Quasi-likelihood Framework. Bayesian Analysis, 10(2):411–439, June 2015. Publisher: International Society for Bayesian Analysis. URL: https://projecteuclid.org/journals/bayesian-analysis/volume-10/issue-2/Approximate-Bayesian-Computation-by-Modelling-Summary-Statistics-in-a-Quasi/10.1214/14-BA921.full (visited on 2021-06-14), doi:10.1214/14-BA921.">192</a>, <a class="reference internal" href="R.Bibliography.html#id194" title="Jarno Lintusaari, Michael U. Gutmann, Ritabrata Dutta, Samuel Kaski, and Jukka Corander. Fundamentals and Recent Developments in Approximate Bayesian Computation. Systematic Biology, 66(1):e66–e82, January 2017. URL: https://doi.org/10.1093/sysbio/syw077 (visited on 2021-06-14), doi:10.1093/sysbio/syw077.">193</a>, <a class="reference internal" href="R.Bibliography.html#id195" title="Mikael Sunnåker, Alberto Giovanni Busetto, Elina Numminen, Jukka Corander, Matthieu Foll, and Christophe Dessimoz. Approximate Bayesian Computation. PLOS Computational Biology, 9(1):e1002803, January 2013. Publisher: Public Library of Science. URL: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002803 (visited on 2021-06-14), doi:10.1371/journal.pcbi.1002803.">194</a>]</span>. ABC is a likelihood-free approach that compares model output to a set of summary statistics. ABC is therefore well-suited for models and residual structures which do not lend themselves to a computationally-tractable likelihood, but the resulting inferences are known to be biased if the set of summary statistics is not sufficient, which can be difficult to know a-priori.</p>
+</section>
+<section id="the-critical-first-step-how-to-choose-a-prior-distribution">
+<span id="critical-first-step"></span><h2><span class="section-number">A.6. </span>The Critical First Step: How to Choose a Prior Distribution<a class="headerlink" href="#the-critical-first-step-how-to-choose-a-prior-distribution" title="Link to this heading">#</a></h2>
+<p>Prior distributions play an important role in Bayesian uncertainty quantification, particularly when data is limited relative to the dimension of the model. Bayesian updating can be thought of as an information filter, where each additional datum is added to the information contained in the prior; eventually, the prior makes relatively little impact. In real world problems, it can be extremely difficult to assess how much data is required for the choice of prior to become less relevant. The choice of prior can also be influential when conducting SA prior to or without UQ. This is because a prior distribution for a sensitivity analysis for a given parameter which is much wider than the region where the model response surface is sensitive to the parameter value might cause the sensitivity calculation to underestimate the response in that potentially critical region. Similarly, a prior which is too narrow may miss regions where the model responds to the parameter altogether.</p>
+<p>Ideally, prior distributions are constructed independently of any analysis of the new data considered. This is because using data to inform the prior as well as to compute the likelihood reuses information in a potentially inappropriate way, which can lead to overconfident inferences. Following <span id="id31">Jaynes [<a class="reference internal" href="R.Bibliography.html#id196" title="Edwin T. Jaynes. Probability theory: the logic of science. Washington University St. Louis, MO, 1996.">195</a>], Gelman <em>et al.</em> [<a class="reference internal" href="R.Bibliography.html#id197" title="Andrew Gelman, Daniel Simpson, and Michael Betancourt. The Prior Can Often Only Be Understood in the Context of the Likelihood. Entropy, 19(10):555, October 2017. Number: 10 Publisher: Multidisciplinary Digital Publishing Institute. URL: https://www.mdpi.com/1099-4300/19/10/555 (visited on 2021-06-14), doi:10.3390/e19100555.">196</a>]</span> refers to the ideal prior as one which encodes all available information about the model. For practical reasons (difficulty of construction or computational inconvenience), most priors fail to achieve this ideal. These compromises mean that priors should be transparently articulated and justified, so that the impact of the choice of prior can be fully understood. When there is ambiguity about an appropriate prior, such as how fat the tails should be, an analyst should examine how sensitive the UQ results are to the choice of prior.</p>
+<p>Priors can also be classified in terms of the information encoded by them, demonstrated in <a class="reference internal" href="#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a>. Non-informative priors (illustrated in <a class="reference internal" href="#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a> (a)) allegedly correspond to (and are frequently justified by) a position of ignorance. A classic example is the use of a uniform distribution. A uniform prior can, however, be problematic, as it can lead to improper inferences by giving extremely large values the same prior probability as values which may seem more likely <span id="id32">[<a class="reference internal" href="R.Bibliography.html#id197" title="Andrew Gelman, Daniel Simpson, and Michael Betancourt. The Prior Can Often Only Be Understood in the Context of the Likelihood. Entropy, 19(10):555, October 2017. Number: 10 Publisher: Multidisciplinary Digital Publishing Institute. URL: https://www.mdpi.com/1099-4300/19/10/555 (visited on 2021-06-14), doi:10.3390/e19100555.">196</a>, <a class="reference internal" href="R.Bibliography.html#id198" title="Christian Robert. The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer Science &amp; Business Media, 2007.">197</a>]</span>, and therefore does not really reflect a state of complete ignorance. In the extreme case of a uniform prior over the entire real line, every particular region has effectively a prior weight of zero, even though not all regions are a priori unlikely <span id="id33">[<a class="reference internal" href="R.Bibliography.html#id198" title="Christian Robert. The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer Science &amp; Business Media, 2007.">197</a>]</span>. Moreover, a uniform prior which excludes possible parameter values is not actually noninformative, as it assigns zero probability to those values while jumping to a nonzero probability as soon as the boundary is crossed. While a uniform prior can be problematic for the task of uncertainty quantification, it may be useful for an initial sensitivity analysis to identify the boundary of any regions where the model is sensitive to the parameter.</p>
+<figure class="margin-caption align-center" id="id41">
+<span id="figure-a1-5"></span><a class="reference internal image-reference" href="_images/figureA1_5_priors_posteriors.png"><img alt="Figure A1.5" src="_images/figureA1_5_priors_posteriors.png" style="width: 700px;" />
+</a>
+<figcaption>
+<p><span class="caption-number">Fig. A.5 </span><span class="caption-text">Impact of priors on posterior inferences. These plots show the results of inference for a linear regression model with 15 data points. The true value of the parameter is equal to -3. All priors have mean 0. In panel (a), a non-informative prior allows the tails of the posterior to extend freely, which may result in unreasonably large parameter values. In panel (b), a weakly informative prior constrains the tails more, but allows them to extend without too much restriction. In panel (c), an informative prior strongly constrains the tails of the posterior and biases the inference closer towards the prior mean (the posterior mean is -0.89 in this case, and closer to -3 in the other two cases).</span><a class="headerlink" href="#id41" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>Informative priors strongly bound the range of probable values (illustrated in <a class="reference internal" href="#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a> (c)). One example is a Gaussian distribution with a relatively small standard deviation, so that large values are assigned a close to null prior probability. Another example is the jump from zero to non-zero probability occurring at the truncation point of a truncated Gaussian, which could be justified based on information that the parameter cannot take on values beyond this point. Without this type of justification, however, priors may be too informative, failing to allow the information contained in the available data to update them.</p>
+<p>Finally, weakly informative priors (illustrated in <a class="reference internal" href="#figure-a1-5"><span class="std std-numref">Fig. A.5</span></a> (b)) fall in between <span id="id34">[<a class="reference internal" href="R.Bibliography.html#id197" title="Andrew Gelman, Daniel Simpson, and Michael Betancourt. The Prior Can Often Only Be Understood in the Context of the Likelihood. Entropy, 19(10):555, October 2017. Number: 10 Publisher: Multidisciplinary Digital Publishing Institute. URL: https://www.mdpi.com/1099-4300/19/10/555 (visited on 2021-06-14), doi:10.3390/e19100555.">196</a>]</span>. They regularize better than non-informative priors, but allow for more inference flexibility than fully informative priors. An example might be a Gaussian distribution with a moderate standard deviation, which still assigns negligible probability for values far away from the mean, but is less constrained than a narrow Gaussian for a reasonably large area. A key note is that it is not necessarily better to be more informative if this cannot be justified by the available information.</p>
+</section>
+<section id="the-critical-final-step-predictive-checks">
+<h2><span class="section-number">A.7. </span>The Critical Final Step: Predictive Checks<a class="headerlink" href="#the-critical-final-step-predictive-checks" title="Link to this heading">#</a></h2>
+<p>Every UQ workflow requires a number of choices, potentially including selecting prior distributions, the likelihood specification, and any used numerical models. Checking the appropriateness of these choices is an essential step for sound inferences, as misspecification can produce biased results <span id="id35">[<a class="reference internal" href="R.Bibliography.html#id180" title="Jenný Brynjarsdóttir and Anthony O'Hagan. Learning about physical parameters: the importance of model discrepancy. Inverse Problems, 30(11):114007, October 2014. Publisher: IOP Publishing. URL: https://doi.org/10.1088/0266-5611/30/11/114007 (visited on 2021-06-14), doi:10.1088/0266-5611/30/11/114007.">179</a>]</span>. Model checking in this fashion is part of an iterative UQ process, as the results can reveal adjustments to the statistical model or the need to select a different numerical model <span id="id36">[<a class="reference internal" href="R.Bibliography.html#id200" title="Andrew Gelman, Xiao-Li Meng, and Hal Stern. Posterior Predictive Assessment of Model Fitness via Realized Discrepancies. Statistica Sinica, 6(4):733–760, 1996. Publisher: Institute of Statistical Science, Academia Sinica. URL: https://www.jstor.org/stable/24306036 (visited on 2021-06-14).">198</a>, <a class="reference internal" href="R.Bibliography.html#id199" title="Andrew Gelman, Aki Vehtari, Daniel Simpson, Charles C. Margossian, Bob Carpenter, Yuling Yao, Lauren Kennedy, Jonah Gabry, Paul-Christian Bürkner, and Martin Modrák. Bayesian Workflow. arXiv:2011.01808 [stat], November 2020. arXiv: 2011.01808. URL: http://arxiv.org/abs/2011.01808 (visited on 2021-06-14).">199</a>, <a class="reference internal" href="R.Bibliography.html#id201" title="Andrew Gelman and Cosma Rohilla Shalizi. Philosophy and the practice of Bayesian statistics. British Journal of Mathematical and Statistical Psychology, 66(1):8–38, 2013. _eprint: https://bpspsychub.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2044-8317.2011.02037.x. URL: https://bpspsychub.onlinelibrary.wiley.com/doi/abs/10.1111/j.2044-8317.2011.02037.x (visited on 2021-06-14), doi:10.1111/j.2044-8317.2011.02037.x.">200</a>]</span>.</p>
+<p>A classic example is the need to check the structure of residuals for correlations. Many standard statistical models, such as linear regression, assume that the residuals are independent and identically distributed from the error distribution. The presence of correlations, including temporal autocorrelations and spatial correlations, indicates a structural mismatch between the likelihood and the data. In these cases, the likelihood should be adjusted to account for these correlations.</p>
+<p>Checking residuals in this fashion is one example of a predictive check (or a posterior predictive check in the Bayesian setting). One way to view UQ is as a means to recover data-generating processes (associated with each parameter vector) consistent with the observations. Predictive checks compare the inferred data-generating process to the observations to determine whether the model is capable of appropriately capturing uncertainty. After conducting the UQ analysis, alternatively realized datasets are simulated from sampled parameters. These alternative datasets, or their summary statistics, can be tested against the observations to determine adequacy of the fit. Predictive checks are therefore a way of probing various model components to identify shortcomings that might result in biased inferences or poor projections, depending on the goal of the analysis.</p>
+<p>One example of a graphical predictive check for time series models is hindcasting, where predictive intervals are constructed from the alternative datasets and plotted along with the data. Hindcasts demonstrate how well the model is capable of capturing the broader dynamics of the data, as well as whether the parameter distributions produce appropriate levels of output uncertainty. A related quantitative check is the surprise index, which calculates the percentage of data points located within a fixed predictive interval. For example, the 90% predictive interval should contain approximately 90% of the data. More uncertainty than this reflects underconfidence, while less uncertainty reflects overconfidence. This could be the result of priors that are not appropriately informative, or a likelihood that does not account for correlations between data points appropriately. It could also be the result of a numerical model that isn’t sufficiently sensitive to the parameters that are treated as uncertain.</p>
+</section>
+<section id="key-take-home-points">
+<h2><span class="section-number">A.8. </span>Key Take-Home Points<a class="headerlink" href="#key-take-home-points" title="Link to this heading">#</a></h2>
+<p>When appropriate, UQ is an important component of the exploratory modeling workflow. While a number of parameter sets could be consistent with observations, they may result in divergent model outputs when exposed to different future conditions. This can result in identifying risks which are not visible when selecting a “best fit” parameterization. Quantifying uncertainties also allows us to quantify the support for hypotheses, which is an essential part of the scientific process.</p>
+<p>Due to the scale and complexity of the experiments taking place in IM3, UQ has not been extensively used. The tradeoff between the available number of function evaluations and the number of uncertain parameters illustrated in <a class="reference internal" href="#figure-a1-1"><span class="std std-numref">Fig. A.1</span></a> is particularly challenging due to the increasing complexity of state-of-the-art models and the movement towards coupled, multisector models. This tradeoff can be addressed somewhat through the use of emulators and parallelizable methods. In particular, when attempting to navigate this tradeoff by limiting the number of uncertain parameters, it is important to carefully iterate with sensitivity analyses to ensure that critical parameters are identified.</p>
+<p>Specifying prior distributions and likelihoods is an ongoing challenge. Prior distributions, in particular, should be treated as deeply uncertain when appropriate. One key advantage of the methods described in this chapter is that they have the potential for increased transparency. When it is not possible to conduct a sensitivity analysis on a number of critical priors due to limited computational budgets, fully specifying and providing a justification for the utilized distributions allows other researchers to identify key assumptions and build on existing work. The same is true for the specification of likelihoods—while likelihood-free methods avoid the need to specify a likelihood function, they require other assumptions or choices, which should be described and justified as transparently as possible.</p>
+<p>We conclude this appendix with some key recommendations:
+1. UQ analysis does not require full confidence in priors and likelihoods. Rather, UQ should be treated as part of an exploratory modeling workflow, where hypotheses related to model structures, prior distributions, and likelihoods can be tested.
+2. For complex multisectoral models, UQ will typically require the use of a reduced set of parameters, either through emulation or by fixing the others to their best-fit values. These parameters should be selected through a careful sensitivity analysis.
+3. Avoid the use of supposedly “non-informative” priors, such as uniform priors, whenever possible. In the absence of strong information about parameter values, the use of weakly informative priors, such as diffuse normals, is preferable.
+4. Be cognizant of the limitations of conclusions that can be drawn by using each method. The bootstrap, for example, may result in overconfidence if the dataset is limited and is not truly representative of the underlying stochastic process.
+5. When using MCMC, Markov chains can not be shown to have converged to the target distribution, but rather evidence can be collected to demonstrate that it is likely that they have.
+6. Conduct predictive checks based on the assumptions underlying the choices made in the analysis, and iteratively update those choices if the assumptions prove to be ill-suited for the problem at hand.</p>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">A.1. Introduction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#parametric-bootstrap">A.2. Parametric Bootstrap</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pre-calibration">A.3. Pre-Calibration</a></li>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-critical-first-step-how-to-choose-a-prior-distribution">A.6. The Critical First Step: How to Choose a Prior Distribution</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-critical-final-step-predictive-checks">A.7. The Critical Final Step: Predictive Checks</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#key-take-home-points">A.8. Key Take-Home Points</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html">4. Sensitivity Analysis: Diagnostic &amp; Exploratory Modeling</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="A1_Uncertainty_Quantification.html">A. Uncertainty Quantification</a></li>
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+    <h1>Fishery Dynamics Tutorial</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics">Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s get started!</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-identified-solutions-and-explore-performance">Step 1: Load identified solutions and explore performance</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis">Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-evaluate-the-system-over-all-generated-states-of-the-world">Step 3: Evaluate the system over all generated states of the world</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-calculate-sensitivity-indices">Step 4: Calculate sensitivity indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-explore-relationship-between-uncertain-factors-and-performance">Step 5: Explore relationship between uncertain factors and performance</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem">Tips to Apply Sobol SA and Scenario Discovery to your Problem</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
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+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="fishery-dynamics-tutorial">
+<h1>Fishery Dynamics Tutorial<a class="headerlink" href="#fishery-dynamics-tutorial" title="Link to this heading">#</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/fishery_dynamics.ipynb">Fishery Dynamics Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics">
+<h2>Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics<a class="headerlink" href="#tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics" title="Link to this heading">#</a></h2>
+<p>This notebook demonstrates the application of sensitivity analysis to
+discover factors that shape the behavior modes of a socio-ecological
+system with dynamic human action.</p>
+<p>The system of differential equations below represent a system of prey
+(defined in the equation below as x) and predator (defined as y) fish,
+with a human actor harvesting the prey fish. You can read more about
+this system at <a class="reference external" href="https://doi.org/10.1155/2020/4170453">Hadjimichael et
+al. (2020)</a>.</p>
+<img alt="_images/eqn2.png" src="_images/eqn2.png" />
+<p>The table below defines the parameters in the system and also denotes
+the baseline and ranges associated with each uncertain parameter.</p>
+<img alt="_images/table1.png" src="_images/table1.png" />
+<p>The system is simple but very rich in the dynamic behaviors it exhibits.
+This complexity is accompanied by the presence of several equilibria
+that come in and out of existence with different parameter values. The
+equilibria also change in their stability according to different
+parameter values, giving rise to different behavior modes as shown by
+the diverse predator and prey abundace trajectories in the figure below.</p>
+<img alt="_images/Figure_1.png" src="_images/Figure_1.png" />
+<p>In the unharvested system (without the human actor) the stability of
+several of these equilibria can be derived analytically. The task
+becomes significantly more difficult when the adaptive human actor is
+introduced, deciding to harvest the system at different rates according
+to their objectives and preferences.</p>
+<p>Sensitivity analysis methods can help us identify the factors that most
+control these dynamics by exploring the space of parameter values and
+seeing how system outputs change as a result.</p>
+<p>Through previously conducted optimization, there already exists a set of
+potential harvesting strategies that were identified in pursuit of five
+objectives:</p>
+<ul class="simple">
+<li><p>Maximize Harvesting Discounted Profits (Net Present Value)</p></li>
+<li><p>Minimize Prey Population Deficit</p></li>
+<li><p>Minimize Longest Duration of Consecutive Low Harvest</p></li>
+<li><p>Maximize Worst Harvest Instance</p></li>
+<li><p>Minimize Harvest Variance</p></li>
+</ul>
+<p>The identified harvesting strategies also meet the necessary constraint
+of not causing inadvertent predator collapse.</p>
+<p>We will be examining the effects of parametric uncertainty on these
+identified strategies, particularly focusing on two strategies: one
+selected to maximize harvesting profits and one identified through
+previous analysis to perform ‘well enough’ for all objectives across a
+wide range of states of the world (referred to as the ‘robust’
+harvesting policy).</p>
+<section id="lets-get-started">
+<h3>Let’s get started!<a class="headerlink" href="#lets-get-started" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we will be loading in data that has been produced in
+Hadjimichael et al. (2020). Before we start our analysis, we’ll load the
+relevant Python libraries. <strong>NOTE</strong>: To step through the notebook,
+execute each gray (code) box by typing “Shift+Enter”.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Import necessary libraries</span>
+
+<span class="kn">import</span> <span class="nn">msdbook</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">saltelli</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">sobol</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">patheffects</span> <span class="k">as</span> <span class="n">pe</span>
+
+<span class="c1"># load example data</span>
+<span class="n">msdbook</span><span class="o">.</span><span class="n">install_package_data</span><span class="p">()</span>
+
+<span class="o">%</span><span class="k">matplotlib</span> inline
+<span class="o">%</span><span class="k">config</span> InlineBackend.print_figure_kwargs = {&#39;bbox_inches&#39;:None}
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Downloading</span> <span class="n">example</span> <span class="n">data</span> <span class="k">for</span> <span class="n">msdbook</span> <span class="n">version</span> <span class="mf">0.1.5</span><span class="o">...</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">uncertain_params_bounds</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_metric_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">solutions</span><span class="o">.</span><span class="n">resultfile</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LHsamples_original_1000</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">param_values</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_params_256samples</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LeafCatch</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_simulations_256samples</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">Robustness</span><span class="o">.</span><span class="n">txt</span>
+</pre></div>
+</div>
+</section>
+<section id="step-1-load-identified-solutions-and-explore-performance">
+<h3>Step 1: Load identified solutions and explore performance<a class="headerlink" href="#step-1-load-identified-solutions-and-explore-performance" title="Link to this heading">#</a></h3>
+<p>Here we load in the solution set obtained in Hadjimichael et al. (2020).
+The solution set contains the decision variables and objectives
+associated with a variety of harvesting policies. For this tutorial, we
+focus on comparing two policies: harvesting profits and one that
+performs robustly across all objectives. Below, we are reading in the
+decision variables and objectives from an external file that can be
+found within the msdbook package data.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">robustness</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_robustness_data</span><span class="p">()</span>
+<span class="n">results</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_profit_maximization_data</span><span class="p">()</span>
+
+<span class="n">robust_solution</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">robustness</span><span class="p">[:,</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="c1">#pick robust solution</span>
+<span class="n">profit_solution</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">results</span><span class="p">[:,</span><span class="mi">6</span><span class="p">])</span> <span class="c1">#pick profitable solution</span>
+<span class="n">objective_performance</span> <span class="o">=</span> <span class="o">-</span><span class="n">results</span><span class="p">[:,</span><span class="mi">6</span><span class="p">:]</span> <span class="c1">#Retain objective values</span>
+
+<span class="c1"># Get decision variables for each of the policies</span>
+<span class="n">highprofitpolicy</span> <span class="o">=</span> <span class="n">results</span><span class="p">[</span><span class="n">profit_solution</span><span class="p">,</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">]</span>
+<span class="n">mostrobustpolicy</span> <span class="o">=</span> <span class="n">results</span><span class="p">[</span><span class="n">robust_solution</span><span class="p">,</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">]</span>
+</pre></div>
+</div>
+<p>Next we plot the identified solutions with regards to their objective
+performance in a parallel axis plot</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-objective-performance">plot_objective_performance</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ax</span><span class="p">,</span> <span class="n">ax1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_objective_performance</span><span class="p">(</span><span class="n">objective_performance</span><span class="p">,</span> <span class="n">profit_solution</span><span class="p">,</span> <span class="n">robust_solution</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/fishery_output_6_0.png" src="_images/fishery_output_6_0.png" />
+<p>The solution set from the optimization in Hadjimichael et al. (2020) are
+presented in a parallel axis plot where each of the five objectives (and
+one constraint) are represented as an axis. Each solution on the Pareto
+front is represented as a line where the color of the line indicates the
+value of the NPV objective. The preference for objective values is in
+the upward direction. Therefore, the ideal solution would be a line
+straight across the top of the plot that satisfies every objective.
+However, no such line exists because there are tradeoffs when sets of
+objectives are prioritized over the others. When lines cross in between
+axes, this indicates a tradeoff between objectives (as seen in the first
+two axes).The solution that is most robust in the NPV objective has the
+highest value on the first axis and is outlined in dark gold. The
+solution that is most robust across all objectives is outlined in a
+brighter yellow. A parallel axis is an effective visual to characterize
+high-dimensional tradeoffs in the system and visualize differences in
+performance across policies.</p>
+</section>
+<section id="step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis">
+<h3>Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis<a class="headerlink" href="#step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis" title="Link to this heading">#</a></h3>
+<p>In Step 1, we showed how the optimized harvesting policies performed in
+the objective space, which utilized the baseline parameters outlined in
+the table above. Now, we are interested in understanding how sensitive
+our two policies are to alternative states of the world that may be
+characterized by different parameter values. To do so, we first need to
+define the problem dictionary that allows us to generate these
+alternative states of the world.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set up SALib problem</span>
+<span class="n">problem</span> <span class="o">=</span> <span class="p">{</span>
+  <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">9</span><span class="p">,</span>
+  <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;a&#39;</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="s1">&#39;c&#39;</span><span class="p">,</span> <span class="s1">&#39;d&#39;</span><span class="p">,</span> <span class="s1">&#39;h&#39;</span><span class="p">,</span> <span class="s1">&#39;K&#39;</span><span class="p">,</span> <span class="s1">&#39;m&#39;</span><span class="p">,</span> <span class="s1">&#39;sigmaX&#39;</span><span class="p">,</span> <span class="s1">&#39;sigmaY&#39;</span><span class="p">],</span>
+  <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="mf">0.002</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.005</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+             <span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">5000</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">]]</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+<p>Then we use the following command to generate a Saltelli sample from
+these defined ranges:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">param_values</span> <span class="o">=</span> <span class="n">saltelli</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="mi">1024</span><span class="p">,</span> <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Generally, it is a good idea to save the result of the sample since it
+is often reused and regenerating it produces a different sample set. For
+this reason, we will load one from file that was previously generated.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously generated Saltelli sample from our msdbook package data</span>
+<span class="n">param_values</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_saltelli_param_values</span><span class="p">()</span>
+</pre></div>
+</div>
+</section>
+<section id="step-3-evaluate-the-system-over-all-generated-states-of-the-world">
+<h3>Step 3: Evaluate the system over all generated states of the world<a class="headerlink" href="#step-3-evaluate-the-system-over-all-generated-states-of-the-world" title="Link to this heading">#</a></h3>
+<p>Now we re-evaluate how well the policies do in the new states of the
+world. In order to characterize failure of a policy, we identify the
+states where the predator population collapses, as an inadvertent
+consequence of applying the harvesting strategy under a state of the
+world different from the one originally assumed. Due to how long this
+step takes to execute within the tutorial, we will read in the solutions
+from an external file. However, the block of code below shows how
+evaluation can be implemented.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># create array to store collapse values under both policies</span>
+<span class="n">collapse_days</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="nb">len</span><span class="p">(</span><span class="n">param_values</span><span class="p">),</span> <span class="mi">2</span><span class="p">])</span>
+
+<span class="c1"># evaluate performance under every state</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">param_values</span><span class="p">)):</span>
+
+    <span class="n">additional_inputs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="s1">&#39;Previous_Prey&#39;</span><span class="p">],</span>
+                                  <span class="p">[</span><span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">6</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">7</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span>
+
+    <span class="n">collapse_days</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span><span class="n">fish_game</span><span class="p">(</span><span class="n">highprofitpolicy</span><span class="p">,</span> <span class="n">additional_inputs</span><span class="p">)[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+    <span class="n">collapse_days</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">=</span><span class="n">fish_game</span><span class="p">(</span><span class="n">mostrobustpolicy</span><span class="p">,</span> <span class="n">additional_inputs</span><span class="p">)[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load the simulation data from our msdbook package data</span>
+<span class="n">collapse_days</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_collapse_data</span><span class="p">()</span>
+</pre></div>
+</div>
+</section>
+<section id="step-4-calculate-sensitivity-indices">
+<h3>Step 4: Calculate sensitivity indices<a class="headerlink" href="#step-4-calculate-sensitivity-indices" title="Link to this heading">#</a></h3>
+<p>Now we use a Sobol sensitivity analysis to calculate first-order,
+second-order, and total-order sensitivity indices for each parameter and
+for each of the two policies. These indicies help determine which
+factors explain the most variability in the number of days of predator
+population collapse.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Perform the Sobol SA for the profit-maximizing solution</span>
+<span class="n">Si_profit</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="n">collapse_days</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
+                          <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+                          <span class="n">conf_level</span><span class="o">=</span><span class="mf">0.95</span><span class="p">,</span>
+                          <span class="n">print_to_console</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Perform the Sobol SA for the robust solution</span>
+<span class="n">Si_robustness</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span>
+                              <span class="n">collapse_days</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
+                              <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+                              <span class="n">conf_level</span><span class="o">=</span><span class="mf">0.95</span><span class="p">,</span>
+                              <span class="n">print_to_console</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>              <span class="n">ST</span>   <span class="n">ST_conf</span>
+<span class="n">a</span>       <span class="mf">0.226402</span>  <span class="mf">0.036146</span>
+<span class="n">b</span>       <span class="mf">0.066819</span>  <span class="mf">0.013347</span>
+<span class="n">c</span>       <span class="mf">0.004395</span>  <span class="mf">0.004023</span>
+<span class="n">d</span>       <span class="mf">0.024509</span>  <span class="mf">0.006993</span>
+<span class="n">h</span>       <span class="mf">0.009765</span>  <span class="mf">0.005488</span>
+<span class="n">K</span>       <span class="mf">0.020625</span>  <span class="mf">0.009494</span>
+<span class="n">m</span>       <span class="mf">0.897971</span>  <span class="mf">0.066470</span>
+<span class="n">sigmaX</span>  <span class="mf">0.000136</span>  <span class="mf">0.000149</span>
+<span class="n">sigmaY</span>  <span class="mf">0.000739</span>  <span class="mf">0.001040</span>
+              <span class="n">S1</span>   <span class="n">S1_conf</span>
+<span class="n">a</span>       <span class="mf">0.087936</span>  <span class="mf">0.044236</span>
+<span class="n">b</span>       <span class="mf">0.000554</span>  <span class="mf">0.021474</span>
+<span class="n">c</span>      <span class="o">-</span><span class="mf">0.002970</span>  <span class="mf">0.004590</span>
+<span class="n">d</span>       <span class="mf">0.001206</span>  <span class="mf">0.015881</span>
+<span class="n">h</span>       <span class="mf">0.004554</span>  <span class="mf">0.007998</span>
+<span class="n">K</span>       <span class="mf">0.003843</span>  <span class="mf">0.012661</span>
+<span class="n">m</span>       <span class="mf">0.751301</span>  <span class="mf">0.071862</span>
+<span class="n">sigmaX</span> <span class="o">-</span><span class="mf">0.000325</span>  <span class="mf">0.001245</span>
+<span class="n">sigmaY</span> <span class="o">-</span><span class="mf">0.001887</span>  <span class="mf">0.002768</span>
+</pre></div>
+</div>
+<p>Looking at the total-order indices, (ST) factors <span class="math notranslate nohighlight">\(m\)</span>, <span class="math notranslate nohighlight">\(a\)</span>,
+<span class="math notranslate nohighlight">\(b\)</span>, <span class="math notranslate nohighlight">\(d\)</span> and <span class="math notranslate nohighlight">\(K\)</span> explain a non-negligible amount of
+variance therefore have an effect on the stability of this system.
+Looking at the first-order indices (S1), we also see that besides
+factors <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(a\)</span>, all other factors are important in this
+system through their interactions, which make up the difference between
+their S1 and ST indices. This shows the danger of limiting sensitivity
+analyses to first order effects, as factor importance might be
+significantly misjudged.</p>
+<p>These findings are supported by the analytical condition of equilibrium
+stability in this system:</p>
+<img alt="_images/eqn4.png" src="_images/eqn4.png" />
+<p>In an unharvested system, this condition is both necessary and
+sufficient for the equilibrium of the two species coexisting to be
+stable.</p>
+<p>When adaptive human action is introduced however, this condition is
+still necessary, but no longer sufficient, as harvesting reduces the
+numbers of prey fish and as a result reduces the resources for the
+predator fish. Since this harvesting value is not constant, but can
+dynamically adapt according to the harvester’s objectives, it cannot be
+introduced into this simple equation.</p>
+</section>
+<section id="step-5-explore-relationship-between-uncertain-factors-and-performance">
+<h3>Step 5: Explore relationship between uncertain factors and performance<a class="headerlink" href="#step-5-explore-relationship-between-uncertain-factors-and-performance" title="Link to this heading">#</a></h3>
+<p>In the following steps, we will use the results of our sensitivity
+analysis to investigate the relationships between parametric
+uncertainty, equilibrium stability and the performance of the two
+policies.</p>
+<p>We can use the top three factors identified (<span class="math notranslate nohighlight">\(m\)</span>, <span class="math notranslate nohighlight">\(a\)</span>, and
+<span class="math notranslate nohighlight">\(b\)</span>) to visualize the performance of our policies in this
+three-dimensional parametric space.</p>
+<p>We first define the stability condition, as a function of <span class="math notranslate nohighlight">\(b\)</span> and
+<span class="math notranslate nohighlight">\(m\)</span>, and calculate the corresponding values of <span class="math notranslate nohighlight">\(a\)</span>.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">inequality</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
+    <span class="k">return</span> <span class="p">((</span><span class="n">b</span><span class="o">**</span><span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">h</span><span class="o">*</span><span class="n">K</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">m</span><span class="p">))</span>
+
+<span class="c1"># boundary interval that separates successful and failed states of the world</span>
+<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mf">0.005</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">m</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">K</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">2000</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">b</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">inequality</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">clip</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-factor-performance">plot_factor_performance</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># generate plot</span>
+<span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_factor_performance</span><span class="p">(</span><span class="n">param_values</span><span class="p">,</span> <span class="n">collapse_days</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/fishery_output_22_0.png" src="_images/fishery_output_22_0.png" />
+<p>These figures show the combinations of factors that lead to success or
+failure in different states of the world for the profit-maximizing and
+robust policies. Each point is a state of the world, characterized by
+specific values of the parameters, and ideally, we would like the color
+of the point to be blue, to represent that there are a low number of
+days with a predator collapse in that world. The gray curve denotes the
+highly non-linear nature of the boundary, defined by the stability
+condition, that separates successful and failed states of the world. The
+figures demonstrate the following key points:</p>
+<p>First, as asserted above, the policies interact with the system in
+different and complex ways. In the presence of human action, the
+stability condition is not sufficient in determining whether the policy
+will succeed, even though it clearly shapes the system in a fundamental
+manner.</p>
+<p>Secondly, the robust policy manages to avoid collapse in many more of the sampled states of the world, indicated by the number of blue points. The robust policy avoids collapse in 31% of worlds versus 14% in the profit-maximizing policy.This presents a clear tradeoff between profit-maximizing performance androbustness against uncertainty.</p>
+</section>
+<section id="tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem">
+<h3>Tips to Apply Sobol SA and Scenario Discovery to your Problem<a class="headerlink" href="#tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we demonstrated a Sobol SA to identify the most
+important factors driving the behavior of a system (i.e. the number of
+the collapse days). In order to apply this methodology to your problem,
+you will need to have a set of optimized policies for your system that
+you are interested in analyzing. The general workflow is as follows:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters and set up the problem
+dictionary as in Step 2 above.</p></li>
+<li><p>Generate samples, or alternative states of the world using the
+<code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function.</p></li>
+<li><p>Evaluate your policies on the alternative states of the world. For
+your application, you will also need to develop a rule for
+determining success or failure of your policy in a new SOW. In this
+tutorial, success was denoted by a small number of collapse days.
+Ultimately, the rule will be specific to your application and can
+include various satisficing criteria.</p></li>
+<li><p>Calculate the Sobol indices and discover the most important
+parameters driving success and failure.</p></li>
+<li><p>Finally, use a similar plotting procedure as in step 5 to identify
+the combination of parameter values that lead to success and failure
+in the system.</p></li>
+</ol>
+</section>
+</section>
+</section>
+
+
+                </article>
+              
+
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics">Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s get started!</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-identified-solutions-and-explore-performance">Step 1: Load identified solutions and explore performance</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis">Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-evaluate-the-system-over-all-generated-states-of-the-world">Step 3: Evaluate the system over all generated states of the world</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-calculate-sensitivity-indices">Step 4: Calculate sensitivity indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-explore-relationship-between-uncertain-factors-and-performance">Step 5: Explore relationship between uncertain factors and performance</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem">Tips to Apply Sobol SA and Scenario Discovery to your Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html">2. Diagnostic Modeling Overview and Perspectives</a></li>
+<li class="toctree-l1"><a class="reference internal" href="3_sensitivity_analysis_the_basics.html">3. Sensitivity Analysis: The Basics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html">4. Sensitivity Analysis: Diagnostic &amp; Exploratory Modeling</a></li>
+<li class="toctree-l1"><a class="reference internal" href="5_conclusion.html">5. Conclusion</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="A1_Uncertainty_Quantification.html">A. Uncertainty Quantification</a></li>
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+    <h1>Sobol SA Tutorial</h1>
+    <!-- Table of contents -->
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+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa">Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s get started!</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-generate-the-problem-dictionary">Step 1: Generate the problem dictionary</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-generate-samples-using-the-saltelli-sampling-scheme">Step 2: Generate samples using the Saltelli sampling scheme</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-execute-the-ishigami-function-over-our-sample-set">Step 3: Execute the Ishigami function over our sample set</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method">Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-interpret-our-results">Step 5: Interpret our results</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-to-your-own-problem">Tips to Apply Sobol SA to Your Own Problem</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="sobol-sa-tutorial">
+<h1>Sobol SA Tutorial<a class="headerlink" href="#sobol-sa-tutorial" title="Link to this heading">#</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA Tutorial</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa">
+<h2>Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA<a class="headerlink" href="#tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa" title="Link to this heading">#</a></h2>
+<p>In this tutorial, we will set up a workflow to investigate how sensitive
+the output of a function is to its inputs. Why might you want to do
+this? Imagine that this function represents a complex system, such as
+the rainfall-runoff process of a watershed model, and that you, the
+researcher, want to investigate how your choice of input parameter
+values are affecting the model’s characterization of runoff in the
+watershed. Your parameter values are likely uncertain and can take on
+any value in a pre-defined range. Using a Sobol SA will allow you to
+sample different values of your parameters and calculate how sensitive
+your output of interest is to certain parameters. Below, we demonstrate
+Sobol SA for a simple function to illustrate the method, but the
+workflow can be applied to your own problem of interest!</p>
+<p>In order to conduct this analysis, we will use the popular Python
+Sensitivity Analysis Library
+(<a class="reference external" href="https://salib.readthedocs.io/en/latest/index.html">SALib</a>) to:</p>
+<ol class="arabic simple">
+<li><p>Generate a problem set as a dictionary for our Ishigami function that has three inputs</p></li>
+<li><p>Generate 2048 samples for our problem set using the Saltelli <a class="footnote-reference brackets" href="#id7" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id8" id="id2" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a> sampling scheme</p></li>
+<li><p>Execute the Ishigami function for each of our samples and gather the outputs</p></li>
+<li><p>Compute the sensitivity analysis to generate first-order and total-order sensitivity indices using the Sobol <a class="footnote-reference brackets" href="#id9" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a> method</p></li>
+<li><p>Interpret the meaning of our results</p></li>
+</ol>
+<section id="lets-get-started">
+<h3>Let’s get started!<a class="headerlink" href="#lets-get-started" title="Link to this heading">#</a></h3>
+<p><strong>NOTE</strong>: Content from this tutorial is taken directly from the SALib
+<a class="reference external" href="https://salib.readthedocs.io/en/latest/basics.html">“Basics”</a>
+walkthrough. To step through the notebook, execute each gray (code) box
+by typing “Shift+Enter”.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Import relevant libraries</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">from</span> <span class="nn">mpl_toolkits</span> <span class="kn">import</span> <span class="n">mplot3d</span>
+
+<span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">saltelli</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">sobol</span>
+<span class="kn">from</span> <span class="nn">SALib.test_functions</span> <span class="kn">import</span> <span class="n">Ishigami</span>
+</pre></div>
+</div>
+</section>
+<section id="step-1-generate-the-problem-dictionary">
+<h3>Step 1: Generate the problem dictionary<a class="headerlink" href="#step-1-generate-the-problem-dictionary" title="Link to this heading">#</a></h3>
+<p>The Ishigami function is of the form:</p>
+<div class="math notranslate nohighlight">
+\[f(x_1,x_2,x_3) = sin(x_1)+asin^2(x_2)+bx_3^4sin(x_1)\]</div>
+<p>The function has three inputs, 𝑥1, 𝑥2, 𝑥3 where 𝑥𝑖 ∈ [−𝜋, 𝜋]. The
+constants <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are defined as 7.0 and 0.1
+respectively.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Create a problem dictionary. Here we supply the number of variables, the names of each variable, and the bounds of the variables.</span>
+<span class="n">problem</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span>
+    <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;x1&#39;</span><span class="p">,</span> <span class="s1">&#39;x2&#39;</span><span class="p">,</span> <span class="s1">&#39;x3&#39;</span><span class="p">],</span>
+    <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="o">-</span><span class="mf">3.14159265359</span><span class="p">,</span> <span class="mf">3.14159265359</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">3.14159265359</span><span class="p">,</span> <span class="mf">3.14159265359</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">3.14159265359</span><span class="p">,</span> <span class="mf">3.14159265359</span><span class="p">]]</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+</section>
+<section id="step-2-generate-samples-using-the-saltelli-sampling-scheme">
+<h3>Step 2: Generate samples using the Saltelli sampling scheme<a class="headerlink" href="#step-2-generate-samples-using-the-saltelli-sampling-scheme" title="Link to this heading">#</a></h3>
+<p>Sobol SA requires the use of the Saltelli sampling scheme. The output of
+the <code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function is a NumPy array that is of shape 2048
+by 3. The sampler generates 𝑁∗(2𝐷+2) samples, where in this example, N
+is 256 (the argument we supplied) and D is 3 (the number of model
+inputs), yielding 2048 samples. The keyword argument
+<code class="docutils literal notranslate"><span class="pre">calc_second_order=False</span></code> will exclude second-order indices, resulting
+in a smaller sample matrix with 𝑁∗(𝐷+2) rows instead. Below, we plot the
+resulting Saltelli sample.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Generate parmeter values using the saltelli.sample function</span>
+<span class="n">param_values</span> <span class="o">=</span> <span class="n">saltelli</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="mi">256</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;`param_values` shape:  </span><span class="si">{</span><span class="n">param_values</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<pre class="literal-block"><cite>param_values</cite> shape:  (2048, 3)</pre>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Plot the 2048 samples of the parameters</span>
+
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
+<span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">axes</span><span class="p">(</span><span class="n">projection</span> <span class="o">=</span><span class="s2">&quot;3d&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter3D</span><span class="p">(</span><span class="n">param_values</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">param_values</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">param_values</span><span class="p">[:,</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;X1 Parameter&#39;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;X2 Parameter&#39;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s1">&#39;X3 Parameter&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Saltelli Sample of Parameter Values&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/output_7_0.png" src="_images/output_7_0.png" />
+</section>
+<section id="step-3-execute-the-ishigami-function-over-our-sample-set">
+<h3>Step 3: Execute the Ishigami function over our sample set<a class="headerlink" href="#step-3-execute-the-ishigami-function-over-our-sample-set" title="Link to this heading">#</a></h3>
+<p>SALib provides a nice wrapper to the Ishigami function that allows the
+user to directly pass the <code class="docutils literal notranslate"><span class="pre">param_values</span></code> array we just generated into
+the function directly.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Y</span> <span class="o">=</span> <span class="n">Ishigami</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">param_values</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method">
+<h3>Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method<a class="headerlink" href="#step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method" title="Link to this heading">#</a></h3>
+<p>The <code class="docutils literal notranslate"><span class="pre">sobol.analyze</span></code> function will use our problem dictionary and the
+result of the Ishigami runs (<code class="docutils literal notranslate"><span class="pre">Y</span></code>) to compute first-, second-, and
+total-order indicies.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Si</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><code class="docutils literal notranslate"><span class="pre">Si</span></code> is a Python dict with the keys “S1”, “S2”, “ST”, “S1_conf”,
+“S2_conf”, and “ST_conf”. The <code class="docutils literal notranslate"><span class="pre">_conf</span></code> keys store the corresponding
+confidence intervals, typically with a confidence level of 95%. Use the
+keyword argument <code class="docutils literal notranslate"><span class="pre">print_to_console=True</span></code> to print all indices. Or, we
+can print the individual values from <code class="docutils literal notranslate"><span class="pre">Si</span></code> as shown in the next step.</p>
+</section>
+<section id="step-5-interpret-our-results">
+<h3>Step 5: Interpret our results<a class="headerlink" href="#step-5-interpret-our-results" title="Link to this heading">#</a></h3>
+<p>We execute the following code and take a look at our first-order indices
+(<code class="docutils literal notranslate"><span class="pre">S1</span></code>) for each of our three inputs. These indicies can be interpreted
+as the fraction of variance in the output that is explained by each
+input individually.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">first_order</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">]</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;First-order:&#39;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1: </span><span class="si">{</span><span class="n">first_order</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">, x2: </span><span class="si">{</span><span class="n">first_order</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">, x3: </span><span class="si">{</span><span class="n">first_order</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">First</span><span class="o">-</span><span class="n">order</span><span class="p">:</span>
+<span class="n">x1</span><span class="p">:</span> <span class="mf">0.3184242969763115</span><span class="p">,</span> <span class="n">x2</span><span class="p">:</span> <span class="mf">0.4303808201623416</span><span class="p">,</span> <span class="n">x3</span><span class="p">:</span> <span class="mf">0.022687722804980225</span>
+</pre></div>
+</div>
+<p>If we were to rank the importance of the inputs in how much they
+individually explain the variance in the output, we would rank them from
+greatest to least importance as follows: 𝑥2, 𝑥1 and then 𝑥3. Since 𝑥3
+only explains 2% of the output variance, it does not explain output
+variability meaningfully. Thus, this indicates that there is
+contribution to the output variance by 𝑥2 and 𝑥1 independently, whereas
+𝑥3 does not contribute to the output variance. Determining what inputs
+are most important or what index value is meaningful is a common
+question, but one for which there is no general rule or threshold. This
+question is problem and context-dependent, but procedures have been
+identified to rank order influential inputs and which can be used to
+identify the least influential factors. These factors can be fixed to
+simplify the model <a class="footnote-reference brackets" href="#id10" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id12" id="id5" role="doc-noteref"><span class="fn-bracket">[</span>5<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id13" id="id6" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a>.</p>
+<p>Next, we evaluate the total-order indices, which measure the
+contribution to the output variance caused by varying the model input,
+including both its first-order effects (the input varying alone) and all
+higher-order interactions across the input parameters.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">total_order</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">]</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Total-order:&#39;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1: </span><span class="si">{</span><span class="n">total_order</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">, x2: </span><span class="si">{</span><span class="n">total_order</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">, x3: </span><span class="si">{</span><span class="n">total_order</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Total</span><span class="o">-</span><span class="n">order</span><span class="p">:</span>
+<span class="n">x1</span><span class="p">:</span> <span class="mf">0.5184119098161343</span><span class="p">,</span> <span class="n">x2</span><span class="p">:</span> <span class="mf">0.41021260250026054</span><span class="p">,</span> <span class="n">x3</span><span class="p">:</span> <span class="mf">0.2299058431439953</span>
+</pre></div>
+</div>
+<p>The magnitude of the total order indices are substantially larger than
+the first-order indices, which reveals that higher-order interactions
+are occurring, i.e. that the interactions across inputs are also
+explaining some of the total variance in the output. Note that 𝑥3 has
+non-negligible total-order indices, which indicates that it is not a
+consequential parameter when considered in isolation, but becomes
+consequential and explains 25% of variance in the output through its
+interactions with 𝑥1 and 𝑥2.</p>
+<p>Finally, we can investigate these higher order interactions by viewing
+the second-order indices. The second-order indicies measure the
+contribution to the output variance caused by the interaction between
+any two model inputs. Some computing error can appear in these
+sensitivity indices, such as negative values. Typically, these computing
+errors shrink as the number of samples increases.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">second_order</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S2&#39;</span><span class="p">]</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Second-order:&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1-x2:  </span><span class="si">{</span><span class="n">second_order</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1-x3:  </span><span class="si">{</span><span class="n">second_order</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x2-x3:  </span><span class="si">{</span><span class="n">second_order</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Second</span><span class="o">-</span><span class="n">order</span><span class="p">:</span>
+<span class="n">x1</span><span class="o">-</span><span class="n">x2</span><span class="p">:</span>  <span class="o">-</span><span class="mf">0.043237389723234154</span>
+<span class="n">x1</span><span class="o">-</span><span class="n">x3</span><span class="p">:</span>  <span class="mf">0.17506452088709862</span>
+<span class="n">x2</span><span class="o">-</span><span class="n">x3</span><span class="p">:</span>  <span class="o">-</span><span class="mf">0.03430682392607577</span>
+</pre></div>
+</div>
+<p>We can see that there are strong interactions between 𝑥1 and 𝑥3. Note
+that in the Ishigami function, these two variables are multiplied in the
+last term of the function, which leads to interactive effects. If we
+were considering first order indices alone, we would erroneously assume
+that 𝑥3 explains no variance in the output, but the second-order and
+total order indices reveal that this is not the case. It’s easy to
+understand where we might see interactive effects in the case of the
+simple Ishigami function. However, it’s important to remember that in
+more complex systems, there may be many higher-order interactions that
+are not apparent, but could be extremely consequential in explaining the
+variance of the output.</p>
+</section>
+<section id="tips-to-apply-sobol-sa-to-your-own-problem">
+<h3>Tips to Apply Sobol SA to Your Own Problem<a class="headerlink" href="#tips-to-apply-sobol-sa-to-your-own-problem" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we demonstrated how to apply an SA analysis to a
+simple mathematical test function. In order to apply a Sobol SA to your
+own problem, you will follow the same general workflow that we defined
+above. You will need to:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters and set up the problem
+dictionary as in Step 1 above.</p></li>
+<li><p>Generate samples using the <code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function. This step is
+problem-dependent and note that the Sobol method can be
+computationally intensive depending on the model being analyzed. For
+example, for a simple rainfall-runoff model such as HYMOD, it has
+been recommended to run a sample size of at least N = 10,000 (which
+translates to 60,000 model runs). More complex models will be slower
+to run and will also require more samples to calculate accurate
+estimates of Sobol indices. Once you complete this process, pay
+attention to the confidence bounds on your sensitivity indices to see
+whether you need to run more samples.</p></li>
+<li><p>Run the parameter sets through your model. In the example above, the
+Ishigami function could be evaluated through SALib since it is a
+built in function. For your application, you will need to run these
+parameter sets through the problem externally and save the output.
+The output file should contain one row of output values for each
+model run.</p></li>
+<li><p>Calculate the Sobol indices. Now, the Y will be a numpy array with
+your external model output and you will need to include the parameter
+samples as an additional argument.</p></li>
+<li><p>Finally, we interpet the results. If the confidence intervals of your
+dominant indices are larger than roughly 10% of the value itself, you
+may want to consider increasing your sample size as computation
+permits. You should additionally read the references noted in Step 5
+above to understand more about identifying important factors.</p></li>
+</ol>
+<p><strong>References</strong></p>
+<aside class="footnote-list brackets">
+<aside class="footnote brackets" id="id7" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">1</a><span class="fn-bracket">]</span></span>
+<p>Saltelli, A. (2002). “Making best use of model evaluations to compute sensitivity indices.” Computer Physics Communications, 145(2):280-297, doi:<a class="reference external" href="https://doi.org/10.1016/S0010-4655(02)00280-1">10.1016/S0010-4655(02)00280-1</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id8" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">2</a><span class="fn-bracket">]</span></span>
+<p>Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and S. Tarantola (2010). “Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index.” Computer Physics Communications, 181(2):259-270, doi:<a class="reference external" href="https://doi.org/10.1016/j.cpc.2009.09.018">10.1016/j.cpc.2009.09.018</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id9" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">3</a><span class="fn-bracket">]</span></span>
+<p>Sobol, I. M. (2001). “Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates.” Mathematics and Computers in Simulation, 55(1-3):271-280, doi:<a class="reference external" href="https://doi.org/10.1016/S0378-4754(00)00270-6">10.1016/S0378-4754(00)00270-6</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id10" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">4</a><span class="fn-bracket">]</span></span>
+<p>T. H. Andres, “Sampling methods and sensitivity analysis for large parameter sets,” Journal of Statistical Computation and Simulation, vol. 57, no. 1–4, pp. 77–110, Apr. 1997, doi:<a class="reference external" href="https://doi.org/10.1080/00949659708811804">10.1080/00949659708811804</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id12" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id5">5</a><span class="fn-bracket">]</span></span>
+<p>Y. Tang, P. Reed, T. Wagener, and K. van Werkhoven, “Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation,” Hydrology and Earth System Sciences, vol. 11, no. 2, pp. 793–817, Feb. 2007, doi:<a class="reference external" href="https://doi.org/10.5194/hess-11-793-2007">10.5194/hess-11-793-2007</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id13" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id6">6</a><span class="fn-bracket">]</span></span>
+<p>J. Nossent, P. Elsen, and W. Bauwens, “Sobol’ sensitivity analysis of a complex environmental model,” Environmental Modelling &amp; Software, vol. 26, no. 12, pp. 1515–1525, Dec. 2011, doi:<a class="reference external" href="https://doi.org/10.1016/j.envsoft.2011.08.010">10.1016/j.envsoft.2011.08.010</a>.</p>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa">Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s get started!</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-generate-the-problem-dictionary">Step 1: Generate the problem dictionary</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-generate-samples-using-the-saltelli-sampling-scheme">Step 2: Generate samples using the Saltelli sampling scheme</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-execute-the-ishigami-function-over-our-sample-set">Step 3: Execute the Ishigami function over our sample set</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method">Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-interpret-our-results">Step 5: Interpret our results</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-to-your-own-problem">Tips to Apply Sobol SA to Your Own Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="3_sensitivity_analysis_the_basics.html">3. Sensitivity Analysis: The Basics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html">4. Sensitivity Analysis: Diagnostic &amp; Exploratory Modeling</a></li>
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+    <h1>Logistic Regression Tutorial</h1>
+    <!-- Table of contents -->
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+                <h2> Contents </h2>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-logistic-regression-for-factor-mapping">Tutorial: Logistic Regression for Factor Mapping</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#background">Background</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s get started!</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-latin-hypercube-sample-and-set-up-problem">Step 1: Load Latin hypercube sample and set up problem</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-define-decision-relevant-metrics-for-illustration">Step 2: Define decision-relevant metrics for illustration</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-run-the-logistic-regression">Step 3: Run the logistic regression</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-factor-ranking">Step 4: Factor ranking</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-draw-factor-maps">Step 5: Draw factor maps</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-scenario-discovery-to-your-own-problem">Tips to Apply Scenario Discovery to Your Own Problem</a></li>
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+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/basin_users_logistic_regression.ipynb">Logistic Regression Tutorial</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-logistic-regression-for-factor-mapping">
+<h2>Tutorial: Logistic Regression for Factor Mapping<a class="headerlink" href="#tutorial-logistic-regression-for-factor-mapping" title="Link to this heading">#</a></h2>
+<p>This tutorial replicates a scenario discovery analysis performed in
+<a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020EF001503">Hadjimichael et
+al. (2020)</a>.</p>
+<section id="background">
+<h3>Background<a class="headerlink" href="#background" title="Link to this heading">#</a></h3>
+<p>Planners in the the Upper Colorado River Basin (UCRB, shown in the
+figure below) are seeking to understand the vulnerability of water users
+to uncertainties stemming from climate change, population growth and
+water policy changes. The UCRB spans 25,682 km2 in western Colorado and
+is home to approximately 300,000 residents and 1,012 km2 of irrigated
+land. Several thousand irrigation ditches divert water from the main
+river and its tributaties for irrigation (shown as small black dots in
+the figure). Transmountain diversions of approximately 567,400,000 m3
+per year are exported for irrigation, industrial and municipal uses in
+northern and eastern Colorado, serving the major population centers of
+Denver and Colorado Springs. These diversions are carried through
+tunnels, shown as large black dots in the figure.</p>
+<img alt="_images/basin_map.png" src="_images/basin_map.png" />
+<p>An important planning consideration is the water rights of each user,
+defined by seniority across all water uses (irrigation diversions,
+transboundary diversions, power plants etc.) in the basin. To assess the
+vulnerability of users with varying degrees of water rights seniority,
+planners simulate the system across an ensemble of scenarios using the
+state of Colorado’s StateMod platform. The model simulates streamflow,
+diversions, instream demands, and reservoir operations.</p>
+<p>Hadjimichael et al. (2020) employ an exploratory analysis by simulating
+a large ensemble of plausible scenarios using StateMod and then
+identifying consequential decision-relevant combinations of uncertain
+factors, termed scenario discovery. Focusing on decision-relevant
+metrics (metrics that are important to the user, the scenario discovery
+is applied to the water shortages experienced by each individual user
+(i.e., not on a single basin-wide or sector-wide metric). For this
+training example, we’ll be performing scenario discovery for three
+different water users: two irrigation users and one municipal user.</p>
+</section>
+<section id="lets-get-started">
+<h3>Let’s get started!<a class="headerlink" href="#lets-get-started" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we will be loading in data that has been produced in
+Hadjimichael et al. (2020). Before we start our analysis, we’ll load the
+relevant Python libraries, example data, and information for the three
+users.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#import libraries</span>
+<span class="kn">import</span> <span class="nn">msdbook</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="nn">mpl</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="c1"># load example data from Hadjimichael et al. (2020)</span>
+<span class="n">msdbook</span><span class="o">.</span><span class="n">install_package_data</span><span class="p">()</span>
+
+<span class="c1"># Select the IDs for the three users that we will perform the analysis for</span>
+<span class="n">all_IDs</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;7000550&#39;</span><span class="p">,</span><span class="s1">&#39;7200799&#39;</span><span class="p">,</span><span class="s1">&#39;3704614&#39;</span><span class="p">]</span>
+<span class="n">usernames</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Medium seniority irrigation&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Low seniority irrigation&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Transbasin municipal diversion&#39;</span><span class="p">]</span>
+<span class="n">nStructures</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Downloading</span> <span class="n">example</span> <span class="n">data</span> <span class="k">for</span> <span class="n">msdbook</span> <span class="n">version</span> <span class="mf">0.1.5</span><span class="o">...</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">uncertain_params_bounds</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_metric_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">solutions</span><span class="o">.</span><span class="n">resultfile</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LHsamples_original_1000</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">param_values</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_params_256samples</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LeafCatch</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_simulations_256samples</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">Robustness</span><span class="o">.</span><span class="n">txt</span>
+</pre></div>
+</div>
+</section>
+<section id="step-1-load-latin-hypercube-sample-and-set-up-problem">
+<h3>Step 1: Load Latin hypercube sample and set up problem<a class="headerlink" href="#step-1-load-latin-hypercube-sample-and-set-up-problem" title="Link to this heading">#</a></h3>
+<p>To examine regional vulnerability, we generate an ensemble of plausible
+future states of the world (SOWs) using Latin Hypercube Sampling. For
+this tutorial, we’ll load a file containing 1,000 samples across 14
+parameters. The sampled parameters encompass plausible changes to the
+future state of the basin, including changes to hydrology, water demands
+(irrigation, municipal &amp; industry, transbasin), and institutional and
+environmental factors (environmental flows, reservoir storage, operation
+of the Shoshone Power Plant). These samples are taken from ranges
+identified in <code class="docutils literal notranslate"><span class="pre">param_bounds</span></code>. Below we load in the 1000 samples, the
+range of values that the samples can take for each parameter, and the
+parameter names. More information on what each parameter constitutes can
+be found in Table 1 of Hadjimichael et al., 2020.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Identify the bounds for each of the 14 parameters</span>
+<span class="n">param_bounds</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_basin_param_bounds</span><span class="p">()</span>
+
+<span class="c1">#Load in the parameter samples</span>
+<span class="n">LHsamples</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_lhs_basin_sample</span><span class="p">()</span>
+
+<span class="c1">#Create an array of the parameter names</span>
+<span class="n">param_names</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Irrigation demand multiplier&#39;</span><span class="p">,</span><span class="s1">&#39;Reservoir loss&#39;</span><span class="p">,</span><span class="s1">&#39;Transbasin demand multiplier&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Municipal &amp; industrial multiplier&#39;</span><span class="p">,</span> <span class="s1">&#39;Shoshone&#39;</span><span class="p">,</span><span class="s1">&#39;Environmental flows&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Evaporation change&#39;</span><span class="p">,</span><span class="s1">&#39;Mean dry flow&#39;</span><span class="p">,</span><span class="s1">&#39;Dry flow variance&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Mean wet flow&#39;</span><span class="p">,</span><span class="s1">&#39;Wet flow variance&#39;</span><span class="p">,</span><span class="s1">&#39;Dry-dry probability&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Wet-wet probability&#39;</span><span class="p">,</span> <span class="s1">&#39;Snowmelt shift&#39;</span><span class="p">]</span>
+</pre></div>
+</div>
+</section>
+<section id="step-2-define-decision-relevant-metrics-for-illustration">
+<h3>Step 2: Define decision-relevant metrics for illustration<a class="headerlink" href="#step-2-define-decision-relevant-metrics-for-illustration" title="Link to this heading">#</a></h3>
+<p>Scenario discovery attempts to identify parametric regions that lead to
+‘success’ and ‘failure’. For this demonstration we’ll be defining
+‘success’ as states of the world where a shortage level doesn’t exceed
+its historical frequency.</p>
+</section>
+<section id="step-3-run-the-logistic-regression">
+<h3>Step 3: Run the logistic regression<a class="headerlink" href="#step-3-run-the-logistic-regression" title="Link to this heading">#</a></h3>
+<p>Logistic regression estimates the probability that a future SOW will be
+classified as a success or failure given a set of performance criteria.
+A logistic regression model is defined by:</p>
+<div class="math notranslate nohighlight">
+\[ln \bigg (\frac{p_i}{1-p_i} \bigg ) = X^T_i \beta\]</div>
+<p>where <span class="math notranslate nohighlight">\(p_i\)</span> is the probability the performance in the
+<span class="math notranslate nohighlight">\(i^{th}\)</span> SOW will be classified as a success, <span class="math notranslate nohighlight">\(X_i\)</span> is the
+vector of covariates describing the <span class="math notranslate nohighlight">\(i^{th}\)</span> SOW, and
+<span class="math notranslate nohighlight">\(\beta\)</span> is the vector of coefficients describing the relationship
+between the covariates and the response, which here will be estimated
+using maximum likelihood estimation.</p>
+<p>A logistic regression model was fit to the ensemble of SOWs using the
+performance criteria defined in step 2. Logistic regression modeling was
+conducted using the <a class="reference external" href="https://www.statsmodels.org/stable/index.html">Statsmodel
+Python</a> package. The
+data required for the full analysis is too large to include in this
+tutorial, but results can be found in the data file loaded below.</p>
+<p>The results files contain the occurence of different shortage frequency
+and magnitude combinations under the experiment, in increments of 10,
+between 0 and 100. These combinations (100 for each user) are
+alternative decision-relevant metrics that can be used for scenario
+discovery.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set arrays for shortage frequencies and magnitudes</span>
+<span class="n">frequencies</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">110</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+<span class="n">magnitudes</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">110</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+<span class="n">realizations</span> <span class="o">=</span> <span class="mi">10</span>
+
+<span class="c1"># Load performance and pseudo r scores for each of the users</span>
+<span class="n">results</span> <span class="o">=</span> <span class="p">[</span><span class="n">msdbook</span><span class="o">.</span><span class="n">load_user_heatmap_array</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">/</span> <span class="mi">100</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">))]</span>
+</pre></div>
+</div>
+</section>
+<section id="step-4-factor-ranking">
+<h3>Step 4: Factor ranking<a class="headerlink" href="#step-4-factor-ranking" title="Link to this heading">#</a></h3>
+<p>To rank the importance of each uncertain factor, we utilize McFadden’s
+psuedo-R2, a measure that quantifies the improvement of the model when
+utilizing each given predictor as compared to prediction using the mean
+of the data set:</p>
+<div class="math notranslate nohighlight">
+\[R^2_{McFadden}=1-\frac{ln \hat{L}(M_{full})}{ln \hat{L}(M_{intercept})}\]</div>
+<p>Here <span class="math notranslate nohighlight">\(ln \hat{L}(M_{full})\)</span> is the log likelihood of the full
+model (including the predictor) and <span class="math notranslate nohighlight">\(ln \hat{L}(M_{intercept})\)</span> is
+the log likelihood of the intercept model (which predicts the mean
+probability of success across all SOWs).</p>
+<p>Higher values of McFadden’s psuedo-R2 indicate higher factor importance
+(when the likelihood of the full model approaches one, the ratio of the
+likelihood of the full model compared to the intercept model will get
+very small).</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Load the pseudo-R^2 scores</span>
+<span class="n">scores</span> <span class="o">=</span> <span class="p">[</span><span class="n">msdbook</span><span class="o">.</span><span class="n">load_user_pseudo_scores</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">))]</span>
+
+<span class="c1"># Select indices of frequency and magnitudes that will be used for the visualization</span>
+<span class="n">freq</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
+<span class="n">mag</span> <span class="o">=</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">7</span><span class="p">]</span>
+</pre></div>
+</div>
+</section>
+<section id="step-5-draw-factor-maps">
+<h3>Step 5: Draw factor maps<a class="headerlink" href="#step-5-draw-factor-maps" title="Link to this heading">#</a></h3>
+<p>The McFadden’s psuedo-R2 scores files contain preliminary logistic
+regression results on parameter importance for each of these
+combinations. Using these psuedo-R2 scores we will identify the two most
+important factors for each metric which we’ll use to generate the final
+scenario discovery maps (note: there may be more than two important
+metrics for each user, but here we will demonstrate by mapping two).</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># setup figure</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">18</span><span class="p">),</span> <span class="n">tight_layout</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">patch</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">axes</span><span class="o">.</span><span class="n">flat</span><span class="p">)):</span>
+
+    <span class="n">ax</span> <span class="o">=</span> <span class="n">axes</span><span class="o">.</span><span class="n">flat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+    <span class="n">allSOWsperformance</span> <span class="o">=</span> <span class="n">results</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+    <span class="n">all_pseudo_r_scores</span> <span class="o">=</span> <span class="n">scores</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+    <span class="c1"># construct dataframe</span>
+    <span class="n">dta</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">LHsamples</span><span class="p">,</span> <span class="n">realizations</span><span class="p">,</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">),</span> <span class="n">columns</span><span class="o">=</span><span class="n">param_names</span><span class="p">)</span>
+    <span class="n">dta</span><span class="p">[</span><span class="s1">&#39;Success&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">allSOWsperformance</span><span class="p">[</span><span class="n">freq</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">mag</span><span class="p">[</span><span class="n">i</span><span class="p">],:]</span>
+
+    <span class="n">pseudo_r_scores</span> <span class="o">=</span> <span class="n">all_pseudo_r_scores</span><span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">frequencies</span><span class="p">[</span><span class="n">freq</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span><span class="o">+</span><span class="s1">&#39;yrs_&#39;</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">magnitudes</span><span class="p">[</span><span class="n">mag</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span><span class="o">+</span><span class="s1">&#39;prc&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>
+    <span class="n">top_predictors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">pseudo_r_scores</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">][:</span><span class="mi">2</span><span class="p">]</span> <span class="c1">#Sort scores and pick top 2 predictors</span>
+
+    <span class="c1"># define color map for dots representing SOWs in which the policy</span>
+    <span class="c1"># succeeds (light blue) and fails (dark red)</span>
+    <span class="n">dot_cmap</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">ListedColormap</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">227</span><span class="p">,</span><span class="mi">26</span><span class="p">,</span><span class="mi">28</span><span class="p">],[</span><span class="mi">166</span><span class="p">,</span><span class="mi">206</span><span class="p">,</span><span class="mi">227</span><span class="p">]])</span><span class="o">/</span><span class="mf">255.0</span><span class="p">)</span>
+
+    <span class="c1"># define color map for probability contours</span>
+    <span class="n">contour_cmap</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">&#39;RdBu&#39;</span><span class="p">)</span>
+
+    <span class="c1"># define probability contours</span>
+    <span class="n">contour_levels</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.05</span><span class="p">,</span><span class="mf">0.1</span><span class="p">)</span>
+
+    <span class="c1"># define base values of the predictors</span>
+    <span class="n">SOW_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span> <span class="c1"># default parameter values for base SOW</span>
+    <span class="n">base</span> <span class="o">=</span> <span class="n">SOW_values</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">]</span>
+    <span class="n">ranges</span> <span class="o">=</span> <span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">]</span>
+
+    <span class="c1"># define grid of x (1st predictor), and y (2nd predictor) dimensions</span>
+    <span class="c1"># to plot contour map over</span>
+    <span class="n">xgrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="mi">0</span><span class="p">],</span>
+                      <span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="mi">1</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">around</span><span class="p">((</span><span class="n">ranges</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">ranges</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span><span class="o">/</span><span class="mi">500</span><span class="p">,</span><span class="n">decimals</span><span class="o">=</span><span class="mi">4</span><span class="p">))</span>
+    <span class="n">ygrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">]][</span><span class="mi">0</span><span class="p">],</span>
+                      <span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">]][</span><span class="mi">1</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">around</span><span class="p">((</span><span class="n">ranges</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">ranges</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span><span class="o">/</span><span class="mi">500</span><span class="p">,</span><span class="n">decimals</span><span class="o">=</span><span class="mi">4</span><span class="p">))</span>
+    <span class="n">all_predictors</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dta</span><span class="o">.</span><span class="n">columns</span><span class="o">.</span><span class="n">tolist</span><span class="p">()[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">top_predictors</span><span class="p">]</span>
+    <span class="n">dta</span><span class="p">[</span><span class="s1">&#39;Interaction&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">dta</span><span class="p">[</span><span class="n">all_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span><span class="o">*</span><span class="n">dta</span><span class="p">[</span><span class="n">all_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
+
+    <span class="c1"># logistic regression here</span>
+    <span class="n">predictor_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">all_predictors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]]</span>
+    <span class="n">result</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">fit_logit</span><span class="p">(</span><span class="n">dta</span><span class="p">,</span> <span class="n">predictor_list</span><span class="p">)</span>
+
+    <span class="c1"># plot contour map</span>
+    <span class="n">contourset</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_contour_map</span><span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="n">dta</span><span class="p">,</span> <span class="n">contour_cmap</span><span class="p">,</span>
+                                          <span class="n">dot_cmap</span><span class="p">,</span> <span class="n">contour_levels</span><span class="p">,</span> <span class="n">xgrid</span><span class="p">,</span>
+                                          <span class="n">ygrid</span><span class="p">,</span> <span class="n">all_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">all_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">base</span><span class="p">)</span>
+
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">usernames</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+
+<span class="c1"># set up colorbar</span>
+<span class="n">cbar_ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_axes</span><span class="p">([</span><span class="mf">0.98</span><span class="p">,</span> <span class="mf">0.15</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">])</span>
+<span class="n">cbar</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">contourset</span><span class="p">,</span> <span class="n">cax</span><span class="o">=</span><span class="n">cbar_ax</span><span class="p">)</span>
+<span class="n">cbar_ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Probability of Success&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">cbar_ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;y&#39;</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">&#39;major&#39;</span><span class="p">,</span> <span class="n">labelsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">statsmodels</span><span class="o">/</span><span class="n">base</span><span class="o">/</span><span class="n">model</span><span class="o">.</span><span class="n">py</span><span class="p">:</span><span class="mi">127</span><span class="p">:</span> <span class="n">ValueWarning</span><span class="p">:</span> <span class="n">unknown</span> <span class="n">kwargs</span> <span class="p">[</span><span class="s1">&#39;disp&#39;</span><span class="p">]</span>
+  <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="n">msg</span><span class="p">,</span> <span class="n">ValueWarning</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Optimization</span> <span class="n">terminated</span> <span class="n">successfully</span><span class="o">.</span>
+         <span class="n">Current</span> <span class="n">function</span> <span class="n">value</span><span class="p">:</span> <span class="mf">0.378619</span>
+         <span class="n">Iterations</span> <span class="mi">8</span>
+<span class="n">Optimization</span> <span class="n">terminated</span> <span class="n">successfully</span><span class="o">.</span>
+         <span class="n">Current</span> <span class="n">function</span> <span class="n">value</span><span class="p">:</span> <span class="mf">0.397285</span>
+         <span class="n">Iterations</span> <span class="mi">8</span>
+<span class="n">Optimization</span> <span class="n">terminated</span> <span class="n">successfully</span><span class="o">.</span>
+         <span class="n">Current</span> <span class="n">function</span> <span class="n">value</span><span class="p">:</span> <span class="mf">0.377323</span>
+         <span class="n">Iterations</span> <span class="mi">8</span>
+</pre></div>
+</div>
+<img alt="_images/notebook_logistic_output_11_1.png" src="_images/notebook_logistic_output_11_1.png" />
+<p>The figure above demonstrates how different combinations of the
+uncertain factors lead to success or failure in different states of the
+world, which are denoted by the blue and red dots respectively. The
+probability of success and failure are further denoted by the contours
+in the figure. Several insights can be drawn from this figure.</p>
+<p>First, using metrics chosen to be decision-relevant (specific to each
+user) causes different factors to be identified as most important by
+this scenario-discovery exercise (the x- and y-axes for each of the
+subplots). In other words, depending on what the decision makers of this
+system want to prioritize they might choose to monitor different
+uncertain factors to track performance.</p>
+<p>Second, in the top panel, the two identified factors appear to also have
+an interactive effect on the metric used (shortages of a certain level
+and frequency in this example). In terms of scenario discovery, the
+Patient Rule Induction Method (PRIM) or Classification And Regression
+Trees (CART) would not be able to delineate this non-linear space and
+would therefore misclassify parameter combinations as ‘desirable’ when
+they were in fact undesirable, and vice versa.</p>
+<p>Lastly, logistic regression also produces contours of probability of
+success, i.e. different factor-value combinations are assigned different
+probabilities that a shortage level will be exceeded. This allows the
+decision makers to evaluate these insights while considering their risk
+aversion.</p>
+</section>
+<section id="tips-to-apply-scenario-discovery-to-your-own-problem">
+<h3>Tips to Apply Scenario Discovery to Your Own Problem<a class="headerlink" href="#tips-to-apply-scenario-discovery-to-your-own-problem" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we demonstrated how to perform a scenario discovery
+analysis for three different users in the UCRB. The analysis allowed us
+to determine which parameters the users would be most affected by and to
+visualize how different ranges of these parameters lead to success and
+failure for different users. This framework can be applicable to any
+other application where it is of interest to characterize success and
+failure based on uncertain parameter ranges. In order to apply the same
+framework to your own problem:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters of interest, which will
+represent uncertainties that characterize your system.</p></li>
+<li><p>Generate samples for these parameters (this can be done using the
+<code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function or externally).</p></li>
+<li><p>Define what constitutes success and failure in your problem. In this
+tutorial, success was defined based on not surpassing the historical
+drought frequency. Choose a metric that is relevant to your problem
+and decision-makers that might be involved. If your model involves an
+optimization, you can also define metrics based on meeting certain
+values of these objectives.</p></li>
+<li><p>Run the parameter sets through your model and calculate success and
+failure based on your metrics and across different users if
+applicable. This step will allow you to create the scatter plot part
+of the final figure.</p></li>
+<li><p>If it is of interest, the contours on the figure can be created by
+fitting the logistic regression model in a similiar manner as denoted
+in Steps 3 and 5 of the tutorial.</p></li>
+</ol>
+</section>
+</section>
+</section>
+
+
+                </article>
+              
+
+              
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+    <i class="fa-solid fa-list"></i> Contents
+  </div>
+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-logistic-regression-for-factor-mapping">Tutorial: Logistic Regression for Factor Mapping</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#background">Background</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s get started!</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-latin-hypercube-sample-and-set-up-problem">Step 1: Load Latin hypercube sample and set up problem</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-define-decision-relevant-metrics-for-illustration">Step 2: Define decision-relevant metrics for illustration</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-run-the-logistic-regression">Step 3: Run the logistic regression</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-factor-ranking">Step 4: Factor ranking</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-draw-factor-maps">Step 5: Draw factor maps</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-scenario-discovery-to-your-own-problem">Tips to Apply Scenario Discovery to Your Own Problem</a></li>
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+              
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+
+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>HYMOD Dynamics Tutorial</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-of-the-hymod-model">Tutorial: Sensitivity Analysis of the HYMOD Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction-to-hymod">1. Introduction to HYMOD</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#global-sensitivity-analysis">2. Global Sensitivity Analysis</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">1 - Introduction to HYMOD</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">2- Global Sensitivity Analysis</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#running-hymod-model-warm-up">Running HYMOD - Model Warm-up</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#sensitivity-analysis-indices-for-each-month">2.4.1 Sensitivity analysis indices for each month</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#first-order-indices">First-order Indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#total-order-indices">Total-order indices</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#annual-sensitivity-analysis-indices">2.4.2 Annual sensitivity analysis indices</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">First-order indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id4">Total-order indices</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#monthly-time-varying-sensitivity-analysis">2.4.3 Monthly time-varying sensitivity analysis</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id5">First-order indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#total-order-time-varying-sensitivity-analysis">Total order - time varying sensitivity analysis</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-this-methodology-to-your-own-problem">Tips to Apply this methodology to your own problem</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="hymod-dynamics-tutorial">
+<h1>HYMOD Dynamics Tutorial<a class="headerlink" href="#hymod-dynamics-tutorial" title="Link to this heading">#</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">HYMOD Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-sensitivity-analysis-of-the-hymod-model">
+<h2>Tutorial: Sensitivity Analysis of the HYMOD Model<a class="headerlink" href="#tutorial-sensitivity-analysis-of-the-hymod-model" title="Link to this heading">#</a></h2>
+<p>The purpose of this tutorial is to demonstrate the global sensitivity
+analysis concepts and tools established in the Section 3.1 of the main
+text of this eBook. This demonstration will highlight the central role
+of design of experiments (Section 3.3), when implementing global
+sensitivity analysis tools described in Section 3.4.</p>
+<p>We’ll explore these tools and concepts using the HYdrological MODel
+(HYMOD), a rainfall-runoff model developed and used for river flow
+forecasting. HYMOD was chosen for demonstration because its purpose is
+to abstract highly complex and non-linear systems. The methods
+demonstrated in thistutorial can be applied to numerical models that
+simulate other complex non-linear systems.</p>
+<p>This tutorial will first introduce HYMOD and use it to simulate
+streamflows in a river basin. Next, we’ll employ sensitivity analysis
+concepets described in Section 3 of the main text to examine how values
+of HYMOD’s parameters impact streamflow predictions. We’ll then explore
+how the effects of these parameters may change over time using
+time-varying sensitivtiy analysis. Finally, we’ll demonstrate concepts
+presented in Chapter 7 through two ensemble-based methods of uncertainty
+quantification - Generalized Likelihood Uncertainty Estimation (GLUE)
+and Pre-Calibration.</p>
+<p>The tutorial includes the following steps:</p>
+<section id="introduction-to-hymod">
+<h3>1. Introduction to HYMOD<a class="headerlink" href="#introduction-to-hymod" title="Link to this heading">#</a></h3>
+<p>1.1 - Introduction to a simple hydrologic model (HYMOD)</p>
+<p>1.2 - Input Data</p>
+<p>1.3 - Running a basic simulation</p>
+<p>1.4 - Model outputs</p>
+</section>
+<section id="global-sensitivity-analysis">
+<h3>2. Global Sensitivity Analysis<a class="headerlink" href="#global-sensitivity-analysis" title="Link to this heading">#</a></h3>
+<p>2.1 - Design of Experiments</p>
+<p>2.2 - Sensitivity analysis for one output</p>
+<p>2.3 - Sensitivity analysis across multiple outputs</p>
+<p>2.4 - Time-varying sensitivity analysis</p>
+</section>
+</section>
+<section id="id1">
+<h2>1 - Introduction to HYMOD<a class="headerlink" href="#id1" title="Link to this heading">#</a></h2>
+<p id="hymod">1.1 Overview</p>
+<p>HYMOD is a hydrologic model (rainfall-runoff model) that simulates key
+hydrologic fluxes such as infiltration, streamflow and
+evapotranspiration. The model was originally developed and used for
+river flow forecasting, but it has also been been used to explore
+different sensitivity analysis (e.g., <a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/wrcr.20124">Herman et al.,
+2013</a>),
+uncertainty quantification (e.g., <a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2006WR005205">Smith et al.,
+2008</a>),
+and optimization (e.g., <a class="reference external" href="https://www.sciencedirect.com/science/article/pii/S0022169414006362?casa_token=IRqE19Hkfa8AAAAA:_fXOqfwpzxMpchvu8_0njCe0Ok9H29Gyw2F46l9PzG9UVODDTUg6wIOSiyp6uybGevNVnZ7N">Ye et al.,
+2014</a>)
+concepts.</p>
+<p>HYMOD accepts two inputs - daily precepitation and daily potential
+evapotranspiration (PET)- and generates predicitons of daily streamflow.
+HYMOD abstracts the highly non-linear process of runoff routing by
+dividing the flow into two components: quick flow, representing
+precipitation that quickly runs off the surface of the watershed into
+the stream, and slow flow, that moves through the soil and takes much
+longer to arrive at the stream.</p>
+<p>To generate streamflow predictions, HYMOD first models vertical
+processes within the watershed to determine how much water infiltrates
+and evaporates from the soil at a given time step. It then determines
+how much water should be partitioned into quick flow and slow flow
+processes. Within each process it abstracts residence time (the time it
+takes a unit volume of water to move through the watershed and into the
+stream) using a series of “reservoirs” each with a calibrated residence
+time.</p>
+<p>HYMOD’s representation of hydrologic processes are shown Figure 1 below
+and controlled by the following parameters:</p>
+<p><span class="math notranslate nohighlight">\(H_{uz}\)</span>: the maximum water storage capacity of the soil (mm)</p>
+<p><span class="math notranslate nohighlight">\(B_{exp}\)</span>: parameters describing the degree of spatial variability
+within the basin between 0 and Huz</p>
+<p><span class="math notranslate nohighlight">\(Alp\)</span>: Fraction of runoff contributing to quick flow</p>
+<p><span class="math notranslate nohighlight">\(K_q\)</span>: Quick flow residence time of linear infinite reservoir (the
+Kq values of all three linear reservoirs are the same)</p>
+<p><span class="math notranslate nohighlight">\(K_s\)</span>: Slow flow residence time of linear infinite reservoir</p>
+<img alt="_images/hymod_schematic-DAVE.png" src="_images/hymod_schematic-DAVE.png" />
+<p>HYMOD models the fraction of water that is stored in the soil
+<span class="math notranslate nohighlight">\((F(XH_{uz}))\)</span> using the following relationship:</p>
+<div class="math notranslate nohighlight">
+\[F(XH_{uz}) = 1 - (1 - \frac{XH_{uz}}{H_{uz}})^{B}\]</div>
+<p>where <span class="math notranslate nohighlight">\(XH_{uz}\)</span> is the water storage capacity of the soil;
+<span class="math notranslate nohighlight">\(H_{uz}\)</span> is the parameter describing basin maximum water
+storage capacity (mm); and <span class="math notranslate nohighlight">\(B\)</span> is the parameter describing the
+degree of spatial variability within the basin.</p>
+<p>The portion of precipitation that exceeds the water storage capacity is
+treated as runoff.</p>
+<p>To route runoff to streamflow, the excess runoff from the vertical
+processes is split into quick flow and slow flow. The proportion of
+runoff partitioned into quick flow and slow flow is determined by a
+parameter <span class="math notranslate nohighlight">\(Alp\)</span>, which ranges between 0 and 1. Quick flow is
+routed through <span class="math notranslate nohighlight">\(N\)</span> identical quick flow tanks <span class="math notranslate nohighlight">\(Q1, Q2... QN\)</span>
+(shown above as <span class="math notranslate nohighlight">\(N=3\)</span>). The rate at which runoff moves through the
+quick flow system is described by the residence time of the quick frlow
+tanks, <span class="math notranslate nohighlight">\(Kq\)</span> (day). Slow flow is routed through a parallel slow
+flow tank and the rate at which slow flow is routed is described by the
+slow flow residences time, <span class="math notranslate nohighlight">\(Ks\)</span> (day).</p>
+<p>Citation: Wagener, T., Boyle, D. P., Lees, M. J., Wheater, H. S., Gupta,
+H. V., &amp; Sorooshian, S. (2001). A framework for development and
+application of hydrological models. Hydrology and Earth System Sciences,
+5(1), 13-26.</p>
+<p id="inputs">1.2 Input data</p>
+<p>The HYMOD model only requires precipitation and potential
+evapotranspiration as inputs. For this example, we’ll run HYMOD using
+data from the Leaf River, a humid catchment located north of Collins
+Mississippi that has been widely used to explore HYMOD. The dataset also
+includes daily observed runoff that we later use to evaluate the
+performace of each sensitvity analysis sample set.</p>
+<p>In the following section of code, we’ll load the necessary python
+libraries and read in the input file. For this exercise we’ll only use
+the first eleven years of data. The first five rows of the input dataset
+are printed to show what they look like:</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">msdbook</span>
+
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+
+<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">metrics</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
+
+<span class="c1"># load example data</span>
+<span class="n">msdbook</span><span class="o">.</span><span class="n">install_package_data</span><span class="p">()</span>
+
+<span class="c1"># load the Leaf River HYMOD input file</span>
+<span class="n">leaf_data</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_input_file</span><span class="p">()</span>
+
+<span class="c1"># extract the first eleven years of data</span>
+<span class="n">leaf_data</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">4015</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Leaf River Data structure:&#39;</span><span class="p">)</span>
+
+<span class="c1"># There are only three columns in the file including precipitation, potential evapotranspiration and  streamflow</span>
+<span class="n">leaf_data</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Downloading</span> <span class="n">example</span> <span class="n">data</span> <span class="k">for</span> <span class="n">msdbook</span> <span class="n">version</span> <span class="mf">0.1.5</span><span class="o">...</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">uncertain_params_bounds</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_metric_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">solutions</span><span class="o">.</span><span class="n">resultfile</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LHsamples_original_1000</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">param_values</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_params_256samples</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LeafCatch</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_simulations_256samples</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">Robustness</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Leaf</span> <span class="n">River</span> <span class="n">Data</span> <span class="n">structure</span><span class="p">:</span>
+</pre></div>
+</div>
+<div>
+<style scoped>
+    .dataframe tbody tr th:only-of-type {
+        vertical-align: middle;
+    }
+
+    .dataframe tbody tr th {
+        vertical-align: top;
+    }
+
+    .dataframe thead th {
+        text-align: right;
+    }
+</style>
+<table border="1" class="dataframe">
+  <thead>
+    <tr style="text-align: right;">
+      <th></th>
+      <th>Precip</th>
+      <th>Pot_ET</th>
+      <th>Strmflw</th>
+    </tr>
+  </thead>
+  <tbody>
+    <tr>
+      <th>0</th>
+      <td>0.0</td>
+      <td>4.60</td>
+      <td>0.29</td>
+    </tr>
+    <tr>
+      <th>1</th>
+      <td>0.0</td>
+      <td>4.31</td>
+      <td>0.24</td>
+    </tr>
+    <tr>
+      <th>2</th>
+      <td>0.0</td>
+      <td>4.33</td>
+      <td>0.21</td>
+    </tr>
+    <tr>
+      <th>3</th>
+      <td>0.0</td>
+      <td>4.78</td>
+      <td>0.19</td>
+    </tr>
+    <tr>
+      <th>4</th>
+      <td>0.0</td>
+      <td>2.91</td>
+      <td>0.18</td>
+    </tr>
+  </tbody>
+</table>
+</div><p>To visualize catchment hydrology, streamflow and precipitation data are
+usually plotted together as a combined hydrograph (streamflow ) and
+hyetograph (rainfall, from Greek.hyetos, “rain”). Streamflow is plotted
+as a time series, while rainfall is shown as an inverted bar plot along
+the top of the graph. Streamflow labels are shown on the left y-axis,
+while rainfall labels are shown on the right y-axis.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># make an axis for the hydrograph</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">strmflw_ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">12</span><span class="p">,</span><span class="mi">6</span><span class="p">])</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">50</span><span class="p">])</span>
+
+<span class="c1">#make a second y-axis for the hyetograph</span>
+<span class="n">precip_ax</span> <span class="o">=</span> <span class="n">strmflw_ax</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">200</span><span class="p">])</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">invert_yaxis</span><span class="p">()</span>
+
+<span class="n">precip</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">]</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">])),</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;lightcoral&#39;</span><span class="p">)</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Streamflow (mm/day)&#39;</span><span class="p">)</span>
+
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">])),</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">],</span> <span class="n">width</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Rainfall (mm/day)&#39;</span><span class="p">)</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">&#39;Precip&#39;</span><span class="p">],</span> <span class="n">loc</span><span class="o">=</span><span class="s1">&#39;center right&#39;</span><span class="p">)</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">&#39;Streamflow&#39;</span><span class="p">],</span><span class="n">bbox_to_anchor</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.48</span><span class="p">))</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">&lt;</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">legend</span><span class="o">.</span><span class="n">Legend</span> <span class="n">at</span> <span class="mh">0x7f53b95c6850</span><span class="o">&gt;</span>
+</pre></div>
+</div>
+<img alt="_images/hymod1.png" src="_images/hymod1.png" />
+<p id="baseline">1.3 Running a Baseline Model Simulation</p>
+<p>We’ll start our experiment by running HYMOD using its default
+parameters.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># assign input parameters to generate a baseline simulated streamflow</span>
+<span class="n">Nq</span> <span class="o">=</span> <span class="mi">3</span>  <span class="c1"># Number of quickflow routing tanks</span>
+<span class="n">Kq</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="c1"># Quickflow routing tanks&#39; rate parameter</span>
+<span class="n">Ks</span> <span class="o">=</span>  <span class="mf">0.001</span> <span class="c1"># Slowflow routing tank&#39;s rate parameter</span>
+<span class="n">Alp</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="c1"># Quick/slow split parameter</span>
+<span class="n">Huz</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1"># Maximum height of soil moisture accounting tank</span>
+<span class="n">B</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="c1"># Scaled distribution function shape parameter</span>
+
+<span class="c1"># Note that the number of years is 11. One year of model warm-up and ten years are used for actual simulation</span>
+<span class="n">model</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">hymod</span><span class="p">(</span><span class="n">Nq</span><span class="p">,</span> <span class="n">Kq</span><span class="p">,</span> <span class="n">Ks</span><span class="p">,</span> <span class="n">Alp</span><span class="p">,</span> <span class="n">Huz</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">leaf_data</span><span class="p">,</span> <span class="n">ndays</span><span class="o">=</span><span class="mi">4015</span><span class="p">)</span>
+</pre></div>
+</div>
+<p id="outputs">1.4 Model Outputs</p>
+<p>Model outputs include actual evapotranspiration, quick and fast
+streamflow, and combined runoff. In this tutorial we focus on the total
+daily runoff, QQ (<span class="math notranslate nohighlight">\(m^3/s\)</span>). We can use the following script to
+plot simulated streamflow against observed streamflow.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-observed-vs-simulated-streamflow">plot_observed_vs_simulated_streamflow</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ax</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_observed_vs_simulated_streamflow</span><span class="p">(</span><span class="n">df</span><span class="o">=</span><span class="n">leaf_data</span><span class="p">,</span> <span class="n">hymod_dict</span><span class="o">=</span><span class="n">model</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod2.png" src="_images/hymod2.png" />
+<p>So how does our model perform? We can investigate model performance
+across several metrics:</p>
+<p>1: Mean Absolute Error (MAE); MAE conveys how the model performs on
+average across the 10 year simulation period, with smaller values
+indicating better performance. The absolute value is taken so that
+positive and negative errors do not cancel each other out.</p>
+<div class="math notranslate nohighlight">
+\[MAE = \frac{1}{N}\sum_{t=0}^N\left\lvert Q_{sim,t}-Q_{obs,t}\right\rvert\]</div>
+<p>2: Root Mean Square Error (RMSE); RMSE is sum of square errors across
+the 10 year simulation period. RMSE is sensitive to large errors between
+the historical record and the simulated flows, and thus is useful for
+highlighting the model’s ability to capture of extreme flood events.</p>
+<div class="math notranslate nohighlight">
+\[RMSE = \sqrt{\frac{1}{N}\sum_{t=1}^{N}(Q_{sim,t}-Q_{obs,t})^2}\]</div>
+<p>3: Log-Root Mean Square Error (Log(RMSE)) LOG(RMSE) focuses on model
+performance during low-flow events.</p>
+<div class="math notranslate nohighlight">
+\[LOG(RMSE) = log(RMSE)\]</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">mae</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">]</span> <span class="o">-</span> <span class="n">model</span><span class="p">[</span><span class="s1">&#39;Q&#39;</span><span class="p">]))</span>
+<span class="n">mse</span> <span class="o">=</span> <span class="n">metrics</span><span class="o">.</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">model</span><span class="p">[</span><span class="s1">&#39;Q&#39;</span><span class="p">],</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">])</span>
+<span class="n">rmse</span> <span class="o">=</span> <span class="n">mse</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;MAE: &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">mae</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">RMSE: &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">mse</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">LOG(RMSE): &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">rmse</span><span class="p">))</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">MAE</span><span class="p">:</span> <span class="mf">1.0787471470460999</span>
+<span class="n">RMSE</span><span class="p">:</span> <span class="mf">4.375695937555197</span>
+<span class="n">LOG</span><span class="p">(</span><span class="n">RMSE</span><span class="p">):</span> <span class="mf">2.0918164206151544</span>
+</pre></div>
+</div>
+<p>The error metrics show that HYMOD performs reasonably well, the MAE is
+around 1 <span class="math notranslate nohighlight">\(m^3/s\)</span>, the RMSE is on the order of 10% of the largest
+observed streamflow and the LOG(RMSE) is fairly low.</p>
+</section>
+<section id="id2">
+<h2>2- Global Sensitivity Analysis<a class="headerlink" href="#id2" title="Link to this heading">#</a></h2>
+<p id="sensitivity">2.1 Experimental Design and Setup</p>
+<p>Now we’ll examine how sensitive streamflow simulations generated by
+HYMOD are to the model’s input parameters. We’ll perform global
+sensitivity analysis (see Section 3.1 of the main text) using the SALib
+Python library.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">saltelli</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">sobol</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">delta</span>
+</pre></div>
+</div>
+<p>A first and critical step when conducting sensitivity analysis is
+determining the experimental design (see Design of Experiments, Section
+3.4 of the main text). Our experimental design involves defining the
+uncertainties that we’ll be examining, the output of interest, the
+ranges of each uncertainty that will be explored and the strategy for
+sampling the uncertainty space.</p>
+<p>For this experiment we’ll explore the five parameters highlighted in
+Figure 1. We’ll draw their ranges from existing literature on the model
+(note Jon H. paper). We’ll use a Sobol sampling an a quasi-random
+sampling with low sequences approach to sample the uncertainty space
+(Section 3.3.4).</p>
+<p>In this demonstration we’ll utilize Sobol Sensitivity Analysis, a
+variance based method (Section 3.4.5).</p>
+<p>To explore HYMOD’s behavoir, we’ll examine the sensitivity of four model
+ouputs to input parameters: 1) predicted flow, 2) Mean Absolute Error
+(compared with the calibaration data set), 3) Root Mean Square Error and
+4) Log Root Mean Square Error.</p>
+<p>This analysis will employ SALib, a Python implementation also utilized
+in the other SA tutorial (make this more formal).</p>
+<p>To start our analysis, we’ll create a dictionary that describes our
+model uncertainties and their ranges, this dictionary is named
+“problem_hymod” (SALib refers to these dictionaries as “problems”).</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">problem_hymod</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">5</span><span class="p">,</span>
+    <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">],</span>
+    <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>  <span class="c1"># Kq</span>
+               <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">],</span>  <span class="c1"># Ks</span>
+               <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>    <span class="c1"># Alp</span>
+               <span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">500</span><span class="p">],</span>  <span class="c1"># Huz</span>
+               <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">1.9</span><span class="p">]]</span>  <span class="c1"># B</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+<p>After defining our uncertainites and ranges, we’ll use SALib to sample
+the uncertainty space and run the model for each of the sample sets. We
+will load a sample that has already been created <code class="docutils literal notranslate"><span class="pre">param_values_hymod</span></code>
+for demonstration purposes. For HYMOD, literature recommends running at
+least N = 10,000 samples, to keep this demonstration easy to run
+however, we utilize only 256 sobol samples of uncertainties. To generate
+accurate approximations of second order sensitivity indicies SALib
+generates N*(2k+2) sets of samples, where N=256 and k=5 (number of
+uncertainties). For the math behind why this is needed, see (Saltelli,
+A., 2002. Making best use of model evaluations to compute sensitivity
+indices. Computer Physics Communications 145, 280–297.
+<a class="reference external" href="https://doi.org/10.1016/S0010-4655(02)00280-1">https://doi.org/10.1016/S0010-4655(02)00280-1</a>).</p>
+<p>The actual model simulation takes an extended period, so we also load
+the simulation data from a previous run. The following demonstrates how
+to conduct this analysis:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># generate 256 samples.</span>
+<span class="n">param_values_hymod</span> <span class="o">=</span> <span class="n">saltelli</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="mi">256</span><span class="p">)</span>
+
+<span class="c1"># dictionary to store outputs in</span>
+<span class="n">d_outputs</span> <span class="o">=</span> <span class="p">{}</span>
+
+<span class="c1"># run simulation for each parameter sample</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">param_values_hymod</span><span class="p">)):</span>
+
+    <span class="c1"># run model for each sensitivity analysis parameter sets</span>
+    <span class="n">hymod_output</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">hymod</span><span class="p">(</span><span class="n">Nq</span><span class="p">,</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>
+                                 <span class="n">leaf_data</span><span class="p">,</span>
+                                 <span class="n">ndays</span><span class="o">=</span><span class="mi">4015</span><span class="p">)</span>
+
+    <span class="c1"># store the simulated total flow discharge</span>
+    <span class="n">d_outputs</span><span class="p">[</span><span class="sa">f</span><span class="s2">&quot;Q</span><span class="si">{</span><span class="n">i</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">hymod_output</span><span class="p">[</span><span class="s2">&quot;Q&quot;</span><span class="p">]</span>
+
+
+<span class="n">Q_df_bw</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d_outputs</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously generated parameter values</span>
+<span class="n">param_values_hymod</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_params</span><span class="p">()</span>
+
+<span class="c1"># number of samples</span>
+<span class="n">n_samples</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">param_values_hymod</span><span class="p">)</span>
+
+<span class="c1"># load previously generated hymod simulated outputs</span>
+<span class="n">Q_df_bw</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_simulation</span><span class="p">()</span>
+
+<span class="c1"># column names of each sample simulation number</span>
+<span class="n">sample_column_names</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">Q_df_bw</span><span class="o">.</span><span class="n">columns</span> <span class="k">if</span> <span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s1">&#39;Q&#39;</span><span class="p">]</span>
+</pre></div>
+</div>
+<section id="running-hymod-model-warm-up">
+<h3>Running HYMOD - Model Warm-up<a class="headerlink" href="#running-hymod-model-warm-up" title="Link to this heading">#</a></h3>
+<p>A hydrological model such as HYMOD usually includes ordinary
+differential equations that are sensitive to their initial condition.
+They also have components in their underlying formulation that have long
+memory such that prior time steps can affect their current simulations.
+For example, soil moisture or groundwater can hold water for a long time
+and therefore they are often considered to exhibit a long memory. This
+can affect the partitioning of water to runoff and infiltration, while
+also controlling the generation of base flow. Therefore, it is important
+to have a reasonable initial value for them. To achieve this,
+hydrologists usually extend their simulation period and after the
+simulations, they remove that extended time period that has unreasonable
+groundwater or surface water values. This time period is called the
+warm-up time period.</p>
+<p>Here we extended our simulation for one year (from 10 years to 11 years)
+and we removed the first year of simulation, therefore our warm-up
+period is one year.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># exclude the first year of simulation from the simulations and reset the index</span>
+<span class="n">Q_df</span> <span class="o">=</span> <span class="n">Q_df_bw</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">365</span><span class="p">:</span><span class="mi">4015</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">reset_index</span><span class="p">(</span><span class="n">drop</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="c1"># exclude the first year of the input data and reset the index</span>
+<span class="n">leaf_data</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">365</span><span class="p">:</span><span class="mi">4015</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">reset_index</span><span class="p">(</span><span class="n">drop</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now that HYMOD has been warmed up, we’ll examine how HYMOD’s streamflow
+outputs vary under different sample sets, and compare them with the
+observed streamflow.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># add date columns to our simulation data frame; for this data our start date is 1/1/2000</span>
+<span class="n">date_ts</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">date_range</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="s1">&#39;1/1/2000&#39;</span><span class="p">,</span> <span class="n">periods</span><span class="o">=</span><span class="mi">3650</span><span class="p">,</span> <span class="n">freq</span><span class="o">=</span><span class="s1">&#39;D&#39;</span><span class="p">)</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;date&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">year</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;month&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">month</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;day&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">day</span>
+
+<span class="c1"># aggregate the simulated observed streamflow to monthly mean</span>
+<span class="n">df_sim_mth_mean</span> <span class="o">=</span> <span class="n">Q_df</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">,</span> <span class="s1">&#39;month&#39;</span><span class="p">])[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+
+<span class="c1"># do the same for the observed data</span>
+<span class="n">date_ts</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">date_range</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="s1">&#39;1/1/2000&#39;</span><span class="p">,</span> <span class="n">periods</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">),</span> <span class="n">freq</span><span class="o">=</span><span class="s1">&#39;D&#39;</span><span class="p">)</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;date&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">year</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;month&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">month</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;day&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">day</span>
+
+<span class="c1"># aggregate the daily observed streamflow to monthly mean</span>
+<span class="n">df_obs_mth_mean</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">,</span> <span class="s1">&#39;month&#39;</span><span class="p">])</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-observed-vs-sensitivity-streamflow">plot_observed_vs_sensitivity_streamflow</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ax</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_observed_vs_sensitivity_streamflow</span><span class="p">(</span><span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_mth_mean</span><span class="p">,</span>
+                                                     <span class="n">df_sim</span><span class="o">=</span><span class="n">df_sim_mth_mean</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod3.png" src="_images/hymod3.png" />
+<p id="sametrics">2.2 Sensitivity of streamflows to model parameters</p>
+<p>Now we’ll examine how each of HYMOD’s parameters impact the variance of
+simulated streamflows. Using SALib we’ll calculate the first order and
+total order sensitivity indicies of each model parameter. The first
+order sensitivity index measure’s the individual impact that a given
+parameter has on the variance of the simulated streamflows. The total
+order index measures the impact of a given parameter along with all
+interactions that other parameters have with the given parameter on
+simulated streamflows.</p>
+<p>We’ll start with an matrix, Y, which contains our simulated streamflows
+for every uncertainty sample. We’ll then use the sobol.analyze function
+from SALib to calculate the sensitivity indicies (Si). The arguments for
+this function are the problem dictionary defined in part 2.2 of this
+tutorial, and the matrix of simulated streamflows, Y.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># overall aggregated indices</span>
+<span class="n">Y</span> <span class="o">=</span> <span class="n">Q_df</span><span class="p">[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+<span class="c1"># Perform analysis</span>
+<span class="n">Si</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now we can examine our results, we’ll print the first order and total
+order Si’s for each parameter, then visualize the results with bar plots</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;First order indices = &#39;</span><span class="p">,</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">])</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Total order indicies = &#39;</span><span class="p">,</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">])</span>
+
+<span class="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+<span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s1">&#39;&#39;</span><span class="p">,</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;First order Si&#39;</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s1">&#39;&#39;</span><span class="p">,</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Total order Si&#39;</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">First</span> <span class="n">order</span> <span class="n">indices</span> <span class="o">=</span>  <span class="p">[</span><span class="mf">9.55550001e-05</span> <span class="mf">7.49249463e-04</span> <span class="mf">5.62386413e-04</span> <span class="mf">7.03327551e-01</span>
+ <span class="mf">2.53701895e-01</span><span class="p">]</span>
+<span class="n">Total</span> <span class="n">order</span> <span class="n">indicies</span> <span class="o">=</span>  <span class="p">[</span><span class="mf">1.76174200e-06</span> <span class="mf">1.63288175e-03</span> <span class="mf">3.41378460e-04</span> <span class="mf">6.88983864e-01</span>
+ <span class="mf">2.53922146e-01</span><span class="p">]</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">ipykernel_launcher</span><span class="o">.</span><span class="n">py</span><span class="p">:</span><span class="mi">9</span><span class="p">:</span> <span class="ne">UserWarning</span><span class="p">:</span> <span class="n">FixedFormatter</span> <span class="n">should</span> <span class="n">only</span> <span class="n">be</span> <span class="n">used</span> <span class="n">together</span> <span class="k">with</span> <span class="n">FixedLocator</span>
+  <span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">&#39;__main__&#39;</span><span class="p">:</span>
+<span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">ipykernel_launcher</span><span class="o">.</span><span class="n">py</span><span class="p">:</span><span class="mi">15</span><span class="p">:</span> <span class="ne">UserWarning</span><span class="p">:</span> <span class="n">FixedFormatter</span> <span class="n">should</span> <span class="n">only</span> <span class="n">be</span> <span class="n">used</span> <span class="n">together</span> <span class="k">with</span> <span class="n">FixedLocator</span>
+  <span class="kn">from</span> <span class="nn">ipykernel</span> <span class="kn">import</span> <span class="n">kernelapp</span> <span class="k">as</span> <span class="n">app</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod4.png" src="_images/hymod4.png" />
+<p>Our findings indicate that in this instance, the streamflow estimate
+from HYMOD is highly sensitive to soil moisture parameters Huz and B and
+hardly affected by the routing parameters. Notably, there is very little
+interactions between parameters causing the total order indicies to be
+nearly identical to the first order indicies.</p>
+<p id="diffperformance">2.3 How do different performance metrics affect the results of our
+sensitivity analysis?</p>
+<p>Streamflow has many different properties. In this section, we discuss
+how the selection of metrics can lead to fundamentally different
+sensitivity analysis results. For example, one can only focus on
+aggregated streamflow metrics such as mean (what has been presented so
+far), or only on extreme events such as drought or floods.</p>
+<p>Here we compare three different metrics: 1- Mean error (ME) 2- Root Mean
+Square Error (RMSE) 3- Log-Root Mean Square Error (Log(RMSE))</p>
+<p>Each of these metrics focuses on a specific attribute of streamflow. For
+example, RMSE highlights the impacts of extreme flood events, while
+LOG(RMSE) focuses on model performance during low-flow events.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># calculate error metrics</span>
+<span class="n">mae</span> <span class="o">=</span> <span class="n">Q_df</span><span class="p">[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">leaf_data</span><span class="p">[</span><span class="s2">&quot;Strmflw&quot;</span><span class="p">]),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="n">mse</span> <span class="o">=</span> <span class="n">Q_df</span><span class="p">[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">metrics</span><span class="o">.</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s2">&quot;Strmflw&quot;</span><span class="p">]),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="n">rmse</span> <span class="o">=</span> <span class="n">mse</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="c1"># add error metrics to a dictionary</span>
+<span class="n">d_metrics</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;MAE&#39;</span><span class="p">:</span> <span class="n">mae</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">.</span><span class="n">values</span><span class="p">,</span>
+             <span class="s1">&#39;RMSE&#39;</span><span class="p">:</span> <span class="n">rmse</span><span class="o">.</span><span class="n">values</span><span class="p">,</span>
+             <span class="s1">&#39;LOG[RMSE]&#39;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">rmse</span><span class="o">.</span><span class="n">values</span><span class="p">)}</span>
+
+<span class="c1"># convert to a dataframe</span>
+<span class="n">df_metrics_SA</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d_metrics</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>We can use the following to calculate the SA indices for each metric and
+visualize it.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">df_metric_s1_result</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">df_metric_sT_result</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+
+<span class="c1"># conduct sensitivity analysis for each metric</span>
+<span class="k">for</span> <span class="n">index</span><span class="p">,</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">d_metrics</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+
+    <span class="c1"># get the data as a numpy array for the target metric</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">d_metrics</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+    <span class="c1"># use the metric to conduct SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add the sensitivity indices to the output data frame</span>
+    <span class="n">df_metric_s1_result</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">index</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">]</span>
+    <span class="n">df_metric_sT_result</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">index</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">]</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create seaborn heatmap with required labels</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+<span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="c1"># labels for y-axis</span>
+<span class="n">y_axis_labels</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Mean Absoulte Error&#39;</span><span class="p">,</span> <span class="s1">&#39;RSME&#39;</span><span class="p">,</span> <span class="s1">&#39;Log(RMSE)&#39;</span><span class="p">]</span>
+
+<span class="c1"># plot heatmap</span>
+<span class="n">ax1</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">df_metric_s1_result</span><span class="p">,</span> <span class="n">yticklabels</span><span class="o">=</span><span class="n">y_axis_labels</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>  <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;inferno_r&#39;</span><span class="p">,</span> <span class="n">cbar_kws</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;label&#39;</span><span class="p">:</span> <span class="s1">&#39;Si&#39;</span><span class="p">},</span> <span class="n">cbar</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">figure</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">label</span><span class="o">.</span><span class="n">set_size</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;First Order Sensitivity&#39;</span><span class="p">)</span>
+
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">df_metric_sT_result</span><span class="p">,</span> <span class="n">yticklabels</span><span class="o">=</span><span class="n">y_axis_labels</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>  <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;inferno_r&#39;</span><span class="p">,</span> <span class="n">cbar_kws</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;label&#39;</span><span class="p">:</span> <span class="s1">&#39;Si&#39;</span><span class="p">})</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">figure</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">label</span><span class="o">.</span><span class="n">set_size</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Total Order Sensitivity&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Text</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="s1">&#39;Total Order Sensitivity&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod5.png" src="_images/hymod5.png" />
+<p>The first order sensitivity indicies indicate that HYMOD’s sensitivity
+to its parameters is different depending on how its output is measured.
+Unsurprisingly, the mean absolute error is highly sensitive to the soil
+moisture accounting parameters Huz and B, just like the overall
+streamflow predictions above. However, when we examine RMSE and
+log(RMSE), the routing parameters Alp become sensitive, and the
+sensitivity to parameter B is reduced. As described above, RMSE and
+LOG(RMSE) respond to model performance in high-flow and low flow periods
+respectively. Our results indicate that for these flow regimes Alp, the
+parameter that governs the split between quick and slow flow is an
+important factor. While still the parameter with the highest most effect
+on all three measures, Huz is much less influential for RMSE and
+LOG(RMSE) than it is for MAE.</p>
+<p>The total order sensitivity indicies review a different, more complex
+story. While the MAE sensitivity is relatively governed by first order
+effects (like the streamflow predictions above), the RMSE and LOG(RMSE)
+error metrics show significant interactions. Alp has the highest total
+order sensitivity for RMSE and is eqal to Huz for Log(RMSE). Kq, which
+has a relatively low first order sensitivity index, shows strong
+contribution to variance when interactions are taken into account.</p>
+<p>Radial convergence plots are a helpful way to visualize the interactions
+between parameters. These plots array the model parameters in a circle
+and plot the first order, total order and second order Sobol sensitivity
+indices for each parameter. The first order sensitivity is shown as the
+size of a closed circle, the total order as the size of a larger open
+circle and the second order as the thickness of a line connecting two
+parameters. Below is an example of a radial convergence plot for the
+LOG(RMSE) measure. The plot indicates strong interactions between the
+Huz and Alp parameters, as well as Alp and Kq. There is also an
+interaction between Alp and Ks.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">itertools</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+<span class="kn">import</span> <span class="nn">math</span>
+<span class="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s1">&#39;whitegrid&#39;</span><span class="p">,</span> <span class="p">{</span><span class="s1">&#39;axes_linewidth&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="s1">&#39;axes.edgecolor&#39;</span><span class="p">:</span> <span class="s1">&#39;white&#39;</span><span class="p">})</span>
+
+<span class="k">def</span> <span class="nf">is_significant</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">confidence_interval</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="s2">&quot;conf&quot;</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">threshold</span> <span class="o">==</span> <span class="s2">&quot;conf&quot;</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">value</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">(</span><span class="n">confidence_interval</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">value</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">threshold</span><span class="p">))</span> <span class="o">&gt;</span> <span class="mi">0</span>
+
+<span class="k">def</span> <span class="nf">grouped_radial</span><span class="p">(</span><span class="n">SAresults</span><span class="p">,</span> <span class="n">parameters</span><span class="p">,</span> <span class="n">radSc</span><span class="o">=</span><span class="mf">2.0</span><span class="p">,</span> <span class="n">scaling</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">widthSc</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">STthick</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">varNameMult</span><span class="o">=</span><span class="mf">1.3</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">groups</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">gpNameMult</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="s2">&quot;conf&quot;</span><span class="p">):</span>
+    <span class="c1"># Derived from https://github.com/calvinwhealton/SensitivityAnalysisPlots</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+    <span class="n">color_map</span> <span class="o">=</span> <span class="p">{}</span>
+
+    <span class="c1"># initialize parameters and colors</span>
+    <span class="k">if</span> <span class="n">groups</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+
+        <span class="k">if</span> <span class="n">colors</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">colors</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;k&quot;</span><span class="p">]</span>
+
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">parameter</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+            <span class="n">color_map</span><span class="p">[</span><span class="n">parameter</span><span class="p">]</span> <span class="o">=</span> <span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)]</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">if</span> <span class="n">colors</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">colors</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">color_palette</span><span class="p">(</span><span class="s2">&quot;deep&quot;</span><span class="p">,</span> <span class="nb">max</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">groups</span><span class="p">)))</span>
+
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">groups</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+            <span class="c1">#parameters.extend(groups[key])</span>
+
+            <span class="k">for</span> <span class="n">parameter</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">[</span><span class="n">key</span><span class="p">]:</span>
+                <span class="n">color_map</span><span class="p">[</span><span class="n">parameter</span><span class="p">]</span> <span class="o">=</span> <span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)]</span>
+
+    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">parameters</span><span class="p">)</span>
+    <span class="n">angles</span> <span class="o">=</span> <span class="n">radSc</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">radSc</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angles</span><span class="p">)</span>
+    <span class="n">y</span> <span class="o">=</span> <span class="n">radSc</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angles</span><span class="p">)</span>
+
+    <span class="c1"># plot second-order indices</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">combinations</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="mi">2</span><span class="p">):</span>
+        <span class="c1">#key1 = parameters[i]</span>
+        <span class="c1">#key2 = parameters[j]</span>
+
+        <span class="k">if</span> <span class="n">is_significant</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S2&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S2_conf&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">threshold</span><span class="p">):</span>
+            <span class="n">angle</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">atan</span><span class="p">((</span><span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">-</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
+
+            <span class="k">if</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">-</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
+                <span class="n">angle</span> <span class="o">+=</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span>
+
+            <span class="n">line_hw</span> <span class="o">=</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S2&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">])</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
+
+            <span class="n">coords</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">((</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Polygon</span><span class="p">(</span><span class="n">coords</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;0.75&quot;</span><span class="p">))</span>
+
+    <span class="c1"># plot total order indices</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">is_significant</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST_conf&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">threshold</span><span class="p">):</span>
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Circle</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;w&#39;</span><span class="p">))</span>
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Circle</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="n">STthick</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;0.4&#39;</span><span class="p">,</span> <span class="n">fill</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
+
+    <span class="c1"># plot first-order indices</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">is_significant</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S1&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S1_conf&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">threshold</span><span class="p">):</span>
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Circle</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S1&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;0.4&#39;</span><span class="p">))</span>
+
+    <span class="c1"># add labels</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+        <span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">varNameMult</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">varNameMult</span><span class="o">*</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">key</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span>
+                <span class="n">rotation</span><span class="o">=</span><span class="n">angles</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="mi">360</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">-</span> <span class="mi">90</span><span class="p">,</span>
+                <span class="n">color</span><span class="o">=</span><span class="n">color_map</span><span class="p">[</span><span class="n">key</span><span class="p">])</span>
+
+    <span class="k">if</span> <span class="n">groups</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">group</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">groups</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+            <span class="nb">print</span><span class="p">(</span><span class="n">group</span><span class="p">)</span>
+            <span class="n">group_angle</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">angles</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">if</span> <span class="n">parameters</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">[</span><span class="n">group</span><span class="p">]])</span>
+
+            <span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">gpNameMult</span><span class="o">*</span><span class="n">radSc</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">group_angle</span><span class="p">),</span> <span class="n">gpNameMult</span><span class="o">*</span><span class="n">radSc</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">group_angle</span><span class="p">),</span> <span class="n">group</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span>
+                <span class="n">rotation</span><span class="o">=</span><span class="n">group_angle</span><span class="o">*</span><span class="mi">360</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">-</span> <span class="mi">90</span><span class="p">,</span>
+                <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)])</span>
+
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">([])</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;equal&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">])</span>
+    <span class="c1">#plt.show()</span>
+
+
+    <span class="k">return</span> <span class="n">fig</span>
+
+<span class="c1"># define groups for parameter uncertainties</span>
+<span class="n">groups</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;Soil Moisture&quot;</span> <span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Huz&quot;</span><span class="p">,</span> <span class="s2">&quot;B&quot;</span><span class="p">],</span>
+        <span class="s2">&quot;Routing&quot;</span> <span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Alp&quot;</span><span class="p">,</span> <span class="s2">&quot;Kq&quot;</span><span class="p">,</span> <span class="s2">&quot;Ks&quot;</span><span class="p">]}</span>
+
+
+<span class="n">fig</span> <span class="o">=</span> <span class="n">grouped_radial</span><span class="p">(</span><span class="n">Si</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">],</span> <span class="n">groups</span><span class="o">=</span><span class="n">groups</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="mf">0.025</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Soil</span> <span class="n">Moisture</span>
+<span class="n">Routing</span>
+</pre></div>
+</div>
+<img alt="_images/hymod6.png" src="_images/hymod6.png" />
+<p>2.4 Time-Varying Sensitivity Analysis</p>
+<p id="time">In section 2.5 we saw how performing sensitivity analysis on different
+measurements of model output can yeild in different results on the
+importance of each uncertain input. In this section we’ll examine how
+performing this analysis over time can yeild additional insight into the
+performance of HYMOD. We’ll first examine how model sensitivities vary
+by month, then examine how they change across each year of the
+simulation.</p>
+<p>For this demonstration, we’ll focus only on the monthly streamflow
+predictions generated by HYMOD.</p>
+<section id="sensitivity-analysis-indices-for-each-month">
+<h4>2.4.1 Sensitivity analysis indices for each month<a class="headerlink" href="#sensitivity-analysis-indices-for-each-month" title="Link to this heading">#</a></h4>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># aggregate simulated streamflow data to monthly time series</span>
+<span class="n">df_sim_by_mth_mean</span> <span class="o">=</span> <span class="n">Q_df</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s1">&#39;month&#39;</span><span class="p">)[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+
+<span class="c1"># aggregate observed streamflow data to monthly time series</span>
+<span class="n">df_obs_by_mth_mean</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s1">&#39;month&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+<p>We can use the following to calculate the SA indices for each month and
+visualize it. Results are pre-loaded for efficiency.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># set up dataframes to store outputs</span>
+<span class="n">df_mth_s1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">12</span><span class="p">,</span><span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">df_mth_delta</span> <span class="o">=</span> <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="c1"># iterate through each month</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">12</span><span class="p">):</span>
+
+    <span class="c1"># generate the simulation data</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">df_sim_by_mth_mean</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+    <span class="c1"># run SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">delta</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">param_values_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add to output dataframes</span>
+    <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="n">df_mth_delta</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;delta&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+
+<span class="c1"># convert to arrays</span>
+<span class="n">arr_mth_s1</span> <span class="o">=</span> <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">values</span>
+<span class="n">arr_mth_delta</span> <span class="o">=</span> <span class="n">df_mth_delta</span><span class="o">.</span><span class="n">values</span>
+</pre></div>
+</div>
+</section>
+</section>
+<section id="first-order-indices">
+<h3>First-order Indices<a class="headerlink" href="#first-order-indices" title="Link to this heading">#</a></h3>
+<p>The following can be used to visualize the time-varying first-order
+indices. The first order represents the direct impacts of a specific
+parameter on model outputs.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-monthly-heatmap">plot_monthly_heatmap</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously ran data</span>
+<span class="n">arr_mth_delta</span><span class="p">,</span> <span class="n">arr_mth_s1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_monthly_simulations</span><span class="p">()</span>
+
+<span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_monthly_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_mth_s1</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                       <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_mth_mean</span><span class="p">,</span>
+                                       <span class="n">title</span><span class="o">=</span><span class="s1">&#39;First Order - Mean Monthly SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod7.png" src="_images/hymod7.png" />
+<p>This figure demonstrates the first order sensitivity indices when the
+streamflow data are aggregated by month. The purple line represents the
+observed monthly discharge. The figure indicates that the first order
+indices are highest for B and Huz across all months and lowest for Alp,
+Ks, and Kq. Note that in the months with the highest flow, Ks becomes an
+influential parameter.</p>
+</section>
+<section id="total-order-indices">
+<h3>Total-order indices<a class="headerlink" href="#total-order-indices" title="Link to this heading">#</a></h3>
+<p>We can also focus on the total order sensitivity index that includes
+first-order SA indices and interactions between parameters</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_monthly_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_mth_delta</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                       <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_mth_mean</span><span class="p">,</span>
+                                       <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Total Order - Mean monthly SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod8.png" src="_images/hymod8.png" />
+<p>Notably, the total order sensitivity results are different than the
+first order sensitivity results, which indicates that interactions
+between the parameters (particularly in regards to routing parameters
+<span class="math notranslate nohighlight">\(Kq\)</span>, <span class="math notranslate nohighlight">\(Ks\)</span>, and <span class="math notranslate nohighlight">\(Alp\)</span>) contribute to changes in HYMOD
+output.</p>
+<section id="annual-sensitivity-analysis-indices">
+<h4>2.4.2 Annual sensitivity analysis indices<a class="headerlink" href="#annual-sensitivity-analysis-indices" title="Link to this heading">#</a></h4>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># group by year and get mean</span>
+<span class="n">df_sim_by_yr_mean</span> <span class="o">=</span> <span class="n">Q_df</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">])[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+
+<span class="c1"># group input data and get mean</span>
+<span class="n">df_obs_by_yr_mean</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">])</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+<p>We can also calculate the sensitivity analysis indices for each
+individual year. This will allow us to understand if model control
+changes during different years. The following code first aggregates the
+outputs to annual time steps, and then calculates the SA indices.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># set up dataframes to store outputs</span>
+<span class="n">df_yr_s1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">df_yr_delta</span> <span class="o">=</span> <span class="n">df_yr_s1</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="c1"># iterate through each year</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">):</span>
+
+    <span class="c1"># generate the simulation data</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">df_sim_by_yr_mean</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+    <span class="c1"># run SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">delta</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">param_values_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add to output dataframes</span>
+    <span class="n">df_yr_s1</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="n">df_yr_delta</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;delta&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+
+<span class="c1"># convert to arrays</span>
+<span class="n">arr_yr_s1</span> <span class="o">=</span> <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">values</span>
+<span class="n">arr_yr_delta</span> <span class="o">=</span> <span class="n">df_mth_delta</span><span class="o">.</span><span class="n">values</span>
+</pre></div>
+</div>
+</section>
+</section>
+<section id="id3">
+<h3>First-order indices<a class="headerlink" href="#id3" title="Link to this heading">#</a></h3>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-annual-heatmap">plot_annual_heatmap</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously ran data</span>
+<span class="n">arr_yr_delta</span><span class="p">,</span> <span class="n">arr_yr_s1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_annual_simulations</span><span class="p">()</span>
+
+<span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_annual_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_yr_s1</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_yr_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;First Order - Mean Annual SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod9.png" src="_images/hymod9.png" />
+<p>The first order sensitivities at the annual scale are not unlike the
+first order monthly sensitivities. Once again, sensitivities vary across
+year and Huz and B are the most consequential parameters.</p>
+</section>
+<section id="id4">
+<h3>Total-order indices<a class="headerlink" href="#id4" title="Link to this heading">#</a></h3>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_annual_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_yr_delta</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_yr_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Total Order - Mean Annual SA and Observed flow&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod10.png" src="_images/hymod10.png" />
+<p>Our results indicate that sensitivity analysis indices vary in different
+years and now that interactions are included, the Kq, Ks, and Alp
+variables impact the sensitivity of the streamflow output.</p>
+<section id="monthly-time-varying-sensitivity-analysis">
+<h4>2.4.3 Monthly time-varying sensitivity analysis<a class="headerlink" href="#monthly-time-varying-sensitivity-analysis" title="Link to this heading">#</a></h4>
+<p>Although time-varying sensitivity analysis (TVSA) at average monthly and
+average annual temporal resolutions is informative, TVSA is susceptible
+to the aggregation issue that we discussed earlier in section 3-2. To
+avoid that we can further discretize our time domain to zoom into
+individual months. This will provide us with even more information about
+model behavior and the sensitivity of different parameters in different
+states of the system. The block of code demonstrates how to implement
+the monthly TVSA.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># set up dataframes to store outputs</span>
+<span class="n">df_vary_s1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">df_obs_mth_mean</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">5</span><span class="p">)),</span>
+                          <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+
+<span class="n">df_vary_delta</span> <span class="o">=</span> <span class="n">df_vary_s1</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="c1"># iterate through each month</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">df_obs_mth_mean</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+
+    <span class="c1"># generate the simulation data</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">df_sim_mth_mean</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+    <span class="c1"># run SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">delta</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">param_values_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add to output dataframes</span>
+    <span class="n">df_vary_s1</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="n">df_vary_delta</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;delta&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+
+<span class="c1"># convert to arrays</span>
+<span class="n">arr_vary_s1</span> <span class="o">=</span> <span class="n">df_vary_s1</span><span class="o">.</span><span class="n">values</span>
+<span class="n">arr_vary_delta</span> <span class="o">=</span> <span class="n">df_vary_delta</span><span class="o">.</span><span class="n">values</span>
+</pre></div>
+</div>
+</section>
+</section>
+<section id="id5">
+<h3>First-order indices<a class="headerlink" href="#id5" title="Link to this heading">#</a></h3>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-varying-heatmap">plot_varying_heatmap</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load in previously ran data</span>
+<span class="n">arr_vary_delta</span><span class="p">,</span> <span class="n">arr_vary_s1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_varying_simulations</span><span class="p">()</span>
+
+<span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_varying_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_vary_s1</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_mth_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;First Order - Time-Varying SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod11.png" src="_images/hymod11.png" />
+<p>Compared to the TVSA when streamflow was aggregated, this figure
+suggests that Kq is indeed a relevant parameter for influencing
+streamflow output when individual months are considered.</p>
+</section>
+<section id="total-order-time-varying-sensitivity-analysis">
+<h3>Total order - time varying sensitivity analysis<a class="headerlink" href="#total-order-time-varying-sensitivity-analysis" title="Link to this heading">#</a></h3>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_varying_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_vary_delta</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_mth_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Total Order - Time-Varying SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod12.png" src="_images/hymod12.png" />
+<p>As above, the total order sensitivities further indicate the importance
+of Kq that is not apparent if aggregation is utilized.</p>
+</section>
+</section>
+<section id="tips-to-apply-this-methodology-to-your-own-problem">
+<h2>Tips to Apply this methodology to your own problem<a class="headerlink" href="#tips-to-apply-this-methodology-to-your-own-problem" title="Link to this heading">#</a></h2>
+<p>In this tutorial, we demonstrated how to use global sensitivtiy analysis
+to explore a complex, non-linear model. We showed how measuring
+sensitivity across multiple measures of model performance and temporal
+aggregations yeilding differing results about model
+sensitivity/behavoir. While these results may seem contraditory, they
+provide useful insight into the behavoir of HYMOD. Would we expect the
+same parameters to control high flow and low flow regimes within the
+model? Maybe, depending on the system, but also, maybe not. This
+analysis can provide insight into how the model responds to its input
+parameters, allowing us to compare the results to our expectaions. This
+may allow us to find problems with our intial assumptions, or call
+attention to model features that can be improved or expanded upon.
+Depending on the model and context, it may also yield insight into the
+workings of the underlying system.</p>
+<p>To run this tutorial on your own model you will need to:</p>
+<ol class="arabic simple">
+<li><p>Design your experiment by choosing sampling bounds for your
+parameters and setting up the problem dictionary as in step 2-2</p></li>
+<li><p>Choose the parameters of interest</p></li>
+<li><p>Generate samples using the saltelli.sample function. This step is
+problem-dependent and note that the Sobol method can be
+computationally intensive depending on the model being analyzed. More
+complex models will be slower to run and will also require more
+samples to calculate accurate estimates of Sobol indices. Once you
+complete this process, pay attention to the confidence bounds on your
+sensitivity indices to see whether you need to run more samples.</p></li>
+<li><p>Run the parameter sets through your model and record each of the
+desired model outputs.</p></li>
+<li><p>Calculate the Sobol indices for each performance criteria. Now, the Y
+will be a numpy array with your external model output and you will
+need to include the parameter samples as an additional argument.</p></li>
+<li><p>Follow the procedure in step 2.6 to disaggregate performance across
+time</p></li>
+</ol>
+</section>
+</section>
+
+
+                </article>
+              
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+    <i class="fa-solid fa-list"></i> Contents
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+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-of-the-hymod-model">Tutorial: Sensitivity Analysis of the HYMOD Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction-to-hymod">1. Introduction to HYMOD</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#global-sensitivity-analysis">2. Global Sensitivity Analysis</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">1 - Introduction to HYMOD</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">2- Global Sensitivity Analysis</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#running-hymod-model-warm-up">Running HYMOD - Model Warm-up</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#sensitivity-analysis-indices-for-each-month">2.4.1 Sensitivity analysis indices for each month</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#first-order-indices">First-order Indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#total-order-indices">Total-order indices</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#annual-sensitivity-analysis-indices">2.4.2 Annual sensitivity analysis indices</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">First-order indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id4">Total-order indices</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#monthly-time-varying-sensitivity-analysis">2.4.3 Monthly time-varying sensitivity analysis</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id5">First-order indices</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#total-order-time-varying-sensitivity-analysis">Total order - time varying sensitivity analysis</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-this-methodology-to-your-own-problem">Tips to Apply this methodology to your own problem</a></li>
+</ul>
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+    <h1>Time-evolving scenario discovery for infrastructure pathways</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Time-evolving scenario discovery for infrastructure pathways</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#background">Background</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluating-robustness-over-time">Evaluating Robustness over time</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exploring-performance-evolution">Exploring performance evolution</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#how-do-deep-uncertainties-generate-vulnerability">How do deep uncertainties generate vulnerability</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#which-uncertainties-have-the-most-influence-on-time-evolving-performance">Which uncertainties have the most influence on time-evolving performance?</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#open-exploration">Open exploration</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-this-methodology-to-your-own-problem">Tips to apply this methodology to your own problem</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#references">References</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="time-evolving-scenario-discovery-for-infrastructure-pathways">
+<h1>Time-evolving scenario discovery for infrastructure pathways<a class="headerlink" href="#time-evolving-scenario-discovery-for-infrastructure-pathways" title="Link to this heading">#</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/Bedford_Greene_SD_tutorial.ipynb">Scenario Discovery Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="id1">
+<h2>Time-evolving scenario discovery for infrastructure pathways<a class="headerlink" href="#id1" title="Link to this heading">#</a></h2>
+<p>The purpose of this tutorial is to explore time-evolving vulnerability for systems that dynamically adapt to changing conditions. Using an example from water supply planning, we’ll first examine how performance of a dynamic infrastructure pathway policy changes over time, then use factor mapping (main text Chapter 4.3) to understand which combinations of uncertaities generate vulnerability for two water utilities. Next, we’ll perform factor prioritization (main text Chapter 4.3) to determine which uncertainties have the most influence on water supply performance. Finally, we’ll provide an open platform to explore vulnerability across multiple measures of performance and different combinations of uncertainties.</p>
+<section id="background">
+<h3>Background<a class="headerlink" href="#background" title="Link to this heading">#</a></h3>
+<p>The Bedford-Greene metropolitan area (Figure 1) is a stylized water resources test case where two urban water utilities seek to develop an infrastructure and investment and management strategy to confront growing demands and changing climate. The utilities have agreed to jointly construct a new water treatment plant on Lake Classon, a large regional resource. Both utilities have also identified a set of individual infrastructure options to construct if necessary.</p>
+<figure class="align-default" id="id2">
+<img alt="Figure 1" src="_images/Map_small.png" />
+<figcaption>
+<p><span class="caption-text">Figure 1</span><a class="headerlink" href="#id2" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The utilities are formulating a cooperative and adaptive regional management strategy that uses a risk-of-failure (ROF) metric to trigger both short term drought mitigation actions (water use restrictions and treated transfers between utilities) and long-term infrastructure investment decisions (shown in Figure 2a). Both utilities have specified a set of risk triggers and developed a construction order for available infrastructure options.</p>
+<p>The utilities have run a Monte Carlo simulation to evaluate how these policies respond to a wide array of future States Of the World (SOWs). Each SOW represents a different combinations of thirteen uncertain system inputs including demand growth rates, changes to streamflows and financial variables. In this context, a fully specified SOW is composed of one sample of uncertain model inputs (e.g. one projection of demand growth rate coupled with one future streamflow scenario and one projection of future financial conditions). The water utilities used Latin Hypercube sampling (Chapter 3.3 of the main text) to develop an ensemble of 1,000 plausible future SOWs. The Monte Carlo simulation evaluates each candidate water supply infrastructure investment and management policy across all 1,000 SOWs, as shown in Figure 2b. For more details on the Monte Carlo sampling for this type  of analysis, see Trindade et al., (2019).</p>
+<p>The ROF-based policies respond to each SOW by generating a unique infrastructure pathway - a sequence of infrastructure investment decisions over time. Infrastructure pathways over a set of 1,000 future SOWs are shown in Figure 2c. Infrastructure options are arrayed along the vertical axis and the sequence of infrastructure investments triggered by the policy is plotted as pathways over time.  Since the adaptive rule system generates a unique infrastructure sequence for each scenario, Figure 2c summarizes the ensemble of pathways by clustering SOWs according to infrastructure intensity. Dark green lines represent SOWs where the utilities heavily invest in new infrastructure, light green lines represent SOWs with low infrastructure investment and medium shaded lines represent moderate investment. The shading behind each pathway represents the frequency that each infrastructure option was triggered over time across sampled scenarios</p>
+<figure class="align-default" id="id3">
+<img alt="Figure 2" src="_images/Policy_MonteCarlo_Pathways_small.png" />
+<figcaption>
+<p><span class="caption-text">Figure 2</span><a class="headerlink" href="#id3" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="evaluating-robustness-over-time">
+<h3>Evaluating Robustness over time<a class="headerlink" href="#evaluating-robustness-over-time" title="Link to this heading">#</a></h3>
+<p>The two water utilities are interested in mainting both supply
+reliability and financial stability across the broadest set of plausible
+future SOWs. To measure the performance of the infrastructure pathway
+policy, they’ve defined five critical performance criteria:</p>
+<ul class="simple">
+<li><p>Reliability &gt; 99%</p></li>
+<li><p>Restriction Frequency &lt; 20%</p></li>
+<li><p>Peak Financial Cost &lt; 80% of annual revenue (a measure of debt
+service spending)</p></li>
+<li><p>Worst-case drought management cost &lt; 10% of annual revenue (a measure
+of unexpected drought costs)</p></li>
+<li><p>Unit Cost of Expansion &lt; 5 dollars/kgal</p></li>
+</ul>
+<p>To assess the robustness of the infrastructure pathway policy, the two
+utilities apply a satisficing metric, which measures the percentage of
+sampled SOWs where the pathway policy meets the peformance criteria:</p>
+<p><span class="math notranslate nohighlight">\(R =\frac{1}{N}\sum_{j=1}^{N}\Lambda_{\theta,j}\)</span></p>
+<p>Where, <span class="math notranslate nohighlight">\(\Lambda\_{\theta,j}=\)</span></p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases}
+1, \quad \textrm{if}\ F(\theta)_{j}\leq \Phi_j \\
+0, \quad \textrm{otherwise}
+\end{cases}\end{split}\]</div>
+<p>And <span class="math notranslate nohighlight">\(\Phi\)</span> is a vector of performance criteria for utility
+<span class="math notranslate nohighlight">\(j\)</span>, <span class="math notranslate nohighlight">\(\theta\)</span> is the portfolio and <span class="math notranslate nohighlight">\(N\)</span> is the total
+number of sampled SOWs.</p>
+<p>Below, we’ll visualize how robustness for the two utilities evolves over the 45-year planning horizon. We’ll assess robustness across three time periods, near-term (first 10 years), mid-term (22 years) and long term (45 years).</p>
+<p>We start by loading robustness values for the both utilities. These values are calculated by applying the robustness metric above across 2,000 simulated SOWs. To make this exercise computationally tractable, we’ve precomputed these values, which can be found in the files “short_term_robustness.csv”, “mid_term_robustness.csv” and “long_term_robustness.csv”. These values are calculated using the function “check_rdm_meet_criteria” within the helper functions.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
+<span class="kn">from</span> <span class="nn">functions.eBook_SD_helpers</span> <span class="kn">import</span> <span class="n">check_rdm_meet_criteria</span><span class="p">,</span> <span class="n">create_sd_input</span><span class="p">,</span> <span class="n">plot_selected_tree_maps</span><span class="p">,</span> <span class="n">get_factor_importances</span><span class="p">,</span> <span class="n">open_exploration</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+
+<span class="c1"># load Deeply uncertain factors</span>
+<span class="n">rdm_factors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/DU_Factors.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+
+<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
+<span class="n">short_term_robustness</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/short_term_robustness.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">mid_term_robustness</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/mid_term_robustness.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">long_term_robustness</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/long_term_robustness.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+
+<span class="c1"># plot robustness over time</span>
+<span class="n">fig</span> <span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span><span class="mi">22</span><span class="p">,</span><span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span><span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span> <span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span><span class="mi">22</span><span class="p">,</span><span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span><span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;Time Horizon (yrs)&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Robustness (% SOWs)&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">&#39;Bedford&#39;</span><span class="p">,</span> <span class="s1">&#39;Greene&#39;</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Robustness Over Time&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">107</span><span class="p">])</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">107.0</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_4_1.png" src="_images/discovery_4_1.png" />
+</section>
+<section id="exploring-performance-evolution">
+<h3>Exploring performance evolution<a class="headerlink" href="#exploring-performance-evolution" title="Link to this heading">#</a></h3>
+<p>The figure above reveals that the robustness of both water utilities degrades over time, with Bedford’s robustness declining further than Greene. This suggests that the proposed pathway policy is likely insufficient to meet the long-term needs of the two utilities. But how is the current policy insufficient? To answer that question we examine the performance measures that fail to meet performance criteria for each utility across the three planning horizons.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot the type of vulnerability over time</span>
+
+<span class="c1">### Bedford ###</span>
+<span class="n">plot_robustness_1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
+<span class="c1"># Determine the percentage of failure SOWs that violate each criterion (note some SOWS fail multiple criteria, so this may some to &gt;1)</span>
+<span class="n">criteria</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Reliability&#39;</span><span class="p">,</span> <span class="s1">&#39;Restriction Frequency&#39;</span><span class="p">,</span> <span class="s1">&#39;Peak Financial Cost&#39;</span><span class="p">,</span> <span class="s1">&#39;Worst-case drought</span><span class="se">\n</span><span class="s1">Management Cost&#39;</span><span class="p">,</span> <span class="s1">&#39;Stranded Assets&#39;</span><span class="p">]</span>
+<span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span>
+<span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">1</span><span class="p">,:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span>
+<span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">2</span><span class="p">,:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span>
+
+<span class="c1"># Plot over time</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">0</span><span class="p">,:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Fraction of failure SOWs&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">1</span><span class="p">,:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">2</span><span class="p">,:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Bedford&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="c1">### Greene ###</span>
+<span class="c1"># Determine the percentage of failure SOWs that violate each criterion (note some SOWS fail multiple criteria, so this may some to &gt;1)</span>
+<span class="n">plot_robustness_2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
+<span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">11</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">])</span>
+<span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">11</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">])</span>
+<span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">11</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">])</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Fraction of failure SOWs&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="p">:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Greene&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_6_0.png" src="_images/discovery_6_0.png" />
+<img alt="_images/discovery_6_1.png" src="_images/discovery_6_1.png" />
+<p>In the figures above, we observe that the vulnerability of both utilities changes in different ways. Early in the simulation period, Bedford is vulnerable to failures in reliability (though the robustness figure created in step B5.2 reveals that these failures are very rare). As the simulation period progresses, Bedford’s vulnerability expands to include failures in restriction frequency and worst-case cost. These failures indicate that the utility has an overall inability to manage drought conditions and future conditions progress.</p>
+<p>Greene shows a very different evolution in vulnerability. Early in the simulation period, failures manifest in the restriction frequency objective, suggesting that the utility must rely on water use restrictions to maintain supply reliability. As the simulation progresses however, the vulnerability evolves. When evaluated across the 45-year planning horizon, a new failure modes emerges - financial failure manifesting in peak financial cost and stranded assets. This suggests that the proposed pathway policy may be over-investing in new infrastructure, straining the utility’s budget with large debt payments that are unnecessary to maintain supply reliability.</p>
+</section>
+<section id="how-do-deep-uncertainties-generate-vulnerability">
+<h3>How do deep uncertainties generate vulnerability<a class="headerlink" href="#how-do-deep-uncertainties-generate-vulnerability" title="Link to this heading">#</a></h3>
+<p>While the evolution of robustness provides insight into how the system
+evolves over time, it does not reveal <em>why</em> each utility is vulnerable.
+To examine how deep uncertainties generate vulnerability over time for
+the two utilities, we perform scenario discovery (factor mapping,
+Chapter 4.3). Here we’ll utilize gradient boosted trees to identify
+regions of the uncertainty space that cause the utilities to fail to
+meet performance criteria.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># import the performance data across 2000 SOWs for three time periods</span>
+<span class="n">short_term_performance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/short_term_performance.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">mid_term_performance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/mid_term_performance.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">long_term_performance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/long_term_performance.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+
+<span class="n">satisficing_criteria</span> <span class="o">=</span> <span class="p">[</span><span class="mf">.98</span><span class="p">,</span> <span class="mf">.2</span><span class="p">,</span> <span class="mf">.8</span><span class="p">,</span> <span class="mf">.1</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>
+
+<span class="c1"># transform into scenario discovery input</span>
+<span class="n">short_term_SD_input</span> <span class="o">=</span> <span class="n">create_sd_input</span><span class="p">(</span><span class="n">short_term_performance</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">)</span>
+<span class="n">mid_term_SD_input</span> <span class="o">=</span> <span class="n">create_sd_input</span><span class="p">(</span><span class="n">mid_term_performance</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">)</span>
+<span class="n">long_term_SD_input</span> <span class="o">=</span> <span class="n">create_sd_input</span><span class="p">(</span><span class="n">long_term_performance</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">)</span>
+
+<span class="c1"># factor mapping Bedford</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;short_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;mid_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;long_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Bedford Factor Maps&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="c1"># factor mapping Greene</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="s1">&#39;short_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="s1">&#39;mid_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="s1">&#39;long_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Greene Factor Maps&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Greene</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Greene</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Greene</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_9_1.png" src="_images/discovery_9_1.png" />
+<img alt="_images/discovery_9_2.png" src="_images/discovery_9_2.png" />
+<p>In the figures above, we learn more about how the vulnerability of the two utilities evolves over time. Bedford begins with very few possible failures but appears vulnerable to high demand growth scenarios under future scenarios with high demands. When evaluated across a 22-year planning horizon, Bedford is vulnerable when the near-term demand growth is high and water use restrictions are less effective than predicted. Under the full 45-year planning horizon, Bedford is vulnerable to sustained high levels of demand growth, failing if either near-term or mid-term demand growth exceeds expected levels.</p>
+<p>Greene’s vulnerability evolves differently. It begins with vulnerability to high demand growth, but as the simulation progresses (and infrastructure is constructed), the utility becomes vulnerable to low-demand growth futures which cause the failures in financial criteria shown in section B.5.3. This indicates that the pathway policy over-builds in many SOWs, and becomes financially unstable if demand does not grow sufficiently to provide revenue to cover debt service payments.</p>
+</section>
+<section id="which-uncertainties-have-the-most-influence-on-time-evolving-performance">
+<h3>Which uncertainties have the most influence on time-evolving performance?<a class="headerlink" href="#which-uncertainties-have-the-most-influence-on-time-evolving-performance" title="Link to this heading">#</a></h3>
+<p>The factor maps generated in B.5.4 present the vulnerability generated by the two most important deep uncertainties as determined by Gradient Boosted Trees. Yet the factor prioritization shows that more than two uncertainties are influential to regional performance. Further, we can observe that individual uncertainties have different impacts on each performance obejctive, and these impacts may change over time. In the cells below, explore the impact of deep uncertainty by generating factor maps for different combinations of deep uncertain factors, objectives and time horizons.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+<span class="n">uncertainties</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;D1&#39;</span><span class="p">,</span> <span class="s1">&#39;D2&#39;</span><span class="p">,</span> <span class="s1">&#39;D3&#39;</span><span class="p">,</span> <span class="s1">&#39;BT&#39;</span><span class="p">,</span> <span class="s1">&#39;BM&#39;</span><span class="p">,</span> <span class="s1">&#39;DR&#39;</span><span class="p">,</span> <span class="s1">&#39;RE&#39;</span><span class="p">,</span> <span class="s1">&#39;EV&#39;</span><span class="p">,</span> <span class="s1">&#39;PM&#39;</span><span class="p">,</span> <span class="s1">&#39;CT&#39;</span><span class="p">,</span> <span class="s1">&#39;IA&#39;</span><span class="p">,</span> <span class="s1">&#39;IF&#39;</span><span class="p">,</span> <span class="s1">&#39;IP&#39;</span><span class="p">]</span>
+<span class="n">uncertainties</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Near-term demand&#39;</span><span class="p">,</span> <span class="s1">&#39;Mid-term demand&#39;</span><span class="p">,</span> <span class="s1">&#39;Long-term demand&#39;</span><span class="p">,</span> <span class="s1">&#39;Bond Term&#39;</span><span class="p">,</span> <span class="s1">&#39;Bond Rate&#39;</span><span class="p">,</span> <span class="s1">&#39;Discount Rate&#39;</span><span class="p">,</span> <span class="s1">&#39;Restriction Effectiveness&#39;</span><span class="p">,</span> <span class="s1">&#39;Evaporation Rate&#39;</span><span class="p">,</span> <span class="s1">&#39;Permitting time&#39;</span><span class="p">,</span> <span class="s1">&#39;Construction time&#39;</span><span class="p">,</span> <span class="s1">&#39;Inflow Amplitude&#39;</span><span class="p">,</span> <span class="s1">&#39;Inflow Frequency&#39;</span><span class="p">,</span> <span class="s1">&#39;Inflow Period&#39;</span><span class="p">]</span>
+
+<span class="n">u1_st_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">short_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="n">u1_mt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">mid_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="n">u1_lt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">long_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+
+<span class="n">u1_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">u1_st_FI</span><span class="p">,</span><span class="n">u1_mt_FI</span><span class="p">,</span> <span class="n">u1_lt_FI</span><span class="p">])</span>
+<span class="n">u1_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">u1_all</span><span class="p">)</span>
+
+<span class="c1"># factor ranking -- utility 2</span>
+<span class="n">u2_st_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">short_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
+<span class="n">u2_mt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">mid_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
+<span class="n">u2_lt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">long_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
+<span class="n">u2_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">u2_st_FI</span><span class="p">,</span><span class="n">u2_mt_FI</span><span class="p">,</span> <span class="n">u2_lt_FI</span><span class="p">])</span>
+<span class="n">u2_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">u2_all</span><span class="p">)</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span><span class="p">,</span> <span class="n">cax</span><span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span>
+                  <span class="n">gridspec_kw</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;width_ratios&quot;</span><span class="p">:[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">]})</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">wspace</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">im</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">u1_all</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;Reds&#39;</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mf">.3</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">13</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">(</span><span class="n">uncertainties</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Time Horizon&#39;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Bedford&#39;</span><span class="p">)</span>
+
+<span class="n">im1</span> <span class="o">=</span> <span class="n">ax2</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">u2_all</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;Reds&#39;</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mf">.3</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">13</span><span class="p">))</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">([])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Time Horizon&#39;</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Greene&#39;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="n">cax</span><span class="o">=</span><span class="n">cax</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Factor Importance&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_12_0.png" src="_images/discovery_12_0.png" />
+<p>The Figure above shows the factor importance as determined by gradient boosted trees for both utilities across the three planning horizons. While near-term demand growth is important for both utilities under all three planning horizons, the importance of other factors evolves over time. For example, restriction effectiveness plays an important role for Greene under the 22-year planning horizon but disappears under the 45-year planning horizon. In contrast, the bond interest rate is important for predicting success over the 45-year planning horizon, but does not appear important over the 10- or 22-year planning horizons. These findings highlight how assumptions about the planning period can have a large impact on modeling outcomes.</p>
+</section>
+<section id="open-exploration">
+<h3>Open exploration<a class="headerlink" href="#open-exploration" title="Link to this heading">#</a></h3>
+<p>In the cell below, use the function to explore how factor maps change
+for the two utilities based upon the uncertainties plotted, the
+objectives of interest and the time horizon.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># specify the utility (&quot;Bedford&quot; or &quot;Greene&quot;)</span>
+<span class="n">utility</span> <span class="o">=</span> <span class="s2">&quot;Bedford&quot;</span>
+
+<span class="c1"># specify which performance objectives to investigate (note that not all performance objectives have failures, which may result in a blank factor map)</span>
+<span class="c1"># set this to one of the following: &quot;Reliability&quot;, &quot;Restriction Frequency&quot;, &quot;Peak Financial Cost&quot;, &quot;Worst Case Cost&quot; or &quot;Unit Cost&quot;</span>
+<span class="n">objective</span> <span class="o">=</span> <span class="s2">&quot;Reliability&quot;</span>
+
+<span class="c1"># select uncertainties from the following list: &#39;D1&#39;, &#39;D2&#39;, &#39;D3&#39;, &#39;BT&#39;, &#39;BM&#39;, &#39;DR&#39;, &#39;RE&#39;, &#39;EV&#39;, &#39;PM&#39;, &#39;CT&#39;, &#39;IA&#39;, &#39;IF&#39;, &#39;IP&#39;</span>
+<span class="n">uncertainty_1</span> <span class="o">=</span> <span class="s1">&#39;D1&#39;</span>
+<span class="n">uncertainty_2</span> <span class="o">=</span> <span class="s1">&#39;D2&#39;</span>
+
+<span class="c1"># The code below will plot factor maps over the three planning horizons for the information above</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">open_exploration</span><span class="p">(</span><span class="n">utility</span><span class="p">,</span> <span class="n">objective</span><span class="p">,</span> <span class="s1">&#39;short_term&#39;</span><span class="p">,</span> <span class="n">uncertainty_1</span><span class="p">,</span> <span class="n">uncertainty_2</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">open_exploration</span><span class="p">(</span><span class="n">utility</span><span class="p">,</span> <span class="n">objective</span><span class="p">,</span> <span class="s1">&#39;mid_term&#39;</span><span class="p">,</span> <span class="n">uncertainty_1</span><span class="p">,</span> <span class="n">uncertainty_2</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">open_exploration</span><span class="p">(</span><span class="n">utility</span><span class="p">,</span> <span class="n">objective</span><span class="p">,</span> <span class="s1">&#39;long_term&#39;</span><span class="p">,</span> <span class="n">uncertainty_1</span><span class="p">,</span> <span class="n">uncertainty_2</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span><span class="p">,</span> <span class="n">reliability</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span><span class="p">,</span> <span class="n">reliability</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span><span class="p">,</span> <span class="n">reliability</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_16_1.png" src="_images/discovery_16_1.png" />
+</section>
+<section id="tips-to-apply-this-methodology-to-your-own-problem">
+<h3>Tips to apply this methodology to your own problem<a class="headerlink" href="#tips-to-apply-this-methodology-to-your-own-problem" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we demonstrated time-evolving scenario discovery for a
+cooperative water supply system. To apply this workflow to your own
+problem:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters of interest, which will
+represent uncertainties that characterize your system.</p></li>
+<li><p>Generate samples for these parameters (this can be done using the
+saltelli.sample function as in B.2 or done with another package).</p></li>
+<li><p>Define performance criteria for your problem</p></li>
+<li><p>Evaluate parameter sets through your model, and save performance
+measures across multiple time horizons</p></li>
+<li><p>Draw from the supporting code for this tutorial to perform scneario
+discovery and visualize results</p></li>
+</ol>
+</section>
+<section id="references">
+<h3>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h3>
+<p>Trindade, B. C., Reed, P. M., &amp; Characklis, G. W. (2019). Deeply uncertain pathways: Integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134, 103442.</p>
+</section>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Time-evolving scenario discovery for infrastructure pathways</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#background">Background</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluating-robustness-over-time">Evaluating Robustness over time</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exploring-performance-evolution">Exploring performance evolution</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#how-do-deep-uncertainties-generate-vulnerability">How do deep uncertainties generate vulnerability</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#which-uncertainties-have-the-most-influence-on-time-evolving-performance">Which uncertainties have the most influence on time-evolving performance?</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#open-exploration">Open exploration</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-this-methodology-to-your-own-problem">Tips to apply this methodology to your own problem</a></li>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios">A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#background">Background</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">Let’s Get Started!</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#observed-record">Observed Record</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-stationary-generator-to-better-quantify-natural-variability">Synthetic Stationary Generator to Better Quantify Natural Variability</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#non-stationary-synthetic-generator-to-impose-climate-changes">Non-Stationary Synthetic Generator to Impose Climate Changes</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#placing-cmip5-projections-in-the-context-of-non-stationary-flows">Placing CMIP5 Projections in the Context of Non-Stationary Flows</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-create-an-hmm-based-generator-for-your-system">Tips to Create an HMM-Based Generator for your System</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#references">References</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios-tutorial">
+<h1>A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios Tutorial<a class="headerlink" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios-tutorial" title="Link to this heading">#</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/Hidden-Markov_Modeling_Approaches_to_Creating_Synthetic_Streamflow_Scenarios.ipynb">HMM Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios">
+<h2>A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios<a class="headerlink" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios" title="Link to this heading">#</a></h2>
+<p>In this notebook, we will be covering the basics of fitting a Hidden
+Markov Model-based synthetic streamflow generator for a single site in
+the Upper Colorado River Basin. First, we will characterize the observed
+historical flow in the basin from 1909-2013. Then, we will fit a
+synthetic streamflow generator to the observed flows in the basin in
+order to create stationary synthetic flows. Finally, we will create a
+non-stationary version of the generator to create flows that could be
+representative of plausible future climate in the region. We ultimately
+show how to place the synthetically generated flows in the context of
+physically-informed CMIP5 projections to compare the two methods.</p>
+<section id="background">
+<h3>Background<a class="headerlink" href="#background" title="Link to this heading">#</a></h3>
+<p>In the Western United States (US), and particularly the Colorado River
+Basin, a recent study used tree-ring reconstructions to suggest that the
+megadrought that has been occurring in the Southwest over the past 22
+years is the region’s worst drought since about 800 AD (Williams et al.,
+2022). The study’s lead author, UCLA climatologist Park Williams,
+suggested that had the sequence of wet-dry years occurred without
+anthropogenic forcing, the 2000s would have likely still been dry, but
+not on the same level as the worst of the last millennium’s
+megadroughts.</p>
+<p>The recent trend of warming and reduced soil moisture in the Southwest
+US is highly challenging from a water systems planning and management
+perspective for the Colorado River Basin. Given the wide recognition
+that the river is over-allocated, the most recent drought highlights the
+difficulty of sustaining the flow requirements as dictated by the
+Colorado Compact. Thus, there has been an increasing focus in
+exploratory modeling efforts to clarify how vulnerable water systems in
+this region are to plausible drought streamflow scenarios for the
+future. In this tutorial, we’ll discuss how to create these scenarios
+using a Hidden Markov Model (HMM)- based streamflow synthetic generator.
+As discussed in <a class="reference external" href="https://uc-ebook.org/docs/html/2_diagnostic_modeling_overview_and_perspectives.html#overview-of-model-diagnostics">Section
+2.1</a>
+and
+<a class="reference external" href="https://uc-ebook.org/docs/html/4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-dynamics-what-is-controlling-model-behaviors-of-interest">4.2</a>
+of the eBook, future climate conditions in the basin represent a deep
+uncertainty that can lead to highly consequential water scarcity
+outcomes. It is advantageous to create a model such as the HMM-based
+generator in order to facilitate the creation of many ensembles of
+streamflow that can ultimately be used to force regional water systems
+models to understand how variability and drought extremes affect
+regional water shortages, operations, and policies.</p>
+<figure>
+<center><img src="_static/1-lede_pix_-_usbr-powell-8517152024_aa66437c2e_o-cropped.jpg" style="width:50%"></center>
+<figcaption style="text-align:center"> Lake Powell shows persistent effects from drought (Source: U.S. Bureau of Reclamation) </figcaption>
+</figure></section>
+</section>
+<section id="lets-get-started">
+<h2>Let’s Get Started!<a class="headerlink" href="#lets-get-started" title="Link to this heading">#</a></h2>
+<section id="observed-record">
+<h3>Observed Record<a class="headerlink" href="#observed-record" title="Link to this heading">#</a></h3>
+<p>First, let’s take a look at the observed data from 1909-2013 for a
+specific site. In this example, we use the outlet gauge of the Upper
+Colorado River (USGS Gauge 09163500 at the Colorado-Utah state line).
+Below, we create a plot of the annual streamflow.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Import libraries</span>
+<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="nn">mpl</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">matplotlib.patches</span> <span class="k">as</span> <span class="nn">patches</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span>
+<span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">latin</span>
+<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">stats</span> <span class="k">as</span> <span class="n">ss</span>
+<span class="kn">import</span> <span class="nn">statistics</span>
+<span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>
+
+
+<span class="c1"># Import helper functions from local package</span>
+<span class="kn">from</span> <span class="nn">functions</span> <span class="kn">import</span> <span class="n">fitmodel</span>
+<span class="kn">from</span> <span class="nn">functions</span> <span class="kn">import</span> <span class="n">plotstates</span>
+<span class="kn">from</span> <span class="nn">functions</span> <span class="kn">import</span> <span class="n">plotdist</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Read in annual historical data</span>
+<span class="n">AnnualQ</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;data/uc_historical.csv&#39;</span><span class="p">)</span>
+<span class="n">AnnualQ</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+<span class="c1"># Plot a line graph</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper right&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_9_0.png" src="_images/hmm_9_0.png" />
+<p>Let’s calculate an 11-year rolling mean of the same data to get a sense
+of long-term trends.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+
+<span class="c1"># Plot the original line graph</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+         <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+         <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+         <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1"># Plot an 11-year rolling mean</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+         <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+         <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+         <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_11_0.png" src="_images/hmm_11_0.png" />
+<p>The Colorado Compact, which prescribes flows between the Upper and Lower
+Colorado Basins, was negotiated using data prior to 1922, a time period
+revealed by the above figure to be one of the consistently wetter
+periods on record. It’s clear today that since the 1980s, the Southwest
+US has been experiencing aridification (Overpeck et al., 2020) and that
+this observed record alone isn’t an accurate representation of what
+future climate might look like in this region.</p>
+<p>Let’s get a little more specific and formally quantify decadal droughts
+that have occurred in the observed period. We use a metric proposed in
+Ault et al. (2014). The authors define a decadal drought as when the
+11-year rolling mean falls below a threshold that is 1/2 a standard
+deviation below the overall mean of the record. We can then highlight
+the block of years that fall in a decadal drought using yellow
+rectangles below.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define drought threshold</span>
+<span class="n">std</span> <span class="o">=</span> <span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="n">threshold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+<span class="c1"># Find where the rolling mean dip below the threshold?</span>
+<span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+<span class="n">drought_years</span> <span class="o">=</span> <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">drought_instances</span><span class="p">]</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span><span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span><span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">threshold</span><span class="p">,</span>
+           <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
+           <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span>
+           <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Drought Threshold&#39;</span><span class="p">)</span>
+
+<span class="c1"># Visualize the drought periods as yellow rectangles</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">drought_years</span><span class="p">:</span>
+
+    <span class="c1"># Plot a box centered around those values and with 5 years on either side.</span>
+    <span class="n">rect</span> <span class="o">=</span> <span class="n">patches</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">((</span><span class="n">i</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="mi">11</span><span class="p">,</span><span class="mf">2e7</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">,</span> <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Add the patch to the Axes</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_14_0.png" src="_images/hmm_14_0.png" />
+<p>By this metric, the Upper Colorado Basin region has experienced two
+decadal droughts over the last century.</p>
+</section>
+<section id="synthetic-stationary-generator-to-better-quantify-natural-variability">
+<h3>Synthetic Stationary Generator to Better Quantify Natural Variability<a class="headerlink" href="#synthetic-stationary-generator-to-better-quantify-natural-variability" title="Link to this heading">#</a></h3>
+<p>It is important to remember that the streamflow that we have observed in
+the region over the last century is only one instance of the hydrology
+that could occur since the atmosphere is an inherently stochastic
+system. Thus, we require a tool that will allow us to see multiple
+plausible realizations of the streamflow record to understand the
+internal variability that characterizes the historical period. One
+observed realization of historical streamflow is limited in its ability
+to capture rare extremes; plausible (but not observed) alternative
+instances of streamflow records can help to fill this gap. The tool that
+we use to develop synthetic flows for the region is a Gaussian Hidden
+Markov Model (HMM). If a system follows a Markov process, it switches
+between a number of “hidden states” dictated by a transition matrix.
+Each state has its own Gaussian probability distribution (defined by a
+mean and standard deviation) and one can draw from this distribution to
+create synthetic flows that fit the properties of the historical
+distribution. HMMs are an attractive choice for this region because they
+can simulate persistence (i.e., long duration droughts), which is a
+characteristic of the region’s hydro-climatology. The figure below shows
+an example of a 2-state Gaussian HMM that we will be fitting for this
+example.</p>
+<figure>
+    <center><img src="_static/HMM_example.png" style="width:75%"></center>
+<figcaption style="text-align:center"> Two-state Gaussian HMM with mean and standard deviation parameters</figcaption>
+</figure><p>Below is the code that fits the HMM model to the last 2/3 of the
+historical record of log annual flows at the CO-UT stateline gauge and
+creates an alternative trace of 105 years. A subset of the dataset is
+chosen in order to minimize overfitting and to retain a set of data for
+validation of the model. When we fit our model, we utilize the
+Baum-Welch algorithm (a special version of the expectation-maximization
+algorithm) to find the optimal parameters that maximize the likelihood
+of seeing the observed flows. Ultimately, the algorithm will return a
+mean and standard deviation associated with each state (mus and sigmas
+defined below) and a 2x2 transition probability matrix that captures the
+likelihood of transitioning between states (P). We can also retrieve the
+annual hidden states across the observed series, also known as the
+Viterbi sequence of states, which classifies each year in a “wet” or
+“dry” state.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Number of years for alternative trace</span>
+<span class="n">n_years</span> <span class="o">=</span> <span class="mi">105</span>
+
+<span class="c1"># Import historical data that it used to fit HMM model</span>
+<span class="n">AnnualQ_h</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;data/uc_historical.csv&#39;</span><span class="p">)</span>
+
+<span class="c1"># Fit the model and pull out relevant parameters and samples</span>
+<span class="n">logQ</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">AnnualQ_h</span><span class="p">)</span>
+<span class="n">hidden_states</span><span class="p">,</span> <span class="n">mus</span><span class="p">,</span> <span class="n">sigmas</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">logProb</span><span class="p">,</span> <span class="n">samples</span><span class="p">,</span> <span class="n">model</span> <span class="o">=</span> <span class="n">fitmodel</span><span class="o">.</span><span class="n">fitHMM</span><span class="p">(</span><span class="n">logQ</span><span class="p">,</span> <span class="n">n_years</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>We’ve fit our HMM, but what does the model look like? Let’s plot the
+annual time series of hidden states, or the Viterbi sequence. In the
+code, above, we have defined that the drier state is always represented
+by state 0. Thus, we know that hidden_states = 0 corresponds to the dry
+state and hidden_states = 1 to the wet state.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot Vitebi sequence</span>
+<span class="n">plotstates</span><span class="o">.</span><span class="n">plotTimeSeries</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]),</span> <span class="n">hidden_states</span><span class="p">,</span> <span class="s1">&#39;Annual Flow (cube feet per year)&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_21_0.png" src="_images/hmm_21_0.png" />
+<p>In the figure above, we see that the years with the higher log flows
+tend to be classified in a “wet” state and the opposite is true of the
+“dry” state. We can also print the transition matrix, which shows the
+likelihood of transitioning between states. Note that the system has a
+high likelihood of persisting in the same state.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">transmat_</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[[</span><span class="mf">0.65095026</span> <span class="mf">0.34904974</span><span class="p">]</span>
+ <span class="p">[</span><span class="mf">0.3205531</span>  <span class="mf">0.6794469</span> <span class="p">]]</span>
+</pre></div>
+</div>
+<p>Let’s also plot the distribution of log annual flows associated with the
+wet and dry states.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot wet and dry state distributions</span>
+<span class="n">plotdist</span><span class="o">.</span><span class="n">plotDistribution</span><span class="p">(</span><span class="n">logQ</span><span class="p">,</span> <span class="n">mus</span><span class="p">,</span> <span class="n">sigmas</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_25_0.png" src="_images/hmm_25_0.png" />
+<p>The wet state distribution is characterized by a greater mean flow, but
+note that there is significant overlap in the tails of the distributions
+below which demonstrates why years with similiar flows can be classified
+in different states.</p>
+<p>Now let’s see what the drought dynamics look like in the synthetic
+scenario that we created using the same definition that we had used for
+the historical period.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Retrieve samples and back-transform out of log space</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">samples</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="p">)</span>
+<span class="n">AnnualQ_s</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+<span class="c1"># Define drought threshold</span>
+<span class="n">std</span><span class="o">=</span><span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="n">threshold</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+<span class="c1"># Where does the rolling mean dip below the threshold</span>
+<span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+<span class="n">drought_years</span> <span class="o">=</span> <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">drought_instances</span><span class="p">]</span>
+
+<span class="c1">#Visualize the streamflow scenario</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+
+<span class="c1">#Plot the original line graph</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1">#Plot a 11-year rolling mean</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">threshold</span><span class="p">,</span>
+           <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
+           <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span>
+           <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Drought Threshold&#39;</span><span class="p">)</span>
+
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">drought_years</span><span class="p">:</span>
+
+    <span class="c1">#Plot a box centered around those values and with 5 years on either side.</span>
+    <span class="n">rect</span> <span class="o">=</span> <span class="n">patches</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">((</span><span class="n">i</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span>
+                              <span class="mi">0</span><span class="p">),</span>
+                              <span class="mi">11</span><span class="p">,</span>
+                              <span class="mf">2e7</span><span class="p">,</span>
+                              <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+                              <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">,</span>
+                              <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Add the patch to the Axes</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow (Synthetic Stationary)&quot;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_28_0.png" src="_images/hmm_28_0.png" />
+<p>You can sample from the model and create more 105-year traces and note
+how the location and number of decadal droughts changes. This
+demonstrates how different the historical record can look just within
+the range of natural variability. It’s also important to remember that
+when droughts occur can also define the ultimate effect of the drought
+(i.e. is it a time when there is a large population growth or a time
+when humans can adapt by conserving or building more infrastructure?). A
+hydrologic drought need not manifest into an agricultural or operational
+drought of the same magnitude if stored surface water is available.</p>
+<p>We externally run the HMM many times to create a dataset of 100
+instances of the 105-year traces and 1000 instances of the 105-year
+traces that are available in the package
+(“synthetic_stationary_small_sample_100.csv”,“synthetic_stationary_large_sample_1000”).
+The shaded green lines correspond to the flow duration curves (FDCs) for
+the generated streamflow traces in comparison with the FDC of the
+historical record in beige.</p>
+<figure>
+    <center><img src="_static/stationary_synthetic_FDC.png" style="width:75%"></center>
+<figcaption style="text-align:center">Generated streamflow traces in comparison with the FDC of the historical record.</figcaption>
+</figure><p>As expected, the stationary synthetic FDCs envelope the historical FDC
+and particularly, the synthetic traces offer many more instances of low
+flow conditions that could lead to more extreme drought conditions than
+what has been observed historically. It is also useful to check for
+convergence of samples and to determine how many samples are needed to
+fully represent internal variability. Above we see that the extension to
+1000 instances of 105-year traces fills out regions of the FDC,
+including creating some more extreme drought conditions, but that
+additional samples will likely not fill out the FDC substantially more.</p>
+</section>
+<section id="non-stationary-synthetic-generator-to-impose-climate-changes">
+<h3>Non-Stationary Synthetic Generator to Impose Climate Changes<a class="headerlink" href="#non-stationary-synthetic-generator-to-impose-climate-changes" title="Link to this heading">#</a></h3>
+<p>Now, we create flows under non-stationary conditions to get a better
+understanding of what flows can look like under climate changes. In
+order to create flows under non-stationary conditions, we can toggle the
+parameters of the HMM model in order to create systematic changes to the
+model that can represent a changing climate. The HMM has 6 parameters
+that define it. When we fit the historical model, the parameters that
+are fit represent a baseline parameter value. In this non-stationary
+generator, we define a range to sample these parameters from.</p>
+<div class="pst-scrollable-table-container"><table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Parameter</p></th>
+<th class="head"><p>Current Value</p></th>
+<th class="head"><p>Lower Bound</p></th>
+<th class="head"><p>Upper Bound</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>Log-Space Wet State
+Mean Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.98</p></td>
+<td><p>1.02</p></td>
+</tr>
+<tr class="row-odd"><td><p>Log-Space Dry State
+Mean Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.98</p></td>
+<td><p>1.02</p></td>
+</tr>
+<tr class="row-even"><td><p>Log-Space Wet State
+Standard Deviation
+Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.75</p></td>
+<td><p>1.25</p></td>
+</tr>
+<tr class="row-odd"><td><p>Log-Space Dry State
+Standard Deviation
+Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.75</p></td>
+<td><p>1.25</p></td>
+</tr>
+<tr class="row-even"><td><p>Change in Dry-Dry
+Transition
+Probability</p></td>
+<td><p>0.00</p></td>
+<td><p>-0.30</p></td>
+<td><p>+0.30</p></td>
+</tr>
+<tr class="row-odd"><td><p>Change in Wet-Wet
+Transition
+Probability</p></td>
+<td><p>0.00</p></td>
+<td><p>-0.30</p></td>
+<td><p>+0.30</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>Now let’s sample 1000 times from these bounds to create 1000 new
+parameterizations of the model. Here we use SALib and the Latin
+Hypercube sample function.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create problem structure with parameters that we want to sample</span>
+<span class="n">problem</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">6</span><span class="p">,</span>
+    <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;wet_mu&#39;</span><span class="p">,</span> <span class="s1">&#39;dry_mu&#39;</span><span class="p">,</span> <span class="s1">&#39;wet_std&#39;</span><span class="p">,</span><span class="s1">&#39;dry_std&#39;</span><span class="p">,</span><span class="s1">&#39;dry_tp&#39;</span><span class="p">,</span><span class="s2">&quot;wet_tp&quot;</span><span class="p">],</span>
+    <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="mf">0.98</span><span class="p">,</span> <span class="mf">1.02</span><span class="p">],</span>
+               <span class="p">[</span><span class="mf">0.98</span><span class="p">,</span> <span class="mf">1.02</span><span class="p">],</span>
+               <span class="p">[</span><span class="mf">0.75</span><span class="p">,</span><span class="mf">1.25</span><span class="p">],</span>
+               <span class="p">[</span><span class="mf">0.75</span><span class="p">,</span><span class="mf">1.25</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.3</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.3</span><span class="p">]]</span>
+<span class="p">}</span>
+
+<span class="c1"># generate 1000 parameterizations</span>
+<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">1000</span>
+
+<span class="c1"># set random seed for reproducibility</span>
+<span class="n">seed_value</span> <span class="o">=</span> <span class="mi">123</span>
+
+<span class="c1"># Generate our samples</span>
+<span class="n">LHsamples</span> <span class="o">=</span> <span class="n">latin</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="n">n_samples</span><span class="p">,</span> <span class="n">seed_value</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now let’s look at what some of the traces look like in our
+non-stationary generator. Let’s choose a random instance from the
+1000-member space and adjust the parameters accordingly.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define static parameters</span>
+<span class="n">n_years</span> <span class="o">=</span> <span class="mi">105</span>
+
+<span class="c1"># Sample parameter; Adjust to any sample number from 0-999</span>
+<span class="n">sample</span> <span class="o">=</span> <span class="mi">215</span>
+
+<span class="c1"># Create empty arrays to store the new Gaussian HMM parameters for each SOW</span>
+<span class="n">Pnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">musNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">sigmasNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">logAnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+
+<span class="c1"># Calculate new transition matrix and stationary distribution of SOW at last node as well as new means and standard deviations</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">4</span><span class="p">]))</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">5</span><span class="p">]))</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+<span class="n">eigenvals</span><span class="p">,</span> <span class="n">eigenvecs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">))</span>
+<span class="n">one_eigval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">eigenvals</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
+<span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">divide</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span> <span class="n">one_eigval</span><span class="p">]),</span>
+                  <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span><span class="n">one_eigval</span><span class="p">])))</span>
+
+<span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
+<span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+<span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">3</span><span class="p">]</span>
+<span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
+
+<span class="c1"># Generate first state and log-space annual flow at last node</span>
+<span class="n">states</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+<span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">piNew</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
+    <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="k">else</span><span class="p">:</span>
+    <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+    <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="c1"># Generate remaining state trajectory and log space flows at last node</span>
+<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_years</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">Pnew</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">])]:</span>
+        <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+
+    <span class="k">if</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="c1"># Convert log-space flows to real-space flows</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">logAnnualQ_s</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
+</pre></div>
+</div>
+<p>Now let’s see what this synthetic trace looks like.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Retrieve samples and back-transform out of log space</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="p">)</span>
+<span class="n">AnnualQ_s</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+<span class="c1"># Define drought threshold</span>
+<span class="n">std</span> <span class="o">=</span> <span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="n">threshold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+<span class="c1"># Where does the rolling mean dip below the threshold</span>
+<span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+<span class="n">drought_years</span> <span class="o">=</span> <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">drought_instances</span><span class="p">]</span>
+
+<span class="c1"># Visualize the streamflow scenario</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+
+<span class="c1"># Plot the original line graph</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1"># Plot a 11-year rolling mean</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">threshold</span><span class="p">,</span>
+           <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
+           <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span>
+           <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Drought Threshold&#39;</span><span class="p">)</span>
+
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">drought_years</span><span class="p">:</span>
+
+    <span class="c1"># Plot a box centered around those values and with 5 years on either side.</span>
+    <span class="n">rect</span> <span class="o">=</span> <span class="n">patches</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">((</span><span class="n">i</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span>
+                             <span class="mi">11</span><span class="p">,</span>
+                             <span class="mf">2e7</span><span class="p">,</span>
+                             <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+                             <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">,</span>
+                             <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Add the patch to the Axes</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Annual Flow (Synthetic Non-Stationary)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper right&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_40_0.png" src="_images/hmm_40_0.png" />
+<p>Above is the example trace from the new non-stationary model. You may
+see fewer or more decadal drought instances. We can further summarize
+overall decadal drought characteristics across the samples. Let’s plot a
+histogram of the total number of times we go below the drought threshold
+across these realizations.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">decadal_drought_occurence</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">1000</span><span class="p">])</span>
+
+<span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">):</span>
+
+    <span class="c1"># Create empty arrays to store the new Gaussian HMM parameters for each SOW</span>
+    <span class="n">Pnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
+    <span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+    <span class="n">musNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+    <span class="n">sigmasNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+    <span class="n">logAnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+
+    <span class="c1"># Calculate new transition matrix and stationary distribution of SOW at last node</span>
+    <span class="c1"># as well as new means and standard deviations</span>
+
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">4</span><span class="p">]))</span>
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">5</span><span class="p">]))</span>
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+    <span class="n">eigenvals</span><span class="p">,</span> <span class="n">eigenvecs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">))</span>
+    <span class="n">one_eigval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">eigenvals</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
+    <span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">divide</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span> <span class="n">one_eigval</span><span class="p">]),</span>
+                      <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span> <span class="n">one_eigval</span><span class="p">])))</span>
+
+    <span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
+    <span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+    <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">3</span><span class="p">]</span>
+    <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
+
+    <span class="c1"># Generate first state and log-space annual flow at last node</span>
+    <span class="n">states</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+    <span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">piNew</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
+        <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+    <span class="c1"># generate remaining state trajectory and log space flows at last node</span>
+    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_years</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">Pnew</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">])]:</span>
+            <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+
+        <span class="k">if</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+    <span class="c1"># Convert log-space flows to real-space flows</span>
+    <span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">logAnnualQ_s</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="p">)</span>
+    <span class="n">AnnualQ_s</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+    <span class="c1"># Define drought threshold</span>
+    <span class="n">std</span> <span class="o">=</span> <span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+    <span class="n">threshold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+    <span class="c1"># Where does the rolling mean dip below the threshold</span>
+    <span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+    <span class="n">decadal_drought_occurence</span><span class="p">[</span><span class="n">y</span><span class="p">]</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">drought_instances</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">decadal_drought_occurence</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Non-Stationary generator&#39;</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s2">&quot;#005F73&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Number of Instances of Decadal Drought&#39;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Frequency&#39;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;-&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Observed&#39;</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span> <span class="o">=</span> <span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span> <span class="o">=</span> <span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_43_0.png" src="_images/hmm_43_0.png" />
+<p>Note how many more instances of the decadal droughts we are creating
+with the non-stationary generator than our observed 105-year trace which
+creates a rich space in which we can test our models. Just as we did
+with the stationary generator, we can externally run the non-stationary
+generator to create 10,000 instances of the 105-year traces that are
+available in the package
+(“synthetic_nonstationary_large_sample_10000.csv”). The shaded green and
+blue lines correspond to the FDCs for the stationary and non-stationary
+generated streamflow traces in comparison with the FDC of the historical
+record in beige. Note how the non-stationary generator produces even
+more drought extremes than the stationary non-synthetic traces.</p>
+<figure>
+    <center><img src="_static/nonstationary_synthetic_FDC.png" style="width:75%"></center>
+<figcaption style="text-align:center">Generated streamflow traces in comparison with the FDC of the historical record.</figcaption>
+</figure></section>
+<section id="placing-cmip5-projections-in-the-context-of-non-stationary-flows">
+<h3>Placing CMIP5 Projections in the Context of Non-Stationary Flows<a class="headerlink" href="#placing-cmip5-projections-in-the-context-of-non-stationary-flows" title="Link to this heading">#</a></h3>
+<p>We have broadened the drought conditions that we are creating which that
+can be very useful to understand how our water systems model performs
+under potentially extreme scenarios. However, it’s useful to compare our
+bottom-up synthetically generated flows in the context of global
+physically-driven CMIP5 projections to get a better understanding of how
+the two approaches compare. We first aquire 97 CMIP5 projections from
+the Colorado River Water Availability Study (CWCB, 2012). In each of
+these projections, monthly precipitation factor changes and temperature
+delta changes were computed between mean projected 2035–2065 climate
+statistics and mean historical climate statistics from 1950–2013. These
+97 different combinations of 12 monthly precipitation multipliers and 12
+monthly temperature delta shifts were applied to historical
+precipitation and temperature time series from 1950–2013. The resulting
+climate time series were run through a Variable Infiltration Capacity
+(VIC) model of the UCRB, resulting in 97 time series of projected future
+streamflows at the Colorado‐Utah state line.</p>
+<p>We fit an HMM to each trace of projected streamflow and get a set of
+corresponding HMM parameters. Then we take the ratio between these
+parameters and the baseline HMM parameters that we calculated earlier in
+the notebook in order to calculate the multipliers associated with each
+CMIP5 projection. This is all done externally, so we import the
+resulting multipliers in the next line.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Read in CMIP5 and paleo multipliers</span>
+<span class="n">CMIP5_multipliers</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;data/CMIP5_SOWs.txt&#39;</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s2">&quot; &quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Let’s plot a response surface that will allow us to see how combinations
+of HMM parameters tend to influence decadal drought. In order to get a
+continuous surface, we’ll fit a non-linear regression to the parameter
+values and then predict the decadal drought over a set of grid points.
+We fit the response surface for two parameters that should have an
+affect on decadal drought: the dry distribution mean and the dry-dry
+transition probabilites.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Choose two parameters to fit the response surface for</span>
+<span class="n">mu_dry</span><span class="o">=</span><span class="p">[</span><span class="n">i</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">LHsamples</span><span class="p">]</span>
+<span class="n">tp_dry</span><span class="o">=</span><span class="p">[</span><span class="n">i</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">LHsamples</span><span class="p">]</span>
+
+<span class="c1"># Create an interpolation grid</span>
+<span class="n">xgrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">),</span>
+                  <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">),</span>
+                  <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">))</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span>
+
+<span class="n">ygrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">),</span>
+                  <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">),</span>
+                  <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">))</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span>
+
+<span class="c1"># Fit regression</span>
+<span class="n">d</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;Dry_Tp&#39;</span><span class="p">:</span> <span class="n">tp_dry</span><span class="p">,</span>
+     <span class="s1">&#39;Dry_Mu&#39;</span><span class="p">:</span> <span class="n">mu_dry</span><span class="p">,</span>
+     <span class="s1">&#39;Drought_Occurrence&#39;</span><span class="p">:</span><span class="n">decadal_drought_occurence</span><span class="p">}</span>
+
+<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
+<span class="n">df</span><span class="p">[</span><span class="s1">&#39;Intercept&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">shape</span><span class="p">(</span><span class="n">df</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">df</span><span class="p">[</span><span class="s1">&#39;Interaction&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s1">&#39;Dry_Tp&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">df</span><span class="p">[</span><span class="s1">&#39;Dry_Mu&#39;</span><span class="p">]</span>
+<span class="n">cols</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Intercept&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s1">&#39;Dry_Mu&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s1">&#39;Dry_Tp&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s1">&#39;Interaction&#39;</span><span class="p">]</span>
+<span class="n">ols</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Drought_Occurrence&#39;</span><span class="p">],</span> <span class="n">df</span><span class="p">[</span><span class="n">cols</span><span class="p">])</span>
+<span class="n">result</span> <span class="o">=</span> <span class="n">ols</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
+
+<span class="c1"># Calculate drought occurence for each grid point</span>
+<span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">xgrid</span><span class="p">,</span> <span class="n">ygrid</span><span class="p">)</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
+<span class="n">y</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
+<span class="n">grid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">column_stack</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">*</span> <span class="n">y</span><span class="p">])</span>
+<span class="n">z</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
+<span class="n">z</span><span class="p">[</span><span class="n">z</span> <span class="o">&lt;</span> <span class="mf">0.0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="c1"># replace negative shortage predictions with 0</span>
+</pre></div>
+</div>
+<p>Let’s plot our results:</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set color gradient for response surface</span>
+<span class="n">drought_map</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">&#39;RdBu_r&#39;</span><span class="p">)</span>
+
+<span class="c1"># Reshape our predicted drought occurrence and define bounds of colors</span>
+<span class="n">Z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">shape</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
+<span class="n">vmin</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Drought_Occurrence&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">)])</span>
+<span class="n">vmax</span> <span class="o">=</span> <span class="mi">15</span>
+<span class="n">norm</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="p">,</span> <span class="n">vmax</span><span class="p">)</span>
+
+<span class="c1"># Plot response surface and CMIP5 projections</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">drought_map</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">norm</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">CMIP5_multipliers</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">7</span><span class="p">],</span>
+           <span class="n">CMIP5_multipliers</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">12</span><span class="p">],</span>
+           <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#ffffb3&#39;</span><span class="p">,</span>
+           <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;none&#39;</span><span class="p">,</span>
+           <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span>
+<span class="n">cbar</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">figure</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">ScalarMappable</span><span class="p">(</span><span class="n">norm</span><span class="o">=</span><span class="n">norm</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">drought_map</span><span class="p">),</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">nanmin</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">nanmax</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">nanmin</span><span class="p">(</span><span class="n">Y</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">nanmax</span><span class="p">(</span><span class="n">Y</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Dry State Mu&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Dry-Dry Transition Probability&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="n">labelsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Decadal Drought Occurrence&#39;</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=-</span><span class="mi">90</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">,</span> <span class="n">labelpad</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
+<span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;y&#39;</span><span class="p">,</span><span class="n">labelsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_53_0.png" src="_images/hmm_53_0.png" />
+<p>We see the influence of the dry state mean and dry-dry transition
+parameters. We’re likely to see more decadal droughts when we (1)
+increase the dry-dry transition probability, which inherently will
+increase persistence of the dry state, and (2) when we make the dry
+state log mean drier. Note that the CMIP5 scenarios tend to span the
+extent of the dry mean sample space, but are less representative of the
+dry transition probability sample space, which suggests that the types
+of hydrological droughts represented in the projections tend to only be
+wetter to slightly drier than our baseline. Both methods of producing
+these scenarios are valid, though studies have suggested that
+globally-resolved GCMs may be inappropriate to represent regional
+extremes. Ultimately, if your goal is to produce a variety of ensembles
+that are characterized by many different drought characteristics, you
+will likely find that a generator approach will serve this purpose
+better.</p>
+</section>
+<section id="tips-to-create-an-hmm-based-generator-for-your-system">
+<h3>Tips to Create an HMM-Based Generator for your System<a class="headerlink" href="#tips-to-create-an-hmm-based-generator-for-your-system" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we demonstrated how to fit an HMM-based generator for
+a single gauge located in the Upper Colorado River Basin. In order to
+apply this methodology to your problem, you will need to first ask:</p>
+<ol class="arabic simple">
+<li><p>Is this model appropriate for my location of interest? We have
+applied this style of generator to locations where persistent wet
+and dry states are characteristic, which tends to be in the Western
+US. Ultimately the best way to judge if an HMM is useful for your
+application is to fit the model and explore the resulting
+distributions. Are there two (or more) distinct states that emerge?
+If not, then your location may not exhibit the type of persistence
+that an HMM-based generator is useful for. You can consider
+exploring other styles of generators such as the Kirsch-Nowak
+generator (Kirsch et al., 2013).</p></li>
+<li><p>Do I have the right datasets? We use annual data for our location of
+interest. In this notebook, the HMM is fit to log annual flows.
+Ultimately, it can be disaggregated to daily flows (using a
+reference historical daily dataset) to be useful in water resources
+operational applications. You could also disaggregate to a finer
+resolution than daily if the historical dataset exists.</p></li>
+</ol>
+<p>If you meet these requirements, feel free to proceed through fitting the
+model using the code available in the notebook. Be sure to consider the
+appropirate number of samples to generate (both in a stationary and
+non-stationary case). Make sure that you test multiple sample sizes and
+continue to increase your sample size until you converge to a consistent
+representation of extremes. What is the appropriate number of LHS
+samples of the parameters to use? In this experiment we used 1,000
+samples of parameters due to extensive stability tests described in
+Quinn et al. (2020).</p>
+<p>Finally, to learn more about this test case refer to Hadmichael et
+al. (2020a) and Hadmichael et al. (2020b). For another study on
+synthetic drought generation to support vulnerability assessments in the
+Research Triangle region of North Carolina, please refer to Herman et
+al. (2016)</p>
+</section>
+<section id="references">
+<h3>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h3>
+<p>Ault, T. R., Cole, J. E., Overpeck, J. T., Pederson, G. T., &amp; Meko, D.
+M. (2014). Assessing the risk of persistent drought using climate model
+simulations and paleoclimate data. Journal of Climate, 27(20),
+7529-7549.</p>
+<p>CWCB (2012).Colorado River Water Availability Study Phase I Report.
+Colorado Water Conservation Board</p>
+<p>Hadjimichael, A., Quinn, J., Wilson, E., Reed, P., Basdekas, L., Yates,
+D., &amp; Garrison, M. (2020a). Defining robustness, vulnerabilities, and
+consequential scenarios for diverse stakeholder interests in
+institutionally complex river basins. Earth’s Future, 8(7),
+e2020EF001503.</p>
+<p>Hadjimichael, A., Quinn, J., &amp; Reed, P. (2020). Advancing diagnostic
+model evaluation to better understand water shortage mechanisms in
+institutionally complex river basins. Water Resources Research, 56(10),
+e2020WR028079.</p>
+<p>Herman, J. D., Zeff, H. B., Lamontagne, J. R., Reed, P. M., &amp;
+Characklis, G. W. (2016). Synthetic drought scenario generation to
+support bottom-up water supply vulnerability assessments. Journal of
+Water Resources Planning and Management, (11), 04016050.</p>
+<p>Kirsch, B. R., Characklis, G. W., &amp; Zeff, H. B. (2013). Evaluating the
+impact of alternative hydro-climate scenarios on transfer agreements:
+Practical improvement for generating synthetic streamflows. Journal of
+Water Resources Planning and Management, 139(4), 396-406.</p>
+<p>Overpeck, J.T. &amp; Udall, B. (2020) “Climate change and the aridification
+of North America.” Proceedings of the national academy of sciences
+117.22 11856-11858.</p>
+<p>Quinn, J. D., Hadjimichael, A., Reed,P. M., &amp; Steinschneider, S. (2020).
+Canexploratory modeling of water scarcity vulnerabilities and robustness
+bescenario neutral?Earth’s Future,8,e2020EF001650.
+<a class="reference external" href="https://doi.org/10.1029/2020EF001650Received">https://doi.org/10.1029/2020EF001650Received</a></p>
+<p>Williams, A. P., Cook, B. I., &amp; Smerdon, J. E. (2022). Rapid
+intensification of the emerging southwestern North American megadrought
+in 2020–2021. Nature Climate Change, 12(3), 232-234.</p>
+</section>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios">A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#background">Background</a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-stationary-generator-to-better-quantify-natural-variability">Synthetic Stationary Generator to Better Quantify Natural Variability</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#non-stationary-synthetic-generator-to-impose-climate-changes">Non-Stationary Synthetic Generator to Impose Climate Changes</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#placing-cmip5-projections-in-the-context-of-non-stationary-flows">Placing CMIP5 Projections in the Context of Non-Stationary Flows</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-create-an-hmm-based-generator-for-your-system">Tips to Create an HMM-Based Generator for your System</a></li>
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+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fishery-dynamics-tutorial">B.1. Fishery Dynamics Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics">B.1.1. Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">B.1.1.1. Let’s get started!</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-identified-solutions-and-explore-performance">B.1.1.2. Step 1: Load identified solutions and explore performance</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis">B.1.1.3. Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-evaluate-the-system-over-all-generated-states-of-the-world">B.1.1.4. Step 3: Evaluate the system over all generated states of the world</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-calculate-sensitivity-indices">B.1.1.5. Step 4: Calculate sensitivity indices</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-explore-relationship-between-uncertain-factors-and-performance">B.1.1.6. Step 5: Explore relationship between uncertain factors and performance</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem">B.1.1.7. Tips to Apply Sobol SA and Scenario Discovery to your Problem</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sobol-sa-tutorial">B.2. Sobol SA Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa">B.2.1. Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id6">B.2.1.1. Let’s get started!</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-generate-the-problem-dictionary">B.2.1.2. Step 1: Generate the problem dictionary</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-generate-samples-using-the-saltelli-sampling-scheme">B.2.1.3. Step 2: Generate samples using the Saltelli sampling scheme</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-execute-the-ishigami-function-over-our-sample-set">B.2.1.4. Step 3: Execute the Ishigami function over our sample set</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method">B.2.1.5. Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-interpret-our-results">B.2.1.6. Step 5: Interpret our results</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-to-your-own-problem">B.2.1.7. Tips to Apply Sobol SA to Your Own Problem</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#logistic-regression-tutorial">B.3. Logistic Regression Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-logistic-regression-for-factor-mapping">B.3.1. Tutorial: Logistic Regression for Factor Mapping</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#background">B.3.1.1. Background</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id19">B.3.1.2. Let’s get started!</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-latin-hypercube-sample-and-set-up-problem">B.3.1.3. Step 1: Load Latin hypercube sample and set up problem</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-define-decision-relevant-metrics-for-illustration">B.3.1.4. Step 2: Define decision-relevant metrics for illustration</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-run-the-logistic-regression">B.3.1.5. Step 3: Run the logistic regression</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-factor-ranking">B.3.1.6. Step 4: Factor ranking</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-draw-factor-maps">B.3.1.7. Step 5: Draw factor maps</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-scenario-discovery-to-your-own-problem">B.3.1.8. Tips to Apply Scenario Discovery to Your Own Problem</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#hymod-dynamics-tutorial">B.4. HYMOD Dynamics Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-of-the-hymod-model">B.4.1. Tutorial: Sensitivity Analysis of the HYMOD Model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction-to-hymod">B.4.2. 1 - Introduction to HYMOD</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#global-sensitivity-analysis">B.4.3. 2- Global Sensitivity Analysis</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-this-methodology-to-your-own-problem">B.4.4. Tips to Apply this methodology to your own problem</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#time-evolving-scenario-discovery-for-infrastructure-pathways">B.5. Time-evolving scenario discovery for infrastructure pathways</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id24">B.5.1. Time-evolving scenario discovery for infrastructure pathways</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id25">B.5.1.1. Background</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluating-robustness-over-time">B.5.1.2. Evaluating Robustness over time</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#exploring-performance-evolution">B.5.1.3. Exploring performance evolution</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#how-do-deep-uncertainties-generate-vulnerability">B.5.1.4. How do deep uncertainties generate vulnerability</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#which-uncertainties-have-the-most-influence-on-time-evolving-performance">B.5.1.5. Which uncertainties have the most influence on time-evolving performance?</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#open-exploration">B.5.1.6. Open exploration</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id26">B.5.1.7. Tips to apply this methodology to your own problem</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#references">B.5.1.8. References</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios-tutorial">B.6. A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios">B.6.1. A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id29">B.6.1.1. Background</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id30">B.6.2. Let’s Get Started!</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#observed-record">B.6.2.1. Observed Record</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-stationary-generator-to-better-quantify-natural-variability">B.6.2.2. Synthetic Stationary Generator to Better Quantify Natural Variability</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#non-stationary-synthetic-generator-to-impose-climate-changes">B.6.2.3. Non-Stationary Synthetic Generator to Impose Climate Changes</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#placing-cmip5-projections-in-the-context-of-non-stationary-flows">B.6.2.4. Placing CMIP5 Projections in the Context of Non-Stationary Flows</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-create-an-hmm-based-generator-for-your-system">B.6.2.5. Tips to Create an HMM-Based Generator for your System</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id31">B.6.2.6. References</a></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="jupyter-notebook-tutorials">
+<span id="a2-jupyter-notebooks"></span><h1><span class="section-number">B. </span>Jupyter Notebook Tutorials<a class="headerlink" href="#jupyter-notebook-tutorials" title="Link to this heading">#</a></h1>
+<section id="fishery-dynamics-tutorial">
+<h2><span class="section-number">B.1. </span>Fishery Dynamics Tutorial<a class="headerlink" href="#fishery-dynamics-tutorial" title="Link to this heading">#</a></h2>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/fishery_dynamics.ipynb">Fishery Dynamics Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics">
+<h3><span class="section-number">B.1.1. </span>Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics<a class="headerlink" href="#tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics" title="Link to this heading">#</a></h3>
+<p>This notebook demonstrates the application of sensitivity analysis to
+discover factors that shape the behavior modes of a socio-ecological
+system with dynamic human action.</p>
+<p>The system of differential equations below represent a system of prey
+(defined in the equation below as x) and predator (defined as y) fish,
+with a human actor harvesting the prey fish. You can read more about
+this system at <a class="reference external" href="https://doi.org/10.1155/2020/4170453">Hadjimichael et
+al. (2020)</a>.</p>
+<img alt="_images/eqn2.png" src="_images/eqn2.png" />
+<p>The table below defines the parameters in the system and also denotes
+the baseline and ranges associated with each uncertain parameter.</p>
+<img alt="_images/table1.png" src="_images/table1.png" />
+<p>The system is simple but very rich in the dynamic behaviors it exhibits.
+This complexity is accompanied by the presence of several equilibria
+that come in and out of existence with different parameter values. The
+equilibria also change in their stability according to different
+parameter values, giving rise to different behavior modes as shown by
+the diverse predator and prey abundace trajectories in the figure below.</p>
+<img alt="_images/Figure_1.png" src="_images/Figure_1.png" />
+<p>In the unharvested system (without the human actor) the stability of
+several of these equilibria can be derived analytically. The task
+becomes significantly more difficult when the adaptive human actor is
+introduced, deciding to harvest the system at different rates according
+to their objectives and preferences.</p>
+<p>Sensitivity analysis methods can help us identify the factors that most
+control these dynamics by exploring the space of parameter values and
+seeing how system outputs change as a result.</p>
+<p>Through previously conducted optimization, there already exists a set of
+potential harvesting strategies that were identified in pursuit of five
+objectives:</p>
+<ul class="simple">
+<li><p>Maximize Harvesting Discounted Profits (Net Present Value)</p></li>
+<li><p>Minimize Prey Population Deficit</p></li>
+<li><p>Minimize Longest Duration of Consecutive Low Harvest</p></li>
+<li><p>Maximize Worst Harvest Instance</p></li>
+<li><p>Minimize Harvest Variance</p></li>
+</ul>
+<p>The identified harvesting strategies also meet the necessary constraint
+of not causing inadvertent predator collapse.</p>
+<p>We will be examining the effects of parametric uncertainty on these
+identified strategies, particularly focusing on two strategies: one
+selected to maximize harvesting profits and one identified through
+previous analysis to perform ‘well enough’ for all objectives across a
+wide range of states of the world (referred to as the ‘robust’
+harvesting policy).</p>
+<section id="lets-get-started">
+<h4><span class="section-number">B.1.1.1. </span>Let’s get started!<a class="headerlink" href="#lets-get-started" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we will be loading in data that has been produced in
+Hadjimichael et al. (2020). Before we start our analysis, we’ll load the
+relevant Python libraries. <strong>NOTE</strong>: To step through the notebook,
+execute each gray (code) box by typing “Shift+Enter”.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Import necessary libraries</span>
+
+<span class="kn">import</span> <span class="nn">msdbook</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">saltelli</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">sobol</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">patheffects</span> <span class="k">as</span> <span class="n">pe</span>
+
+<span class="c1"># load example data</span>
+<span class="n">msdbook</span><span class="o">.</span><span class="n">install_package_data</span><span class="p">()</span>
+
+<span class="o">%</span><span class="k">matplotlib</span> inline
+<span class="o">%</span><span class="k">config</span> InlineBackend.print_figure_kwargs = {&#39;bbox_inches&#39;:None}
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Downloading</span> <span class="n">example</span> <span class="n">data</span> <span class="k">for</span> <span class="n">msdbook</span> <span class="n">version</span> <span class="mf">0.1.5</span><span class="o">...</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">uncertain_params_bounds</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_metric_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">solutions</span><span class="o">.</span><span class="n">resultfile</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LHsamples_original_1000</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">param_values</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_params_256samples</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LeafCatch</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_simulations_256samples</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">Robustness</span><span class="o">.</span><span class="n">txt</span>
+</pre></div>
+</div>
+</section>
+<section id="step-1-load-identified-solutions-and-explore-performance">
+<h4><span class="section-number">B.1.1.2. </span>Step 1: Load identified solutions and explore performance<a class="headerlink" href="#step-1-load-identified-solutions-and-explore-performance" title="Link to this heading">#</a></h4>
+<p>Here we load in the solution set obtained in Hadjimichael et al. (2020).
+The solution set contains the decision variables and objectives
+associated with a variety of harvesting policies. For this tutorial, we
+focus on comparing two policies: harvesting profits and one that
+performs robustly across all objectives. Below, we are reading in the
+decision variables and objectives from an external file that can be
+found within the msdbook package data.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">robustness</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_robustness_data</span><span class="p">()</span>
+<span class="n">results</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_profit_maximization_data</span><span class="p">()</span>
+
+<span class="n">robust_solution</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">robustness</span><span class="p">[:,</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="c1">#pick robust solution</span>
+<span class="n">profit_solution</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">results</span><span class="p">[:,</span><span class="mi">6</span><span class="p">])</span> <span class="c1">#pick profitable solution</span>
+<span class="n">objective_performance</span> <span class="o">=</span> <span class="o">-</span><span class="n">results</span><span class="p">[:,</span><span class="mi">6</span><span class="p">:]</span> <span class="c1">#Retain objective values</span>
+
+<span class="c1"># Get decision variables for each of the policies</span>
+<span class="n">highprofitpolicy</span> <span class="o">=</span> <span class="n">results</span><span class="p">[</span><span class="n">profit_solution</span><span class="p">,</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">]</span>
+<span class="n">mostrobustpolicy</span> <span class="o">=</span> <span class="n">results</span><span class="p">[</span><span class="n">robust_solution</span><span class="p">,</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">]</span>
+</pre></div>
+</div>
+<p>Next we plot the identified solutions with regards to their objective
+performance in a parallel axis plot</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-objective-performance">plot_objective_performance</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ax</span><span class="p">,</span> <span class="n">ax1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_objective_performance</span><span class="p">(</span><span class="n">objective_performance</span><span class="p">,</span> <span class="n">profit_solution</span><span class="p">,</span> <span class="n">robust_solution</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/fishery_output_6_0.png" src="_images/fishery_output_6_0.png" />
+<p>The solution set from the optimization in Hadjimichael et al. (2020) are
+presented in a parallel axis plot where each of the five objectives (and
+one constraint) are represented as an axis. Each solution on the Pareto
+front is represented as a line where the color of the line indicates the
+value of the NPV objective. The preference for objective values is in
+the upward direction. Therefore, the ideal solution would be a line
+straight across the top of the plot that satisfies every objective.
+However, no such line exists because there are tradeoffs when sets of
+objectives are prioritized over the others. When lines cross in between
+axes, this indicates a tradeoff between objectives (as seen in the first
+two axes).The solution that is most robust in the NPV objective has the
+highest value on the first axis and is outlined in dark gold. The
+solution that is most robust across all objectives is outlined in a
+brighter yellow. A parallel axis is an effective visual to characterize
+high-dimensional tradeoffs in the system and visualize differences in
+performance across policies.</p>
+</section>
+<section id="step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis">
+<h4><span class="section-number">B.1.1.3. </span>Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis<a class="headerlink" href="#step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis" title="Link to this heading">#</a></h4>
+<p>In Step 1, we showed how the optimized harvesting policies performed in
+the objective space, which utilized the baseline parameters outlined in
+the table above. Now, we are interested in understanding how sensitive
+our two policies are to alternative states of the world that may be
+characterized by different parameter values. To do so, we first need to
+define the problem dictionary that allows us to generate these
+alternative states of the world.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set up SALib problem</span>
+<span class="n">problem</span> <span class="o">=</span> <span class="p">{</span>
+  <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">9</span><span class="p">,</span>
+  <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;a&#39;</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="s1">&#39;c&#39;</span><span class="p">,</span> <span class="s1">&#39;d&#39;</span><span class="p">,</span> <span class="s1">&#39;h&#39;</span><span class="p">,</span> <span class="s1">&#39;K&#39;</span><span class="p">,</span> <span class="s1">&#39;m&#39;</span><span class="p">,</span> <span class="s1">&#39;sigmaX&#39;</span><span class="p">,</span> <span class="s1">&#39;sigmaY&#39;</span><span class="p">],</span>
+  <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="mf">0.002</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.005</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+             <span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">5000</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">]]</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+<p>Then we use the following command to generate a Saltelli sample from
+these defined ranges:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">param_values</span> <span class="o">=</span> <span class="n">saltelli</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="mi">1024</span><span class="p">,</span> <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Generally, it is a good idea to save the result of the sample since it
+is often reused and regenerating it produces a different sample set. For
+this reason, we will load one from file that was previously generated.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously generated Saltelli sample from our msdbook package data</span>
+<span class="n">param_values</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_saltelli_param_values</span><span class="p">()</span>
+</pre></div>
+</div>
+</section>
+<section id="step-3-evaluate-the-system-over-all-generated-states-of-the-world">
+<h4><span class="section-number">B.1.1.4. </span>Step 3: Evaluate the system over all generated states of the world<a class="headerlink" href="#step-3-evaluate-the-system-over-all-generated-states-of-the-world" title="Link to this heading">#</a></h4>
+<p>Now we re-evaluate how well the policies do in the new states of the
+world. In order to characterize failure of a policy, we identify the
+states where the predator population collapses, as an inadvertent
+consequence of applying the harvesting strategy under a state of the
+world different from the one originally assumed. Due to how long this
+step takes to execute within the tutorial, we will read in the solutions
+from an external file. However, the block of code below shows how
+evaluation can be implemented.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># create array to store collapse values under both policies</span>
+<span class="n">collapse_days</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="nb">len</span><span class="p">(</span><span class="n">param_values</span><span class="p">),</span> <span class="mi">2</span><span class="p">])</span>
+
+<span class="c1"># evaluate performance under every state</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">param_values</span><span class="p">)):</span>
+
+    <span class="n">additional_inputs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="s1">&#39;Previous_Prey&#39;</span><span class="p">],</span>
+                                  <span class="p">[</span><span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">6</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">7</span><span class="p">],</span>
+                                   <span class="n">param_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span>
+
+    <span class="n">collapse_days</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span><span class="n">fish_game</span><span class="p">(</span><span class="n">highprofitpolicy</span><span class="p">,</span> <span class="n">additional_inputs</span><span class="p">)[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+    <span class="n">collapse_days</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">=</span><span class="n">fish_game</span><span class="p">(</span><span class="n">mostrobustpolicy</span><span class="p">,</span> <span class="n">additional_inputs</span><span class="p">)[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load the simulation data from our msdbook package data</span>
+<span class="n">collapse_days</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_collapse_data</span><span class="p">()</span>
+</pre></div>
+</div>
+</section>
+<section id="step-4-calculate-sensitivity-indices">
+<h4><span class="section-number">B.1.1.5. </span>Step 4: Calculate sensitivity indices<a class="headerlink" href="#step-4-calculate-sensitivity-indices" title="Link to this heading">#</a></h4>
+<p>Now we use a Sobol sensitivity analysis to calculate first-order,
+second-order, and total-order sensitivity indices for each parameter and
+for each of the two policies. These indicies help determine which
+factors explain the most variability in the number of days of predator
+population collapse.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Perform the Sobol SA for the profit-maximizing solution</span>
+<span class="n">Si_profit</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="n">collapse_days</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
+                          <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+                          <span class="n">conf_level</span><span class="o">=</span><span class="mf">0.95</span><span class="p">,</span>
+                          <span class="n">print_to_console</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Perform the Sobol SA for the robust solution</span>
+<span class="n">Si_robustness</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span>
+                              <span class="n">collapse_days</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
+                              <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+                              <span class="n">conf_level</span><span class="o">=</span><span class="mf">0.95</span><span class="p">,</span>
+                              <span class="n">print_to_console</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>              <span class="n">ST</span>   <span class="n">ST_conf</span>
+<span class="n">a</span>       <span class="mf">0.226402</span>  <span class="mf">0.036146</span>
+<span class="n">b</span>       <span class="mf">0.066819</span>  <span class="mf">0.013347</span>
+<span class="n">c</span>       <span class="mf">0.004395</span>  <span class="mf">0.004023</span>
+<span class="n">d</span>       <span class="mf">0.024509</span>  <span class="mf">0.006993</span>
+<span class="n">h</span>       <span class="mf">0.009765</span>  <span class="mf">0.005488</span>
+<span class="n">K</span>       <span class="mf">0.020625</span>  <span class="mf">0.009494</span>
+<span class="n">m</span>       <span class="mf">0.897971</span>  <span class="mf">0.066470</span>
+<span class="n">sigmaX</span>  <span class="mf">0.000136</span>  <span class="mf">0.000149</span>
+<span class="n">sigmaY</span>  <span class="mf">0.000739</span>  <span class="mf">0.001040</span>
+              <span class="n">S1</span>   <span class="n">S1_conf</span>
+<span class="n">a</span>       <span class="mf">0.087936</span>  <span class="mf">0.044236</span>
+<span class="n">b</span>       <span class="mf">0.000554</span>  <span class="mf">0.021474</span>
+<span class="n">c</span>      <span class="o">-</span><span class="mf">0.002970</span>  <span class="mf">0.004590</span>
+<span class="n">d</span>       <span class="mf">0.001206</span>  <span class="mf">0.015881</span>
+<span class="n">h</span>       <span class="mf">0.004554</span>  <span class="mf">0.007998</span>
+<span class="n">K</span>       <span class="mf">0.003843</span>  <span class="mf">0.012661</span>
+<span class="n">m</span>       <span class="mf">0.751301</span>  <span class="mf">0.071862</span>
+<span class="n">sigmaX</span> <span class="o">-</span><span class="mf">0.000325</span>  <span class="mf">0.001245</span>
+<span class="n">sigmaY</span> <span class="o">-</span><span class="mf">0.001887</span>  <span class="mf">0.002768</span>
+</pre></div>
+</div>
+<p>Looking at the total-order indices, (ST) factors <span class="math notranslate nohighlight">\(m\)</span>, <span class="math notranslate nohighlight">\(a\)</span>,
+<span class="math notranslate nohighlight">\(b\)</span>, <span class="math notranslate nohighlight">\(d\)</span> and <span class="math notranslate nohighlight">\(K\)</span> explain a non-negligible amount of
+variance therefore have an effect on the stability of this system.
+Looking at the first-order indices (S1), we also see that besides
+factors <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(a\)</span>, all other factors are important in this
+system through their interactions, which make up the difference between
+their S1 and ST indices. This shows the danger of limiting sensitivity
+analyses to first order effects, as factor importance might be
+significantly misjudged.</p>
+<p>These findings are supported by the analytical condition of equilibrium
+stability in this system:</p>
+<img alt="_images/eqn4.png" src="_images/eqn4.png" />
+<p>In an unharvested system, this condition is both necessary and
+sufficient for the equilibrium of the two species coexisting to be
+stable.</p>
+<p>When adaptive human action is introduced however, this condition is
+still necessary, but no longer sufficient, as harvesting reduces the
+numbers of prey fish and as a result reduces the resources for the
+predator fish. Since this harvesting value is not constant, but can
+dynamically adapt according to the harvester’s objectives, it cannot be
+introduced into this simple equation.</p>
+</section>
+<section id="step-5-explore-relationship-between-uncertain-factors-and-performance">
+<h4><span class="section-number">B.1.1.6. </span>Step 5: Explore relationship between uncertain factors and performance<a class="headerlink" href="#step-5-explore-relationship-between-uncertain-factors-and-performance" title="Link to this heading">#</a></h4>
+<p>In the following steps, we will use the results of our sensitivity
+analysis to investigate the relationships between parametric
+uncertainty, equilibrium stability and the performance of the two
+policies.</p>
+<p>We can use the top three factors identified (<span class="math notranslate nohighlight">\(m\)</span>, <span class="math notranslate nohighlight">\(a\)</span>, and
+<span class="math notranslate nohighlight">\(b\)</span>) to visualize the performance of our policies in this
+three-dimensional parametric space.</p>
+<p>We first define the stability condition, as a function of <span class="math notranslate nohighlight">\(b\)</span> and
+<span class="math notranslate nohighlight">\(m\)</span>, and calculate the corresponding values of <span class="math notranslate nohighlight">\(a\)</span>.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">inequality</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
+    <span class="k">return</span> <span class="p">((</span><span class="n">b</span><span class="o">**</span><span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">h</span><span class="o">*</span><span class="n">K</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">m</span><span class="p">))</span>
+
+<span class="c1"># boundary interval that separates successful and failed states of the world</span>
+<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mf">0.005</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">m</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">K</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">2000</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
+<span class="n">b</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">inequality</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">clip</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-factor-performance">plot_factor_performance</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># generate plot</span>
+<span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_factor_performance</span><span class="p">(</span><span class="n">param_values</span><span class="p">,</span> <span class="n">collapse_days</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/fishery_output_22_0.png" src="_images/fishery_output_22_0.png" />
+<p>These figures show the combinations of factors that lead to success or
+failure in different states of the world for the profit-maximizing and
+robust policies. Each point is a state of the world, characterized by
+specific values of the parameters, and ideally, we would like the color
+of the point to be blue, to represent that there are a low number of
+days with a predator collapse in that world. The gray curve denotes the
+highly non-linear nature of the boundary, defined by the stability
+condition, that separates successful and failed states of the world. The
+figures demonstrate the following key points:</p>
+<p>First, as asserted above, the policies interact with the system in
+different and complex ways. In the presence of human action, the
+stability condition is not sufficient in determining whether the policy
+will succeed, even though it clearly shapes the system in a fundamental
+manner.</p>
+<p>Secondly, the robust policy manages to avoid collapse in many more of the sampled states of the world, indicated by the number of blue points. The robust policy avoids collapse in 31% of worlds versus 14% in the profit-maximizing policy.This presents a clear tradeoff between profit-maximizing performance androbustness against uncertainty.</p>
+</section>
+<section id="tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem">
+<h4><span class="section-number">B.1.1.7. </span>Tips to Apply Sobol SA and Scenario Discovery to your Problem<a class="headerlink" href="#tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we demonstrated a Sobol SA to identify the most
+important factors driving the behavior of a system (i.e. the number of
+the collapse days). In order to apply this methodology to your problem,
+you will need to have a set of optimized policies for your system that
+you are interested in analyzing. The general workflow is as follows:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters and set up the problem
+dictionary as in Step 2 above.</p></li>
+<li><p>Generate samples, or alternative states of the world using the
+<code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function.</p></li>
+<li><p>Evaluate your policies on the alternative states of the world. For
+your application, you will also need to develop a rule for
+determining success or failure of your policy in a new SOW. In this
+tutorial, success was denoted by a small number of collapse days.
+Ultimately, the rule will be specific to your application and can
+include various satisficing criteria.</p></li>
+<li><p>Calculate the Sobol indices and discover the most important
+parameters driving success and failure.</p></li>
+<li><p>Finally, use a similar plotting procedure as in step 5 to identify
+the combination of parameter values that lead to success and failure
+in the system.</p></li>
+</ol>
+</section>
+</section>
+</section>
+<section id="sobol-sa-tutorial">
+<h2><span class="section-number">B.2. </span>Sobol SA Tutorial<a class="headerlink" href="#sobol-sa-tutorial" title="Link to this heading">#</a></h2>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol SA Tutorial</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa">
+<h3><span class="section-number">B.2.1. </span>Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA<a class="headerlink" href="#tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we will set up a workflow to investigate how sensitive
+the output of a function is to its inputs. Why might you want to do
+this? Imagine that this function represents a complex system, such as
+the rainfall-runoff process of a watershed model, and that you, the
+researcher, want to investigate how your choice of input parameter
+values are affecting the model’s characterization of runoff in the
+watershed. Your parameter values are likely uncertain and can take on
+any value in a pre-defined range. Using a Sobol SA will allow you to
+sample different values of your parameters and calculate how sensitive
+your output of interest is to certain parameters. Below, we demonstrate
+Sobol SA for a simple function to illustrate the method, but the
+workflow can be applied to your own problem of interest!</p>
+<p>In order to conduct this analysis, we will use the popular Python
+Sensitivity Analysis Library
+(<a class="reference external" href="https://salib.readthedocs.io/en/latest/index.html">SALib</a>) to:</p>
+<ol class="arabic simple">
+<li><p>Generate a problem set as a dictionary for our Ishigami function that has three inputs</p></li>
+<li><p>Generate 2048 samples for our problem set using the Saltelli <a class="footnote-reference brackets" href="#id10" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id11" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a> sampling scheme</p></li>
+<li><p>Execute the Ishigami function for each of our samples and gather the outputs</p></li>
+<li><p>Compute the sensitivity analysis to generate first-order and total-order sensitivity indices using the Sobol <a class="footnote-reference brackets" href="#id12" id="id5" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a> method</p></li>
+<li><p>Interpret the meaning of our results</p></li>
+</ol>
+<section id="id6">
+<h4><span class="section-number">B.2.1.1. </span>Let’s get started!<a class="headerlink" href="#id6" title="Link to this heading">#</a></h4>
+<p><strong>NOTE</strong>: Content from this tutorial is taken directly from the SALib
+<a class="reference external" href="https://salib.readthedocs.io/en/latest/basics.html">“Basics”</a>
+walkthrough. To step through the notebook, execute each gray (code) box
+by typing “Shift+Enter”.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Import relevant libraries</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">from</span> <span class="nn">mpl_toolkits</span> <span class="kn">import</span> <span class="n">mplot3d</span>
+
+<span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">saltelli</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">sobol</span>
+<span class="kn">from</span> <span class="nn">SALib.test_functions</span> <span class="kn">import</span> <span class="n">Ishigami</span>
+</pre></div>
+</div>
+</section>
+<section id="step-1-generate-the-problem-dictionary">
+<h4><span class="section-number">B.2.1.2. </span>Step 1: Generate the problem dictionary<a class="headerlink" href="#step-1-generate-the-problem-dictionary" title="Link to this heading">#</a></h4>
+<p>The Ishigami function is of the form:</p>
+<div class="math notranslate nohighlight">
+\[f(x_1,x_2,x_3) = sin(x_1)+asin^2(x_2)+bx_3^4sin(x_1)\]</div>
+<p>The function has three inputs, 𝑥1, 𝑥2, 𝑥3 where 𝑥𝑖 ∈ [−𝜋, 𝜋]. The
+constants <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are defined as 7.0 and 0.1
+respectively.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Create a problem dictionary. Here we supply the number of variables, the names of each variable, and the bounds of the variables.</span>
+<span class="n">problem</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span>
+    <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;x1&#39;</span><span class="p">,</span> <span class="s1">&#39;x2&#39;</span><span class="p">,</span> <span class="s1">&#39;x3&#39;</span><span class="p">],</span>
+    <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="o">-</span><span class="mf">3.14159265359</span><span class="p">,</span> <span class="mf">3.14159265359</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">3.14159265359</span><span class="p">,</span> <span class="mf">3.14159265359</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">3.14159265359</span><span class="p">,</span> <span class="mf">3.14159265359</span><span class="p">]]</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+</section>
+<section id="step-2-generate-samples-using-the-saltelli-sampling-scheme">
+<h4><span class="section-number">B.2.1.3. </span>Step 2: Generate samples using the Saltelli sampling scheme<a class="headerlink" href="#step-2-generate-samples-using-the-saltelli-sampling-scheme" title="Link to this heading">#</a></h4>
+<p>Sobol SA requires the use of the Saltelli sampling scheme. The output of
+the <code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function is a NumPy array that is of shape 2048
+by 3. The sampler generates 𝑁∗(2𝐷+2) samples, where in this example, N
+is 256 (the argument we supplied) and D is 3 (the number of model
+inputs), yielding 2048 samples. The keyword argument
+<code class="docutils literal notranslate"><span class="pre">calc_second_order=False</span></code> will exclude second-order indices, resulting
+in a smaller sample matrix with 𝑁∗(𝐷+2) rows instead. Below, we plot the
+resulting Saltelli sample.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Generate parmeter values using the saltelli.sample function</span>
+<span class="n">param_values</span> <span class="o">=</span> <span class="n">saltelli</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="mi">256</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;`param_values` shape:  </span><span class="si">{</span><span class="n">param_values</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<pre class="literal-block"><cite>param_values</cite> shape:  (2048, 3)</pre>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Plot the 2048 samples of the parameters</span>
+
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
+<span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">axes</span><span class="p">(</span><span class="n">projection</span> <span class="o">=</span><span class="s2">&quot;3d&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter3D</span><span class="p">(</span><span class="n">param_values</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">param_values</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">param_values</span><span class="p">[:,</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;X1 Parameter&#39;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;X2 Parameter&#39;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s1">&#39;X3 Parameter&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Saltelli Sample of Parameter Values&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/output_7_0.png" src="_images/output_7_0.png" />
+</section>
+<section id="step-3-execute-the-ishigami-function-over-our-sample-set">
+<h4><span class="section-number">B.2.1.4. </span>Step 3: Execute the Ishigami function over our sample set<a class="headerlink" href="#step-3-execute-the-ishigami-function-over-our-sample-set" title="Link to this heading">#</a></h4>
+<p>SALib provides a nice wrapper to the Ishigami function that allows the
+user to directly pass the <code class="docutils literal notranslate"><span class="pre">param_values</span></code> array we just generated into
+the function directly.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Y</span> <span class="o">=</span> <span class="n">Ishigami</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">param_values</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method">
+<h4><span class="section-number">B.2.1.5. </span>Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method<a class="headerlink" href="#step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method" title="Link to this heading">#</a></h4>
+<p>The <code class="docutils literal notranslate"><span class="pre">sobol.analyze</span></code> function will use our problem dictionary and the
+result of the Ishigami runs (<code class="docutils literal notranslate"><span class="pre">Y</span></code>) to compute first-, second-, and
+total-order indicies.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Si</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><code class="docutils literal notranslate"><span class="pre">Si</span></code> is a Python dict with the keys “S1”, “S2”, “ST”, “S1_conf”,
+“S2_conf”, and “ST_conf”. The <code class="docutils literal notranslate"><span class="pre">_conf</span></code> keys store the corresponding
+confidence intervals, typically with a confidence level of 95%. Use the
+keyword argument <code class="docutils literal notranslate"><span class="pre">print_to_console=True</span></code> to print all indices. Or, we
+can print the individual values from <code class="docutils literal notranslate"><span class="pre">Si</span></code> as shown in the next step.</p>
+</section>
+<section id="step-5-interpret-our-results">
+<h4><span class="section-number">B.2.1.6. </span>Step 5: Interpret our results<a class="headerlink" href="#step-5-interpret-our-results" title="Link to this heading">#</a></h4>
+<p>We execute the following code and take a look at our first-order indices
+(<code class="docutils literal notranslate"><span class="pre">S1</span></code>) for each of our three inputs. These indicies can be interpreted
+as the fraction of variance in the output that is explained by each
+input individually.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">first_order</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">]</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;First-order:&#39;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1: </span><span class="si">{</span><span class="n">first_order</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">, x2: </span><span class="si">{</span><span class="n">first_order</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">, x3: </span><span class="si">{</span><span class="n">first_order</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">First</span><span class="o">-</span><span class="n">order</span><span class="p">:</span>
+<span class="n">x1</span><span class="p">:</span> <span class="mf">0.3184242969763115</span><span class="p">,</span> <span class="n">x2</span><span class="p">:</span> <span class="mf">0.4303808201623416</span><span class="p">,</span> <span class="n">x3</span><span class="p">:</span> <span class="mf">0.022687722804980225</span>
+</pre></div>
+</div>
+<p>If we were to rank the importance of the inputs in how much they
+individually explain the variance in the output, we would rank them from
+greatest to least importance as follows: 𝑥2, 𝑥1 and then 𝑥3. Since 𝑥3
+only explains 2% of the output variance, it does not explain output
+variability meaningfully. Thus, this indicates that there is
+contribution to the output variance by 𝑥2 and 𝑥1 independently, whereas
+𝑥3 does not contribute to the output variance. Determining what inputs
+are most important or what index value is meaningful is a common
+question, but one for which there is no general rule or threshold. This
+question is problem and context-dependent, but procedures have been
+identified to rank order influential inputs and which can be used to
+identify the least influential factors. These factors can be fixed to
+simplify the model <a class="footnote-reference brackets" href="#id13" id="id7" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id15" id="id8" role="doc-noteref"><span class="fn-bracket">[</span>5<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id16" id="id9" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a>.</p>
+<p>Next, we evaluate the total-order indices, which measure the
+contribution to the output variance caused by varying the model input,
+including both its first-order effects (the input varying alone) and all
+higher-order interactions across the input parameters.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">total_order</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">]</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Total-order:&#39;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1: </span><span class="si">{</span><span class="n">total_order</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">, x2: </span><span class="si">{</span><span class="n">total_order</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">, x3: </span><span class="si">{</span><span class="n">total_order</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Total</span><span class="o">-</span><span class="n">order</span><span class="p">:</span>
+<span class="n">x1</span><span class="p">:</span> <span class="mf">0.5184119098161343</span><span class="p">,</span> <span class="n">x2</span><span class="p">:</span> <span class="mf">0.41021260250026054</span><span class="p">,</span> <span class="n">x3</span><span class="p">:</span> <span class="mf">0.2299058431439953</span>
+</pre></div>
+</div>
+<p>The magnitude of the total order indices are substantially larger than
+the first-order indices, which reveals that higher-order interactions
+are occurring, i.e. that the interactions across inputs are also
+explaining some of the total variance in the output. Note that 𝑥3 has
+non-negligible total-order indices, which indicates that it is not a
+consequential parameter when considered in isolation, but becomes
+consequential and explains 25% of variance in the output through its
+interactions with 𝑥1 and 𝑥2.</p>
+<p>Finally, we can investigate these higher order interactions by viewing
+the second-order indices. The second-order indicies measure the
+contribution to the output variance caused by the interaction between
+any two model inputs. Some computing error can appear in these
+sensitivity indices, such as negative values. Typically, these computing
+errors shrink as the number of samples increases.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">second_order</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S2&#39;</span><span class="p">]</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Second-order:&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1-x2:  </span><span class="si">{</span><span class="n">second_order</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x1-x3:  </span><span class="si">{</span><span class="n">second_order</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x2-x3:  </span><span class="si">{</span><span class="n">second_order</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Second</span><span class="o">-</span><span class="n">order</span><span class="p">:</span>
+<span class="n">x1</span><span class="o">-</span><span class="n">x2</span><span class="p">:</span>  <span class="o">-</span><span class="mf">0.043237389723234154</span>
+<span class="n">x1</span><span class="o">-</span><span class="n">x3</span><span class="p">:</span>  <span class="mf">0.17506452088709862</span>
+<span class="n">x2</span><span class="o">-</span><span class="n">x3</span><span class="p">:</span>  <span class="o">-</span><span class="mf">0.03430682392607577</span>
+</pre></div>
+</div>
+<p>We can see that there are strong interactions between 𝑥1 and 𝑥3. Note
+that in the Ishigami function, these two variables are multiplied in the
+last term of the function, which leads to interactive effects. If we
+were considering first order indices alone, we would erroneously assume
+that 𝑥3 explains no variance in the output, but the second-order and
+total order indices reveal that this is not the case. It’s easy to
+understand where we might see interactive effects in the case of the
+simple Ishigami function. However, it’s important to remember that in
+more complex systems, there may be many higher-order interactions that
+are not apparent, but could be extremely consequential in explaining the
+variance of the output.</p>
+</section>
+<section id="tips-to-apply-sobol-sa-to-your-own-problem">
+<h4><span class="section-number">B.2.1.7. </span>Tips to Apply Sobol SA to Your Own Problem<a class="headerlink" href="#tips-to-apply-sobol-sa-to-your-own-problem" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we demonstrated how to apply an SA analysis to a
+simple mathematical test function. In order to apply a Sobol SA to your
+own problem, you will follow the same general workflow that we defined
+above. You will need to:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters and set up the problem
+dictionary as in Step 1 above.</p></li>
+<li><p>Generate samples using the <code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function. This step is
+problem-dependent and note that the Sobol method can be
+computationally intensive depending on the model being analyzed. For
+example, for a simple rainfall-runoff model such as HYMOD, it has
+been recommended to run a sample size of at least N = 10,000 (which
+translates to 60,000 model runs). More complex models will be slower
+to run and will also require more samples to calculate accurate
+estimates of Sobol indices. Once you complete this process, pay
+attention to the confidence bounds on your sensitivity indices to see
+whether you need to run more samples.</p></li>
+<li><p>Run the parameter sets through your model. In the example above, the
+Ishigami function could be evaluated through SALib since it is a
+built in function. For your application, you will need to run these
+parameter sets through the problem externally and save the output.
+The output file should contain one row of output values for each
+model run.</p></li>
+<li><p>Calculate the Sobol indices. Now, the Y will be a numpy array with
+your external model output and you will need to include the parameter
+samples as an additional argument.</p></li>
+<li><p>Finally, we interpet the results. If the confidence intervals of your
+dominant indices are larger than roughly 10% of the value itself, you
+may want to consider increasing your sample size as computation
+permits. You should additionally read the references noted in Step 5
+above to understand more about identifying important factors.</p></li>
+</ol>
+<p><strong>References</strong></p>
+<aside class="footnote-list brackets">
+<aside class="footnote brackets" id="id10" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">1</a><span class="fn-bracket">]</span></span>
+<p>Saltelli, A. (2002). “Making best use of model evaluations to compute sensitivity indices.” Computer Physics Communications, 145(2):280-297, doi:<a class="reference external" href="https://doi.org/10.1016/S0010-4655(02)00280-1">10.1016/S0010-4655(02)00280-1</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id11" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">2</a><span class="fn-bracket">]</span></span>
+<p>Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and S. Tarantola (2010). “Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index.” Computer Physics Communications, 181(2):259-270, doi:<a class="reference external" href="https://doi.org/10.1016/j.cpc.2009.09.018">10.1016/j.cpc.2009.09.018</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id12" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id5">3</a><span class="fn-bracket">]</span></span>
+<p>Sobol, I. M. (2001). “Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates.” Mathematics and Computers in Simulation, 55(1-3):271-280, doi:<a class="reference external" href="https://doi.org/10.1016/S0378-4754(00)00270-6">10.1016/S0378-4754(00)00270-6</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id13" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id7">4</a><span class="fn-bracket">]</span></span>
+<p>T. H. Andres, “Sampling methods and sensitivity analysis for large parameter sets,” Journal of Statistical Computation and Simulation, vol. 57, no. 1–4, pp. 77–110, Apr. 1997, doi:<a class="reference external" href="https://doi.org/10.1080/00949659708811804">10.1080/00949659708811804</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id15" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id8">5</a><span class="fn-bracket">]</span></span>
+<p>Y. Tang, P. Reed, T. Wagener, and K. van Werkhoven, “Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation,” Hydrology and Earth System Sciences, vol. 11, no. 2, pp. 793–817, Feb. 2007, doi:<a class="reference external" href="https://doi.org/10.5194/hess-11-793-2007">10.5194/hess-11-793-2007</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id16" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id9">6</a><span class="fn-bracket">]</span></span>
+<p>J. Nossent, P. Elsen, and W. Bauwens, “Sobol’ sensitivity analysis of a complex environmental model,” Environmental Modelling &amp; Software, vol. 26, no. 12, pp. 1515–1525, Dec. 2011, doi:<a class="reference external" href="https://doi.org/10.1016/j.envsoft.2011.08.010">10.1016/j.envsoft.2011.08.010</a>.</p>
+</aside>
+</aside>
+</section>
+</section>
+</section>
+<section id="logistic-regression-tutorial">
+<h2><span class="section-number">B.3. </span>Logistic Regression Tutorial<a class="headerlink" href="#logistic-regression-tutorial" title="Link to this heading">#</a></h2>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/basin_users_logistic_regression.ipynb">Logistic Regression Tutorial</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-logistic-regression-for-factor-mapping">
+<h3><span class="section-number">B.3.1. </span>Tutorial: Logistic Regression for Factor Mapping<a class="headerlink" href="#tutorial-logistic-regression-for-factor-mapping" title="Link to this heading">#</a></h3>
+<p>This tutorial replicates a scenario discovery analysis performed in
+<a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020EF001503">Hadjimichael et
+al. (2020)</a>.</p>
+<section id="background">
+<h4><span class="section-number">B.3.1.1. </span>Background<a class="headerlink" href="#background" title="Link to this heading">#</a></h4>
+<p>Planners in the the Upper Colorado River Basin (UCRB, shown in the
+figure below) are seeking to understand the vulnerability of water users
+to uncertainties stemming from climate change, population growth and
+water policy changes. The UCRB spans 25,682 km2 in western Colorado and
+is home to approximately 300,000 residents and 1,012 km2 of irrigated
+land. Several thousand irrigation ditches divert water from the main
+river and its tributaties for irrigation (shown as small black dots in
+the figure). Transmountain diversions of approximately 567,400,000 m3
+per year are exported for irrigation, industrial and municipal uses in
+northern and eastern Colorado, serving the major population centers of
+Denver and Colorado Springs. These diversions are carried through
+tunnels, shown as large black dots in the figure.</p>
+<img alt="_images/basin_map.png" src="_images/basin_map.png" />
+<p>An important planning consideration is the water rights of each user,
+defined by seniority across all water uses (irrigation diversions,
+transboundary diversions, power plants etc.) in the basin. To assess the
+vulnerability of users with varying degrees of water rights seniority,
+planners simulate the system across an ensemble of scenarios using the
+state of Colorado’s StateMod platform. The model simulates streamflow,
+diversions, instream demands, and reservoir operations.</p>
+<p>Hadjimichael et al. (2020) employ an exploratory analysis by simulating
+a large ensemble of plausible scenarios using StateMod and then
+identifying consequential decision-relevant combinations of uncertain
+factors, termed scenario discovery. Focusing on decision-relevant
+metrics (metrics that are important to the user, the scenario discovery
+is applied to the water shortages experienced by each individual user
+(i.e., not on a single basin-wide or sector-wide metric). For this
+training example, we’ll be performing scenario discovery for three
+different water users: two irrigation users and one municipal user.</p>
+</section>
+<section id="id19">
+<h4><span class="section-number">B.3.1.2. </span>Let’s get started!<a class="headerlink" href="#id19" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we will be loading in data that has been produced in
+Hadjimichael et al. (2020). Before we start our analysis, we’ll load the
+relevant Python libraries, example data, and information for the three
+users.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#import libraries</span>
+<span class="kn">import</span> <span class="nn">msdbook</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="nn">mpl</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="c1"># load example data from Hadjimichael et al. (2020)</span>
+<span class="n">msdbook</span><span class="o">.</span><span class="n">install_package_data</span><span class="p">()</span>
+
+<span class="c1"># Select the IDs for the three users that we will perform the analysis for</span>
+<span class="n">all_IDs</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;7000550&#39;</span><span class="p">,</span><span class="s1">&#39;7200799&#39;</span><span class="p">,</span><span class="s1">&#39;3704614&#39;</span><span class="p">]</span>
+<span class="n">usernames</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Medium seniority irrigation&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Low seniority irrigation&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Transbasin municipal diversion&#39;</span><span class="p">]</span>
+<span class="n">nStructures</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Downloading</span> <span class="n">example</span> <span class="n">data</span> <span class="k">for</span> <span class="n">msdbook</span> <span class="n">version</span> <span class="mf">0.1.5</span><span class="o">...</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">uncertain_params_bounds</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_metric_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">solutions</span><span class="o">.</span><span class="n">resultfile</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LHsamples_original_1000</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">param_values</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_params_256samples</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LeafCatch</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_simulations_256samples</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">Robustness</span><span class="o">.</span><span class="n">txt</span>
+</pre></div>
+</div>
+</section>
+<section id="step-1-load-latin-hypercube-sample-and-set-up-problem">
+<h4><span class="section-number">B.3.1.3. </span>Step 1: Load Latin hypercube sample and set up problem<a class="headerlink" href="#step-1-load-latin-hypercube-sample-and-set-up-problem" title="Link to this heading">#</a></h4>
+<p>To examine regional vulnerability, we generate an ensemble of plausible
+future states of the world (SOWs) using Latin Hypercube Sampling. For
+this tutorial, we’ll load a file containing 1,000 samples across 14
+parameters. The sampled parameters encompass plausible changes to the
+future state of the basin, including changes to hydrology, water demands
+(irrigation, municipal &amp; industry, transbasin), and institutional and
+environmental factors (environmental flows, reservoir storage, operation
+of the Shoshone Power Plant). These samples are taken from ranges
+identified in <code class="docutils literal notranslate"><span class="pre">param_bounds</span></code>. Below we load in the 1000 samples, the
+range of values that the samples can take for each parameter, and the
+parameter names. More information on what each parameter constitutes can
+be found in Table 1 of Hadjimichael et al., 2020.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Identify the bounds for each of the 14 parameters</span>
+<span class="n">param_bounds</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_basin_param_bounds</span><span class="p">()</span>
+
+<span class="c1">#Load in the parameter samples</span>
+<span class="n">LHsamples</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_lhs_basin_sample</span><span class="p">()</span>
+
+<span class="c1">#Create an array of the parameter names</span>
+<span class="n">param_names</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Irrigation demand multiplier&#39;</span><span class="p">,</span><span class="s1">&#39;Reservoir loss&#39;</span><span class="p">,</span><span class="s1">&#39;Transbasin demand multiplier&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Municipal &amp; industrial multiplier&#39;</span><span class="p">,</span> <span class="s1">&#39;Shoshone&#39;</span><span class="p">,</span><span class="s1">&#39;Environmental flows&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Evaporation change&#39;</span><span class="p">,</span><span class="s1">&#39;Mean dry flow&#39;</span><span class="p">,</span><span class="s1">&#39;Dry flow variance&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Mean wet flow&#39;</span><span class="p">,</span><span class="s1">&#39;Wet flow variance&#39;</span><span class="p">,</span><span class="s1">&#39;Dry-dry probability&#39;</span><span class="p">,</span>
+             <span class="s1">&#39;Wet-wet probability&#39;</span><span class="p">,</span> <span class="s1">&#39;Snowmelt shift&#39;</span><span class="p">]</span>
+</pre></div>
+</div>
+</section>
+<section id="step-2-define-decision-relevant-metrics-for-illustration">
+<h4><span class="section-number">B.3.1.4. </span>Step 2: Define decision-relevant metrics for illustration<a class="headerlink" href="#step-2-define-decision-relevant-metrics-for-illustration" title="Link to this heading">#</a></h4>
+<p>Scenario discovery attempts to identify parametric regions that lead to
+‘success’ and ‘failure’. For this demonstration we’ll be defining
+‘success’ as states of the world where a shortage level doesn’t exceed
+its historical frequency.</p>
+</section>
+<section id="step-3-run-the-logistic-regression">
+<h4><span class="section-number">B.3.1.5. </span>Step 3: Run the logistic regression<a class="headerlink" href="#step-3-run-the-logistic-regression" title="Link to this heading">#</a></h4>
+<p>Logistic regression estimates the probability that a future SOW will be
+classified as a success or failure given a set of performance criteria.
+A logistic regression model is defined by:</p>
+<div class="math notranslate nohighlight">
+\[ln \bigg (\frac{p_i}{1-p_i} \bigg ) = X^T_i \beta\]</div>
+<p>where <span class="math notranslate nohighlight">\(p_i\)</span> is the probability the performance in the
+<span class="math notranslate nohighlight">\(i^{th}\)</span> SOW will be classified as a success, <span class="math notranslate nohighlight">\(X_i\)</span> is the
+vector of covariates describing the <span class="math notranslate nohighlight">\(i^{th}\)</span> SOW, and
+<span class="math notranslate nohighlight">\(\beta\)</span> is the vector of coefficients describing the relationship
+between the covariates and the response, which here will be estimated
+using maximum likelihood estimation.</p>
+<p>A logistic regression model was fit to the ensemble of SOWs using the
+performance criteria defined in step 2. Logistic regression modeling was
+conducted using the <a class="reference external" href="https://www.statsmodels.org/stable/index.html">Statsmodel
+Python</a> package. The
+data required for the full analysis is too large to include in this
+tutorial, but results can be found in the data file loaded below.</p>
+<p>The results files contain the occurence of different shortage frequency
+and magnitude combinations under the experiment, in increments of 10,
+between 0 and 100. These combinations (100 for each user) are
+alternative decision-relevant metrics that can be used for scenario
+discovery.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set arrays for shortage frequencies and magnitudes</span>
+<span class="n">frequencies</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">110</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+<span class="n">magnitudes</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">110</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+<span class="n">realizations</span> <span class="o">=</span> <span class="mi">10</span>
+
+<span class="c1"># Load performance and pseudo r scores for each of the users</span>
+<span class="n">results</span> <span class="o">=</span> <span class="p">[</span><span class="n">msdbook</span><span class="o">.</span><span class="n">load_user_heatmap_array</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">/</span> <span class="mi">100</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">))]</span>
+</pre></div>
+</div>
+</section>
+<section id="step-4-factor-ranking">
+<h4><span class="section-number">B.3.1.6. </span>Step 4: Factor ranking<a class="headerlink" href="#step-4-factor-ranking" title="Link to this heading">#</a></h4>
+<p>To rank the importance of each uncertain factor, we utilize McFadden’s
+psuedo-R2, a measure that quantifies the improvement of the model when
+utilizing each given predictor as compared to prediction using the mean
+of the data set:</p>
+<div class="math notranslate nohighlight">
+\[R^2_{McFadden}=1-\frac{ln \hat{L}(M_{full})}{ln \hat{L}(M_{intercept})}\]</div>
+<p>Here <span class="math notranslate nohighlight">\(ln \hat{L}(M_{full})\)</span> is the log likelihood of the full
+model (including the predictor) and <span class="math notranslate nohighlight">\(ln \hat{L}(M_{intercept})\)</span> is
+the log likelihood of the intercept model (which predicts the mean
+probability of success across all SOWs).</p>
+<p>Higher values of McFadden’s psuedo-R2 indicate higher factor importance
+(when the likelihood of the full model approaches one, the ratio of the
+likelihood of the full model compared to the intercept model will get
+very small).</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Load the pseudo-R^2 scores</span>
+<span class="n">scores</span> <span class="o">=</span> <span class="p">[</span><span class="n">msdbook</span><span class="o">.</span><span class="n">load_user_pseudo_scores</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">all_IDs</span><span class="p">))]</span>
+
+<span class="c1"># Select indices of frequency and magnitudes that will be used for the visualization</span>
+<span class="n">freq</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
+<span class="n">mag</span> <span class="o">=</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">7</span><span class="p">]</span>
+</pre></div>
+</div>
+</section>
+<section id="step-5-draw-factor-maps">
+<h4><span class="section-number">B.3.1.7. </span>Step 5: Draw factor maps<a class="headerlink" href="#step-5-draw-factor-maps" title="Link to this heading">#</a></h4>
+<p>The McFadden’s psuedo-R2 scores files contain preliminary logistic
+regression results on parameter importance for each of these
+combinations. Using these psuedo-R2 scores we will identify the two most
+important factors for each metric which we’ll use to generate the final
+scenario discovery maps (note: there may be more than two important
+metrics for each user, but here we will demonstrate by mapping two).</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># setup figure</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">18</span><span class="p">),</span> <span class="n">tight_layout</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">patch</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">axes</span><span class="o">.</span><span class="n">flat</span><span class="p">)):</span>
+
+    <span class="n">ax</span> <span class="o">=</span> <span class="n">axes</span><span class="o">.</span><span class="n">flat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+    <span class="n">allSOWsperformance</span> <span class="o">=</span> <span class="n">results</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+    <span class="n">all_pseudo_r_scores</span> <span class="o">=</span> <span class="n">scores</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+    <span class="c1"># construct dataframe</span>
+    <span class="n">dta</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">LHsamples</span><span class="p">,</span> <span class="n">realizations</span><span class="p">,</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">),</span> <span class="n">columns</span><span class="o">=</span><span class="n">param_names</span><span class="p">)</span>
+    <span class="n">dta</span><span class="p">[</span><span class="s1">&#39;Success&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">allSOWsperformance</span><span class="p">[</span><span class="n">freq</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">mag</span><span class="p">[</span><span class="n">i</span><span class="p">],:]</span>
+
+    <span class="n">pseudo_r_scores</span> <span class="o">=</span> <span class="n">all_pseudo_r_scores</span><span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">frequencies</span><span class="p">[</span><span class="n">freq</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span><span class="o">+</span><span class="s1">&#39;yrs_&#39;</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">magnitudes</span><span class="p">[</span><span class="n">mag</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span><span class="o">+</span><span class="s1">&#39;prc&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>
+    <span class="n">top_predictors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">pseudo_r_scores</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">][:</span><span class="mi">2</span><span class="p">]</span> <span class="c1">#Sort scores and pick top 2 predictors</span>
+
+    <span class="c1"># define color map for dots representing SOWs in which the policy</span>
+    <span class="c1"># succeeds (light blue) and fails (dark red)</span>
+    <span class="n">dot_cmap</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">ListedColormap</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">227</span><span class="p">,</span><span class="mi">26</span><span class="p">,</span><span class="mi">28</span><span class="p">],[</span><span class="mi">166</span><span class="p">,</span><span class="mi">206</span><span class="p">,</span><span class="mi">227</span><span class="p">]])</span><span class="o">/</span><span class="mf">255.0</span><span class="p">)</span>
+
+    <span class="c1"># define color map for probability contours</span>
+    <span class="n">contour_cmap</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">&#39;RdBu&#39;</span><span class="p">)</span>
+
+    <span class="c1"># define probability contours</span>
+    <span class="n">contour_levels</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.05</span><span class="p">,</span><span class="mf">0.1</span><span class="p">)</span>
+
+    <span class="c1"># define base values of the predictors</span>
+    <span class="n">SOW_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span> <span class="c1"># default parameter values for base SOW</span>
+    <span class="n">base</span> <span class="o">=</span> <span class="n">SOW_values</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">]</span>
+    <span class="n">ranges</span> <span class="o">=</span> <span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">]</span>
+
+    <span class="c1"># define grid of x (1st predictor), and y (2nd predictor) dimensions</span>
+    <span class="c1"># to plot contour map over</span>
+    <span class="n">xgrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="mi">0</span><span class="p">],</span>
+                      <span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="mi">1</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">around</span><span class="p">((</span><span class="n">ranges</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">ranges</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span><span class="o">/</span><span class="mi">500</span><span class="p">,</span><span class="n">decimals</span><span class="o">=</span><span class="mi">4</span><span class="p">))</span>
+    <span class="n">ygrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">]][</span><span class="mi">0</span><span class="p">],</span>
+                      <span class="n">param_bounds</span><span class="p">[</span><span class="n">top_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">]][</span><span class="mi">1</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">around</span><span class="p">((</span><span class="n">ranges</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">ranges</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span><span class="o">/</span><span class="mi">500</span><span class="p">,</span><span class="n">decimals</span><span class="o">=</span><span class="mi">4</span><span class="p">))</span>
+    <span class="n">all_predictors</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dta</span><span class="o">.</span><span class="n">columns</span><span class="o">.</span><span class="n">tolist</span><span class="p">()[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">top_predictors</span><span class="p">]</span>
+    <span class="n">dta</span><span class="p">[</span><span class="s1">&#39;Interaction&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">dta</span><span class="p">[</span><span class="n">all_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span><span class="o">*</span><span class="n">dta</span><span class="p">[</span><span class="n">all_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
+
+    <span class="c1"># logistic regression here</span>
+    <span class="n">predictor_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">all_predictors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]]</span>
+    <span class="n">result</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">fit_logit</span><span class="p">(</span><span class="n">dta</span><span class="p">,</span> <span class="n">predictor_list</span><span class="p">)</span>
+
+    <span class="c1"># plot contour map</span>
+    <span class="n">contourset</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_contour_map</span><span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="n">dta</span><span class="p">,</span> <span class="n">contour_cmap</span><span class="p">,</span>
+                                          <span class="n">dot_cmap</span><span class="p">,</span> <span class="n">contour_levels</span><span class="p">,</span> <span class="n">xgrid</span><span class="p">,</span>
+                                          <span class="n">ygrid</span><span class="p">,</span> <span class="n">all_predictors</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">all_predictors</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">base</span><span class="p">)</span>
+
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">usernames</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+
+<span class="c1"># set up colorbar</span>
+<span class="n">cbar_ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_axes</span><span class="p">([</span><span class="mf">0.98</span><span class="p">,</span> <span class="mf">0.15</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">])</span>
+<span class="n">cbar</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">contourset</span><span class="p">,</span> <span class="n">cax</span><span class="o">=</span><span class="n">cbar_ax</span><span class="p">)</span>
+<span class="n">cbar_ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Probability of Success&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">cbar_ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;y&#39;</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">&#39;major&#39;</span><span class="p">,</span> <span class="n">labelsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">statsmodels</span><span class="o">/</span><span class="n">base</span><span class="o">/</span><span class="n">model</span><span class="o">.</span><span class="n">py</span><span class="p">:</span><span class="mi">127</span><span class="p">:</span> <span class="n">ValueWarning</span><span class="p">:</span> <span class="n">unknown</span> <span class="n">kwargs</span> <span class="p">[</span><span class="s1">&#39;disp&#39;</span><span class="p">]</span>
+  <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="n">msg</span><span class="p">,</span> <span class="n">ValueWarning</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Optimization</span> <span class="n">terminated</span> <span class="n">successfully</span><span class="o">.</span>
+         <span class="n">Current</span> <span class="n">function</span> <span class="n">value</span><span class="p">:</span> <span class="mf">0.378619</span>
+         <span class="n">Iterations</span> <span class="mi">8</span>
+<span class="n">Optimization</span> <span class="n">terminated</span> <span class="n">successfully</span><span class="o">.</span>
+         <span class="n">Current</span> <span class="n">function</span> <span class="n">value</span><span class="p">:</span> <span class="mf">0.397285</span>
+         <span class="n">Iterations</span> <span class="mi">8</span>
+<span class="n">Optimization</span> <span class="n">terminated</span> <span class="n">successfully</span><span class="o">.</span>
+         <span class="n">Current</span> <span class="n">function</span> <span class="n">value</span><span class="p">:</span> <span class="mf">0.377323</span>
+         <span class="n">Iterations</span> <span class="mi">8</span>
+</pre></div>
+</div>
+<img alt="_images/notebook_logistic_output_11_1.png" src="_images/notebook_logistic_output_11_1.png" />
+<p>The figure above demonstrates how different combinations of the
+uncertain factors lead to success or failure in different states of the
+world, which are denoted by the blue and red dots respectively. The
+probability of success and failure are further denoted by the contours
+in the figure. Several insights can be drawn from this figure.</p>
+<p>First, using metrics chosen to be decision-relevant (specific to each
+user) causes different factors to be identified as most important by
+this scenario-discovery exercise (the x- and y-axes for each of the
+subplots). In other words, depending on what the decision makers of this
+system want to prioritize they might choose to monitor different
+uncertain factors to track performance.</p>
+<p>Second, in the top panel, the two identified factors appear to also have
+an interactive effect on the metric used (shortages of a certain level
+and frequency in this example). In terms of scenario discovery, the
+Patient Rule Induction Method (PRIM) or Classification And Regression
+Trees (CART) would not be able to delineate this non-linear space and
+would therefore misclassify parameter combinations as ‘desirable’ when
+they were in fact undesirable, and vice versa.</p>
+<p>Lastly, logistic regression also produces contours of probability of
+success, i.e. different factor-value combinations are assigned different
+probabilities that a shortage level will be exceeded. This allows the
+decision makers to evaluate these insights while considering their risk
+aversion.</p>
+</section>
+<section id="tips-to-apply-scenario-discovery-to-your-own-problem">
+<h4><span class="section-number">B.3.1.8. </span>Tips to Apply Scenario Discovery to Your Own Problem<a class="headerlink" href="#tips-to-apply-scenario-discovery-to-your-own-problem" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we demonstrated how to perform a scenario discovery
+analysis for three different users in the UCRB. The analysis allowed us
+to determine which parameters the users would be most affected by and to
+visualize how different ranges of these parameters lead to success and
+failure for different users. This framework can be applicable to any
+other application where it is of interest to characterize success and
+failure based on uncertain parameter ranges. In order to apply the same
+framework to your own problem:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters of interest, which will
+represent uncertainties that characterize your system.</p></li>
+<li><p>Generate samples for these parameters (this can be done using the
+<code class="docutils literal notranslate"><span class="pre">saltelli.sample</span></code> function or externally).</p></li>
+<li><p>Define what constitutes success and failure in your problem. In this
+tutorial, success was defined based on not surpassing the historical
+drought frequency. Choose a metric that is relevant to your problem
+and decision-makers that might be involved. If your model involves an
+optimization, you can also define metrics based on meeting certain
+values of these objectives.</p></li>
+<li><p>Run the parameter sets through your model and calculate success and
+failure based on your metrics and across different users if
+applicable. This step will allow you to create the scatter plot part
+of the final figure.</p></li>
+<li><p>If it is of interest, the contours on the figure can be created by
+fitting the logistic regression model in a similiar manner as denoted
+in Steps 3 and 5 of the tutorial.</p></li>
+</ol>
+</section>
+</section>
+</section>
+<section id="hymod-dynamics-tutorial">
+<h2><span class="section-number">B.4. </span>HYMOD Dynamics Tutorial<a class="headerlink" href="#hymod-dynamics-tutorial" title="Link to this heading">#</a></h2>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">HYMOD Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="tutorial-sensitivity-analysis-of-the-hymod-model">
+<h3><span class="section-number">B.4.1. </span>Tutorial: Sensitivity Analysis of the HYMOD Model<a class="headerlink" href="#tutorial-sensitivity-analysis-of-the-hymod-model" title="Link to this heading">#</a></h3>
+<p>The purpose of this tutorial is to demonstrate the global sensitivity
+analysis concepts and tools established in the Section 3.1 of the main
+text of this eBook. This demonstration will highlight the central role
+of design of experiments (Section 3.3), when implementing global
+sensitivity analysis tools described in Section 3.4.</p>
+<p>We’ll explore these tools and concepts using the HYdrological MODel
+(HYMOD), a rainfall-runoff model developed and used for river flow
+forecasting. HYMOD was chosen for demonstration because its purpose is
+to abstract highly complex and non-linear systems. The methods
+demonstrated in thistutorial can be applied to numerical models that
+simulate other complex non-linear systems.</p>
+<p>This tutorial will first introduce HYMOD and use it to simulate
+streamflows in a river basin. Next, we’ll employ sensitivity analysis
+concepets described in Section 3 of the main text to examine how values
+of HYMOD’s parameters impact streamflow predictions. We’ll then explore
+how the effects of these parameters may change over time using
+time-varying sensitivtiy analysis. Finally, we’ll demonstrate concepts
+presented in Chapter 7 through two ensemble-based methods of uncertainty
+quantification - Generalized Likelihood Uncertainty Estimation (GLUE)
+and Pre-Calibration.</p>
+<p>The tutorial includes the following steps:</p>
+<p>1.1 - Introduction to a simple hydrologic model (HYMOD)</p>
+<p>1.2 - Input Data</p>
+<p>1.3 - Running a basic simulation</p>
+<p>1.4 - Model outputs</p>
+<p>2.1 - Design of Experiments</p>
+<p>2.2 - Sensitivity analysis for one output</p>
+<p>2.3 - Sensitivity analysis across multiple outputs</p>
+<p>2.4 - Time-varying sensitivity analysis</p>
+</section>
+<section id="introduction-to-hymod">
+<h3><span class="section-number">B.4.2. </span>1 - Introduction to HYMOD<a class="headerlink" href="#introduction-to-hymod" title="Link to this heading">#</a></h3>
+<p id="hymod">1.1 Overview</p>
+<p>HYMOD is a hydrologic model (rainfall-runoff model) that simulates key
+hydrologic fluxes such as infiltration, streamflow and
+evapotranspiration. The model was originally developed and used for
+river flow forecasting, but it has also been been used to explore
+different sensitivity analysis (e.g., <a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/wrcr.20124">Herman et al.,
+2013</a>),
+uncertainty quantification (e.g., <a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2006WR005205">Smith et al.,
+2008</a>),
+and optimization (e.g., <a class="reference external" href="https://www.sciencedirect.com/science/article/pii/S0022169414006362?casa_token=IRqE19Hkfa8AAAAA:_fXOqfwpzxMpchvu8_0njCe0Ok9H29Gyw2F46l9PzG9UVODDTUg6wIOSiyp6uybGevNVnZ7N">Ye et al.,
+2014</a>)
+concepts.</p>
+<p>HYMOD accepts two inputs - daily precepitation and daily potential
+evapotranspiration (PET)- and generates predicitons of daily streamflow.
+HYMOD abstracts the highly non-linear process of runoff routing by
+dividing the flow into two components: quick flow, representing
+precipitation that quickly runs off the surface of the watershed into
+the stream, and slow flow, that moves through the soil and takes much
+longer to arrive at the stream.</p>
+<p>To generate streamflow predictions, HYMOD first models vertical
+processes within the watershed to determine how much water infiltrates
+and evaporates from the soil at a given time step. It then determines
+how much water should be partitioned into quick flow and slow flow
+processes. Within each process it abstracts residence time (the time it
+takes a unit volume of water to move through the watershed and into the
+stream) using a series of “reservoirs” each with a calibrated residence
+time.</p>
+<p>HYMOD’s representation of hydrologic processes are shown Figure 1 below
+and controlled by the following parameters:</p>
+<p><span class="math notranslate nohighlight">\(H_{uz}\)</span>: the maximum water storage capacity of the soil (mm)</p>
+<p><span class="math notranslate nohighlight">\(B_{exp}\)</span>: parameters describing the degree of spatial variability
+within the basin between 0 and Huz</p>
+<p><span class="math notranslate nohighlight">\(Alp\)</span>: Fraction of runoff contributing to quick flow</p>
+<p><span class="math notranslate nohighlight">\(K_q\)</span>: Quick flow residence time of linear infinite reservoir (the
+Kq values of all three linear reservoirs are the same)</p>
+<p><span class="math notranslate nohighlight">\(K_s\)</span>: Slow flow residence time of linear infinite reservoir</p>
+<img alt="_images/hymod_schematic-DAVE.png" src="_images/hymod_schematic-DAVE.png" />
+<p>HYMOD models the fraction of water that is stored in the soil
+<span class="math notranslate nohighlight">\((F(XH_{uz}))\)</span> using the following relationship:</p>
+<div class="math notranslate nohighlight">
+\[F(XH_{uz}) = 1 - (1 - \frac{XH_{uz}}{H_{uz}})^{B}\]</div>
+<p>where <span class="math notranslate nohighlight">\(XH_{uz}\)</span> is the water storage capacity of the soil;
+<span class="math notranslate nohighlight">\(H_{uz}\)</span> is the parameter describing basin maximum water
+storage capacity (mm); and <span class="math notranslate nohighlight">\(B\)</span> is the parameter describing the
+degree of spatial variability within the basin.</p>
+<p>The portion of precipitation that exceeds the water storage capacity is
+treated as runoff.</p>
+<p>To route runoff to streamflow, the excess runoff from the vertical
+processes is split into quick flow and slow flow. The proportion of
+runoff partitioned into quick flow and slow flow is determined by a
+parameter <span class="math notranslate nohighlight">\(Alp\)</span>, which ranges between 0 and 1. Quick flow is
+routed through <span class="math notranslate nohighlight">\(N\)</span> identical quick flow tanks <span class="math notranslate nohighlight">\(Q1, Q2... QN\)</span>
+(shown above as <span class="math notranslate nohighlight">\(N=3\)</span>). The rate at which runoff moves through the
+quick flow system is described by the residence time of the quick frlow
+tanks, <span class="math notranslate nohighlight">\(Kq\)</span> (day). Slow flow is routed through a parallel slow
+flow tank and the rate at which slow flow is routed is described by the
+slow flow residences time, <span class="math notranslate nohighlight">\(Ks\)</span> (day).</p>
+<p>Citation: Wagener, T., Boyle, D. P., Lees, M. J., Wheater, H. S., Gupta,
+H. V., &amp; Sorooshian, S. (2001). A framework for development and
+application of hydrological models. Hydrology and Earth System Sciences,
+5(1), 13-26.</p>
+<p id="inputs">1.2 Input data</p>
+<p>The HYMOD model only requires precipitation and potential
+evapotranspiration as inputs. For this example, we’ll run HYMOD using
+data from the Leaf River, a humid catchment located north of Collins
+Mississippi that has been widely used to explore HYMOD. The dataset also
+includes daily observed runoff that we later use to evaluate the
+performace of each sensitvity analysis sample set.</p>
+<p>In the following section of code, we’ll load the necessary python
+libraries and read in the input file. For this exercise we’ll only use
+the first eleven years of data. The first five rows of the input dataset
+are printed to show what they look like:</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">msdbook</span>
+
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+
+<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">metrics</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
+
+<span class="c1"># load example data</span>
+<span class="n">msdbook</span><span class="o">.</span><span class="n">install_package_data</span><span class="p">()</span>
+
+<span class="c1"># load the Leaf River HYMOD input file</span>
+<span class="n">leaf_data</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_input_file</span><span class="p">()</span>
+
+<span class="c1"># extract the first eleven years of data</span>
+<span class="n">leaf_data</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">4015</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Leaf River Data structure:&#39;</span><span class="p">)</span>
+
+<span class="c1"># There are only three columns in the file including precipitation, potential evapotranspiration and  streamflow</span>
+<span class="n">leaf_data</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Downloading</span> <span class="n">example</span> <span class="n">data</span> <span class="k">for</span> <span class="n">msdbook</span> <span class="n">version</span> <span class="mf">0.1.5</span><span class="o">...</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">uncertain_params_bounds</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_metric_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">solutions</span><span class="o">.</span><span class="n">resultfile</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LHsamples_original_1000</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">3704614</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">param_values</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_mth_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_params_256samples</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_vary_s1</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7000550</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_heatmap</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">sa_by_yr_delta</span><span class="o">.</span><span class="n">npy</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="mi">7200799</span><span class="n">_pseudo_r_scores</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">LeafCatch</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">hymod_simulations_256samples</span><span class="o">.</span><span class="n">csv</span>
+<span class="n">Unzipped</span><span class="p">:</span> <span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">msdbook</span><span class="o">/</span><span class="n">data</span><span class="o">/</span><span class="n">Robustness</span><span class="o">.</span><span class="n">txt</span>
+<span class="n">Leaf</span> <span class="n">River</span> <span class="n">Data</span> <span class="n">structure</span><span class="p">:</span>
+</pre></div>
+</div>
+<div>
+<style scoped>
+    .dataframe tbody tr th:only-of-type {
+        vertical-align: middle;
+    }
+
+    .dataframe tbody tr th {
+        vertical-align: top;
+    }
+
+    .dataframe thead th {
+        text-align: right;
+    }
+</style>
+<table border="1" class="dataframe">
+  <thead>
+    <tr style="text-align: right;">
+      <th></th>
+      <th>Precip</th>
+      <th>Pot_ET</th>
+      <th>Strmflw</th>
+    </tr>
+  </thead>
+  <tbody>
+    <tr>
+      <th>0</th>
+      <td>0.0</td>
+      <td>4.60</td>
+      <td>0.29</td>
+    </tr>
+    <tr>
+      <th>1</th>
+      <td>0.0</td>
+      <td>4.31</td>
+      <td>0.24</td>
+    </tr>
+    <tr>
+      <th>2</th>
+      <td>0.0</td>
+      <td>4.33</td>
+      <td>0.21</td>
+    </tr>
+    <tr>
+      <th>3</th>
+      <td>0.0</td>
+      <td>4.78</td>
+      <td>0.19</td>
+    </tr>
+    <tr>
+      <th>4</th>
+      <td>0.0</td>
+      <td>2.91</td>
+      <td>0.18</td>
+    </tr>
+  </tbody>
+</table>
+</div><p>To visualize catchment hydrology, streamflow and precipitation data are
+usually plotted together as a combined hydrograph (streamflow ) and
+hyetograph (rainfall, from Greek.hyetos, “rain”). Streamflow is plotted
+as a time series, while rainfall is shown as an inverted bar plot along
+the top of the graph. Streamflow labels are shown on the left y-axis,
+while rainfall labels are shown on the right y-axis.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># make an axis for the hydrograph</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">strmflw_ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">12</span><span class="p">,</span><span class="mi">6</span><span class="p">])</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">50</span><span class="p">])</span>
+
+<span class="c1">#make a second y-axis for the hyetograph</span>
+<span class="n">precip_ax</span> <span class="o">=</span> <span class="n">strmflw_ax</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">200</span><span class="p">])</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">invert_yaxis</span><span class="p">()</span>
+
+<span class="n">precip</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">]</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">])),</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;lightcoral&#39;</span><span class="p">)</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Streamflow (mm/day)&#39;</span><span class="p">)</span>
+
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">])),</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Precip&#39;</span><span class="p">],</span> <span class="n">width</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Rainfall (mm/day)&#39;</span><span class="p">)</span>
+<span class="n">precip_ax</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">&#39;Precip&#39;</span><span class="p">],</span> <span class="n">loc</span><span class="o">=</span><span class="s1">&#39;center right&#39;</span><span class="p">)</span>
+<span class="n">strmflw_ax</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">&#39;Streamflow&#39;</span><span class="p">],</span><span class="n">bbox_to_anchor</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.48</span><span class="p">))</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">&lt;</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">legend</span><span class="o">.</span><span class="n">Legend</span> <span class="n">at</span> <span class="mh">0x7f53b95c6850</span><span class="o">&gt;</span>
+</pre></div>
+</div>
+<img alt="_images/hymod1.png" src="_images/hymod1.png" />
+<p id="baseline">1.3 Running a Baseline Model Simulation</p>
+<p>We’ll start our experiment by running HYMOD using its default
+parameters.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># assign input parameters to generate a baseline simulated streamflow</span>
+<span class="n">Nq</span> <span class="o">=</span> <span class="mi">3</span>  <span class="c1"># Number of quickflow routing tanks</span>
+<span class="n">Kq</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="c1"># Quickflow routing tanks&#39; rate parameter</span>
+<span class="n">Ks</span> <span class="o">=</span>  <span class="mf">0.001</span> <span class="c1"># Slowflow routing tank&#39;s rate parameter</span>
+<span class="n">Alp</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="c1"># Quick/slow split parameter</span>
+<span class="n">Huz</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1"># Maximum height of soil moisture accounting tank</span>
+<span class="n">B</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="c1"># Scaled distribution function shape parameter</span>
+
+<span class="c1"># Note that the number of years is 11. One year of model warm-up and ten years are used for actual simulation</span>
+<span class="n">model</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">hymod</span><span class="p">(</span><span class="n">Nq</span><span class="p">,</span> <span class="n">Kq</span><span class="p">,</span> <span class="n">Ks</span><span class="p">,</span> <span class="n">Alp</span><span class="p">,</span> <span class="n">Huz</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">leaf_data</span><span class="p">,</span> <span class="n">ndays</span><span class="o">=</span><span class="mi">4015</span><span class="p">)</span>
+</pre></div>
+</div>
+<p id="outputs">1.4 Model Outputs</p>
+<p>Model outputs include actual evapotranspiration, quick and fast
+streamflow, and combined runoff. In this tutorial we focus on the total
+daily runoff, QQ (<span class="math notranslate nohighlight">\(m^3/s\)</span>). We can use the following script to
+plot simulated streamflow against observed streamflow.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-observed-vs-simulated-streamflow">plot_observed_vs_simulated_streamflow</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ax</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_observed_vs_simulated_streamflow</span><span class="p">(</span><span class="n">df</span><span class="o">=</span><span class="n">leaf_data</span><span class="p">,</span> <span class="n">hymod_dict</span><span class="o">=</span><span class="n">model</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod2.png" src="_images/hymod2.png" />
+<p>So how does our model perform? We can investigate model performance
+across several metrics:</p>
+<p>1: Mean Absolute Error (MAE); MAE conveys how the model performs on
+average across the 10 year simulation period, with smaller values
+indicating better performance. The absolute value is taken so that
+positive and negative errors do not cancel each other out.</p>
+<div class="math notranslate nohighlight">
+\[MAE = \frac{1}{N}\sum_{t=0}^N\left\lvert Q_{sim,t}-Q_{obs,t}\right\rvert\]</div>
+<p>2: Root Mean Square Error (RMSE); RMSE is sum of square errors across
+the 10 year simulation period. RMSE is sensitive to large errors between
+the historical record and the simulated flows, and thus is useful for
+highlighting the model’s ability to capture of extreme flood events.</p>
+<div class="math notranslate nohighlight">
+\[RMSE = \sqrt{\frac{1}{N}\sum_{t=1}^{N}(Q_{sim,t}-Q_{obs,t})^2}\]</div>
+<p>3: Log-Root Mean Square Error (Log(RMSE)) LOG(RMSE) focuses on model
+performance during low-flow events.</p>
+<div class="math notranslate nohighlight">
+\[LOG(RMSE) = log(RMSE)\]</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">mae</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">]</span> <span class="o">-</span> <span class="n">model</span><span class="p">[</span><span class="s1">&#39;Q&#39;</span><span class="p">]))</span>
+<span class="n">mse</span> <span class="o">=</span> <span class="n">metrics</span><span class="o">.</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">model</span><span class="p">[</span><span class="s1">&#39;Q&#39;</span><span class="p">],</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">])</span>
+<span class="n">rmse</span> <span class="o">=</span> <span class="n">mse</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;MAE: &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">mae</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">RMSE: &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">mse</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">LOG(RMSE): &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">rmse</span><span class="p">))</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">MAE</span><span class="p">:</span> <span class="mf">1.0787471470460999</span>
+<span class="n">RMSE</span><span class="p">:</span> <span class="mf">4.375695937555197</span>
+<span class="n">LOG</span><span class="p">(</span><span class="n">RMSE</span><span class="p">):</span> <span class="mf">2.0918164206151544</span>
+</pre></div>
+</div>
+<p>The error metrics show that HYMOD performs reasonably well, the MAE is
+around 1 <span class="math notranslate nohighlight">\(m^3/s\)</span>, the RMSE is on the order of 10% of the largest
+observed streamflow and the LOG(RMSE) is fairly low.</p>
+</section>
+<section id="global-sensitivity-analysis">
+<h3><span class="section-number">B.4.3. </span>2- Global Sensitivity Analysis<a class="headerlink" href="#global-sensitivity-analysis" title="Link to this heading">#</a></h3>
+<p id="sensitivity">2.1 Experimental Design and Setup</p>
+<p>Now we’ll examine how sensitive streamflow simulations generated by
+HYMOD are to the model’s input parameters. We’ll perform global
+sensitivity analysis (see Section 3.1 of the main text) using the SALib
+Python library.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">saltelli</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">sobol</span>
+<span class="kn">from</span> <span class="nn">SALib.analyze</span> <span class="kn">import</span> <span class="n">delta</span>
+</pre></div>
+</div>
+<p>A first and critical step when conducting sensitivity analysis is
+determining the experimental design (see Design of Experiments, Section
+3.4 of the main text). Our experimental design involves defining the
+uncertainties that we’ll be examining, the output of interest, the
+ranges of each uncertainty that will be explored and the strategy for
+sampling the uncertainty space.</p>
+<p>For this experiment we’ll explore the five parameters highlighted in
+Figure 1. We’ll draw their ranges from existing literature on the model
+(note Jon H. paper). We’ll use a Sobol sampling an a quasi-random
+sampling with low sequences approach to sample the uncertainty space
+(Section 3.3.4).</p>
+<p>In this demonstration we’ll utilize Sobol Sensitivity Analysis, a
+variance based method (Section 3.4.5).</p>
+<p>To explore HYMOD’s behavoir, we’ll examine the sensitivity of four model
+ouputs to input parameters: 1) predicted flow, 2) Mean Absolute Error
+(compared with the calibaration data set), 3) Root Mean Square Error and
+4) Log Root Mean Square Error.</p>
+<p>This analysis will employ SALib, a Python implementation also utilized
+in the other SA tutorial (make this more formal).</p>
+<p>To start our analysis, we’ll create a dictionary that describes our
+model uncertainties and their ranges, this dictionary is named
+“problem_hymod” (SALib refers to these dictionaries as “problems”).</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">problem_hymod</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">5</span><span class="p">,</span>
+    <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">],</span>
+    <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>  <span class="c1"># Kq</span>
+               <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">],</span>  <span class="c1"># Ks</span>
+               <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>    <span class="c1"># Alp</span>
+               <span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">500</span><span class="p">],</span>  <span class="c1"># Huz</span>
+               <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">1.9</span><span class="p">]]</span>  <span class="c1"># B</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+<p>After defining our uncertainites and ranges, we’ll use SALib to sample
+the uncertainty space and run the model for each of the sample sets. We
+will load a sample that has already been created <code class="docutils literal notranslate"><span class="pre">param_values_hymod</span></code>
+for demonstration purposes. For HYMOD, literature recommends running at
+least N = 10,000 samples, to keep this demonstration easy to run
+however, we utilize only 256 sobol samples of uncertainties. To generate
+accurate approximations of second order sensitivity indicies SALib
+generates N*(2k+2) sets of samples, where N=256 and k=5 (number of
+uncertainties). For the math behind why this is needed, see (Saltelli,
+A., 2002. Making best use of model evaluations to compute sensitivity
+indices. Computer Physics Communications 145, 280–297.
+<a class="reference external" href="https://doi.org/10.1016/S0010-4655(02)00280-1">https://doi.org/10.1016/S0010-4655(02)00280-1</a>).</p>
+<p>The actual model simulation takes an extended period, so we also load
+the simulation data from a previous run. The following demonstrates how
+to conduct this analysis:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># generate 256 samples.</span>
+<span class="n">param_values_hymod</span> <span class="o">=</span> <span class="n">saltelli</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="mi">256</span><span class="p">)</span>
+
+<span class="c1"># dictionary to store outputs in</span>
+<span class="n">d_outputs</span> <span class="o">=</span> <span class="p">{}</span>
+
+<span class="c1"># run simulation for each parameter sample</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">param_values_hymod</span><span class="p">)):</span>
+
+    <span class="c1"># run model for each sensitivity analysis parameter sets</span>
+    <span class="n">hymod_output</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">hymod</span><span class="p">(</span><span class="n">Nq</span><span class="p">,</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span>
+                                 <span class="n">param_values_hymod</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>
+                                 <span class="n">leaf_data</span><span class="p">,</span>
+                                 <span class="n">ndays</span><span class="o">=</span><span class="mi">4015</span><span class="p">)</span>
+
+    <span class="c1"># store the simulated total flow discharge</span>
+    <span class="n">d_outputs</span><span class="p">[</span><span class="sa">f</span><span class="s2">&quot;Q</span><span class="si">{</span><span class="n">i</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">hymod_output</span><span class="p">[</span><span class="s2">&quot;Q&quot;</span><span class="p">]</span>
+
+
+<span class="n">Q_df_bw</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d_outputs</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously generated parameter values</span>
+<span class="n">param_values_hymod</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_params</span><span class="p">()</span>
+
+<span class="c1"># number of samples</span>
+<span class="n">n_samples</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">param_values_hymod</span><span class="p">)</span>
+
+<span class="c1"># load previously generated hymod simulated outputs</span>
+<span class="n">Q_df_bw</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_simulation</span><span class="p">()</span>
+
+<span class="c1"># column names of each sample simulation number</span>
+<span class="n">sample_column_names</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">Q_df_bw</span><span class="o">.</span><span class="n">columns</span> <span class="k">if</span> <span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s1">&#39;Q&#39;</span><span class="p">]</span>
+</pre></div>
+</div>
+<p>A hydrological model such as HYMOD usually includes ordinary
+differential equations that are sensitive to their initial condition.
+They also have components in their underlying formulation that have long
+memory such that prior time steps can affect their current simulations.
+For example, soil moisture or groundwater can hold water for a long time
+and therefore they are often considered to exhibit a long memory. This
+can affect the partitioning of water to runoff and infiltration, while
+also controlling the generation of base flow. Therefore, it is important
+to have a reasonable initial value for them. To achieve this,
+hydrologists usually extend their simulation period and after the
+simulations, they remove that extended time period that has unreasonable
+groundwater or surface water values. This time period is called the
+warm-up time period.</p>
+<p>Here we extended our simulation for one year (from 10 years to 11 years)
+and we removed the first year of simulation, therefore our warm-up
+period is one year.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># exclude the first year of simulation from the simulations and reset the index</span>
+<span class="n">Q_df</span> <span class="o">=</span> <span class="n">Q_df_bw</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">365</span><span class="p">:</span><span class="mi">4015</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">reset_index</span><span class="p">(</span><span class="n">drop</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="c1"># exclude the first year of the input data and reset the index</span>
+<span class="n">leaf_data</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">365</span><span class="p">:</span><span class="mi">4015</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">reset_index</span><span class="p">(</span><span class="n">drop</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now that HYMOD has been warmed up, we’ll examine how HYMOD’s streamflow
+outputs vary under different sample sets, and compare them with the
+observed streamflow.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># add date columns to our simulation data frame; for this data our start date is 1/1/2000</span>
+<span class="n">date_ts</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">date_range</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="s1">&#39;1/1/2000&#39;</span><span class="p">,</span> <span class="n">periods</span><span class="o">=</span><span class="mi">3650</span><span class="p">,</span> <span class="n">freq</span><span class="o">=</span><span class="s1">&#39;D&#39;</span><span class="p">)</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;date&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">year</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;month&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">month</span>
+<span class="n">Q_df</span><span class="p">[</span><span class="s1">&#39;day&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">day</span>
+
+<span class="c1"># aggregate the simulated observed streamflow to monthly mean</span>
+<span class="n">df_sim_mth_mean</span> <span class="o">=</span> <span class="n">Q_df</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">,</span> <span class="s1">&#39;month&#39;</span><span class="p">])[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+
+<span class="c1"># do the same for the observed data</span>
+<span class="n">date_ts</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">date_range</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="s1">&#39;1/1/2000&#39;</span><span class="p">,</span> <span class="n">periods</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">leaf_data</span><span class="p">),</span> <span class="n">freq</span><span class="o">=</span><span class="s1">&#39;D&#39;</span><span class="p">)</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;date&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">year</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;month&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">month</span>
+<span class="n">leaf_data</span><span class="p">[</span><span class="s1">&#39;day&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">date_ts</span><span class="o">.</span><span class="n">day</span>
+
+<span class="c1"># aggregate the daily observed streamflow to monthly mean</span>
+<span class="n">df_obs_mth_mean</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">,</span> <span class="s1">&#39;month&#39;</span><span class="p">])</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-observed-vs-sensitivity-streamflow">plot_observed_vs_sensitivity_streamflow</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ax</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_observed_vs_sensitivity_streamflow</span><span class="p">(</span><span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_mth_mean</span><span class="p">,</span>
+                                                     <span class="n">df_sim</span><span class="o">=</span><span class="n">df_sim_mth_mean</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod3.png" src="_images/hymod3.png" />
+<p id="sametrics">2.2 Sensitivity of streamflows to model parameters</p>
+<p>Now we’ll examine how each of HYMOD’s parameters impact the variance of
+simulated streamflows. Using SALib we’ll calculate the first order and
+total order sensitivity indicies of each model parameter. The first
+order sensitivity index measure’s the individual impact that a given
+parameter has on the variance of the simulated streamflows. The total
+order index measures the impact of a given parameter along with all
+interactions that other parameters have with the given parameter on
+simulated streamflows.</p>
+<p>We’ll start with an matrix, Y, which contains our simulated streamflows
+for every uncertainty sample. We’ll then use the sobol.analyze function
+from SALib to calculate the sensitivity indicies (Si). The arguments for
+this function are the problem dictionary defined in part 2.2 of this
+tutorial, and the matrix of simulated streamflows, Y.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># overall aggregated indices</span>
+<span class="n">Y</span> <span class="o">=</span> <span class="n">Q_df</span><span class="p">[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+<span class="c1"># Perform analysis</span>
+<span class="n">Si</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now we can examine our results, we’ll print the first order and total
+order Si’s for each parameter, then visualize the results with bar plots</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;First order indices = &#39;</span><span class="p">,</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">])</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Total order indicies = &#39;</span><span class="p">,</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">])</span>
+
+<span class="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+<span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s1">&#39;&#39;</span><span class="p">,</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;First order Si&#39;</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s1">&#39;&#39;</span><span class="p">,</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Total order Si&#39;</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">First</span> <span class="n">order</span> <span class="n">indices</span> <span class="o">=</span>  <span class="p">[</span><span class="mf">9.55550001e-05</span> <span class="mf">7.49249463e-04</span> <span class="mf">5.62386413e-04</span> <span class="mf">7.03327551e-01</span>
+ <span class="mf">2.53701895e-01</span><span class="p">]</span>
+<span class="n">Total</span> <span class="n">order</span> <span class="n">indicies</span> <span class="o">=</span>  <span class="p">[</span><span class="mf">1.76174200e-06</span> <span class="mf">1.63288175e-03</span> <span class="mf">3.41378460e-04</span> <span class="mf">6.88983864e-01</span>
+ <span class="mf">2.53922146e-01</span><span class="p">]</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">ipykernel_launcher</span><span class="o">.</span><span class="n">py</span><span class="p">:</span><span class="mi">9</span><span class="p">:</span> <span class="ne">UserWarning</span><span class="p">:</span> <span class="n">FixedFormatter</span> <span class="n">should</span> <span class="n">only</span> <span class="n">be</span> <span class="n">used</span> <span class="n">together</span> <span class="k">with</span> <span class="n">FixedLocator</span>
+  <span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">&#39;__main__&#39;</span><span class="p">:</span>
+<span class="o">/</span><span class="n">srv</span><span class="o">/</span><span class="n">conda</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">notebook</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="mf">.7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">ipykernel_launcher</span><span class="o">.</span><span class="n">py</span><span class="p">:</span><span class="mi">15</span><span class="p">:</span> <span class="ne">UserWarning</span><span class="p">:</span> <span class="n">FixedFormatter</span> <span class="n">should</span> <span class="n">only</span> <span class="n">be</span> <span class="n">used</span> <span class="n">together</span> <span class="k">with</span> <span class="n">FixedLocator</span>
+  <span class="kn">from</span> <span class="nn">ipykernel</span> <span class="kn">import</span> <span class="n">kernelapp</span> <span class="k">as</span> <span class="n">app</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod4.png" src="_images/hymod4.png" />
+<p>Our findings indicate that in this instance, the streamflow estimate
+from HYMOD is highly sensitive to soil moisture parameters Huz and B and
+hardly affected by the routing parameters. Notably, there is very little
+interactions between parameters causing the total order indicies to be
+nearly identical to the first order indicies.</p>
+<p id="diffperformance">2.3 How do different performance metrics affect the results of our
+sensitivity analysis?</p>
+<p>Streamflow has many different properties. In this section, we discuss
+how the selection of metrics can lead to fundamentally different
+sensitivity analysis results. For example, one can only focus on
+aggregated streamflow metrics such as mean (what has been presented so
+far), or only on extreme events such as drought or floods.</p>
+<p>Here we compare three different metrics: 1- Mean error (ME) 2- Root Mean
+Square Error (RMSE) 3- Log-Root Mean Square Error (Log(RMSE))</p>
+<p>Each of these metrics focuses on a specific attribute of streamflow. For
+example, RMSE highlights the impacts of extreme flood events, while
+LOG(RMSE) focuses on model performance during low-flow events.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># calculate error metrics</span>
+<span class="n">mae</span> <span class="o">=</span> <span class="n">Q_df</span><span class="p">[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">leaf_data</span><span class="p">[</span><span class="s2">&quot;Strmflw&quot;</span><span class="p">]),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="n">mse</span> <span class="o">=</span> <span class="n">Q_df</span><span class="p">[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">metrics</span><span class="o">.</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">leaf_data</span><span class="p">[</span><span class="s2">&quot;Strmflw&quot;</span><span class="p">]),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="n">rmse</span> <span class="o">=</span> <span class="n">mse</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="c1"># add error metrics to a dictionary</span>
+<span class="n">d_metrics</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;MAE&#39;</span><span class="p">:</span> <span class="n">mae</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">.</span><span class="n">values</span><span class="p">,</span>
+             <span class="s1">&#39;RMSE&#39;</span><span class="p">:</span> <span class="n">rmse</span><span class="o">.</span><span class="n">values</span><span class="p">,</span>
+             <span class="s1">&#39;LOG[RMSE]&#39;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">rmse</span><span class="o">.</span><span class="n">values</span><span class="p">)}</span>
+
+<span class="c1"># convert to a dataframe</span>
+<span class="n">df_metrics_SA</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d_metrics</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>We can use the following to calculate the SA indices for each metric and
+visualize it.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">df_metric_s1_result</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">df_metric_sT_result</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+
+<span class="c1"># conduct sensitivity analysis for each metric</span>
+<span class="k">for</span> <span class="n">index</span><span class="p">,</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">d_metrics</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+
+    <span class="c1"># get the data as a numpy array for the target metric</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">d_metrics</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+    <span class="c1"># use the metric to conduct SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">sobol</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add the sensitivity indices to the output data frame</span>
+    <span class="n">df_metric_s1_result</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">index</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">]</span>
+    <span class="n">df_metric_sT_result</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">index</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">Si</span><span class="p">[</span><span class="s1">&#39;ST&#39;</span><span class="p">]</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create seaborn heatmap with required labels</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+<span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="c1"># labels for y-axis</span>
+<span class="n">y_axis_labels</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Mean Absoulte Error&#39;</span><span class="p">,</span> <span class="s1">&#39;RSME&#39;</span><span class="p">,</span> <span class="s1">&#39;Log(RMSE)&#39;</span><span class="p">]</span>
+
+<span class="c1"># plot heatmap</span>
+<span class="n">ax1</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">df_metric_s1_result</span><span class="p">,</span> <span class="n">yticklabels</span><span class="o">=</span><span class="n">y_axis_labels</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>  <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;inferno_r&#39;</span><span class="p">,</span> <span class="n">cbar_kws</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;label&#39;</span><span class="p">:</span> <span class="s1">&#39;Si&#39;</span><span class="p">},</span> <span class="n">cbar</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">figure</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">label</span><span class="o">.</span><span class="n">set_size</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;First Order Sensitivity&#39;</span><span class="p">)</span>
+
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">df_metric_sT_result</span><span class="p">,</span> <span class="n">yticklabels</span><span class="o">=</span><span class="n">y_axis_labels</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>  <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;inferno_r&#39;</span><span class="p">,</span> <span class="n">cbar_kws</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;label&#39;</span><span class="p">:</span> <span class="s1">&#39;Si&#39;</span><span class="p">})</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">figure</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">label</span><span class="o">.</span><span class="n">set_size</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Total Order Sensitivity&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Text</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="s1">&#39;Total Order Sensitivity&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod5.png" src="_images/hymod5.png" />
+<p>The first order sensitivity indicies indicate that HYMOD’s sensitivity
+to its parameters is different depending on how its output is measured.
+Unsurprisingly, the mean absolute error is highly sensitive to the soil
+moisture accounting parameters Huz and B, just like the overall
+streamflow predictions above. However, when we examine RMSE and
+log(RMSE), the routing parameters Alp become sensitive, and the
+sensitivity to parameter B is reduced. As described above, RMSE and
+LOG(RMSE) respond to model performance in high-flow and low flow periods
+respectively. Our results indicate that for these flow regimes Alp, the
+parameter that governs the split between quick and slow flow is an
+important factor. While still the parameter with the highest most effect
+on all three measures, Huz is much less influential for RMSE and
+LOG(RMSE) than it is for MAE.</p>
+<p>The total order sensitivity indicies review a different, more complex
+story. While the MAE sensitivity is relatively governed by first order
+effects (like the streamflow predictions above), the RMSE and LOG(RMSE)
+error metrics show significant interactions. Alp has the highest total
+order sensitivity for RMSE and is eqal to Huz for Log(RMSE). Kq, which
+has a relatively low first order sensitivity index, shows strong
+contribution to variance when interactions are taken into account.</p>
+<p>Radial convergence plots are a helpful way to visualize the interactions
+between parameters. These plots array the model parameters in a circle
+and plot the first order, total order and second order Sobol sensitivity
+indices for each parameter. The first order sensitivity is shown as the
+size of a closed circle, the total order as the size of a larger open
+circle and the second order as the thickness of a line connecting two
+parameters. Below is an example of a radial convergence plot for the
+LOG(RMSE) measure. The plot indicates strong interactions between the
+Huz and Alp parameters, as well as Alp and Kq. There is also an
+interaction between Alp and Ks.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">itertools</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+<span class="kn">import</span> <span class="nn">math</span>
+<span class="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s1">&#39;whitegrid&#39;</span><span class="p">,</span> <span class="p">{</span><span class="s1">&#39;axes_linewidth&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="s1">&#39;axes.edgecolor&#39;</span><span class="p">:</span> <span class="s1">&#39;white&#39;</span><span class="p">})</span>
+
+<span class="k">def</span> <span class="nf">is_significant</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">confidence_interval</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="s2">&quot;conf&quot;</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">threshold</span> <span class="o">==</span> <span class="s2">&quot;conf&quot;</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">value</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">(</span><span class="n">confidence_interval</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">value</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">threshold</span><span class="p">))</span> <span class="o">&gt;</span> <span class="mi">0</span>
+
+<span class="k">def</span> <span class="nf">grouped_radial</span><span class="p">(</span><span class="n">SAresults</span><span class="p">,</span> <span class="n">parameters</span><span class="p">,</span> <span class="n">radSc</span><span class="o">=</span><span class="mf">2.0</span><span class="p">,</span> <span class="n">scaling</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">widthSc</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">STthick</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">varNameMult</span><span class="o">=</span><span class="mf">1.3</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">groups</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">gpNameMult</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="s2">&quot;conf&quot;</span><span class="p">):</span>
+    <span class="c1"># Derived from https://github.com/calvinwhealton/SensitivityAnalysisPlots</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+    <span class="n">color_map</span> <span class="o">=</span> <span class="p">{}</span>
+
+    <span class="c1"># initialize parameters and colors</span>
+    <span class="k">if</span> <span class="n">groups</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+
+        <span class="k">if</span> <span class="n">colors</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">colors</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;k&quot;</span><span class="p">]</span>
+
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">parameter</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+            <span class="n">color_map</span><span class="p">[</span><span class="n">parameter</span><span class="p">]</span> <span class="o">=</span> <span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)]</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">if</span> <span class="n">colors</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">colors</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">color_palette</span><span class="p">(</span><span class="s2">&quot;deep&quot;</span><span class="p">,</span> <span class="nb">max</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">groups</span><span class="p">)))</span>
+
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">groups</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+            <span class="c1">#parameters.extend(groups[key])</span>
+
+            <span class="k">for</span> <span class="n">parameter</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">[</span><span class="n">key</span><span class="p">]:</span>
+                <span class="n">color_map</span><span class="p">[</span><span class="n">parameter</span><span class="p">]</span> <span class="o">=</span> <span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)]</span>
+
+    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">parameters</span><span class="p">)</span>
+    <span class="n">angles</span> <span class="o">=</span> <span class="n">radSc</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">radSc</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angles</span><span class="p">)</span>
+    <span class="n">y</span> <span class="o">=</span> <span class="n">radSc</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angles</span><span class="p">)</span>
+
+    <span class="c1"># plot second-order indices</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">combinations</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="mi">2</span><span class="p">):</span>
+        <span class="c1">#key1 = parameters[i]</span>
+        <span class="c1">#key2 = parameters[j]</span>
+
+        <span class="k">if</span> <span class="n">is_significant</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S2&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S2_conf&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">threshold</span><span class="p">):</span>
+            <span class="n">angle</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">atan</span><span class="p">((</span><span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">-</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
+
+            <span class="k">if</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">-</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
+                <span class="n">angle</span> <span class="o">+=</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span>
+
+            <span class="n">line_hw</span> <span class="o">=</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S2&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">])</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
+
+            <span class="n">coords</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">((</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+            <span class="n">coords</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">line_hw</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
+
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Polygon</span><span class="p">(</span><span class="n">coords</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;0.75&quot;</span><span class="p">))</span>
+
+    <span class="c1"># plot total order indices</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">is_significant</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST_conf&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">threshold</span><span class="p">):</span>
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Circle</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;w&#39;</span><span class="p">))</span>
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Circle</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;ST&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="n">STthick</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;0.4&#39;</span><span class="p">,</span> <span class="n">fill</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
+
+    <span class="c1"># plot first-order indices</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">is_significant</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S1&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S1_conf&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">threshold</span><span class="p">):</span>
+            <span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">Circle</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="n">scaling</span><span class="o">*</span><span class="p">(</span><span class="n">SAresults</span><span class="p">[</span><span class="s2">&quot;S1&quot;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">widthSc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;0.4&#39;</span><span class="p">))</span>
+
+    <span class="c1"># add labels</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">parameters</span><span class="p">):</span>
+        <span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">varNameMult</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">varNameMult</span><span class="o">*</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">key</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span>
+                <span class="n">rotation</span><span class="o">=</span><span class="n">angles</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="mi">360</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">-</span> <span class="mi">90</span><span class="p">,</span>
+                <span class="n">color</span><span class="o">=</span><span class="n">color_map</span><span class="p">[</span><span class="n">key</span><span class="p">])</span>
+
+    <span class="k">if</span> <span class="n">groups</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">group</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">groups</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+            <span class="nb">print</span><span class="p">(</span><span class="n">group</span><span class="p">)</span>
+            <span class="n">group_angle</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">angles</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">if</span> <span class="n">parameters</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">[</span><span class="n">group</span><span class="p">]])</span>
+
+            <span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">gpNameMult</span><span class="o">*</span><span class="n">radSc</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">group_angle</span><span class="p">),</span> <span class="n">gpNameMult</span><span class="o">*</span><span class="n">radSc</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">group_angle</span><span class="p">),</span> <span class="n">group</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span>
+                <span class="n">rotation</span><span class="o">=</span><span class="n">group_angle</span><span class="o">*</span><span class="mi">360</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">-</span> <span class="mi">90</span><span class="p">,</span>
+                <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)])</span>
+
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">([])</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;equal&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">radSc</span><span class="p">])</span>
+    <span class="c1">#plt.show()</span>
+
+
+    <span class="k">return</span> <span class="n">fig</span>
+
+<span class="c1"># define groups for parameter uncertainties</span>
+<span class="n">groups</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;Soil Moisture&quot;</span> <span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Huz&quot;</span><span class="p">,</span> <span class="s2">&quot;B&quot;</span><span class="p">],</span>
+        <span class="s2">&quot;Routing&quot;</span> <span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Alp&quot;</span><span class="p">,</span> <span class="s2">&quot;Kq&quot;</span><span class="p">,</span> <span class="s2">&quot;Ks&quot;</span><span class="p">]}</span>
+
+
+<span class="n">fig</span> <span class="o">=</span> <span class="n">grouped_radial</span><span class="p">(</span><span class="n">Si</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">],</span> <span class="n">groups</span><span class="o">=</span><span class="n">groups</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="mf">0.025</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Soil</span> <span class="n">Moisture</span>
+<span class="n">Routing</span>
+</pre></div>
+</div>
+<img alt="_images/hymod6.png" src="_images/hymod6.png" />
+<p>2.4 Time-Varying Sensitivity Analysis</p>
+<p id="time">In section 2.5 we saw how performing sensitivity analysis on different
+measurements of model output can yeild in different results on the
+importance of each uncertain input. In this section we’ll examine how
+performing this analysis over time can yeild additional insight into the
+performance of HYMOD. We’ll first examine how model sensitivities vary
+by month, then examine how they change across each year of the
+simulation.</p>
+<p>For this demonstration, we’ll focus only on the monthly streamflow
+predictions generated by HYMOD.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># aggregate simulated streamflow data to monthly time series</span>
+<span class="n">df_sim_by_mth_mean</span> <span class="o">=</span> <span class="n">Q_df</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s1">&#39;month&#39;</span><span class="p">)[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+
+<span class="c1"># aggregate observed streamflow data to monthly time series</span>
+<span class="n">df_obs_by_mth_mean</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s1">&#39;month&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+<p>We can use the following to calculate the SA indices for each month and
+visualize it. Results are pre-loaded for efficiency.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># set up dataframes to store outputs</span>
+<span class="n">df_mth_s1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">12</span><span class="p">,</span><span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">df_mth_delta</span> <span class="o">=</span> <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="c1"># iterate through each month</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">12</span><span class="p">):</span>
+
+    <span class="c1"># generate the simulation data</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">df_sim_by_mth_mean</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+    <span class="c1"># run SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">delta</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">param_values_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add to output dataframes</span>
+    <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="n">df_mth_delta</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;delta&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+
+<span class="c1"># convert to arrays</span>
+<span class="n">arr_mth_s1</span> <span class="o">=</span> <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">values</span>
+<span class="n">arr_mth_delta</span> <span class="o">=</span> <span class="n">df_mth_delta</span><span class="o">.</span><span class="n">values</span>
+</pre></div>
+</div>
+<p>The following can be used to visualize the time-varying first-order
+indices. The first order represents the direct impacts of a specific
+parameter on model outputs.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-monthly-heatmap">plot_monthly_heatmap</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously ran data</span>
+<span class="n">arr_mth_delta</span><span class="p">,</span> <span class="n">arr_mth_s1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_monthly_simulations</span><span class="p">()</span>
+
+<span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_monthly_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_mth_s1</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                       <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_mth_mean</span><span class="p">,</span>
+                                       <span class="n">title</span><span class="o">=</span><span class="s1">&#39;First Order - Mean Monthly SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod7.png" src="_images/hymod7.png" />
+<p>This figure demonstrates the first order sensitivity indices when the
+streamflow data are aggregated by month. The purple line represents the
+observed monthly discharge. The figure indicates that the first order
+indices are highest for B and Huz across all months and lowest for Alp,
+Ks, and Kq. Note that in the months with the highest flow, Ks becomes an
+influential parameter.</p>
+<p>We can also focus on the total order sensitivity index that includes
+first-order SA indices and interactions between parameters</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_monthly_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_mth_delta</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                       <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_mth_mean</span><span class="p">,</span>
+                                       <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Total Order - Mean monthly SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod8.png" src="_images/hymod8.png" />
+<p>Notably, the total order sensitivity results are different than the
+first order sensitivity results, which indicates that interactions
+between the parameters (particularly in regards to routing parameters
+<span class="math notranslate nohighlight">\(Kq\)</span>, <span class="math notranslate nohighlight">\(Ks\)</span>, and <span class="math notranslate nohighlight">\(Alp\)</span>) contribute to changes in HYMOD
+output.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># group by year and get mean</span>
+<span class="n">df_sim_by_yr_mean</span> <span class="o">=</span> <span class="n">Q_df</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">])[</span><span class="n">sample_column_names</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+
+<span class="c1"># group input data and get mean</span>
+<span class="n">df_obs_by_yr_mean</span> <span class="o">=</span> <span class="n">leaf_data</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">&#39;year&#39;</span><span class="p">])</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+<p>We can also calculate the sensitivity analysis indices for each
+individual year. This will allow us to understand if model control
+changes during different years. The following code first aggregates the
+outputs to annual time steps, and then calculates the SA indices.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># set up dataframes to store outputs</span>
+<span class="n">df_yr_s1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+<span class="n">df_yr_delta</span> <span class="o">=</span> <span class="n">df_yr_s1</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="c1"># iterate through each year</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">):</span>
+
+    <span class="c1"># generate the simulation data</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">df_sim_by_yr_mean</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+    <span class="c1"># run SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">delta</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">param_values_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add to output dataframes</span>
+    <span class="n">df_yr_s1</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="n">df_yr_delta</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;delta&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+
+<span class="c1"># convert to arrays</span>
+<span class="n">arr_yr_s1</span> <span class="o">=</span> <span class="n">df_mth_s1</span><span class="o">.</span><span class="n">values</span>
+<span class="n">arr_yr_delta</span> <span class="o">=</span> <span class="n">df_mth_delta</span><span class="o">.</span><span class="n">values</span>
+</pre></div>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-annual-heatmap">plot_annual_heatmap</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load previously ran data</span>
+<span class="n">arr_yr_delta</span><span class="p">,</span> <span class="n">arr_yr_s1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_annual_simulations</span><span class="p">()</span>
+
+<span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_annual_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_yr_s1</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_yr_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;First Order - Mean Annual SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod9.png" src="_images/hymod9.png" />
+<p>The first order sensitivities at the annual scale are not unlike the
+first order monthly sensitivities. Once again, sensitivities vary across
+year and Huz and B are the most consequential parameters.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_annual_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_yr_delta</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_by_yr_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Total Order - Mean Annual SA and Observed flow&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod10.png" src="_images/hymod10.png" />
+<p>Our results indicate that sensitivity analysis indices vary in different
+years and now that interactions are included, the Kq, Ks, and Alp
+variables impact the sensitivity of the streamflow output.</p>
+<p>Although time-varying sensitivity analysis (TVSA) at average monthly and
+average annual temporal resolutions is informative, TVSA is susceptible
+to the aggregation issue that we discussed earlier in section 3-2. To
+avoid that we can further discretize our time domain to zoom into
+individual months. This will provide us with even more information about
+model behavior and the sensitivity of different parameters in different
+states of the system. The block of code demonstrates how to implement
+the monthly TVSA.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># set up dataframes to store outputs</span>
+<span class="n">df_vary_s1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">df_obs_mth_mean</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">5</span><span class="p">)),</span>
+                          <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+
+<span class="n">df_vary_delta</span> <span class="o">=</span> <span class="n">df_vary_s1</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+<span class="c1"># iterate through each month</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">df_obs_mth_mean</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+
+    <span class="c1"># generate the simulation data</span>
+    <span class="n">Y</span> <span class="o">=</span> <span class="n">df_sim_mth_mean</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
+
+    <span class="c1"># run SA</span>
+    <span class="n">Si</span> <span class="o">=</span> <span class="n">delta</span><span class="o">.</span><span class="n">analyze</span><span class="p">(</span><span class="n">problem_hymod</span><span class="p">,</span> <span class="n">param_values_hymod</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">print_to_console</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="c1"># add to output dataframes</span>
+    <span class="n">df_vary_s1</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;S1&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="n">df_vary_delta</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">Si</span><span class="p">[</span><span class="s1">&#39;delta&#39;</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
+
+<span class="c1"># convert to arrays</span>
+<span class="n">arr_vary_s1</span> <span class="o">=</span> <span class="n">df_vary_s1</span><span class="o">.</span><span class="n">values</span>
+<span class="n">arr_vary_delta</span> <span class="o">=</span> <span class="n">df_vary_delta</span><span class="o">.</span><span class="n">values</span>
+</pre></div>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>View the source code used to create this plot here: <a class="reference external" href="https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-varying-heatmap">plot_varying_heatmap</a></p>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># load in previously ran data</span>
+<span class="n">arr_vary_delta</span><span class="p">,</span> <span class="n">arr_vary_s1</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">load_hymod_varying_simulations</span><span class="p">()</span>
+
+<span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_varying_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_vary_s1</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_mth_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;First Order - Time-Varying SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod11.png" src="_images/hymod11.png" />
+<p>Compared to the TVSA when streamflow was aggregated, this figure
+suggests that Kq is indeed a relevant parameter for influencing
+streamflow output when individual months are considered.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plot figure</span>
+<span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">msdbook</span><span class="o">.</span><span class="n">plot_varying_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="o">=</span><span class="n">arr_vary_delta</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
+                                      <span class="n">df_obs</span><span class="o">=</span><span class="n">df_obs_mth_mean</span><span class="p">,</span>
+                                      <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Total Order - Time-Varying SA&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hymod12.png" src="_images/hymod12.png" />
+<p>As above, the total order sensitivities further indicate the importance
+of Kq that is not apparent if aggregation is utilized.</p>
+</section>
+<section id="tips-to-apply-this-methodology-to-your-own-problem">
+<h3><span class="section-number">B.4.4. </span>Tips to Apply this methodology to your own problem<a class="headerlink" href="#tips-to-apply-this-methodology-to-your-own-problem" title="Link to this heading">#</a></h3>
+<p>In this tutorial, we demonstrated how to use global sensitivtiy analysis
+to explore a complex, non-linear model. We showed how measuring
+sensitivity across multiple measures of model performance and temporal
+aggregations yeilding differing results about model
+sensitivity/behavoir. While these results may seem contraditory, they
+provide useful insight into the behavoir of HYMOD. Would we expect the
+same parameters to control high flow and low flow regimes within the
+model? Maybe, depending on the system, but also, maybe not. This
+analysis can provide insight into how the model responds to its input
+parameters, allowing us to compare the results to our expectaions. This
+may allow us to find problems with our intial assumptions, or call
+attention to model features that can be improved or expanded upon.
+Depending on the model and context, it may also yield insight into the
+workings of the underlying system.</p>
+<p>To run this tutorial on your own model you will need to:</p>
+<ol class="arabic simple">
+<li><p>Design your experiment by choosing sampling bounds for your
+parameters and setting up the problem dictionary as in step 2-2</p></li>
+<li><p>Choose the parameters of interest</p></li>
+<li><p>Generate samples using the saltelli.sample function. This step is
+problem-dependent and note that the Sobol method can be
+computationally intensive depending on the model being analyzed. More
+complex models will be slower to run and will also require more
+samples to calculate accurate estimates of Sobol indices. Once you
+complete this process, pay attention to the confidence bounds on your
+sensitivity indices to see whether you need to run more samples.</p></li>
+<li><p>Run the parameter sets through your model and record each of the
+desired model outputs.</p></li>
+<li><p>Calculate the Sobol indices for each performance criteria. Now, the Y
+will be a numpy array with your external model output and you will
+need to include the parameter samples as an additional argument.</p></li>
+<li><p>Follow the procedure in step 2.6 to disaggregate performance across
+time</p></li>
+</ol>
+</section>
+</section>
+<section id="time-evolving-scenario-discovery-for-infrastructure-pathways">
+<h2><span class="section-number">B.5. </span>Time-evolving scenario discovery for infrastructure pathways<a class="headerlink" href="#time-evolving-scenario-discovery-for-infrastructure-pathways" title="Link to this heading">#</a></h2>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/Bedford_Greene_SD_tutorial.ipynb">Scenario Discovery Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="id24">
+<h3><span class="section-number">B.5.1. </span>Time-evolving scenario discovery for infrastructure pathways<a class="headerlink" href="#id24" title="Link to this heading">#</a></h3>
+<p>The purpose of this tutorial is to explore time-evolving vulnerability for systems that dynamically adapt to changing conditions. Using an example from water supply planning, we’ll first examine how performance of a dynamic infrastructure pathway policy changes over time, then use factor mapping (main text Chapter 4.3) to understand which combinations of uncertaities generate vulnerability for two water utilities. Next, we’ll perform factor prioritization (main text Chapter 4.3) to determine which uncertainties have the most influence on water supply performance. Finally, we’ll provide an open platform to explore vulnerability across multiple measures of performance and different combinations of uncertainties.</p>
+<section id="id25">
+<h4><span class="section-number">B.5.1.1. </span>Background<a class="headerlink" href="#id25" title="Link to this heading">#</a></h4>
+<p>The Bedford-Greene metropolitan area (Figure 1) is a stylized water resources test case where two urban water utilities seek to develop an infrastructure and investment and management strategy to confront growing demands and changing climate. The utilities have agreed to jointly construct a new water treatment plant on Lake Classon, a large regional resource. Both utilities have also identified a set of individual infrastructure options to construct if necessary.</p>
+<figure class="align-default" id="id32">
+<img alt="Figure 1" src="_images/Map_small.png" />
+<figcaption>
+<p><span class="caption-number">Fig. B.1 </span><span class="caption-text">Figure 1</span><a class="headerlink" href="#id32" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+<p>The utilities are formulating a cooperative and adaptive regional management strategy that uses a risk-of-failure (ROF) metric to trigger both short term drought mitigation actions (water use restrictions and treated transfers between utilities) and long-term infrastructure investment decisions (shown in Figure 2a). Both utilities have specified a set of risk triggers and developed a construction order for available infrastructure options.</p>
+<p>The utilities have run a Monte Carlo simulation to evaluate how these policies respond to a wide array of future States Of the World (SOWs). Each SOW represents a different combinations of thirteen uncertain system inputs including demand growth rates, changes to streamflows and financial variables. In this context, a fully specified SOW is composed of one sample of uncertain model inputs (e.g. one projection of demand growth rate coupled with one future streamflow scenario and one projection of future financial conditions). The water utilities used Latin Hypercube sampling (Chapter 3.3 of the main text) to develop an ensemble of 1,000 plausible future SOWs. The Monte Carlo simulation evaluates each candidate water supply infrastructure investment and management policy across all 1,000 SOWs, as shown in Figure 2b. For more details on the Monte Carlo sampling for this type  of analysis, see Trindade et al., (2019).</p>
+<p>The ROF-based policies respond to each SOW by generating a unique infrastructure pathway - a sequence of infrastructure investment decisions over time. Infrastructure pathways over a set of 1,000 future SOWs are shown in Figure 2c. Infrastructure options are arrayed along the vertical axis and the sequence of infrastructure investments triggered by the policy is plotted as pathways over time.  Since the adaptive rule system generates a unique infrastructure sequence for each scenario, Figure 2c summarizes the ensemble of pathways by clustering SOWs according to infrastructure intensity. Dark green lines represent SOWs where the utilities heavily invest in new infrastructure, light green lines represent SOWs with low infrastructure investment and medium shaded lines represent moderate investment. The shading behind each pathway represents the frequency that each infrastructure option was triggered over time across sampled scenarios</p>
+<figure class="align-default" id="id33">
+<img alt="Figure 2" src="_images/Policy_MonteCarlo_Pathways_small.png" />
+<figcaption>
+<p><span class="caption-number">Fig. B.2 </span><span class="caption-text">Figure 2</span><a class="headerlink" href="#id33" title="Link to this image">#</a></p>
+</figcaption>
+</figure>
+</section>
+<section id="evaluating-robustness-over-time">
+<h4><span class="section-number">B.5.1.2. </span>Evaluating Robustness over time<a class="headerlink" href="#evaluating-robustness-over-time" title="Link to this heading">#</a></h4>
+<p>The two water utilities are interested in mainting both supply
+reliability and financial stability across the broadest set of plausible
+future SOWs. To measure the performance of the infrastructure pathway
+policy, they’ve defined five critical performance criteria:</p>
+<ul class="simple">
+<li><p>Reliability &gt; 99%</p></li>
+<li><p>Restriction Frequency &lt; 20%</p></li>
+<li><p>Peak Financial Cost &lt; 80% of annual revenue (a measure of debt
+service spending)</p></li>
+<li><p>Worst-case drought management cost &lt; 10% of annual revenue (a measure
+of unexpected drought costs)</p></li>
+<li><p>Unit Cost of Expansion &lt; 5 dollars/kgal</p></li>
+</ul>
+<p>To assess the robustness of the infrastructure pathway policy, the two
+utilities apply a satisficing metric, which measures the percentage of
+sampled SOWs where the pathway policy meets the peformance criteria:</p>
+<p><span class="math notranslate nohighlight">\(R =\frac{1}{N}\sum_{j=1}^{N}\Lambda_{\theta,j}\)</span></p>
+<p>Where, <span class="math notranslate nohighlight">\(\Lambda\_{\theta,j}=\)</span></p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases}
+1, \quad \textrm{if}\ F(\theta)_{j}\leq \Phi_j \\
+0, \quad \textrm{otherwise}
+\end{cases}\end{split}\]</div>
+<p>And <span class="math notranslate nohighlight">\(\Phi\)</span> is a vector of performance criteria for utility
+<span class="math notranslate nohighlight">\(j\)</span>, <span class="math notranslate nohighlight">\(\theta\)</span> is the portfolio and <span class="math notranslate nohighlight">\(N\)</span> is the total
+number of sampled SOWs.</p>
+<p>Below, we’ll visualize how robustness for the two utilities evolves over the 45-year planning horizon. We’ll assess robustness across three time periods, near-term (first 10 years), mid-term (22 years) and long term (45 years).</p>
+<p>We start by loading robustness values for the both utilities. These values are calculated by applying the robustness metric above across 2,000 simulated SOWs. To make this exercise computationally tractable, we’ve precomputed these values, which can be found in the files “short_term_robustness.csv”, “mid_term_robustness.csv” and “long_term_robustness.csv”. These values are calculated using the function “check_rdm_meet_criteria” within the helper functions.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
+<span class="kn">from</span> <span class="nn">functions.eBook_SD_helpers</span> <span class="kn">import</span> <span class="n">check_rdm_meet_criteria</span><span class="p">,</span> <span class="n">create_sd_input</span><span class="p">,</span> <span class="n">plot_selected_tree_maps</span><span class="p">,</span> <span class="n">get_factor_importances</span><span class="p">,</span> <span class="n">open_exploration</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+
+<span class="c1"># load Deeply uncertain factors</span>
+<span class="n">rdm_factors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/DU_Factors.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+
+<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
+<span class="n">short_term_robustness</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/short_term_robustness.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">mid_term_robustness</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/mid_term_robustness.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">long_term_robustness</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/long_term_robustness.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+
+<span class="c1"># plot robustness over time</span>
+<span class="n">fig</span> <span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span><span class="mi">22</span><span class="p">,</span><span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span><span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span> <span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span><span class="mi">22</span><span class="p">,</span><span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span><span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">45</span><span class="p">],</span> <span class="p">[</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="mi">100</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;Time Horizon (yrs)&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Robustness (% SOWs)&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">&#39;Bedford&#39;</span><span class="p">,</span> <span class="s1">&#39;Greene&#39;</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Robustness Over Time&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">107</span><span class="p">])</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">107.0</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_4_1.png" src="_images/discovery_4_1.png" />
+</section>
+<section id="exploring-performance-evolution">
+<h4><span class="section-number">B.5.1.3. </span>Exploring performance evolution<a class="headerlink" href="#exploring-performance-evolution" title="Link to this heading">#</a></h4>
+<p>The figure above reveals that the robustness of both water utilities degrades over time, with Bedford’s robustness declining further than Greene. This suggests that the proposed pathway policy is likely insufficient to meet the long-term needs of the two utilities. But how is the current policy insufficient? To answer that question we examine the performance measures that fail to meet performance criteria for each utility across the three planning horizons.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot the type of vulnerability over time</span>
+
+<span class="c1">### Bedford ###</span>
+<span class="n">plot_robustness_1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
+<span class="c1"># Determine the percentage of failure SOWs that violate each criterion (note some SOWS fail multiple criteria, so this may some to &gt;1)</span>
+<span class="n">criteria</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Reliability&#39;</span><span class="p">,</span> <span class="s1">&#39;Restriction Frequency&#39;</span><span class="p">,</span> <span class="s1">&#39;Peak Financial Cost&#39;</span><span class="p">,</span> <span class="s1">&#39;Worst-case drought</span><span class="se">\n</span><span class="s1">Management Cost&#39;</span><span class="p">,</span> <span class="s1">&#39;Stranded Assets&#39;</span><span class="p">]</span>
+<span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span>
+<span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">1</span><span class="p">,:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span>
+<span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">2</span><span class="p">,:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span>
+
+<span class="c1"># Plot over time</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">0</span><span class="p">,:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Fraction of failure SOWs&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">1</span><span class="p">,:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_1</span><span class="p">[</span><span class="mi">2</span><span class="p">,:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#B19CD9&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Bedford&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="c1">### Greene ###</span>
+<span class="c1"># Determine the percentage of failure SOWs that violate each criterion (note some SOWS fail multiple criteria, so this may some to &gt;1)</span>
+<span class="n">plot_robustness_2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
+<span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">11</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">short_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">])</span>
+<span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">11</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">mid_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">])</span>
+<span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">11</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">long_term_robustness</span><span class="p">[</span><span class="mi">11</span><span class="p">])</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Fraction of failure SOWs&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">plot_robustness_2</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="p">:],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#43b284&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">criteria</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Greene&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_6_0.png" src="_images/discovery_6_0.png" />
+<img alt="_images/discovery_6_1.png" src="_images/discovery_6_1.png" />
+<p>In the figures above, we observe that the vulnerability of both utilities changes in different ways. Early in the simulation period, Bedford is vulnerable to failures in reliability (though the robustness figure created in step B5.2 reveals that these failures are very rare). As the simulation period progresses, Bedford’s vulnerability expands to include failures in restriction frequency and worst-case cost. These failures indicate that the utility has an overall inability to manage drought conditions and future conditions progress.</p>
+<p>Greene shows a very different evolution in vulnerability. Early in the simulation period, failures manifest in the restriction frequency objective, suggesting that the utility must rely on water use restrictions to maintain supply reliability. As the simulation progresses however, the vulnerability evolves. When evaluated across the 45-year planning horizon, a new failure modes emerges - financial failure manifesting in peak financial cost and stranded assets. This suggests that the proposed pathway policy may be over-investing in new infrastructure, straining the utility’s budget with large debt payments that are unnecessary to maintain supply reliability.</p>
+</section>
+<section id="how-do-deep-uncertainties-generate-vulnerability">
+<h4><span class="section-number">B.5.1.4. </span>How do deep uncertainties generate vulnerability<a class="headerlink" href="#how-do-deep-uncertainties-generate-vulnerability" title="Link to this heading">#</a></h4>
+<p>While the evolution of robustness provides insight into how the system
+evolves over time, it does not reveal <em>why</em> each utility is vulnerable.
+To examine how deep uncertainties generate vulnerability over time for
+the two utilities, we perform scenario discovery (factor mapping,
+Chapter 4.3). Here we’ll utilize gradient boosted trees to identify
+regions of the uncertainty space that cause the utilities to fail to
+meet performance criteria.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># import the performance data across 2000 SOWs for three time periods</span>
+<span class="n">short_term_performance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/short_term_performance.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">mid_term_performance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/mid_term_performance.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+<span class="n">long_term_performance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s1">&#39;data/long_term_performance.csv&#39;</span><span class="p">,</span> <span class="n">delimiter</span> <span class="o">=</span> <span class="s1">&#39;,&#39;</span><span class="p">)</span>
+
+<span class="n">satisficing_criteria</span> <span class="o">=</span> <span class="p">[</span><span class="mf">.98</span><span class="p">,</span> <span class="mf">.2</span><span class="p">,</span> <span class="mf">.8</span><span class="p">,</span> <span class="mf">.1</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>
+
+<span class="c1"># transform into scenario discovery input</span>
+<span class="n">short_term_SD_input</span> <span class="o">=</span> <span class="n">create_sd_input</span><span class="p">(</span><span class="n">short_term_performance</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">)</span>
+<span class="n">mid_term_SD_input</span> <span class="o">=</span> <span class="n">create_sd_input</span><span class="p">(</span><span class="n">mid_term_performance</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">)</span>
+<span class="n">long_term_SD_input</span> <span class="o">=</span> <span class="n">create_sd_input</span><span class="p">(</span><span class="n">long_term_performance</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">)</span>
+
+<span class="c1"># factor mapping Bedford</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;short_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;mid_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;long_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Bedford Factor Maps&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="c1"># factor mapping Greene</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="s1">&#39;short_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;10-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="s1">&#39;mid_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;22-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">plot_selected_tree_maps</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="s1">&#39;long_term&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">satisficing_criteria</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;45-year Horizon&#39;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Greene Factor Maps&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Greene</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Greene</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Greene</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_9_1.png" src="_images/discovery_9_1.png" />
+<img alt="_images/discovery_9_2.png" src="_images/discovery_9_2.png" />
+<p>In the figures above, we learn more about how the vulnerability of the two utilities evolves over time. Bedford begins with very few possible failures but appears vulnerable to high demand growth scenarios under future scenarios with high demands. When evaluated across a 22-year planning horizon, Bedford is vulnerable when the near-term demand growth is high and water use restrictions are less effective than predicted. Under the full 45-year planning horizon, Bedford is vulnerable to sustained high levels of demand growth, failing if either near-term or mid-term demand growth exceeds expected levels.</p>
+<p>Greene’s vulnerability evolves differently. It begins with vulnerability to high demand growth, but as the simulation progresses (and infrastructure is constructed), the utility becomes vulnerable to low-demand growth futures which cause the failures in financial criteria shown in section B.5.3. This indicates that the pathway policy over-builds in many SOWs, and becomes financially unstable if demand does not grow sufficiently to provide revenue to cover debt service payments.</p>
+</section>
+<section id="which-uncertainties-have-the-most-influence-on-time-evolving-performance">
+<h4><span class="section-number">B.5.1.5. </span>Which uncertainties have the most influence on time-evolving performance?<a class="headerlink" href="#which-uncertainties-have-the-most-influence-on-time-evolving-performance" title="Link to this heading">#</a></h4>
+<p>The factor maps generated in B.5.4 present the vulnerability generated by the two most important deep uncertainties as determined by Gradient Boosted Trees. Yet the factor prioritization shows that more than two uncertainties are influential to regional performance. Further, we can observe that individual uncertainties have different impacts on each performance obejctive, and these impacts may change over time. In the cells below, explore the impact of deep uncertainty by generating factor maps for different combinations of deep uncertain factors, objectives and time horizons.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+<span class="n">uncertainties</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;D1&#39;</span><span class="p">,</span> <span class="s1">&#39;D2&#39;</span><span class="p">,</span> <span class="s1">&#39;D3&#39;</span><span class="p">,</span> <span class="s1">&#39;BT&#39;</span><span class="p">,</span> <span class="s1">&#39;BM&#39;</span><span class="p">,</span> <span class="s1">&#39;DR&#39;</span><span class="p">,</span> <span class="s1">&#39;RE&#39;</span><span class="p">,</span> <span class="s1">&#39;EV&#39;</span><span class="p">,</span> <span class="s1">&#39;PM&#39;</span><span class="p">,</span> <span class="s1">&#39;CT&#39;</span><span class="p">,</span> <span class="s1">&#39;IA&#39;</span><span class="p">,</span> <span class="s1">&#39;IF&#39;</span><span class="p">,</span> <span class="s1">&#39;IP&#39;</span><span class="p">]</span>
+<span class="n">uncertainties</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Near-term demand&#39;</span><span class="p">,</span> <span class="s1">&#39;Mid-term demand&#39;</span><span class="p">,</span> <span class="s1">&#39;Long-term demand&#39;</span><span class="p">,</span> <span class="s1">&#39;Bond Term&#39;</span><span class="p">,</span> <span class="s1">&#39;Bond Rate&#39;</span><span class="p">,</span> <span class="s1">&#39;Discount Rate&#39;</span><span class="p">,</span> <span class="s1">&#39;Restriction Effectiveness&#39;</span><span class="p">,</span> <span class="s1">&#39;Evaporation Rate&#39;</span><span class="p">,</span> <span class="s1">&#39;Permitting time&#39;</span><span class="p">,</span> <span class="s1">&#39;Construction time&#39;</span><span class="p">,</span> <span class="s1">&#39;Inflow Amplitude&#39;</span><span class="p">,</span> <span class="s1">&#39;Inflow Frequency&#39;</span><span class="p">,</span> <span class="s1">&#39;Inflow Period&#39;</span><span class="p">]</span>
+
+<span class="n">u1_st_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">short_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="n">u1_mt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">mid_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="n">u1_lt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">long_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+
+<span class="n">u1_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">u1_st_FI</span><span class="p">,</span><span class="n">u1_mt_FI</span><span class="p">,</span> <span class="n">u1_lt_FI</span><span class="p">])</span>
+<span class="n">u1_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">u1_all</span><span class="p">)</span>
+
+<span class="c1"># factor ranking -- utility 2</span>
+<span class="n">u2_st_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">short_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
+<span class="n">u2_mt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">mid_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
+<span class="n">u2_lt_FI</span> <span class="o">=</span> <span class="n">get_factor_importances</span><span class="p">(</span><span class="n">long_term_SD_input</span><span class="p">,</span> <span class="n">rdm_factors</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
+<span class="n">u2_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">u2_st_FI</span><span class="p">,</span><span class="n">u2_mt_FI</span><span class="p">,</span> <span class="n">u2_lt_FI</span><span class="p">])</span>
+<span class="n">u2_all</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">u2_all</span><span class="p">)</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span><span class="p">,</span> <span class="n">cax</span><span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span>
+                  <span class="n">gridspec_kw</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;width_ratios&quot;</span><span class="p">:[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">]})</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">wspace</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">im</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">u1_all</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;Reds&#39;</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mf">.3</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">13</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">(</span><span class="n">uncertainties</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Time Horizon&#39;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Bedford&#39;</span><span class="p">)</span>
+
+<span class="n">im1</span> <span class="o">=</span> <span class="n">ax2</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">u2_all</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;Reds&#39;</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mf">.3</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">13</span><span class="p">))</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">([])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Time Horizon&#39;</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Greene&#39;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="n">cax</span><span class="o">=</span><span class="n">cax</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Factor Importance&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_12_0.png" src="_images/discovery_12_0.png" />
+<p>The Figure above shows the factor importance as determined by gradient boosted trees for both utilities across the three planning horizons. While near-term demand growth is important for both utilities under all three planning horizons, the importance of other factors evolves over time. For example, restriction effectiveness plays an important role for Greene under the 22-year planning horizon but disappears under the 45-year planning horizon. In contrast, the bond interest rate is important for predicting success over the 45-year planning horizon, but does not appear important over the 10- or 22-year planning horizons. These findings highlight how assumptions about the planning period can have a large impact on modeling outcomes.</p>
+</section>
+<section id="open-exploration">
+<h4><span class="section-number">B.5.1.6. </span>Open exploration<a class="headerlink" href="#open-exploration" title="Link to this heading">#</a></h4>
+<p>In the cell below, use the function to explore how factor maps change
+for the two utilities based upon the uncertainties plotted, the
+objectives of interest and the time horizon.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># specify the utility (&quot;Bedford&quot; or &quot;Greene&quot;)</span>
+<span class="n">utility</span> <span class="o">=</span> <span class="s2">&quot;Bedford&quot;</span>
+
+<span class="c1"># specify which performance objectives to investigate (note that not all performance objectives have failures, which may result in a blank factor map)</span>
+<span class="c1"># set this to one of the following: &quot;Reliability&quot;, &quot;Restriction Frequency&quot;, &quot;Peak Financial Cost&quot;, &quot;Worst Case Cost&quot; or &quot;Unit Cost&quot;</span>
+<span class="n">objective</span> <span class="o">=</span> <span class="s2">&quot;Reliability&quot;</span>
+
+<span class="c1"># select uncertainties from the following list: &#39;D1&#39;, &#39;D2&#39;, &#39;D3&#39;, &#39;BT&#39;, &#39;BM&#39;, &#39;DR&#39;, &#39;RE&#39;, &#39;EV&#39;, &#39;PM&#39;, &#39;CT&#39;, &#39;IA&#39;, &#39;IF&#39;, &#39;IP&#39;</span>
+<span class="n">uncertainty_1</span> <span class="o">=</span> <span class="s1">&#39;D1&#39;</span>
+<span class="n">uncertainty_2</span> <span class="o">=</span> <span class="s1">&#39;D2&#39;</span>
+
+<span class="c1"># The code below will plot factor maps over the three planning horizons for the information above</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="n">open_exploration</span><span class="p">(</span><span class="n">utility</span><span class="p">,</span> <span class="n">objective</span><span class="p">,</span> <span class="s1">&#39;short_term&#39;</span><span class="p">,</span> <span class="n">uncertainty_1</span><span class="p">,</span> <span class="n">uncertainty_2</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">open_exploration</span><span class="p">(</span><span class="n">utility</span><span class="p">,</span> <span class="n">objective</span><span class="p">,</span> <span class="s1">&#39;mid_term&#39;</span><span class="p">,</span> <span class="n">uncertainty_1</span><span class="p">,</span> <span class="n">uncertainty_2</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">open_exploration</span><span class="p">(</span><span class="n">utility</span><span class="p">,</span> <span class="n">objective</span><span class="p">,</span> <span class="s1">&#39;long_term&#39;</span><span class="p">,</span> <span class="n">uncertainty_1</span><span class="p">,</span> <span class="n">uncertainty_2</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span><span class="p">,</span> <span class="n">reliability</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span><span class="p">,</span> <span class="n">reliability</span>
+<span class="n">Factor</span> <span class="nb">map</span> <span class="k">for</span> <span class="n">Bedford</span><span class="p">,</span> <span class="n">reliability</span>
+</pre></div>
+</div>
+<img alt="_images/discovery_16_1.png" src="_images/discovery_16_1.png" />
+</section>
+<section id="id26">
+<h4><span class="section-number">B.5.1.7. </span>Tips to apply this methodology to your own problem<a class="headerlink" href="#id26" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we demonstrated time-evolving scenario discovery for a
+cooperative water supply system. To apply this workflow to your own
+problem:</p>
+<ol class="arabic simple">
+<li><p>Choose sampling bounds for your parameters of interest, which will
+represent uncertainties that characterize your system.</p></li>
+<li><p>Generate samples for these parameters (this can be done using the
+saltelli.sample function as in B.2 or done with another package).</p></li>
+<li><p>Define performance criteria for your problem</p></li>
+<li><p>Evaluate parameter sets through your model, and save performance
+measures across multiple time horizons</p></li>
+<li><p>Draw from the supporting code for this tutorial to perform scneario
+discovery and visualize results</p></li>
+</ol>
+</section>
+<section id="references">
+<h4><span class="section-number">B.5.1.8. </span>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h4>
+<p>Trindade, B. C., Reed, P. M., &amp; Characklis, G. W. (2019). Deeply uncertain pathways: Integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134, 103442.</p>
+</section>
+</section>
+</section>
+<section id="a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios-tutorial">
+<h2><span class="section-number">B.6. </span>A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios Tutorial<a class="headerlink" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios-tutorial" title="Link to this heading">#</a></h2>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<div class="line-block">
+<div class="line">Run the tutorial interactively:  <a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/Hidden-Markov_Modeling_Approaches_to_Creating_Synthetic_Streamflow_Scenarios.ipynb">HMM Notebook</a>.</div>
+<div class="line">Please be aware that notebooks can take a couple minutes to launch.</div>
+<div class="line">To run the notebooks yourself, download the files <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks">here</a> and use these <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt">requirements</a>.</div>
+</div>
+</div>
+<section id="a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios">
+<h3><span class="section-number">B.6.1. </span>A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios<a class="headerlink" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios" title="Link to this heading">#</a></h3>
+<p>In this notebook, we will be covering the basics of fitting a Hidden
+Markov Model-based synthetic streamflow generator for a single site in
+the Upper Colorado River Basin. First, we will characterize the observed
+historical flow in the basin from 1909-2013. Then, we will fit a
+synthetic streamflow generator to the observed flows in the basin in
+order to create stationary synthetic flows. Finally, we will create a
+non-stationary version of the generator to create flows that could be
+representative of plausible future climate in the region. We ultimately
+show how to place the synthetically generated flows in the context of
+physically-informed CMIP5 projections to compare the two methods.</p>
+<section id="id29">
+<h4><span class="section-number">B.6.1.1. </span>Background<a class="headerlink" href="#id29" title="Link to this heading">#</a></h4>
+<p>In the Western United States (US), and particularly the Colorado River
+Basin, a recent study used tree-ring reconstructions to suggest that the
+megadrought that has been occurring in the Southwest over the past 22
+years is the region’s worst drought since about 800 AD (Williams et al.,
+2022). The study’s lead author, UCLA climatologist Park Williams,
+suggested that had the sequence of wet-dry years occurred without
+anthropogenic forcing, the 2000s would have likely still been dry, but
+not on the same level as the worst of the last millennium’s
+megadroughts.</p>
+<p>The recent trend of warming and reduced soil moisture in the Southwest
+US is highly challenging from a water systems planning and management
+perspective for the Colorado River Basin. Given the wide recognition
+that the river is over-allocated, the most recent drought highlights the
+difficulty of sustaining the flow requirements as dictated by the
+Colorado Compact. Thus, there has been an increasing focus in
+exploratory modeling efforts to clarify how vulnerable water systems in
+this region are to plausible drought streamflow scenarios for the
+future. In this tutorial, we’ll discuss how to create these scenarios
+using a Hidden Markov Model (HMM)- based streamflow synthetic generator.
+As discussed in <a class="reference external" href="https://uc-ebook.org/docs/html/2_diagnostic_modeling_overview_and_perspectives.html#overview-of-model-diagnostics">Section
+2.1</a>
+and
+<a class="reference external" href="https://uc-ebook.org/docs/html/4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-dynamics-what-is-controlling-model-behaviors-of-interest">4.2</a>
+of the eBook, future climate conditions in the basin represent a deep
+uncertainty that can lead to highly consequential water scarcity
+outcomes. It is advantageous to create a model such as the HMM-based
+generator in order to facilitate the creation of many ensembles of
+streamflow that can ultimately be used to force regional water systems
+models to understand how variability and drought extremes affect
+regional water shortages, operations, and policies.</p>
+<figure>
+<center><img src="_static/1-lede_pix_-_usbr-powell-8517152024_aa66437c2e_o-cropped.jpg" style="width:50%"></center>
+<figcaption style="text-align:center"> Lake Powell shows persistent effects from drought (Source: U.S. Bureau of Reclamation) </figcaption>
+</figure></section>
+</section>
+<section id="id30">
+<h3><span class="section-number">B.6.2. </span>Let’s Get Started!<a class="headerlink" href="#id30" title="Link to this heading">#</a></h3>
+<section id="observed-record">
+<h4><span class="section-number">B.6.2.1. </span>Observed Record<a class="headerlink" href="#observed-record" title="Link to this heading">#</a></h4>
+<p>First, let’s take a look at the observed data from 1909-2013 for a
+specific site. In this example, we use the outlet gauge of the Upper
+Colorado River (USGS Gauge 09163500 at the Colorado-Utah state line).
+Below, we create a plot of the annual streamflow.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Import libraries</span>
+<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="nn">mpl</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">matplotlib.patches</span> <span class="k">as</span> <span class="nn">patches</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span>
+<span class="kn">from</span> <span class="nn">SALib.sample</span> <span class="kn">import</span> <span class="n">latin</span>
+<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">stats</span> <span class="k">as</span> <span class="n">ss</span>
+<span class="kn">import</span> <span class="nn">statistics</span>
+<span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>
+
+
+<span class="c1"># Import helper functions from local package</span>
+<span class="kn">from</span> <span class="nn">functions</span> <span class="kn">import</span> <span class="n">fitmodel</span>
+<span class="kn">from</span> <span class="nn">functions</span> <span class="kn">import</span> <span class="n">plotstates</span>
+<span class="kn">from</span> <span class="nn">functions</span> <span class="kn">import</span> <span class="n">plotdist</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Read in annual historical data</span>
+<span class="n">AnnualQ</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;data/uc_historical.csv&#39;</span><span class="p">)</span>
+<span class="n">AnnualQ</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+<span class="c1"># Plot a line graph</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper right&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_9_0.png" src="_images/hmm_9_0.png" />
+<p>Let’s calculate an 11-year rolling mean of the same data to get a sense
+of long-term trends.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+
+<span class="c1"># Plot the original line graph</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+         <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+         <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+         <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1"># Plot an 11-year rolling mean</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+         <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+         <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+         <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_11_0.png" src="_images/hmm_11_0.png" />
+<p>The Colorado Compact, which prescribes flows between the Upper and Lower
+Colorado Basins, was negotiated using data prior to 1922, a time period
+revealed by the above figure to be one of the consistently wetter
+periods on record. It’s clear today that since the 1980s, the Southwest
+US has been experiencing aridification (Overpeck et al., 2020) and that
+this observed record alone isn’t an accurate representation of what
+future climate might look like in this region.</p>
+<p>Let’s get a little more specific and formally quantify decadal droughts
+that have occurred in the observed period. We use a metric proposed in
+Ault et al. (2014). The authors define a decadal drought as when the
+11-year rolling mean falls below a threshold that is 1/2 a standard
+deviation below the overall mean of the record. We can then highlight
+the block of years that fall in a decadal drought using yellow
+rectangles below.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define drought threshold</span>
+<span class="n">std</span> <span class="o">=</span> <span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="n">threshold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+<span class="c1"># Find where the rolling mean dip below the threshold?</span>
+<span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+<span class="n">drought_years</span> <span class="o">=</span> <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">drought_instances</span><span class="p">]</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span><span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span><span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">threshold</span><span class="p">,</span>
+           <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
+           <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span>
+           <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Drought Threshold&#39;</span><span class="p">)</span>
+
+<span class="c1"># Visualize the drought periods as yellow rectangles</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">drought_years</span><span class="p">:</span>
+
+    <span class="c1"># Plot a box centered around those values and with 5 years on either side.</span>
+    <span class="n">rect</span> <span class="o">=</span> <span class="n">patches</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">((</span><span class="n">i</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="mi">11</span><span class="p">,</span><span class="mf">2e7</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">,</span> <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Add the patch to the Axes</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_14_0.png" src="_images/hmm_14_0.png" />
+<p>By this metric, the Upper Colorado Basin region has experienced two
+decadal droughts over the last century.</p>
+</section>
+<section id="synthetic-stationary-generator-to-better-quantify-natural-variability">
+<h4><span class="section-number">B.6.2.2. </span>Synthetic Stationary Generator to Better Quantify Natural Variability<a class="headerlink" href="#synthetic-stationary-generator-to-better-quantify-natural-variability" title="Link to this heading">#</a></h4>
+<p>It is important to remember that the streamflow that we have observed in
+the region over the last century is only one instance of the hydrology
+that could occur since the atmosphere is an inherently stochastic
+system. Thus, we require a tool that will allow us to see multiple
+plausible realizations of the streamflow record to understand the
+internal variability that characterizes the historical period. One
+observed realization of historical streamflow is limited in its ability
+to capture rare extremes; plausible (but not observed) alternative
+instances of streamflow records can help to fill this gap. The tool that
+we use to develop synthetic flows for the region is a Gaussian Hidden
+Markov Model (HMM). If a system follows a Markov process, it switches
+between a number of “hidden states” dictated by a transition matrix.
+Each state has its own Gaussian probability distribution (defined by a
+mean and standard deviation) and one can draw from this distribution to
+create synthetic flows that fit the properties of the historical
+distribution. HMMs are an attractive choice for this region because they
+can simulate persistence (i.e., long duration droughts), which is a
+characteristic of the region’s hydro-climatology. The figure below shows
+an example of a 2-state Gaussian HMM that we will be fitting for this
+example.</p>
+<figure>
+    <center><img src="_static/HMM_example.png" style="width:75%"></center>
+<figcaption style="text-align:center"> Two-state Gaussian HMM with mean and standard deviation parameters</figcaption>
+</figure><p>Below is the code that fits the HMM model to the last 2/3 of the
+historical record of log annual flows at the CO-UT stateline gauge and
+creates an alternative trace of 105 years. A subset of the dataset is
+chosen in order to minimize overfitting and to retain a set of data for
+validation of the model. When we fit our model, we utilize the
+Baum-Welch algorithm (a special version of the expectation-maximization
+algorithm) to find the optimal parameters that maximize the likelihood
+of seeing the observed flows. Ultimately, the algorithm will return a
+mean and standard deviation associated with each state (mus and sigmas
+defined below) and a 2x2 transition probability matrix that captures the
+likelihood of transitioning between states (P). We can also retrieve the
+annual hidden states across the observed series, also known as the
+Viterbi sequence of states, which classifies each year in a “wet” or
+“dry” state.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Number of years for alternative trace</span>
+<span class="n">n_years</span> <span class="o">=</span> <span class="mi">105</span>
+
+<span class="c1"># Import historical data that it used to fit HMM model</span>
+<span class="n">AnnualQ_h</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;data/uc_historical.csv&#39;</span><span class="p">)</span>
+
+<span class="c1"># Fit the model and pull out relevant parameters and samples</span>
+<span class="n">logQ</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">AnnualQ_h</span><span class="p">)</span>
+<span class="n">hidden_states</span><span class="p">,</span> <span class="n">mus</span><span class="p">,</span> <span class="n">sigmas</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">logProb</span><span class="p">,</span> <span class="n">samples</span><span class="p">,</span> <span class="n">model</span> <span class="o">=</span> <span class="n">fitmodel</span><span class="o">.</span><span class="n">fitHMM</span><span class="p">(</span><span class="n">logQ</span><span class="p">,</span> <span class="n">n_years</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>We’ve fit our HMM, but what does the model look like? Let’s plot the
+annual time series of hidden states, or the Viterbi sequence. In the
+code, above, we have defined that the drier state is always represented
+by state 0. Thus, we know that hidden_states = 0 corresponds to the dry
+state and hidden_states = 1 to the wet state.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot Vitebi sequence</span>
+<span class="n">plotstates</span><span class="o">.</span><span class="n">plotTimeSeries</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">AnnualQ</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]),</span> <span class="n">hidden_states</span><span class="p">,</span> <span class="s1">&#39;Annual Flow (cube feet per year)&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_21_0.png" src="_images/hmm_21_0.png" />
+<p>In the figure above, we see that the years with the higher log flows
+tend to be classified in a “wet” state and the opposite is true of the
+“dry” state. We can also print the transition matrix, which shows the
+likelihood of transitioning between states. Note that the system has a
+high likelihood of persisting in the same state.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">transmat_</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[[</span><span class="mf">0.65095026</span> <span class="mf">0.34904974</span><span class="p">]</span>
+ <span class="p">[</span><span class="mf">0.3205531</span>  <span class="mf">0.6794469</span> <span class="p">]]</span>
+</pre></div>
+</div>
+<p>Let’s also plot the distribution of log annual flows associated with the
+wet and dry states.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot wet and dry state distributions</span>
+<span class="n">plotdist</span><span class="o">.</span><span class="n">plotDistribution</span><span class="p">(</span><span class="n">logQ</span><span class="p">,</span> <span class="n">mus</span><span class="p">,</span> <span class="n">sigmas</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_25_0.png" src="_images/hmm_25_0.png" />
+<p>The wet state distribution is characterized by a greater mean flow, but
+note that there is significant overlap in the tails of the distributions
+below which demonstrates why years with similiar flows can be classified
+in different states.</p>
+<p>Now let’s see what the drought dynamics look like in the synthetic
+scenario that we created using the same definition that we had used for
+the historical period.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Retrieve samples and back-transform out of log space</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">samples</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="p">)</span>
+<span class="n">AnnualQ_s</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+<span class="c1"># Define drought threshold</span>
+<span class="n">std</span><span class="o">=</span><span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="n">threshold</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+<span class="c1"># Where does the rolling mean dip below the threshold</span>
+<span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+<span class="n">drought_years</span> <span class="o">=</span> <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">drought_instances</span><span class="p">]</span>
+
+<span class="c1">#Visualize the streamflow scenario</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+
+<span class="c1">#Plot the original line graph</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1">#Plot a 11-year rolling mean</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">threshold</span><span class="p">,</span>
+           <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
+           <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span>
+           <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Drought Threshold&#39;</span><span class="p">)</span>
+
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">drought_years</span><span class="p">:</span>
+
+    <span class="c1">#Plot a box centered around those values and with 5 years on either side.</span>
+    <span class="n">rect</span> <span class="o">=</span> <span class="n">patches</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">((</span><span class="n">i</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span>
+                              <span class="mi">0</span><span class="p">),</span>
+                              <span class="mi">11</span><span class="p">,</span>
+                              <span class="mf">2e7</span><span class="p">,</span>
+                              <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+                              <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">,</span>
+                              <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Add the patch to the Axes</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Upper Colorado Annual Flow (Synthetic Stationary)&quot;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_28_0.png" src="_images/hmm_28_0.png" />
+<p>You can sample from the model and create more 105-year traces and note
+how the location and number of decadal droughts changes. This
+demonstrates how different the historical record can look just within
+the range of natural variability. It’s also important to remember that
+when droughts occur can also define the ultimate effect of the drought
+(i.e. is it a time when there is a large population growth or a time
+when humans can adapt by conserving or building more infrastructure?). A
+hydrologic drought need not manifest into an agricultural or operational
+drought of the same magnitude if stored surface water is available.</p>
+<p>We externally run the HMM many times to create a dataset of 100
+instances of the 105-year traces and 1000 instances of the 105-year
+traces that are available in the package
+(“synthetic_stationary_small_sample_100.csv”,“synthetic_stationary_large_sample_1000”).
+The shaded green lines correspond to the flow duration curves (FDCs) for
+the generated streamflow traces in comparison with the FDC of the
+historical record in beige.</p>
+<figure>
+    <center><img src="_static/stationary_synthetic_FDC.png" style="width:75%"></center>
+<figcaption style="text-align:center">Generated streamflow traces in comparison with the FDC of the historical record.</figcaption>
+</figure><p>As expected, the stationary synthetic FDCs envelope the historical FDC
+and particularly, the synthetic traces offer many more instances of low
+flow conditions that could lead to more extreme drought conditions than
+what has been observed historically. It is also useful to check for
+convergence of samples and to determine how many samples are needed to
+fully represent internal variability. Above we see that the extension to
+1000 instances of 105-year traces fills out regions of the FDC,
+including creating some more extreme drought conditions, but that
+additional samples will likely not fill out the FDC substantially more.</p>
+</section>
+<section id="non-stationary-synthetic-generator-to-impose-climate-changes">
+<h4><span class="section-number">B.6.2.3. </span>Non-Stationary Synthetic Generator to Impose Climate Changes<a class="headerlink" href="#non-stationary-synthetic-generator-to-impose-climate-changes" title="Link to this heading">#</a></h4>
+<p>Now, we create flows under non-stationary conditions to get a better
+understanding of what flows can look like under climate changes. In
+order to create flows under non-stationary conditions, we can toggle the
+parameters of the HMM model in order to create systematic changes to the
+model that can represent a changing climate. The HMM has 6 parameters
+that define it. When we fit the historical model, the parameters that
+are fit represent a baseline parameter value. In this non-stationary
+generator, we define a range to sample these parameters from.</p>
+<div class="pst-scrollable-table-container"><table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Parameter</p></th>
+<th class="head"><p>Current Value</p></th>
+<th class="head"><p>Lower Bound</p></th>
+<th class="head"><p>Upper Bound</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>Log-Space Wet State
+Mean Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.98</p></td>
+<td><p>1.02</p></td>
+</tr>
+<tr class="row-odd"><td><p>Log-Space Dry State
+Mean Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.98</p></td>
+<td><p>1.02</p></td>
+</tr>
+<tr class="row-even"><td><p>Log-Space Wet State
+Standard Deviation
+Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.75</p></td>
+<td><p>1.25</p></td>
+</tr>
+<tr class="row-odd"><td><p>Log-Space Dry State
+Standard Deviation
+Multiplier</p></td>
+<td><p>1.00</p></td>
+<td><p>0.75</p></td>
+<td><p>1.25</p></td>
+</tr>
+<tr class="row-even"><td><p>Change in Dry-Dry
+Transition
+Probability</p></td>
+<td><p>0.00</p></td>
+<td><p>-0.30</p></td>
+<td><p>+0.30</p></td>
+</tr>
+<tr class="row-odd"><td><p>Change in Wet-Wet
+Transition
+Probability</p></td>
+<td><p>0.00</p></td>
+<td><p>-0.30</p></td>
+<td><p>+0.30</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>Now let’s sample 1000 times from these bounds to create 1000 new
+parameterizations of the model. Here we use SALib and the Latin
+Hypercube sample function.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create problem structure with parameters that we want to sample</span>
+<span class="n">problem</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s1">&#39;num_vars&#39;</span><span class="p">:</span> <span class="mi">6</span><span class="p">,</span>
+    <span class="s1">&#39;names&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s1">&#39;wet_mu&#39;</span><span class="p">,</span> <span class="s1">&#39;dry_mu&#39;</span><span class="p">,</span> <span class="s1">&#39;wet_std&#39;</span><span class="p">,</span><span class="s1">&#39;dry_std&#39;</span><span class="p">,</span><span class="s1">&#39;dry_tp&#39;</span><span class="p">,</span><span class="s2">&quot;wet_tp&quot;</span><span class="p">],</span>
+    <span class="s1">&#39;bounds&#39;</span><span class="p">:</span> <span class="p">[[</span><span class="mf">0.98</span><span class="p">,</span> <span class="mf">1.02</span><span class="p">],</span>
+               <span class="p">[</span><span class="mf">0.98</span><span class="p">,</span> <span class="mf">1.02</span><span class="p">],</span>
+               <span class="p">[</span><span class="mf">0.75</span><span class="p">,</span><span class="mf">1.25</span><span class="p">],</span>
+               <span class="p">[</span><span class="mf">0.75</span><span class="p">,</span><span class="mf">1.25</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.3</span><span class="p">],</span>
+               <span class="p">[</span><span class="o">-</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.3</span><span class="p">]]</span>
+<span class="p">}</span>
+
+<span class="c1"># generate 1000 parameterizations</span>
+<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">1000</span>
+
+<span class="c1"># set random seed for reproducibility</span>
+<span class="n">seed_value</span> <span class="o">=</span> <span class="mi">123</span>
+
+<span class="c1"># Generate our samples</span>
+<span class="n">LHsamples</span> <span class="o">=</span> <span class="n">latin</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">problem</span><span class="p">,</span> <span class="n">n_samples</span><span class="p">,</span> <span class="n">seed_value</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now let’s look at what some of the traces look like in our
+non-stationary generator. Let’s choose a random instance from the
+1000-member space and adjust the parameters accordingly.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define static parameters</span>
+<span class="n">n_years</span> <span class="o">=</span> <span class="mi">105</span>
+
+<span class="c1"># Sample parameter; Adjust to any sample number from 0-999</span>
+<span class="n">sample</span> <span class="o">=</span> <span class="mi">215</span>
+
+<span class="c1"># Create empty arrays to store the new Gaussian HMM parameters for each SOW</span>
+<span class="n">Pnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">musNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">sigmasNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+<span class="n">logAnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+
+<span class="c1"># Calculate new transition matrix and stationary distribution of SOW at last node as well as new means and standard deviations</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">4</span><span class="p">]))</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">5</span><span class="p">]))</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
+<span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+<span class="n">eigenvals</span><span class="p">,</span> <span class="n">eigenvecs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">))</span>
+<span class="n">one_eigval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">eigenvals</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
+<span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">divide</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span> <span class="n">one_eigval</span><span class="p">]),</span>
+                  <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span><span class="n">one_eigval</span><span class="p">])))</span>
+
+<span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
+<span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+<span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">3</span><span class="p">]</span>
+<span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">sample</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
+
+<span class="c1"># Generate first state and log-space annual flow at last node</span>
+<span class="n">states</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+<span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">piNew</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
+    <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="k">else</span><span class="p">:</span>
+    <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+    <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="c1"># Generate remaining state trajectory and log space flows at last node</span>
+<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_years</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">Pnew</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">])]:</span>
+        <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+
+    <span class="k">if</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="c1"># Convert log-space flows to real-space flows</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">logAnnualQ_s</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
+</pre></div>
+</div>
+<p>Now let’s see what this synthetic trace looks like.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Retrieve samples and back-transform out of log space</span>
+<span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="p">)</span>
+<span class="n">AnnualQ_s</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+<span class="c1"># Define drought threshold</span>
+<span class="n">std</span> <span class="o">=</span> <span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="n">threshold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+<span class="c1"># Where does the rolling mean dip below the threshold</span>
+<span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+<span class="n">drought_years</span> <span class="o">=</span> <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">drought_instances</span><span class="p">]</span>
+
+<span class="c1"># Visualize the streamflow scenario</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+
+<span class="c1"># Plot the original line graph</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#005F73&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Annual&#39;</span><span class="p">)</span>
+
+<span class="c1"># Plot a 11-year rolling mean</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
+        <span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+        <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#183A2E&#39;</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s1">&#39;11-Year Rolling Mean&#39;</span><span class="p">)</span>
+
+<span class="c1"># Add labels and title</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">threshold</span><span class="p">,</span>
+           <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
+           <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span>
+           <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Drought Threshold&#39;</span><span class="p">)</span>
+
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">drought_years</span><span class="p">:</span>
+
+    <span class="c1"># Plot a box centered around those values and with 5 years on either side.</span>
+    <span class="n">rect</span> <span class="o">=</span> <span class="n">patches</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">((</span><span class="n">i</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span>
+                             <span class="mi">11</span><span class="p">,</span>
+                             <span class="mf">2e7</span><span class="p">,</span>
+                             <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+                             <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">,</span>
+                             <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;#EFE2BE&#39;</span><span class="p">)</span>
+
+    <span class="c1"># Add the patch to the Axes</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Annual Flow (Synthetic Non-Stationary)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Year&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Annual Flow (cubic feet per year)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">legend</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper right&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_40_0.png" src="_images/hmm_40_0.png" />
+<p>Above is the example trace from the new non-stationary model. You may
+see fewer or more decadal drought instances. We can further summarize
+overall decadal drought characteristics across the samples. Let’s plot a
+histogram of the total number of times we go below the drought threshold
+across these realizations.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">decadal_drought_occurence</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">1000</span><span class="p">])</span>
+
+<span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">):</span>
+
+    <span class="c1"># Create empty arrays to store the new Gaussian HMM parameters for each SOW</span>
+    <span class="n">Pnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
+    <span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+    <span class="n">musNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+    <span class="n">sigmasNew_HMM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
+    <span class="n">logAnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+
+    <span class="c1"># Calculate new transition matrix and stationary distribution of SOW at last node</span>
+    <span class="c1"># as well as new means and standard deviations</span>
+
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">4</span><span class="p">]))</span>
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="nb">min</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">5</span><span class="p">]))</span>
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
+    <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Pnew</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+    <span class="n">eigenvals</span><span class="p">,</span> <span class="n">eigenvecs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">))</span>
+    <span class="n">one_eigval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">eigenvals</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
+    <span class="n">piNew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">divide</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span> <span class="n">one_eigval</span><span class="p">]),</span>
+                      <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">Pnew</span><span class="p">),</span> <span class="n">eigenvecs</span><span class="p">[:,</span> <span class="n">one_eigval</span><span class="p">])))</span>
+
+    <span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
+    <span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">mus</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+    <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">3</span><span class="p">]</span>
+    <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigmas</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">LHsamples</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
+
+    <span class="c1"># Generate first state and log-space annual flow at last node</span>
+    <span class="n">states</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">([</span><span class="n">n_years</span><span class="p">])</span>
+    <span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">piNew</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
+        <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+        <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+    <span class="c1"># generate remaining state trajectory and log space flows at last node</span>
+    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_years</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">random</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">Pnew</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="nb">int</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">])]:</span>
+            <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+
+        <span class="k">if</span> <span class="n">states</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">logAnnualQ_s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">musNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigmasNew_HMM</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+    <span class="c1"># Convert log-space flows to real-space flows</span>
+    <span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">logAnnualQ_s</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">AnnualQ_s</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="p">)</span>
+    <span class="n">AnnualQ_s</span><span class="p">[</span><span class="s1">&#39;Year&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1909</span><span class="p">,</span> <span class="mi">2014</span><span class="p">))</span>
+
+    <span class="c1"># Define drought threshold</span>
+    <span class="n">std</span> <span class="o">=</span> <span class="n">statistics</span><span class="o">.</span><span class="n">stdev</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
+    <span class="n">threshold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">std</span><span class="p">))</span>
+
+    <span class="c1"># Where does the rolling mean dip below the threshold</span>
+    <span class="n">drought_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">AnnualQ_s</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rolling</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="k">if</span> <span class="n">v</span> <span class="o">&lt;</span> <span class="n">threshold</span><span class="p">]</span>
+    <span class="n">decadal_drought_occurence</span><span class="p">[</span><span class="n">y</span><span class="p">]</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">drought_instances</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">decadal_drought_occurence</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Non-Stationary generator&#39;</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s2">&quot;#005F73&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Number of Instances of Decadal Drought&#39;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Frequency&#39;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;-&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Observed&#39;</span><span class="p">)</span>
+<span class="n">mpl</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">&#39;legend&#39;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span> <span class="o">=</span> <span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span> <span class="o">=</span> <span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_43_0.png" src="_images/hmm_43_0.png" />
+<p>Note how many more instances of the decadal droughts we are creating
+with the non-stationary generator than our observed 105-year trace which
+creates a rich space in which we can test our models. Just as we did
+with the stationary generator, we can externally run the non-stationary
+generator to create 10,000 instances of the 105-year traces that are
+available in the package
+(“synthetic_nonstationary_large_sample_10000.csv”). The shaded green and
+blue lines correspond to the FDCs for the stationary and non-stationary
+generated streamflow traces in comparison with the FDC of the historical
+record in beige. Note how the non-stationary generator produces even
+more drought extremes than the stationary non-synthetic traces.</p>
+<figure>
+    <center><img src="_static/nonstationary_synthetic_FDC.png" style="width:75%"></center>
+<figcaption style="text-align:center">Generated streamflow traces in comparison with the FDC of the historical record.</figcaption>
+</figure></section>
+<section id="placing-cmip5-projections-in-the-context-of-non-stationary-flows">
+<h4><span class="section-number">B.6.2.4. </span>Placing CMIP5 Projections in the Context of Non-Stationary Flows<a class="headerlink" href="#placing-cmip5-projections-in-the-context-of-non-stationary-flows" title="Link to this heading">#</a></h4>
+<p>We have broadened the drought conditions that we are creating which that
+can be very useful to understand how our water systems model performs
+under potentially extreme scenarios. However, it’s useful to compare our
+bottom-up synthetically generated flows in the context of global
+physically-driven CMIP5 projections to get a better understanding of how
+the two approaches compare. We first aquire 97 CMIP5 projections from
+the Colorado River Water Availability Study (CWCB, 2012). In each of
+these projections, monthly precipitation factor changes and temperature
+delta changes were computed between mean projected 2035–2065 climate
+statistics and mean historical climate statistics from 1950–2013. These
+97 different combinations of 12 monthly precipitation multipliers and 12
+monthly temperature delta shifts were applied to historical
+precipitation and temperature time series from 1950–2013. The resulting
+climate time series were run through a Variable Infiltration Capacity
+(VIC) model of the UCRB, resulting in 97 time series of projected future
+streamflows at the Colorado‐Utah state line.</p>
+<p>We fit an HMM to each trace of projected streamflow and get a set of
+corresponding HMM parameters. Then we take the ratio between these
+parameters and the baseline HMM parameters that we calculated earlier in
+the notebook in order to calculate the multipliers associated with each
+CMIP5 projection. This is all done externally, so we import the
+resulting multipliers in the next line.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Read in CMIP5 and paleo multipliers</span>
+<span class="n">CMIP5_multipliers</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;data/CMIP5_SOWs.txt&#39;</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s2">&quot; &quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Let’s plot a response surface that will allow us to see how combinations
+of HMM parameters tend to influence decadal drought. In order to get a
+continuous surface, we’ll fit a non-linear regression to the parameter
+values and then predict the decadal drought over a set of grid points.
+We fit the response surface for two parameters that should have an
+affect on decadal drought: the dry distribution mean and the dry-dry
+transition probabilites.</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Choose two parameters to fit the response surface for</span>
+<span class="n">mu_dry</span><span class="o">=</span><span class="p">[</span><span class="n">i</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">LHsamples</span><span class="p">]</span>
+<span class="n">tp_dry</span><span class="o">=</span><span class="p">[</span><span class="n">i</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">LHsamples</span><span class="p">]</span>
+
+<span class="c1"># Create an interpolation grid</span>
+<span class="n">xgrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">),</span>
+                  <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">),</span>
+                  <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">mu_dry</span><span class="p">))</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span>
+
+<span class="n">ygrid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">),</span>
+                  <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">),</span>
+                  <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">tp_dry</span><span class="p">))</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span>
+
+<span class="c1"># Fit regression</span>
+<span class="n">d</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;Dry_Tp&#39;</span><span class="p">:</span> <span class="n">tp_dry</span><span class="p">,</span>
+     <span class="s1">&#39;Dry_Mu&#39;</span><span class="p">:</span> <span class="n">mu_dry</span><span class="p">,</span>
+     <span class="s1">&#39;Drought_Occurrence&#39;</span><span class="p">:</span><span class="n">decadal_drought_occurence</span><span class="p">}</span>
+
+<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
+<span class="n">df</span><span class="p">[</span><span class="s1">&#39;Intercept&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">shape</span><span class="p">(</span><span class="n">df</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">df</span><span class="p">[</span><span class="s1">&#39;Interaction&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s1">&#39;Dry_Tp&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">df</span><span class="p">[</span><span class="s1">&#39;Dry_Mu&#39;</span><span class="p">]</span>
+<span class="n">cols</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Intercept&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s1">&#39;Dry_Mu&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s1">&#39;Dry_Tp&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s1">&#39;Interaction&#39;</span><span class="p">]</span>
+<span class="n">ols</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Drought_Occurrence&#39;</span><span class="p">],</span> <span class="n">df</span><span class="p">[</span><span class="n">cols</span><span class="p">])</span>
+<span class="n">result</span> <span class="o">=</span> <span class="n">ols</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
+
+<span class="c1"># Calculate drought occurence for each grid point</span>
+<span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">xgrid</span><span class="p">,</span> <span class="n">ygrid</span><span class="p">)</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
+<span class="n">y</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
+<span class="n">grid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">column_stack</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">*</span> <span class="n">y</span><span class="p">])</span>
+<span class="n">z</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
+<span class="n">z</span><span class="p">[</span><span class="n">z</span> <span class="o">&lt;</span> <span class="mf">0.0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="c1"># replace negative shortage predictions with 0</span>
+</pre></div>
+</div>
+<p>Let’s plot our results:</p>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set color gradient for response surface</span>
+<span class="n">drought_map</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">&#39;RdBu_r&#39;</span><span class="p">)</span>
+
+<span class="c1"># Reshape our predicted drought occurrence and define bounds of colors</span>
+<span class="n">Z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">shape</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
+<span class="n">vmin</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Drought_Occurrence&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">)])</span>
+<span class="n">vmax</span> <span class="o">=</span> <span class="mi">15</span>
+<span class="n">norm</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="p">,</span> <span class="n">vmax</span><span class="p">)</span>
+
+<span class="c1"># Plot response surface and CMIP5 projections</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">drought_map</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">norm</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">CMIP5_multipliers</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">7</span><span class="p">],</span>
+           <span class="n">CMIP5_multipliers</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span><span class="mi">12</span><span class="p">],</span>
+           <span class="n">c</span><span class="o">=</span><span class="s1">&#39;#ffffb3&#39;</span><span class="p">,</span>
+           <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;none&#39;</span><span class="p">,</span>
+           <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span>
+<span class="n">cbar</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">figure</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">ScalarMappable</span><span class="p">(</span><span class="n">norm</span><span class="o">=</span><span class="n">norm</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">drought_map</span><span class="p">),</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">nanmin</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">nanmax</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">nanmin</span><span class="p">(</span><span class="n">Y</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">nanmax</span><span class="p">(</span><span class="n">Y</span><span class="p">))</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Dry State Mu&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Dry-Dry Transition Probability&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="n">labelsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Decadal Drought Occurrence&#39;</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=-</span><span class="mi">90</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">,</span> <span class="n">labelpad</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
+<span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;y&#39;</span><span class="p">,</span><span class="n">labelsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+<img alt="_images/hmm_53_0.png" src="_images/hmm_53_0.png" />
+<p>We see the influence of the dry state mean and dry-dry transition
+parameters. We’re likely to see more decadal droughts when we (1)
+increase the dry-dry transition probability, which inherently will
+increase persistence of the dry state, and (2) when we make the dry
+state log mean drier. Note that the CMIP5 scenarios tend to span the
+extent of the dry mean sample space, but are less representative of the
+dry transition probability sample space, which suggests that the types
+of hydrological droughts represented in the projections tend to only be
+wetter to slightly drier than our baseline. Both methods of producing
+these scenarios are valid, though studies have suggested that
+globally-resolved GCMs may be inappropriate to represent regional
+extremes. Ultimately, if your goal is to produce a variety of ensembles
+that are characterized by many different drought characteristics, you
+will likely find that a generator approach will serve this purpose
+better.</p>
+</section>
+<section id="tips-to-create-an-hmm-based-generator-for-your-system">
+<h4><span class="section-number">B.6.2.5. </span>Tips to Create an HMM-Based Generator for your System<a class="headerlink" href="#tips-to-create-an-hmm-based-generator-for-your-system" title="Link to this heading">#</a></h4>
+<p>In this tutorial, we demonstrated how to fit an HMM-based generator for
+a single gauge located in the Upper Colorado River Basin. In order to
+apply this methodology to your problem, you will need to first ask:</p>
+<ol class="arabic simple">
+<li><p>Is this model appropriate for my location of interest? We have
+applied this style of generator to locations where persistent wet
+and dry states are characteristic, which tends to be in the Western
+US. Ultimately the best way to judge if an HMM is useful for your
+application is to fit the model and explore the resulting
+distributions. Are there two (or more) distinct states that emerge?
+If not, then your location may not exhibit the type of persistence
+that an HMM-based generator is useful for. You can consider
+exploring other styles of generators such as the Kirsch-Nowak
+generator (Kirsch et al., 2013).</p></li>
+<li><p>Do I have the right datasets? We use annual data for our location of
+interest. In this notebook, the HMM is fit to log annual flows.
+Ultimately, it can be disaggregated to daily flows (using a
+reference historical daily dataset) to be useful in water resources
+operational applications. You could also disaggregate to a finer
+resolution than daily if the historical dataset exists.</p></li>
+</ol>
+<p>If you meet these requirements, feel free to proceed through fitting the
+model using the code available in the notebook. Be sure to consider the
+appropirate number of samples to generate (both in a stationary and
+non-stationary case). Make sure that you test multiple sample sizes and
+continue to increase your sample size until you converge to a consistent
+representation of extremes. What is the appropriate number of LHS
+samples of the parameters to use? In this experiment we used 1,000
+samples of parameters due to extensive stability tests described in
+Quinn et al. (2020).</p>
+<p>Finally, to learn more about this test case refer to Hadmichael et
+al. (2020a) and Hadmichael et al. (2020b). For another study on
+synthetic drought generation to support vulnerability assessments in the
+Research Triangle region of North Carolina, please refer to Herman et
+al. (2016)</p>
+</section>
+<section id="id31">
+<h4><span class="section-number">B.6.2.6. </span>References<a class="headerlink" href="#id31" title="Link to this heading">#</a></h4>
+<p>Ault, T. R., Cole, J. E., Overpeck, J. T., Pederson, G. T., &amp; Meko, D.
+M. (2014). Assessing the risk of persistent drought using climate model
+simulations and paleoclimate data. Journal of Climate, 27(20),
+7529-7549.</p>
+<p>CWCB (2012).Colorado River Water Availability Study Phase I Report.
+Colorado Water Conservation Board</p>
+<p>Hadjimichael, A., Quinn, J., Wilson, E., Reed, P., Basdekas, L., Yates,
+D., &amp; Garrison, M. (2020a). Defining robustness, vulnerabilities, and
+consequential scenarios for diverse stakeholder interests in
+institutionally complex river basins. Earth’s Future, 8(7),
+e2020EF001503.</p>
+<p>Hadjimichael, A., Quinn, J., &amp; Reed, P. (2020). Advancing diagnostic
+model evaluation to better understand water shortage mechanisms in
+institutionally complex river basins. Water Resources Research, 56(10),
+e2020WR028079.</p>
+<p>Herman, J. D., Zeff, H. B., Lamontagne, J. R., Reed, P. M., &amp;
+Characklis, G. W. (2016). Synthetic drought scenario generation to
+support bottom-up water supply vulnerability assessments. Journal of
+Water Resources Planning and Management, (11), 04016050.</p>
+<p>Kirsch, B. R., Characklis, G. W., &amp; Zeff, H. B. (2013). Evaluating the
+impact of alternative hydro-climate scenarios on transfer agreements:
+Practical improvement for generating synthetic streamflows. Journal of
+Water Resources Planning and Management, 139(4), 396-406.</p>
+<p>Overpeck, J.T. &amp; Udall, B. (2020) “Climate change and the aridification
+of North America.” Proceedings of the national academy of sciences
+117.22 11856-11858.</p>
+<p>Quinn, J. D., Hadjimichael, A., Reed,P. M., &amp; Steinschneider, S. (2020).
+Canexploratory modeling of water scarcity vulnerabilities and robustness
+bescenario neutral?Earth’s Future,8,e2020EF001650.
+<a class="reference external" href="https://doi.org/10.1029/2020EF001650Received">https://doi.org/10.1029/2020EF001650Received</a></p>
+<p>Williams, A. P., Cook, B. I., &amp; Smerdon, J. E. (2022). Rapid
+intensification of the emerging southwestern North American megadrought
+in 2020–2021. Nature Climate Change, 12(3), 232-234.</p>
+</section>
+</section>
+</section>
+</section>
+
+
+                </article>
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+    <i class="fa-solid fa-list"></i> Contents
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+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fishery-dynamics-tutorial">B.1. Fishery Dynamics Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-to-discover-factors-shaping-consequential-dynamics">B.1.1. Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#lets-get-started">B.1.1.1. Let’s get started!</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-identified-solutions-and-explore-performance">B.1.1.2. Step 1: Load identified solutions and explore performance</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-use-salib-to-generate-a-sample-for-a-sobol-sensitivity-analysis">B.1.1.3. Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-evaluate-the-system-over-all-generated-states-of-the-world">B.1.1.4. Step 3: Evaluate the system over all generated states of the world</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-calculate-sensitivity-indices">B.1.1.5. Step 4: Calculate sensitivity indices</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-explore-relationship-between-uncertain-factors-and-performance">B.1.1.6. Step 5: Explore relationship between uncertain factors and performance</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-and-scenario-discovery-to-your-problem">B.1.1.7. Tips to Apply Sobol SA and Scenario Discovery to your Problem</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sobol-sa-tutorial">B.2. Sobol SA Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-sa-using-the-saltelli-sampling-scheme-with-sobol-sa">B.2.1. Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id6">B.2.1.1. Let’s get started!</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-generate-the-problem-dictionary">B.2.1.2. Step 1: Generate the problem dictionary</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-generate-samples-using-the-saltelli-sampling-scheme">B.2.1.3. Step 2: Generate samples using the Saltelli sampling scheme</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-execute-the-ishigami-function-over-our-sample-set">B.2.1.4. Step 3: Execute the Ishigami function over our sample set</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-compute-first-second-and-total-order-sensitivity-indices-using-the-sobol-method">B.2.1.5. Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-interpret-our-results">B.2.1.6. Step 5: Interpret our results</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-sobol-sa-to-your-own-problem">B.2.1.7. Tips to Apply Sobol SA to Your Own Problem</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#logistic-regression-tutorial">B.3. Logistic Regression Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-logistic-regression-for-factor-mapping">B.3.1. Tutorial: Logistic Regression for Factor Mapping</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#background">B.3.1.1. Background</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id19">B.3.1.2. Let’s get started!</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-load-latin-hypercube-sample-and-set-up-problem">B.3.1.3. Step 1: Load Latin hypercube sample and set up problem</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-define-decision-relevant-metrics-for-illustration">B.3.1.4. Step 2: Define decision-relevant metrics for illustration</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-run-the-logistic-regression">B.3.1.5. Step 3: Run the logistic regression</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-factor-ranking">B.3.1.6. Step 4: Factor ranking</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-draw-factor-maps">B.3.1.7. Step 5: Draw factor maps</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-scenario-discovery-to-your-own-problem">B.3.1.8. Tips to Apply Scenario Discovery to Your Own Problem</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#hymod-dynamics-tutorial">B.4. HYMOD Dynamics Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-sensitivity-analysis-of-the-hymod-model">B.4.1. Tutorial: Sensitivity Analysis of the HYMOD Model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction-to-hymod">B.4.2. 1 - Introduction to HYMOD</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#global-sensitivity-analysis">B.4.3. 2- Global Sensitivity Analysis</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-apply-this-methodology-to-your-own-problem">B.4.4. Tips to Apply this methodology to your own problem</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#time-evolving-scenario-discovery-for-infrastructure-pathways">B.5. Time-evolving scenario discovery for infrastructure pathways</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id24">B.5.1. Time-evolving scenario discovery for infrastructure pathways</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id25">B.5.1.1. Background</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluating-robustness-over-time">B.5.1.2. Evaluating Robustness over time</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#exploring-performance-evolution">B.5.1.3. Exploring performance evolution</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#how-do-deep-uncertainties-generate-vulnerability">B.5.1.4. How do deep uncertainties generate vulnerability</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#which-uncertainties-have-the-most-influence-on-time-evolving-performance">B.5.1.5. Which uncertainties have the most influence on time-evolving performance?</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#open-exploration">B.5.1.6. Open exploration</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id26">B.5.1.7. Tips to apply this methodology to your own problem</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#references">B.5.1.8. References</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios-tutorial">B.6. A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios Tutorial</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#a-hidden-markov-modeling-approach-to-creating-synthetic-streamflow-scenarios">B.6.1. A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id29">B.6.1.1. Background</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id30">B.6.2. Let’s Get Started!</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#observed-record">B.6.2.1. Observed Record</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#synthetic-stationary-generator-to-better-quantify-natural-variability">B.6.2.2. Synthetic Stationary Generator to Better Quantify Natural Variability</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#non-stationary-synthetic-generator-to-impose-climate-changes">B.6.2.3. Non-Stationary Synthetic Generator to Impose Climate Changes</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#placing-cmip5-projections-in-the-context-of-non-stationary-flows">B.6.2.4. Placing CMIP5 Projections in the Context of Non-Stationary Flows</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tips-to-create-an-hmm-based-generator-for-your-system">B.6.2.5. Tips to Create an HMM-Based Generator for your System</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id31">B.6.2.6. References</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="A1_Uncertainty_Quantification.html">A. Uncertainty Quantification</a></li>
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+<li class="toctree-l1 current active"><a class="current reference internal" href="#">C. Plotting Code Samples</a></li>
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+    <h1>Plotting Code Samples</h1>
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+            
+            <div>
+                <h2> Contents </h2>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#hymod-ipynb">C.1. hymod.ipynb</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-observed-vs-simulated-streamflow">C.1.1. <strong>plot_observed_vs_simulated_streamflow()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-observed-vs-sensitivity-streamflow">C.1.2. <strong>plot_observed_vs_sensitivity_streamflow()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-monthly-heatmap">C.1.3. <strong>plot_monthly_heatmap()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-annual-heatmap">C.1.4. <strong>plot_annual_heatmap()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-varying-heatmap">C.1.5. <strong>plot_varying_heatmap()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-precalibration-flow">C.1.6. <strong>plot_precalibration_flow()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-precalibration-glue">C.1.7. <strong>plot_precalibration_glue()</strong></a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fishery-dynamics-ipynb">C.2. fishery_dynamics.ipynb</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-objective-performance">C.2.1. <strong>plot_objective_performance()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-factor-performance">C.2.2. <strong>plot_factor_performance()</strong></a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="plotting-code-samples">
+<h1><span class="section-number">C. </span>Plotting Code Samples<a class="headerlink" href="#plotting-code-samples" title="Link to this heading">#</a></h1>
+<section id="hymod-ipynb">
+<h2><span class="section-number">C.1. </span>hymod.ipynb<a class="headerlink" href="#hymod-ipynb" title="Link to this heading">#</a></h2>
+<p>The following are the plotting functions as described in the <code class="docutils literal notranslate"><span class="pre">hymod.ipynb</span></code> Jupyter notebook tutorial.</p>
+<p>The following are the necessary package imports to run these functions:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="kn">from</span> <span class="nn">matplotlib.lines</span> <span class="kn">import</span> <span class="n">Line2D</span>
+</pre></div>
+</div>
+<section id="plot-observed-vs-simulated-streamflow">
+<h3><span class="section-number">C.1.1. </span><strong>plot_observed_vs_simulated_streamflow()</strong><a class="headerlink" href="#plot-observed-vs-simulated-streamflow" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_observed_vs_simulated_streamflow</span><span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="n">hymod_dict</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">12</span><span class="p">,</span> <span class="mi">6</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot observed versus simulated streamflow.</span>
+
+<span class="sd">    :param df:              Dataframe of hymod input data including columns for precip, potential evapotranspiration,</span>
+<span class="sd">                            and streamflow</span>
+
+<span class="sd">    :param hymod_dict:      A dictionary of hymod outputs</span>
+<span class="sd">    :type hymod_dict:       dict</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set plot style</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">style</span><span class="o">.</span><span class="n">use</span><span class="p">(</span><span class="s1">&#39;seaborn-white&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># plot observed streamflow</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">])),</span> <span class="n">df</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;pink&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plot simulated streamflow</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">])),</span> <span class="n">hymod_dict</span><span class="p">[</span><span class="s1">&#39;Q&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set axis labels</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Streamflow($m^3/s$)&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Days&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set plot title</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Observed vs. Simulated Streamflow&#39;</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">ax</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-observed-vs-sensitivity-streamflow">
+<h3><span class="section-number">C.1.2. </span><strong>plot_observed_vs_sensitivity_streamflow()</strong><a class="headerlink" href="#plot-observed-vs-sensitivity-streamflow" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_observed_vs_sensitivity_streamflow</span><span class="p">(</span><span class="n">df_obs</span><span class="p">,</span> <span class="n">df_sim</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot observed streamflow versus simulations generated from sensitivity analysis.</span>
+
+<span class="sd">    :param df_obs:          Dataframe of mean monthly hymod input data including columns for precip,</span>
+<span class="sd">                            potential evapotranspiration, and streamflow</span>
+
+<span class="sd">    :param df_sim:          Dataframe of mean monthly simulation data from sensitivity analysis</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">month_list</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df_sim</span><span class="p">))</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># set labels</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Days&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Flow Discharge (m^3/s)&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plots all simulated streamflow cases under different sample sets</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">df_sim</span><span class="o">.</span><span class="n">columns</span><span class="p">:</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">month_list</span><span class="p">,</span> <span class="n">df_sim</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;pink&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
+
+    <span class="c1"># plot observed streamflow</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">month_list</span><span class="p">,</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Observed vs. Sensitivity Analysis Outputs&#39;</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">ax</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-monthly-heatmap">
+<h3><span class="section-number">C.1.3. </span><strong>plot_monthly_heatmap()</strong><a class="headerlink" href="#plot-monthly-heatmap" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_monthly_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="p">,</span> <span class="n">df_obs</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s1">&#39;&#39;</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">14</span><span class="p">,</span> <span class="mi">6</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot a sensitivity metric overlain by observed flow.</span>
+
+<span class="sd">    :param arr_sim:         Numpy array of simulated metrics</span>
+
+<span class="sd">    :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis</span>
+
+<span class="sd">    :param title:           Title of plot</span>
+<span class="sd">    :type title:            str</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># plot heatmap</span>
+    <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="p">,</span>
+                <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span>
+                <span class="n">yticklabels</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">],</span>
+                <span class="n">cmap</span><span class="o">=</span><span class="n">sns</span><span class="o">.</span><span class="n">color_palette</span><span class="p">(</span><span class="s2">&quot;ch:s=-.2,r=.6&quot;</span><span class="p">))</span>
+
+    <span class="c1"># setup overlay axis</span>
+    <span class="n">ax2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+
+    <span class="c1"># plot line</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">12.5</span><span class="p">),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;slateblue&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plot points on line</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">12.5</span><span class="p">),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;slateblue&#39;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set axis limits and labels</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">12</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s1">&#39;jan&#39;</span><span class="p">,</span> <span class="s1">&#39;feb&#39;</span><span class="p">,</span> <span class="s1">&#39;mar&#39;</span><span class="p">,</span> <span class="s1">&#39;apr&#39;</span><span class="p">,</span> <span class="s1">&#39;may&#39;</span><span class="p">,</span> <span class="s1">&#39;jun&#39;</span><span class="p">,</span> <span class="s1">&#39;jul&#39;</span><span class="p">,</span> <span class="s1">&#39;aug&#39;</span><span class="p">,</span> <span class="s1">&#39;sep&#39;</span><span class="p">,</span> <span class="s1">&#39;oct&#39;</span><span class="p">,</span> <span class="s1">&#39;nov&#39;</span><span class="p">,</span> <span class="s1">&#39;dec&#39;</span><span class="p">])</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Flow Discharge($m^3/s$)&#39;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+
+    <span class="k">return</span> <span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-annual-heatmap">
+<h3><span class="section-number">C.1.4. </span><strong>plot_annual_heatmap()</strong><a class="headerlink" href="#plot-annual-heatmap" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_annual_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="p">,</span> <span class="n">df_obs</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s1">&#39;&#39;</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">14</span><span class="p">,</span><span class="mi">5</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot a sensitivity metric overlain by observed flow..</span>
+
+<span class="sd">    :param arr_sim:         Numpy array of simulated metrics</span>
+
+<span class="sd">    :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis</span>
+
+<span class="sd">    :param title:           Title of plot</span>
+<span class="sd">    :type title:            str</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># plot heatmap</span>
+    <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">sns</span><span class="o">.</span><span class="n">color_palette</span><span class="p">(</span><span class="s2">&quot;YlOrBr&quot;</span><span class="p">))</span>
+
+    <span class="c1"># setup overlay axis</span>
+    <span class="n">ax2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+
+    <span class="c1"># plot line</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">10.5</span><span class="p">),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;slateblue&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plot points on line</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">10.5</span><span class="p">),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;slateblue&#39;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set up axis lables and limits</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">([</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">2000</span><span class="p">,</span> <span class="mi">2010</span><span class="p">))</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Flow Discharge($m^3/s$)&#39;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-varying-heatmap">
+<h3><span class="section-number">C.1.5. </span><strong>plot_varying_heatmap()</strong><a class="headerlink" href="#plot-varying-heatmap" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_varying_heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="p">,</span> <span class="n">df_obs</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s1">&#39;&#39;</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">14</span><span class="p">,</span><span class="mi">5</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot a sensitivity metric overlain by observed flow..</span>
+
+<span class="sd">    :param arr_sim:         Numpy array of simulated metrics</span>
+
+<span class="sd">    :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis</span>
+
+<span class="sd">    :param title:           Title of plot</span>
+<span class="sd">    :type title:            str</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># plot heatmap</span>
+    <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">arr_sim</span><span class="p">,</span>
+                <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span>
+                <span class="n">yticklabels</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Kq&#39;</span><span class="p">,</span> <span class="s1">&#39;Ks&#39;</span><span class="p">,</span> <span class="s1">&#39;Alp&#39;</span><span class="p">,</span> <span class="s1">&#39;Huz&#39;</span><span class="p">,</span> <span class="s1">&#39;B&#39;</span><span class="p">],</span>
+                <span class="n">cmap</span><span class="o">=</span><span class="n">sns</span><span class="o">.</span><span class="n">light_palette</span><span class="p">(</span><span class="s2">&quot;seagreen&quot;</span><span class="p">,</span> <span class="n">as_cmap</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span>
+
+    <span class="n">n_years</span> <span class="o">=</span> <span class="n">df_obs</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+
+    <span class="c1"># setup overlay axis</span>
+    <span class="n">ax2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+
+    <span class="c1"># plot line</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n_years</span><span class="p">),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;slateblue&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plot points on line</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n_years</span><span class="p">),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;slateblue&#39;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set up axis lables and limits</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">119.5</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Flow Discharge&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Number of Months&#39;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">ax</span><span class="p">,</span> <span class="n">ax2</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-precalibration-flow">
+<h3><span class="section-number">C.1.6. </span><strong>plot_precalibration_flow()</strong><a class="headerlink" href="#plot-precalibration-flow" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_precalibration_flow</span><span class="p">(</span><span class="n">df_sim</span><span class="p">,</span> <span class="n">df_obs</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot flow discharge provided by the ensemble of parameters sets from Pre-Calibration versus the observed</span>
+<span class="sd">    flow data.</span>
+
+<span class="sd">    :param df_sim:          Dataframe of simulated metrics</span>
+
+<span class="sd">    :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># set axis labels</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Days&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Flow Discharge&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plot pre-calibration results</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">df_sim</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df_sim</span><span class="p">)),</span> <span class="n">df_sim</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span>  <span class="n">color</span><span class="o">=</span><span class="s2">&quot;lightgreen&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
+
+    <span class="c1"># plot observed</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df_sim</span><span class="p">)),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span>  <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Observed vs. Pre-Calibration Outputs&#39;</span><span class="p">)</span>
+
+    <span class="c1"># customize legend</span>
+    <span class="n">custom_lines</span> <span class="o">=</span> <span class="p">[</span><span class="n">Line2D</span><span class="p">([</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span>  <span class="n">color</span><span class="o">=</span><span class="s2">&quot;lightgreen&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">),</span>
+                    <span class="n">Line2D</span><span class="p">([</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">)]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">custom_lines</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;Pre-Calibration&#39;</span><span class="p">,</span> <span class="s1">&#39;Observed&#39;</span><span class="p">])</span>
+
+    <span class="k">return</span> <span class="n">ax</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-precalibration-glue">
+<h3><span class="section-number">C.1.7. </span><strong>plot_precalibration_glue()</strong><a class="headerlink" href="#plot-precalibration-glue" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_precalibration_glue</span><span class="p">(</span><span class="n">df_precal</span><span class="p">,</span> <span class="n">df_glue</span><span class="p">,</span> <span class="n">df_obs</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot flow discharge provided by the ensemble of parameters sets from Pre-Calibration versus the observed</span>
+<span class="sd">    flow data.</span>
+
+<span class="sd">    :param df_sim:          Dataframe of simulated metrics</span>
+
+<span class="sd">    :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis</span>
+
+<span class="sd">    :param figsize:         Matplotlib figure size</span>
+<span class="sd">    :type figsize:          list</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set up figure</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># set axis labels</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Days&#39;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Flow Discharge&#39;</span><span class="p">)</span>
+
+    <span class="c1"># plot pre-calibration results</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">df_precal</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df_precal</span><span class="p">)),</span> <span class="n">df_precal</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span>  <span class="n">color</span><span class="o">=</span><span class="s2">&quot;lightgreen&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
+
+    <span class="c1"># plot glue</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">df_glue</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df_glue</span><span class="p">)),</span> <span class="n">df_glue</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;lightblue&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
+
+    <span class="c1"># plot observed</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df_precal</span><span class="p">)),</span> <span class="n">df_obs</span><span class="p">[</span><span class="s1">&#39;Strmflw&#39;</span><span class="p">],</span>  <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Observed vs. Sensitivity Analysis Outputs across GLUE/Pre-Calibration&#39;</span><span class="p">)</span>
+
+    <span class="c1"># customize legend</span>
+    <span class="n">custom_lines</span> <span class="o">=</span> <span class="p">[</span><span class="n">Line2D</span><span class="p">([</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span>  <span class="n">color</span><span class="o">=</span><span class="s2">&quot;lightgreen&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">),</span>
+                    <span class="n">Line2D</span><span class="p">([</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;lightblue&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">),</span>
+                    <span class="n">Line2D</span><span class="p">([</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">)]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">custom_lines</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;Pre-Calibration&#39;</span><span class="p">,</span> <span class="s1">&#39;GLUE&#39;</span><span class="p">,</span> <span class="s1">&#39;Observed&#39;</span><span class="p">])</span>
+
+    <span class="k">return</span> <span class="n">ax</span>
+</pre></div>
+</div>
+</section>
+</section>
+<section id="fishery-dynamics-ipynb">
+<h2><span class="section-number">C.2. </span>fishery_dynamics.ipynb<a class="headerlink" href="#fishery-dynamics-ipynb" title="Link to this heading">#</a></h2>
+<p>The following are the plotting functions as described in the <code class="docutils literal notranslate"><span class="pre">fishery_dynamics.ipynb</span></code> Jupyter notebook tutorial.</p>
+<p>The following are the necessary package imports to run these functions:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">patheffects</span> <span class="k">as</span> <span class="n">pe</span>
+</pre></div>
+</div>
+<section id="plot-objective-performance">
+<h3><span class="section-number">C.2.1. </span><strong>plot_objective_performance()</strong><a class="headerlink" href="#plot-objective-performance" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_objective_performance</span><span class="p">(</span><span class="n">objective_performance</span><span class="p">,</span> <span class="n">profit_solution</span><span class="p">,</span> <span class="n">robust_solution</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">18</span><span class="p">,</span> <span class="mi">9</span><span class="p">)):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Plot the identified solutions with regards to their objective performance</span>
+<span class="sd">    in a parallel axis plot</span>
+
+<span class="sd">    :param objective_performance:               Objective performance array</span>
+<span class="sd">    :param profit_solution:                     Profitable solutions array</span>
+<span class="sd">    :param robust_solution:                     Robust solutions array</span>
+<span class="sd">    :param figsize:                             Figure size</span>
+<span class="sd">    :type figsize:                              tuple</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># create the figure object</span>
+    <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">figsize</span><span class="p">)</span>
+
+    <span class="c1"># set up subplot axis object</span>
+    <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+
+    <span class="c1"># labels where constraint is always 0</span>
+    <span class="n">objs_labels</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Net present</span><span class="se">\n</span><span class="s1">value (NPV)&#39;</span><span class="p">,</span>
+                   <span class="s1">&#39;Prey population deficit&#39;</span><span class="p">,</span>
+                   <span class="s1">&#39;Longest duration</span><span class="se">\n</span><span class="s1">of low harvest&#39;</span><span class="p">,</span>
+                   <span class="s1">&#39;Worst harvest instance&#39;</span><span class="p">,</span>
+                   <span class="s1">&#39;Variance of harvest&#39;</span><span class="p">,</span>
+                   <span class="s1">&#39;Duration of predator</span><span class="se">\n</span><span class="s1">population collapse&#39;</span><span class="p">]</span>
+
+    <span class="c1"># normalization across objectives</span>
+    <span class="n">mins</span> <span class="o">=</span> <span class="n">objective_performance</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+    <span class="n">maxs</span> <span class="o">=</span> <span class="n">objective_performance</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+    <span class="n">norm_reference</span> <span class="o">=</span> <span class="n">objective_performance</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
+        <span class="n">mm</span> <span class="o">=</span> <span class="n">objective_performance</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">min</span><span class="p">()</span>
+        <span class="n">mx</span> <span class="o">=</span> <span class="n">objective_performance</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
+        <span class="k">if</span> <span class="n">mm</span> <span class="o">!=</span> <span class="n">mx</span><span class="p">:</span>
+            <span class="n">norm_reference</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">objective_performance</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">mm</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">mx</span> <span class="o">-</span> <span class="n">mm</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">norm_reference</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+
+    <span class="c1"># colormap from matplotlib</span>
+    <span class="n">cmap</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s2">&quot;Blues&quot;</span><span class="p">)</span>
+
+    <span class="c1"># plot all solutions</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">norm_reference</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])):</span>
+        <span class="n">ys</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">norm_reference</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:],</span> <span class="mf">1.0</span><span class="p">)</span>
+        <span class="n">xs</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ys</span><span class="p">))</span>
+        <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">cmap</span><span class="p">(</span><span class="n">ys</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+
+    <span class="c1"># to highlight robust solutions</span>
+    <span class="n">ys</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">norm_reference</span><span class="p">[</span><span class="n">profit_solution</span><span class="p">,</span> <span class="p">:],</span> <span class="mf">1.0</span><span class="p">)</span>  <span class="c1"># Most profitable</span>
+    <span class="n">xs</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ys</span><span class="p">))</span>
+    <span class="n">l1</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">],</span>
+                 <span class="n">ys</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">],</span>
+                 <span class="n">c</span><span class="o">=</span><span class="n">cmap</span><span class="p">(</span><span class="n">ys</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
+                 <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span>
+                 <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Most robust in NPV&#39;</span><span class="p">,</span>
+                 <span class="n">path_effects</span><span class="o">=</span><span class="p">[</span><span class="n">pe</span><span class="o">.</span><span class="n">Stroke</span><span class="p">(</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="n">foreground</span><span class="o">=</span><span class="s1">&#39;darkgoldenrod&#39;</span><span class="p">),</span> <span class="n">pe</span><span class="o">.</span><span class="n">Normal</span><span class="p">()])</span>
+
+    <span class="n">ys</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">norm_reference</span><span class="p">[</span><span class="n">robust_solution</span><span class="p">,</span> <span class="p">:],</span> <span class="mf">1.0</span><span class="p">)</span>  <span class="c1"># Most robust in all criteria</span>
+    <span class="n">xs</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ys</span><span class="p">))</span>
+    <span class="n">l2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">],</span>
+                 <span class="n">ys</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">6</span><span class="p">],</span>
+                 <span class="n">c</span><span class="o">=</span><span class="n">cmap</span><span class="p">(</span><span class="n">ys</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
+                 <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span>
+                 <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Most robust across criteria&#39;</span><span class="p">,</span>
+                 <span class="n">path_effects</span><span class="o">=</span><span class="p">[</span><span class="n">pe</span><span class="o">.</span><span class="n">Stroke</span><span class="p">(</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="n">foreground</span><span class="o">=</span><span class="s1">&#39;gold&#39;</span><span class="p">),</span> <span class="n">pe</span><span class="o">.</span><span class="n">Normal</span><span class="p">()])</span>
+
+    <span class="c1"># build colorbar</span>
+    <span class="n">sm</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">ScalarMappable</span><span class="p">(</span><span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">)</span>
+    <span class="n">sm</span><span class="o">.</span><span class="n">set_array</span><span class="p">([</span><span class="n">objective_performance</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">objective_performance</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">max</span><span class="p">()])</span>
+    <span class="n">cbar</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">sm</span><span class="p">)</span>
+    <span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">Net present value (NPV)&quot;</span><span class="p">)</span>
+
+    <span class="c1"># tick values</span>
+    <span class="n">minvalues</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;</span><span class="si">{0:.3f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mins</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
+                 <span class="s2">&quot;</span><span class="si">{0:.3f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="o">-</span><span class="n">mins</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
+                 <span class="nb">str</span><span class="p">(</span><span class="o">-</span><span class="n">mins</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span>
+                 <span class="s2">&quot;</span><span class="si">{0:.3f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="o">-</span><span class="n">mins</span><span class="p">[</span><span class="mi">3</span><span class="p">]),</span>
+                 <span class="s2">&quot;</span><span class="si">{0:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="o">-</span><span class="n">mins</span><span class="p">[</span><span class="mi">4</span><span class="p">]),</span>
+                 <span class="nb">str</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span>
+
+    <span class="n">maxvalues</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;</span><span class="si">{0:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">maxs</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
+                 <span class="s2">&quot;</span><span class="si">{0:.3f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="o">-</span><span class="n">maxs</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
+                 <span class="nb">str</span><span class="p">(</span><span class="o">-</span><span class="n">maxs</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span>
+                 <span class="s2">&quot;</span><span class="si">{0:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">maxs</span><span class="p">[</span><span class="mi">3</span><span class="p">]),</span>
+                 <span class="s2">&quot;</span><span class="si">{0:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="o">-</span><span class="n">maxs</span><span class="p">[</span><span class="mi">4</span><span class="p">]),</span>
+                 <span class="nb">str</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span>
+
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Preference -&gt;&quot;</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">([])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="n">minvalues</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span> <span class="o">+</span> <span class="n">objs_labels</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">objs_labels</span><span class="p">))])</span>
+
+    <span class="c1"># make a twin axis for toplabels</span>
+    <span class="n">ax1</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twiny</span><span class="p">()</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">([])</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="n">maxvalues</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">maxs</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
+
+    <span class="k">return</span> <span class="n">ax</span><span class="p">,</span> <span class="n">ax1</span>
+</pre></div>
+</div>
+</section>
+<section id="plot-factor-performance">
+<h3><span class="section-number">C.2.2. </span><strong>plot_factor_performance()</strong><a class="headerlink" href="#plot-factor-performance" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_factor_performance</span><span class="p">(</span><span class="n">param_values</span><span class="p">,</span> <span class="n">collapse_days</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Visualize the performance of our policies in three-dimensional</span>
+<span class="sd">    parametric space.</span>
+
+<span class="sd">    :param param_values:                Saltelli sample array</span>
+<span class="sd">    :param collapse_days:               Simulation array</span>
+<span class="sd">    :param b:                           b parameter boundary interval</span>
+<span class="sd">    :param m:                           m parameter boundary interval</span>
+<span class="sd">    :param a:                           a parameter boundary interval</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># set colormap</span>
+    <span class="n">cmap</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s2">&quot;RdBu_r&quot;</span><span class="p">)</span>
+
+    <span class="c1"># build figure object</span>
+    <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">figaspect</span><span class="p">(</span><span class="mf">0.5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">600</span><span class="p">,</span> <span class="n">constrained_layout</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+    <span class="c1"># set up scalable colormap</span>
+    <span class="n">sm</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">ScalarMappable</span><span class="p">(</span><span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">)</span>
+
+    <span class="c1"># set up subplot for profit maximizing policy</span>
+    <span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">projection</span><span class="o">=</span><span class="s1">&#39;3d&#39;</span><span class="p">)</span>
+
+    <span class="c1"># add point data for profit plot</span>
+    <span class="n">sows</span> <span class="o">=</span> <span class="n">ax1</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">param_values</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+                       <span class="n">param_values</span><span class="p">[:,</span><span class="mi">6</span><span class="p">],</span>
+                       <span class="n">param_values</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+                       <span class="n">c</span><span class="o">=</span><span class="n">collapse_days</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+                       <span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">,</span>
+                       <span class="n">s</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
+
+    <span class="c1"># add surface data for boundary separating successful and failed states of the world</span>
+    <span class="n">pts_ineq</span> <span class="o">=</span> <span class="n">ax1</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+
+    <span class="c1"># add reference point to plot</span>
+    <span class="n">pt_ref</span> <span class="o">=</span> <span class="n">ax1</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.005</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+
+    <span class="c1"># set up plot aesthetics and labels</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;b&quot;</span><span class="p">)</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;m&quot;</span><span class="p">)</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s2">&quot;a&quot;</span><span class="p">)</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_zlim</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">])</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">])</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_view_interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span>  <span class="mf">0.5</span><span class="p">)</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">view_init</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="o">-</span><span class="mi">17</span><span class="p">)</span>
+    <span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Profit maximizing policy&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set up subplot for robust policy</span>
+    <span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">projection</span><span class="o">=</span><span class="s1">&#39;3d&#39;</span><span class="p">)</span>
+
+    <span class="c1"># add point data for robust plot</span>
+    <span class="n">sows</span> <span class="o">=</span> <span class="n">ax2</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">param_values</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+                       <span class="n">param_values</span><span class="p">[:,</span><span class="mi">6</span><span class="p">],</span>
+                       <span class="n">param_values</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span>
+                       <span class="n">c</span><span class="o">=</span><span class="n">collapse_days</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span>
+                       <span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">,</span>
+                       <span class="n">s</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
+
+    <span class="c1"># add surface data for boundary separating successful and failed states of the world</span>
+    <span class="n">pts_ineq</span> <span class="o">=</span> <span class="n">ax2</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+
+    <span class="c1"># add reference point to plot</span>
+    <span class="n">pt_ref</span> <span class="o">=</span> <span class="n">ax2</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.005</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+
+    <span class="c1"># set up plot aesthetics and labels</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;b&quot;</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;m&quot;</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s2">&quot;a&quot;</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_zlim</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">])</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">])</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_view_interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">view_init</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="o">-</span><span class="mi">17</span><span class="p">)</span>
+    <span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Robust policy&#39;</span><span class="p">)</span>
+
+    <span class="c1"># set up colorbar</span>
+    <span class="n">sm</span><span class="o">.</span><span class="n">set_array</span><span class="p">([</span><span class="n">collapse_days</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">collapse_days</span><span class="o">.</span><span class="n">max</span><span class="p">()])</span>
+    <span class="n">cbar</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">sm</span><span class="p">)</span>
+    <span class="n">cbar</span><span class="o">.</span><span class="n">set_label</span><span class="p">(</span><span class="s1">&#39;Days with predator collapse&#39;</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span>
+</pre></div>
+</div>
+</section>
+</section>
+</section>
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+    <i class="fa-solid fa-list"></i> Contents
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#hymod-ipynb">C.1. hymod.ipynb</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-observed-vs-simulated-streamflow">C.1.1. <strong>plot_observed_vs_simulated_streamflow()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-observed-vs-sensitivity-streamflow">C.1.2. <strong>plot_observed_vs_sensitivity_streamflow()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-monthly-heatmap">C.1.3. <strong>plot_monthly_heatmap()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-annual-heatmap">C.1.4. <strong>plot_annual_heatmap()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-varying-heatmap">C.1.5. <strong>plot_varying_heatmap()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-precalibration-flow">C.1.6. <strong>plot_precalibration_flow()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-precalibration-glue">C.1.7. <strong>plot_precalibration_glue()</strong></a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fishery-dynamics-ipynb">C.2. fishery_dynamics.ipynb</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-objective-performance">C.2.1. <strong>plot_objective_performance()</strong></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-factor-performance">C.2.2. <strong>plot_factor_performance()</strong></a></li>
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+.. _introduction:
+
+************
+Introduction
+************
+
+This guidance text has been developed in support of the Integrated Multisector Multiscale Modeling (IM3) Science Focus Area’s objective to formally integrate uncertainty into its research tasks. IM3 is focused on innovative modeling to explore how human and natural system landscapes in the United States co-evolve in response to short-term shocks and long-term influences. The project’s challenging scope is to advance our ability to study the interactions between energy, water, land, and urban systems, at scales ranging from local (~1km) to the contiguous United States, while consistently addressing influences such as population change, technology change, heat waves, and drought. Uncertainty and careful model-driven scientific insights are central to the project's science objectives shown below.
+
+**IM3 Key MSD Science Objectives:**
+
+*Develop flexible, open-source, and integrated modeling capabilities that capture the structure, dynamic behavior, and emergent properties of the multiscale interactions within and between human and natural systems.*
+
+*Use these capabilities to study the evolution, vulnerability, and resilience of interacting human and natural systems and landscapes from local to continental scales, including their responses to the compounding effects of long-term influences and short-term shocks.*
+
+*Understand the implications of uncertainty in data, observations, models, and model coupling approaches for projections of human-natural system dynamics.*
+
+Addressing the objectives above poses a strong transdisciplinary challenge that depends on a diversity of models and, more specifically, a consistent framing for making model-based science inferences. The term transdisciplinary science as used here formally implies a deep integration of disciplines to aid our hypothesis-driven understanding of coupled human-natural systems--bridging differences in theory, hypothesis generation, modeling, and modes of inference :cite:p:`national2014convergence`. The IM3 MSD research foci and questions require a deep integration across disciplines, where new modes of analysis can emerge that rapidly synthesize and exploit advances for making decision-relevant insights that at minimum acknowledge uncertainty and more ideally promote a rigorous quantitative mapping of its effects on the generality of claimed scientific insights. More broadly, diverse scientific disciplines engaged in the science of coupled human-natural systems, ranging from natural sciences to engineering and economics, employ a diversity of numerical computer models to study and understand their underlying systems of focus. The utility of these computer models hinges on their ability to represent the underlying real systems with sufficient fidelity and enable the inference of novel insights. This is particularly challenging in the case of coupled human-natural systems where there exists a multitude of interdependent human and natural processes taking place that could potentially be represented. These processes usually translate into modeled representations that are highly complex, non-linear, and exhibit strong interactions and threshold behaviors :cite:p:`elsawah2020eight, haimes2018risk,helbing2013globally`. Model complexity and detail have also been increasing as a result of our improving understanding of these processes, the availability of data, and the rapid growth in computing power :cite:p:`saltelli2019so`. As model complexity grows, modelers need to specify a lot more information than before: additional model inputs and relationships as more processes are represented, higher resolution data as more observations are collected, new coupling relationships and interactions as diverse models are being used in combination to answer multisector questions (e.g., the land-water-energy nexus). Typically, not all of this information is well known, nor is the impact of these many uncertainties on model outputs well understood. It is further especially difficult to distinguish the effects of individual as well as interacting sources of uncertainty when modeling coupled systems with multisector and multiscale dynamics :cite:p:`wirtz2017rocky`.
+
+Given the challenge and opportunity posed by the disciplinary diversity of IM3, we utilized an informal team-wide survey to understand how the various disciplines typically address uncertainty, emphasizing key literature examples and domain-specific reviews. The feedback received provided perspectives across diverse areas within the Earth sciences, different engineering fields, as well as economics. Although our synthesis of this survey information highlighted some commonality across areas (e.g., the frequent use of scenario-based modeling), we identified key differences in vocabulary, the frequency with which formal uncertainty analysis appears in the disciplinary literature, and technical approaches. The IM3 team’s responses captured a very broad conceptual continuum of methodological traditions, ranging from deterministic (no uncertainty) modeling to the theoretical case of fully engaging in modeling sources of uncertainty. Overall, error-driven analyses that focus on replicating prior observed conditions were reported to be the most prevalent types of studies for all disciplines. It was generally less common for studies to strongly engage with analyzing uncertainty via more formal ensemble analyses and design of experiments, though some areas did show significantly higher levels of activity. Another notable finding from our survey was the apparent lack of focus on understanding how model coupling relationships shape uncertainty. Although these observations are limited to the scope of feedback attained in the team-wide IM3 survey responses and the bodies of literature reported by respondents, we believe they reflect challenges that are common across the MSD community. 
+
+In the IM3 uncertainty-related research that has occurred since this survey, we have observed that differences in terminology and interpretation of terminology across modeling teams can be confounding. One of the goals of this eBook is to provide a common language for uncertainty analysis within IM3 and, hopefully, for the broader MSD community. While individual scientific disciplines would be expected to retain their own terminology, by providing explicit definitions of terms we can facilitate the translation of concepts across transdisciplinary science teams. To begin, we use the term Uncertainty Analysis (UA) as an umbrella phrase covering all methods in this eBook. Next, we distinguish the key terms of uncertainty quantification (UQ) and uncertainty characterization (UC). UQ refers to the formal focus on the full specification of likelihoods as well as the distributional forms necessary to infer the joint probabilistic response across all modeled factors of interest :cite:p:`cooke1991experts`. UC refers to exploratory modeling of alternative hypotheses to understand the co-evolutionary dynamics of influences and stressors, as well as path dependent changes in the form and function of modelled systems :cite:p:`moallemi2020exploratory,walker2003defining`. As discussed in later sections, the choice of UC or UQ depends on the specific goals of studies, the availability of data, the types of uncertainties (e.g., well-characterized or deep), and the complexity of underlying models as well as computational limits. Definitions of key uncertainty analysis terms used in this eBook appear below, and our Glossary (:numref:`glossary`) contains a complete list of terms.
+
+* **Exploratory modeling**: Use of large ensembles of uncertain conditions to discover decision-relevant combinations of uncertain factors
+* **Factor**: Any model component that can affect model outputs: inputs, resolution levels, coupling relationships, model relationships and parameters. In models with acceptable model fidelity these factors may represent elements of the real-world system under study.
+* **Sensitivity analysis**: Model evaluation to understand the factors and processes that most (or least) control a model’s outputs
+    * **Local sensitivity analysis**: Varying uncertain factors around specific reference values
+    * **Global sensitivity analysis**: Varying uncertain factors throughout their entire feasible value space
+* **Uncertainty characterization**: Model evaluation under alternative factor hypotheses to explore their implications for model output uncertainty
+* **Uncertainty quantification**: Representation of model output uncertainty using probability distributions
+
+At present, there is no singular guide for confronting the computational and conceptual challenges of the multi-model, transdisciplinary workflows that characterize ambitious projects such as IM3 :cite:p:`saltelli2015climate`. The primary aim of this text is to begin to address this gap and provide guidance for facing these challenges. :numref:`2_diagnostic_modeling` provides an overview of diagnostic modeling and the different perspectives for how we should evaluate our models, :numref:`3_sensitivity_analysis_the_basics` summarizes basic methods and concepts for sensitivity analysis, and :numref:`4_sensitivity_analysis` delves into more technical applications of sensitivity analysis to support diagnostic model evaluation and exploratory modeling. Finally, :numref:`5_conclusion` provides some concluding remarks across the UC and UQ topics covered in this text. The appendices of this text include a glossary of the key concepts, an overview of UQ methods, and coding-based illustrative examples of key UC concepts discussed in earlier chapters.
diff --git a/docs/html/_sources/2.1_overview_of_model_diagnostics.rst b/docs/html/_sources/2.1_overview_of_model_diagnostics.rst
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+Overview of model diagnostics
+#############################
+
+Model diagnostics provide a rich basis for hypothesis testing, model innovation, and improved inferences when classifying what is controlling highly consequential results (e.g., vulnerability or resilience in coupled human-natural systems). :numref:`Figure_2_1`, adapted from :cite:p:`saltelli2019so`, presents idealized illustrations of the relationship between UC and global sensitivity analysis for two coupled simulation models. The figure illustrates how UC can be used to address how uncertainties in various modeling decisions (e.g., data inputs, parameters, model structures, coupling relationships) can be sampled and simulated to yield the empirical model output distribution(s) of interest. Monte Carlo frameworks allow us to sample and propagate (or integrate) the ensemble response of the model(s) of focus. The first step of any UC analysis is the specification of the initial input distributions as illustrated in :numref:`Figure_2_1`. The second step is to perform the Monte Carlo simulations. The question can then be raised, which of the modeling assumptions in our Monte Carlo experiment are the most responsible for the resulting output uncertainty. We can answer this question using global sensitivity analysis as illustrated in :numref:`Figure_2_1`. Global sensitivity analysis can be defined as a formal Monte Carlo sampling and analysis of modeling choices (structures, parameters, inputs) to quantify their influence on direct model outputs (or output-informed metrics). UC experiments by themselves do not explain why a particular uncertain outcome is produced, but produce distributions of model outcomes, as portrayed by the yellow curve. The pie chart shown in :numref:`Figure_2_1` is a conceptual representation of the results of using a global sensitivity analysis to identify those factors that are most dominantly influencing results, either individually or interactively :cite:`saltelli_global_2008`.
+
+.. _Figure_2_1:
+.. figure:: _static/figure2_1_idealized_uc.png
+    :alt: Figure 2.1
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Idealized uncertainty characterization and global sensitivity analysis for two coupled simulation models. Uncertainty coming from various sources (e.g., inputs, model structures, coupling relationships) is propagated through the coupled model(s) to generate empirical distributions of outputs of interest (uncertainty characterization). This model output uncertainty can be decomposed to its origins, by means of sensitivity analysis. Figure adapted from :cite:t:`saltelli2019so`.
+
+UC and global sensitivity analysis are not independent modeling analyses. As illustrated here, any global sensitivity analysis requires an initial UC hypothesis in the form of statistical assumptions and representations for the modeling choices of focus (structural, parametric, and data inputs). Information from these two model diagnostic tools can then be used to inform data needs for future model runs, experiments to reduce the uncertainty present, or the simplification or enhancement of the model where necessary. Together UC and global sensitivity analysis provide a foundation for diagnostic exploratory modeling that has a consistent focus on the assumptions, structural model forms, alternative parameterizations, and input data sets that are used to characterize the behavioral space of one or more models.
diff --git a/docs/html/_sources/2.2_perspectives_on_diagnostic_model_evaluation.rst b/docs/html/_sources/2.2_perspectives_on_diagnostic_model_evaluation.rst
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+.. _perspectives:
+
+Perspectives on diagnostic model evaluation
+###########################################
+
+When we judge or diagnose models, the terms "verification" and "validation" are commonly used. However, their appropriateness in the context of numerical models representing complex coupled human-natural systems is questionable :cite:`beven2002towards, oreskes_verification_1994`. The core issue relates to the fact that these systems are often not fully known or perfectly implemented when modeled. Rather, they are defined within specific system framings and boundary conditions in an evolving learning process with the goal of making continual progress towards attaining higher levels of fidelity in capturing behaviors or properties of interest. Evaluating the fidelity of a model's performance can be highly challenging. For example, the observations used to evaluate the fidelity of parameterized processes are often measured at a finer resolution than what is represented in the model, creating the challenge of how to manage their relative scales when performing evaluation. In other cases, numerical models may neglect or simplify system processes because sufficient data is not available or the physical mechanisms are not fully known. If sufficient agreement between prediction and observation is not achieved, it is challenging to know whether these types of modeling choices are the cause, or if other issues, such as deficiencies in the input parameters and/or other modeling assumptions are the true cause of errors. Even if there is high agreement between prediction and observation, the model cannot necessarily be considered validated, as it is always possible that the right values were produced for the wrong reasons. For example, low error can stem from a situation where different errors in underlying assumptions or parameters cancel each other out (“compensatory errors”). Furthermore, coupled human-natural system models are often subject to “equifinality”, a situation where multiple parameterized formulations can produce similar outputs or equally acceptable representations of the observed data. There is therefore no uniquely “true” or validated model, and the common practice of selecting “the best” deterministic calibration set is more of an assumption than a finding :cite:`beven1993prophecy, beven_future_1992`. The situation becomes even more tenuous when observational data is limited in its scope and/or quality to be insufficient to distinguish model representations or their performance differences.
+
+These limitations on model verification undermine any purely positivist treatment of model validity: that a model should correctly and precisely represent reality to be valid. Under this perspective, closely related to empiricism, statistical tests should be used to compare the model’s output with observations and only through empirical verification can a model or theory be deemed credible. A criticism to this viewpoint (besides the aforementioned challenges for model verification) is that it reduces the justification of a model to the single criterion of predictive ability and accuracy :cite:`barlas_philosophical_1990`. Authors have argued that this ignores the explanatory power held in models and other procedures, which can also advance scientific knowledge :cite:`toulmin_form_1977`. These views gave rise to relativist perspectives of science, which instead place more value on model utility in terms of fitness for a specific purpose or inquiry, rather than representational accuracy and predictive ability :cite:`kleindorfer_validation_1998`. This viewpoint appears to be most prevalent among practitioners seeking decision-relevant insights (i.e., inspire new views vs. predict future conditions). The relativist perspective argues for the use of models as heuristics that can enhance our understanding and conceptions of system behaviors or possibilities :cite:`eker2018practice`. In contrast, natural sciences favor a positivist perspective, emphasizing similarity between simulation and observation even in application contexts where it is clear that projections are being made for conditions that have never been observed and the system of focus will have evolved structurally beyond the model representation being employed (e.g., decadal to centennial evolution of human-natural systems).
+
+These differences in prevalent perspectives are mirrored in how model validation is defined by the two camps: From the relativist perspective, validation is seen as a process of incremental “confidence building” in a model as a mechanism for insight :cite:`barlas_formal_1996`, whereas in natural sciences validation is framed as a way to classify a model as having an acceptable representation of physical reality :cite:`oreskes_verification_1994`. Even though the relativist viewpoint does not dismiss the importance of representational accuracy, it does place it within a larger process of establishing confidence through a variety of tools. These tools, not necessarily quantitative, include communicating information between practitioners and modelers, interpreting a multitude of model outputs, and contrasting preferences and viewpoints.
+
+On the technical side of the argument, differing views on the methodology of model validation appear as early as in the 1960’s. :cite:t:`naylor_verification_1967` argue that model validation should not be limited to a single metric or test of performance (e.g., a single error metric), but should rather be extended to multiple tests that reflect different aspects of a model’s structure and behavior. This and similar arguments are made in literature to this day :cite:`beven2018hypothesis, gupta2012towards, gupta_reconciling_2008, kumar_typology_2011, nearing_does_2020` and are primarily founded on two premises. First, that even though modelers widely recognize that their models are abstractions of the truth, they still make truth claims based on traditional performance metrics that measure the divergence of their model from observation :cite:`nearing_does_2020`. Second, that the natural systems mimicked by the models contain many processes that exhibit significant heterogeneity at various temporal and spatial scales. This heterogeneity is lost when a single performance measure is used, as a result of the inherent loss of process information occurring when transitioning from a highly dimensional and interactive system to the dimension of a single metric :cite:`beven2002towards`. These arguments are further elaborated in :numref:`4_sensitivity_analysis`.
+
+Multiple authors have proposed that the traditional reliance on single measures of model performance should be replaced by the evaluation of several model signatures (characteristics) to identify model structural errors and achieve a sufficient assessment of model performance :cite:`gupta_toward_1998, gupta_reconciling_2008, pianosi_understanding_2017, rouge_coordination_2019`. There is however a point of departure here, especially when models are used to produce inferences that can inform decisions. When agencies and practitioners use models of their systems for public decisions, those models have already met sufficient conditions for credibility (e.g., acceptable representational fidelity), but may face broader tests on their salience and legitimacy in informing negotiated decisions :cite:`cash_knowledge_2003, eker2018practice, white_credibility_2010`. This presents a new challenge to model validation, that of selecting decision-relevant performance metrics, reflective of the system’s stakeholders' viewpoints, so that the most consequential uncertainties are identified and addressed :cite:`saltelli_when_2014`. For complex multisector models at the intersection of climatic, hydrologic, agricultural, energy, or other processes, the output space is made up of a multitude of states and variables, with very different levels of salience to the system's stakeholders and to their goals being achieved :cite:`wagener_what_2019`. This is further complicated when such systems are also institutionally and dynamically complex. As a result, a broader set of qualitative and quantitative performance metrics is necessary to evaluate models of such complex systems, one that embraces the plurality of value systems, agencies and perspectives present. For IM3, even though the goal is to develop better projections of future vulnerability and resilience in co-evolving human-natural systems and not to provide decision support per se, it is critical for our multisector, multiscale model evaluation processes to represent stakeholders’ adaptive decision processes credibly.
+
+As a final point, when a model is used in a projection mode, its results are also subject to additional uncertainty, as there is no guarantee that the model’s functionality and predictive ability will stay the same as the baseline, where the verification and validation tests were conducted. This challenge requires an additional expansion of the scope of model evaluation: a broader set of uncertain conditions needs to be explored, spanning beyond historical observation and exploring a wide range of unprecedented conditions. This perspective on modeling, termed exploratory :cite:`bankes_exploratory_1993`, views models as computational experiments that can be used to explore vast ensembles of potential scenarios to identify those with consequential effects. Exploratory modeling literature explicitly orients experiments toward stakeholder consequences and decision-relevant inferences and shifts the focus from predicting future conditions to *discovering* which conditions lead to undesirable or desirable consequences.
+
+This evolution in modeling perspectives can be mirrored by the IM3 family of models in a progression from evaluating models relative to observed history to advanced formalized analyses to make inferences on multisector, multiscale vulnerabilities and resilience. Exploratory modeling approaches can help fashion experiments with large numbers of alternative hypotheses on the co-evolutionary dynamics of influences, stressors, as well as path-dependent changes in the form and function of human-natural systems :cite:`weaver_improving_2013`. The aim of this text is to therefore guide the reader through the use of sensitivity analysis (SA) methods across these perspectives on diagnostic and exploratory modeling.
diff --git a/docs/html/_sources/2_diagnostic_modeling_overview_and_perspectives.rst b/docs/html/_sources/2_diagnostic_modeling_overview_and_perspectives.rst
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+.. _2_diagnostic_modeling:
+
+*********************************************
+Diagnostic Modeling Overview and Perspectives
+*********************************************
+
+This text prescribes a formal model diagnostic approach that is a deliberative and iterative combination of state-of-the-art UC and global sensitivity analysis techniques that progresses from observed history-based fidelity evaluations to forward looking resilience and vulnerability inferences :cite:`gupta_reconciling_2008, hadjimichael_advancing_2020`.
+
+.. include:: 2.1_overview_of_model_diagnostics.rst
+
+.. include:: 2.2_perspectives_on_diagnostic_model_evaluation.rst
+
+.. note::
+    The following articles are suggested as fundamental reading for the information presented in this section:
+
+    * Naomi Oreskes, Kristin Shrader–Frechette, and Kenneth Belitz. Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences. *Science*, 263 (5147): 641-646, February 1994. URL: https://science.sciencemag.org/content/263/5147/641. DOI: https://doi.org/10.1126/science.263.5147.641.
+    * Keith Beven. Towards a coherent philosophy for modelling the environment. *Proceedings of the Royal Society of London*. Series A: mathematical, physical and engineering sciences, 458 (2026): 2465-2484, 2002.
+    * Eker, S., Rovenskaya, E., Obersteiner, M., Langan, S., 2018. Practice and perspectives in the validation of resource management models. *Nature Communications* 9, 1–10. https://doi.org/10.1038/s41467-018-07811-9
+
+    The following articles can be used as supplemental reading:
+
+    * Canham, C.D., Cole, J.J., Lauenroth, W.K. (Eds.), 2003. Models in Ecosystem Science. Princeton University Press. https://doi.org/10.2307/j.ctv1dwq0tq
+
+
+
diff --git a/docs/html/_sources/3.1_global_versus_local_sensitivity.rst b/docs/html/_sources/3.1_global_versus_local_sensitivity.rst
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+.. _global_vs_local:
+
+Global Versus Local Sensitivity
+###############################
+
+Out of the several definitions for sensitivity analysis presented in the literature, the most widely used has been proposed by :cite:t:`saltelli2004sensitivity` as “the study of how uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input”. In other words, sensitivity analysis explores the relationship between the model’s :math:`N` input variables, :math:`x=[x_1,x_2,...,x_N]`, and :math:`M` output variables, :math:`y=[y_1,y_2,...,y_M]` with :math:`y=g(x)`, where :math:`g` is the model that maps the model inputs to the outputs :cite:p:`borgonovo2016sensitivity`.
+
+Historically, there have been two broad categories of sensitivity analysis techniques: local and global. Local sensitivity analysis is performed by varying model parameters around specific reference values, with the goal of exploring how small input perturbations influence model performance. Due to its ease-of-use and limited computational demands, this approach has been widely used in literature, but has important limitations :cite:p:`rakovec2014distributed,saltelli2010avoid`. If the model is not linear, the results of local sensitivity analysis can be heavily biased, as they are strongly influenced by independence assumptions and a limited exploration of model inputs (e.g., :cite:t:`tang2007comparing`). If the model’s factors interact, local sensitivity analysis will underestimate their importance, as it does not account for those effects (e.g., :cite:p:`hamm2006variance`). In general, as local sensitivity analysis only partially and locally explores a model's parametric space, it is not considered a valid approach for nonlinear models :cite:p:`saltelli2019`. This is illustrated in :numref:`Figure_3_1` (a-b), presenting contour plots of a model response (:math:`y`) with an additive linear model (a) and with a nonlinear model (b). In a linear model without interactions between the input terms :math:`x_1` and :math:`x_2`, local sensitivity analysis (assuming deviations from some reference values) can produce appropriate sensitivity indices (:numref:`Figure_3_1` (a)). If however, factors :math:`x_1` and :math:`x_2` interact, the local and partial consideration of the space can not properly account for each factor’s effects on the model response (:numref:`Figure_3_1` (b)), as it is only informative at the reference value where it is applied. In contrast, a global sensitivity analysis varies uncertain factors within the entire feasible space of variable model responses (:numref:`Figure_3_1` (c)). This approach reveals the global effects of each parameter on the model output, including any interactive effects. For models that cannot be proven linear, global sensitivity analysis is preferred and this text is primarily discussing global sensitivity analysis methods. In the text that follows, whenever we use the term sensitivity analysis we are referring to its global application.
+
+.. _Figure_3_1:
+.. figure:: _static/figure3_1_global_versus_local.png
+    :alt: Figure 3.1
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Treatment of a two-dimensional space of variability by local (panels a-b) and global (panel c) sensitivity analyses. Panels depict contour plots with the value of a model response (:math:`y`) changing with changes in the values of input terms :math:`x_1` and :math:`x_2`. Local sensitivity analysis is only an appropriate approach to sensitivity in the case of linear models without interactions between terms, for example in panel (a), where :math:`y=3x_1+5x_2`. In the case of more complex models, for example in panels (b-c), where :math:`y={1 \above 1pt e^{x^2_1+x^2_2}} + {50 \above 1pt e^{(0.1x_1)^2+(0.1x_2)^3}}`, local sensitivity will miscalculate sensitivity indices as the assessed changes in the value :math:`y` depend on the assumed base values chose for :math:`x_1` and :math:`x_2` (panel (b)). In these cases, global sensitivity methods should be used instead (panel (c)). The points in panel (c) are generated using a uniform random sample of :math:`n=50`, but many other methods are available.
diff --git a/docs/html/_sources/3.2_why_perform_sensitivity_analysis.rst b/docs/html/_sources/3.2_why_perform_sensitivity_analysis.rst
new file mode 100644
index 0000000..7b7ceea
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+++ b/docs/html/_sources/3.2_why_perform_sensitivity_analysis.rst
@@ -0,0 +1,33 @@
+.. _why_SA:
+
+Why Perform Sensitivity Analysis
+################################
+
+It is important to understand the many ways in which a SA might be of use to your modeling effort. Most commonly, one might be motivated to perform sensitivity analysis for the following reasons:
+
+*Model evaluation*: Sensitivity analysis can be used to gauge model inferences when assumptions about the structure of the model or its parameterization are dubious or have changed. For instance, consider a numerical model that uses a set of calibrated parameter values to produce outputs, which we then use to inform decisions about the real-world system represented. One might like to know if small changes in these parameter values significantly change this model’s output and the decisions it informs or if, instead, our parameter inferences yield stable model behavior regardless of the uncertainty present in the specific parameterized processes or properties. This can either discredit or lend credence to the model at hand, as well as any inferences drawn that are founded on its accurate representation of the system. Sensitivity analysis can identify which uncertain model factors cause this undesirable model behavior.
+
+*Model simplification*: Sensitivity analysis can also be used to identify factors or components of the model that appear to have limited effects on direct outputs or metrics of interest. Consider a model that has been developed in an organization for the purposes of a specific research question and is later used in the context of a different application. Some processes represented in significant detail might no longer be of the same importance while consuming significant data or computational resources, as different outputs might be pertinent to the new application. Sensitivity analysis can be used to identify unimportant model components and simplify them to nominal values and reduced model forms. Model complexity and computational costs can therefore be reduced.
+
+*Model refinement*: Alternatively, sensitivity analysis can reveal the factors or processes that are highly influential to the outputs or metrics of interest, by assessing their relative importance. In the context of model evaluation, this can inform which model components warrant additional investigation or measurement so the uncertainty surrounding them and the resulting model outputs or metrics of interest can be reduced.
+
+*Exploratory modeling*: When sufficient credence has been established in the model, sensitivity analysis can be applied to a host of other inquiries. Inferences about the factors and processes that most (or least) control a model’s outputs of interest can be extrapolated to the real system they represent and be used in a heuristic manner to inform model-based inferences. On this foundation, a model paired with the advanced techniques presented in this text can be used to “discover” decision relevant and highly consequential outcomes (i.e., scenario discovery, discussed in more detail in :numref:`consequential_scenarios` :cite:p:`bankes_exploratory_1993, bryant2010thinking`).
+
+The nature and context of the model shapes the specific objectives of applying a sensitivity analysis, as well as methods and tools most appropriate and defensible for each application setting :cite:`saltelli2004sensitivity, saltelli2002relative, wagener_what_2019`. The three most common sensitivity analysis modes (*Factor Prioritization*, *Factor Fixing*, and *Factor Mapping*) are presented below, but the reader should be aware that other uses have been proposed in the literature (e.g., :cite:`anderson2014uncertainty,borgonovo2010sensitivity`).
+
+*Factor prioritization*: This sensitivity analysis application mode (also referred to as *factor ranking*) refers to when one would like to identify the uncertain factors that have the greatest impact on the variability of the output, and which, when fixed to their true value (i.e., if there were no uncertainty regarding their value), would lead to the greatest reduction in output variability :cite:`saltelli2008global`. Information from this type of analysis can be crucial to model improvement as these factors can become the focus of future measurement campaigns or numerical experiments so that uncertainty in the model output can be reduced. The impact of each uncertain input on the variance of the model output is often used as the criterion for factor prioritization. :numref:`Figure_3_2` (a) shows the effects of three uncertain variables (:math:`X_1`, :math:`X_2`, and :math:`X_3`) on the variance of output :math:`Y`. :math:`V(E(Y|X_i))` indicates the variance in :math:`Y` if factor :math:`X_i` is left to vary freely while all other factors remain fixed to nominal values. In this case, factor :math:`X_2` makes the largest contribution to the variability of output :math:`Y` and it should therefore be prioritized. In the context of risk analysis, factor prioritization can be used to reduce output variance to below a given tolerable threshold (also known as variance cutting).
+
+*Factor fixing*: This mode of sensitivity analysis (also referred to as *factor screening*) aims to identify the model components that have a negligible effect or make no significant contributions to the variability of the outputs or metrics of interest (usually referred to as non-influential :cite:`saltelli2008global`). In the stylized example of :numref:`Figure_3_2` (a), :math:`X_1` makes the smallest contribution to the variability of output :math:`Y` suggesting that the uncertainty in its value could be negligible and the factor itself fixed in subsequent model executions. Eliminating these factors or processes in the model or fixing them to a nominal value can help reduce model complexity as well as the unnecessary computational burden of subsequent model runs, results processing, or other sensitivity analyses (the fewer uncertain factors considered, the fewer runs are necessary to illuminate their effects on the output). Significance of the outcome can be gauged in a variety of manners, depending on the application. For instance, if applying a variance-based method, a minimum threshold value of contribution to the variance could be considered as a significance ‘cutoff’, and factors with indices below that value can be considered non-influential. Conclusions about factor fixing should be made carefully, considering all of the effects a factor has, individually and in interaction with other factors (explained in more detail in the :numref:`variance-based_methods`).
+
+*Factor mapping*: Finally, factor mapping can be used to pinpoint which values of uncertain factors lead to model outputs within a given range of the output space :cite:`saltelli2008global`. In the context of model diagnostics, it is possible that the model’s output changes in ways considered impossible based on the represented processes, or other observed evidence. In this situation, factor mapping can be used to identify which uncertain model factors cause this undesirable model behavior by ‘filtering’ model runs that are considered ‘non-behavioral’ :cite:`edwards2011precalibrating,pianosi2016sensitivity,spear1980eutrophication`. In :numref:`Figure_3_2` (b), region :math:`B` of the output space :math:`Y` denotes the set of behavioral model outcomes and region :math:`\bar{B}` denotes the set of non-behavioral outcomes, resulting from the entirety of input space :math:`X`. Factor mapping refers to the process of tracing which factor values of input space :math:`X` produce the behavioral model outcomes in the output space.
+
+.. _Figure_3_2:
+.. figure:: _static/figure3_2_factor_mapping.png
+    :alt: Figure 3.2
+    :width: 500px
+    :figclass: margin-caption
+    :align: center
+
+    Factor prioritization, factor fixing and factor mapping settings of sensitivity analysis.
+
+The language used above reflects a use of sensitivity analysis for model fidelity evaluation and refinement. However, as previously mentioned, when a model has been established as a sufficiently accurate representation of the system, sensitivity analysis can produce additional inferences (i.e., exploratory modeling and scenario discovery). For instance, under the factor mapping use, the analyst can now focus on undesirable system states and discover which factors are most responsible for them: for instance, “population growth of above 25% would be responsible for unacceptably high energy demands”. Factor prioritization and factor fixing can be used to make equivalent inferences, such as “growing populations and increasing temperatures are the leading factors for changing energy demands” (prioritizing of factors) or “changing dietary needs are inconsequential to increasing energy demands for this region” (a factor that can be fixed in subsequent model runs). All these inferences hinge on the assumption that the real system’s stakeholders consider the model states faithful enough representations of system states. As elaborated in :numref:`perspectives`, this view on sensitivity analysis is founded on a relativist perspective on modeling, which tends to place more value on model usefulness rather than strict accuracy of representation in terms of error. As such, sensitivity analysis performed with decision-making relevance in mind will focus on model outputs or metrics that are consequential and decision relevant (e.g., energy demand in the examples above).
\ No newline at end of file
diff --git a/docs/html/_sources/3.3.1_one_at_a_time_oat.rst b/docs/html/_sources/3.3.1_one_at_a_time_oat.rst
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+++ b/docs/html/_sources/3.3.1_one_at_a_time_oat.rst
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+One-At-a-Time (OAT)
+*******************
+
+In this approach, only one model factor is changed at a time while all others are kept fixed across each iteration in a sampling sequence. The OAT method assumes that model factors of focus are linearly independent (i.e., there are no interactions) and can analyze how factors individually influence model outputs or metrics of interest. While popular given its ease of implementation, OAT  is ultimately limited in its exploration of a model’s sensitivities :cite:p:`saltelli2008global`. It is primarily used with local sensitivity techniques with similar criticisms: applying this sampling scheme on a system with nonlinear and interactive processes will miss important information on the effect uncertain factors have on the model. OAT samplings can be repeated multiple times in a more sophisticated manner and across different locations of the parameter space to overcome some of these challenges, which would increase computational costs and negate the main reasons for its selection. Given these limitations OAT methods could be used as preliminary, low-cost analyses of the factors' individual effects, but should ultimately be complemented with more sophisticated methods.
diff --git a/docs/html/_sources/3.3.2_full_fractional_factorial_sampling.rst b/docs/html/_sources/3.3.2_full_fractional_factorial_sampling.rst
new file mode 100644
index 0000000..d7a5a45
--- /dev/null
+++ b/docs/html/_sources/3.3.2_full_fractional_factorial_sampling.rst
@@ -0,0 +1,15 @@
+Full and Fractional Factorial Sampling
+**************************************
+
+In full factorial sampling each factor is treated as being discrete by considering two or more levels (or intervals) of its values. The sampling process then generates samples within each possible combination of levels, corresponding to each parameter. This scheme produces a more comprehensive sampling of the factors’ variability space, as it accounts for all candidate combinations of factor levels (:numref:`Figure_3_3` (a)). If the number of levels is the same across all factors, the number of generated samples is estimated using :math:`n^k`, where :math:`n` is the number of levels and :math:`k` is the number of factors. For example, :numref:`Figure_3_3` (a) presents a full factorial sampling of three uncertain factors :math:`(x_1,` :math:`x_2,` and :math:`x_3)`, each considered as having four discrete levels. The total number of samples necessary for such an experiment is :math:`4^3=64`. As the number of factors increases, the number of simulations necessary will also grow exponentially, making full factorial sampling computationally burdensome (:numref:`Figure_3_3` (b)). As a result, it is common in the literature to apply full factorial sampling at only two levels per factor, typically the two extremes :cite:`montgomery2017design`. This significantly reduces computational burden but is only considered appropriate in cases where factors can indeed only assume two discrete values (e.g., when testing the effects of epistemic uncertainty and comparing between model structure A and model structure B). In the case of physical parameters on continuous distributions (e.g., when considering the effects of measurement uncertainty in a temperature sensor), discretizing the range of a factor to only extreme levels can bias its estimated importance.
+
+Fractional factorial sampling is a widely used alternative to full factorial sampling that allows the analyst to significantly reduce the number of simulations by focusing on the main effects of a factor and seeking to avoid model runs that yield redundant response information :cite:p:`saltelli2008global`. In other words, if one can reasonably assume that higher-order interactions are negligible, information about the most significant effects and lower-order interactions (e.g., effects from pairs of factors) can be obtained using a fraction of the full factorial design. Traditionally, fractional factorial design has also been limited to two levels :cite:p:`montgomery2017design`, referred to as Fractional Factorial designs 2k-p :cite:p:`box19612`. Recently, Generalized Fractional Factorial designs have also been proposed that allow for the structured generation of samples at more than two levels per factor :cite:p:`surowiec2017generalized`. Consider a case where the modeling team dealing with the problem in :numref:`Figure_3_3` (a) cannot afford to perform 64 simulations of their model. They can afford 32 runs for their experiment and instead decide to fractionally sample the variability space of their factors. A potential design of such a sampling strategy is presented in :numref:`Figure_3_3` (c).
+
+.. _Figure_3_3:
+.. figure:: _static/figure3_3_alternative_designs.png
+    :alt: Figure 3.3
+    :width: 500px
+    :figclass: margin-caption
+    :align: center
+
+    Alternative designs of experiments and their computational costs for three uncertain factors :math:`(x_1,` :math:`x_2,` and :math:`x_3)`. (a) Full factorial design sampling of three factors at four levels, at a total of 64 samples; (b) exponential growth of necessary number of samples when applying full factorial design at four levels; (c) fractional factorial design of three factors at four levels, at a total of 32 samples; and (d) Latin Hypercube sample of three factors with uniform distributions, at a total of 32 samples.
diff --git a/docs/html/_sources/3.3.3_latin_hypercube_sampling.rst b/docs/html/_sources/3.3.3_latin_hypercube_sampling.rst
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+++ b/docs/html/_sources/3.3.3_latin_hypercube_sampling.rst
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+Latin Hypercube Sampling (LHS)
+*****************************
+
+Latin hypercube sampling (LHS) :cite:p:`mckay1979` is one of the most common methods in space-filling experimental designs. With this sampling technique, for :math:`N` uncertain factors, an :math:`N`-dimensional hypercube is generated, with each factor divided into an equal number of levels depending on the total number of samples to be generated. Equal numbers of samples are then randomly generated at each level, across all factors. In this manner, latin hypercube design guarantees sampling from every level of the variability space and without any overlaps. When the number of samples generated is much larger than the number of uncertain factors, LHS can be very effective in examining the effects of each factor :cite:p:`saltelli2008global`. LHS is an attractive technique, because it guarantees a diverse coverage of the space, through the use of subintervals, without being constrained to discrete levels for each factor - compare :numref:`Figure_3_3` (c) with :numref:`Figure_3_3` (d) for the same number of samples.
+
+LHS is less effective when the number of samples is not much larger than the number of uncertain factors, and the effects of each factor cannot be appropriately distinguished. The samples between factors can also be highly correlated, biasing any subsequent sensitivity analysis results. To address this, the sampling scheme can be modified to control for the correlation in parameters while maximizing the information derived. An example of such modification is through the use of orthogonal arrays :cite:p:`tang1993orthogonal`.
diff --git a/docs/html/_sources/3.3.4_low_discrepancy_sequences.rst b/docs/html/_sources/3.3.4_low_discrepancy_sequences.rst
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+++ b/docs/html/_sources/3.3.4_low_discrepancy_sequences.rst
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+Low-Discrepancy Sequences
+*************************
+
+Low-discrepancy sequences is another sampling technique that employs a pseudo-random generator for Monte Carlo sampling :cite:p:`dalal2008low, zaremba1968mathematical`. These quasi-Monte Carlo methods eliminate 'lumpiness' across samples (i.e, the presence of gaps and clusters) by minimizing discrepancy across the hypercube samples. Discrepancy can be quantitatively measured using the deviations of sampled points from a uniform distribution :cite:`kucherenko2015exploring, dalal2008low`. Low-discrepancy sequences ensure that the number of samples in any subspace of the variability hypercube is approximately the same. This is not something guaranteed by Latin Hypercube sampling, and even though its design can be improved through optimization with various criteria, such adjustments are limited to small sample sizes and low dimensions :cite:`iooss2010numerical,jin2003efficient,kucherenko2015exploring,morris1995exploratory,park1994optimal`. In contrast, the Sobol sequence :cite:p:`sobol1976uniformly,sobol1967distribution`, one of the most widely used sampling techniques, utilizes the low-discrepancy approach to uniformly fill the sampled factor space. A core advantage of this style of sampling is that it takes far fewer samples (i.e., simulations) to attain a much lower level of error in estimating model output statistics (e.g., the mean and variance of outputs).
+
+.. note::
+
+    Put this into practice! Click the following link to try out an interactive tutorial which uses Sobol sequence sampling for the purposes of a Sobol sensitivity analysis:  `Sobol SA using SALib Jupyter Notebook <nb_saltelli_sobol_>`_
diff --git a/docs/html/_sources/3.3.5_other_types_of_sampling.rst b/docs/html/_sources/3.3.5_other_types_of_sampling.rst
new file mode 100644
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+++ b/docs/html/_sources/3.3.5_other_types_of_sampling.rst
@@ -0,0 +1,4 @@
+Other types of sampling
+***********************
+
+The sampling techniques mentioned so far are general sampling methods useful for a variety of applications beyond sensitivity analysis. There are however techniques that have been developed for specific sensitivity analysis methods. Examples of these methods include the Morris One-At-a-Time :cite:p:`morris1991factorial`, Fourier Amplitude Sensitivity Test (FAST; :cite:p:`cukier1973study`), Extended FAST :cite:p:`saltelli1999quantitative`, and Extended Sobol methods :cite:p:`sobol2001global`. For example, the Morris sampling strategy builds a number of trajectories (usually referred to as repetitions and denoted by :math:`r`) in the input space each composed of :math:`N+1` factor points, where :math:`N` is the number of uncertain factors. The first point of the trajectory is selected randomly and the subsequent :math:`N` points are generated by moving one factor at a time by a fixed amount. Each factor is perturbed once along the trajectory, while the starting points of all of the trajectories are randomly and uniformly distributed. Several variations of this strategy also exist in the literature; for more details on each approach and their differences the reader is directed to :cite:t:`pianosi2016sensitivity`.
diff --git a/docs/html/_sources/3.3.6_synthetic_generation_of_input_time_series.rst b/docs/html/_sources/3.3.6_synthetic_generation_of_input_time_series.rst
new file mode 100644
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--- /dev/null
+++ b/docs/html/_sources/3.3.6_synthetic_generation_of_input_time_series.rst
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+Synthetic generation of input time series
+*****************************************
+
+Models often have input time series or processes with strong temporal and/or spatial correlations (e.g., streamflow, energy demand, pricing of commodities, etc.) that, while they might not immediately come to mind as factors to be examined in sensitivity analysis, can be treated as such. Synthetic input time series are used for a variety of reasons, for example, when observations are not available or are limited, or when past observations are not considered sufficiently representative to capture rare or extreme events of interest :cite:p:`herman2016synthetic,milly2008stationarity`. Synthetic generation of input time series provides a valuable tool to consider non-stationarity and incorporate potential stressors, such as climate change impacts into input time series :cite:p:`borgomeo2015numerical`. For example, a century of record will be insufficient to capture very high impact rare extreme events (e.g., persistent multi-year droughts). A large body of statistical literature exists focusing on the topics of synthetic weather :cite:p:`herrera2017review,wilks1999weather` and streamflow :cite:p:`lamontagne2018generating,medda2019comparison` generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound (hot, severe drought) extremes. It is beyond the scope of this text to review these methods, but readers are encouraged to explore the studies cited above as well as the following publications for discussions and comparisons of these methods: :cite:p:`borgomeo2015numerical,herman2016synthetic,kirsch2013evaluating,loucks2017water,steinschneider2015integrated,vogel2017stochastic,vogel1988value`. The use of these methods for the purposes of exploratory modeling, especially in the context of well-characterized versus deep uncertainty, is further discussed in :numref:`consequential_scenarios`.
diff --git a/docs/html/_sources/3.3_design_of_experiments.rst b/docs/html/_sources/3.3_design_of_experiments.rst
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+++ b/docs/html/_sources/3.3_design_of_experiments.rst
@@ -0,0 +1,46 @@
+.. _design_of_experiments:
+
+Design of Experiments
+#####################
+
+Before conducting a sensitivity analysis, the first element that needs to be clarified is the uncertainty space of the model :cite:p:`helton2006survey,pianosi2016sensitivity`. In other words, how many and which factors making up the mathematical model are considered uncertain and can potentially affect the model output and the inferences drawn from it. Uncertain factors can be model parameters, model structures, inputs, or alternative model resolution levels (scales), all of which can be assessed through the tools presented in this text. Depending on the kind of factor, its variability can be elicited through various means: expert opinion, values reported in the literature, historical observations, its physical meaning (e.g., population values in a city can never be negative), or through the use of more formal UQ methods (:numref:`A1_uncertainty_quantification`). The model uncertainty space represents the entire space of variability present in each of the uncertain factors of a model. The complexity of most real-world models means that the response function, :math:`y=g(x)`, mapping inputs to outputs, is hardly ever available in an analytical form and therefore analytically computing the sensitivity of the output to each uncertain factor becomes impossible. In these cases, sensitivity analysis is only feasible through numerical procedures that employ different strategies to sample the uncertainty space and calculate sensitivity indices.
+
+A sampling strategy is often referred to as a *design of experiments* and represents a methodological choice made before conducting any sensitivity analysis. Experimental design was first introduced by :cite:t:`fisher1936design` in the context of laboratory or field-based experiments. Its application in sensitivity analysis is similar to setting up a physical experiment in that it is used to discover the behavior of a system under specific conditions. An ideal design of experiments should provide a framework for the extraction of all plausible information about the impact of each factor on the output of the model. The design of experiments is used to set up a simulation platform with the minimum computational cost to answer specific questions that cannot be readily drawn from the data through analytical or common data mining techniques. Models representing coupled human-natural systems usually have a large number of inputs, state variables and parameters, but not all of them exert fundamental control over the numerical process, despite their uncertainty, nor have substantial impacts on the model output, either independently or through their interactions. Each factor influences the model output in different ways that need to be discovered. For example, the influence of a parameter on model output can be linear or non-linear and can be continuous or only be active during specific times or at particular states of the system :cite:p:`herman2013time, massmann2014new`. An effective and efficient design of experiments allows the analyst to explore these complex relationships and evaluate different behaviors of the model for various scientific questions :cite:p:`van2016sensitivity`. The rest of this section overviews some of the most commonly used designs of experiments. Table 1 summarizes the designs discussed.
+
+.. list-table:: Summary of designs of experiments overviewed in this section. * Depends on the sample size.
+  :header-rows: 1
+
+  * - *Design of experiments*
+    - *Factor interactions considered*
+    - *Treatment of factor domains*
+  * - One-At-a-Time (OAT)
+    - No - main effects only
+    - Continuous (distributions)
+  * - Full Factorial Sampling
+    - Yes - including total effects
+    - Discrete (levels)
+  * - Fractional Factorial Sampling
+    - Yes - only lower-order effects*
+    - Discrete (levels)
+  * - Latin Hypercube (LH) Sampling
+    - Yes - including total effects*
+    - Continuous (distributions)
+  * - Quasi-Random Sampling with Low-Discrepancy Sequences
+    - Yes - including total effects*
+    - Continuous (distributions)
+
+There are a few different approaches to the design of experiments, closely related to the chosen sensitivity analysis approach, which is in turn shaped by the research motivations, scientific questions, and computational constraints at hand (additional discussion of this can be found at the end of :numref:`3_sensitivity_analysis_the_basics`). For example, in a sensitivity analysis using perturbation and derivatives methods, the model input parameters vary from their nominal values one at a time, something that the design of experiments needs to reflect. If, instead, one were to perform sensitivity analysis using a multiple-starts perturbation method, the design of experiments needs to consider that multiple points across the factor space are used. The design of experiments specifically defines two key characteristics of samples that are fed to the numerical model: the number of samples and the range of each factor.
+
+Generally, sampling can be performed randomly or by applying a stratifying approach. In random sampling, such as Monte Carlo :cite:p:`metropolis1949monte`, samples are randomly generated by a pseudo-random number generator with an a-priori assumption about the distribution of parameters and their possible ranges. Random seeds can also be used to ensure consistency and higher control over the random process. However, this method could leave some gaps in the parameter space and cause clustering in some spaces, especially for a large number of parameters :cite:p:`norton2015introduction`. Most sampling strategies use stratified sampling to mitigate these disadvantages. Stratified sampling techniques divide the domain of each factor into subintervals, often of equal lengths. From each subinterval, an equal number of samples is drawn randomly, or based on the specific locations within the subintervals :cite:p:`saltelli2008global`.
+
+.. include:: 3.3.1_one_at_a_time_oat.rst
+
+.. include:: 3.3.2_full_fractional_factorial_sampling.rst
+
+.. include:: 3.3.3_latin_hypercube_sampling.rst
+
+.. include:: 3.3.4_low_discrepancy_sequences.rst
+
+.. include:: 3.3.5_other_types_of_sampling.rst
+
+.. include:: 3.3.6_synthetic_generation_of_input_time_series.rst
diff --git a/docs/html/_sources/3.4.1_derivative_based_methods.rst b/docs/html/_sources/3.4.1_derivative_based_methods.rst
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+Derivative-based Methods
+************************
+
+Derivative-based methods explore how model outputs are affected by perturbations in a single model input around a particular input value. These methods are local and are performed using OAT sampling. For simplicity of mathematical notations, let us assume that the model :math:`g(X)` only returns one output. Following :cite:p:`borgonovo2008sensitivity` and :cite:p:`pianosi2016sensitivity`, the sensitivity index, :math:`S_i` , of the model’s *i*-th input factor, :math:`x_i` , can be measured using the partial derivative evaluated at a nominal value, :math:`\bar{x}`, of the vector of inputs:
+
+.. math::
+
+  S_i (\bar{x}) = \frac{\partial g}{\partial x} |_{\bar{x}{^c{_i}}}
+
+where *c*\ :sub:`i`\  is the scaling factor. In most applications however, the relationship :math:`g(X)` is not fully known in its analytical form, and therefore the above partial derivative is usually approximated: 
+
+.. math::
+
+  S_i (\bar{x}) = \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i
+
+Using this approximation, the *i*-th input factor is perturbed by a magnitude of :math:`\Delta_i`, and its relative importance is calculated. Derivative-based methods are some of the oldest sensitivity analysis methods as they only require :math:`N+1` model evaluations to estimate indices for :math:`N` uncertain factors. As described above, being computationally very cheap comes at the cost of not being able to explore the entire input space, but only (local) perturbations to the nominal value. Additionally, as these methods examine the effects of each input factor one at a time, they cannot assess parametric interactions or capture the interacting nature of many real systems and the models that abstract them.
diff --git a/docs/html/_sources/3.4.2_elementary_effect_methods.rst b/docs/html/_sources/3.4.2_elementary_effect_methods.rst
new file mode 100644
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@@ -0,0 +1,29 @@
+Elementary Effect Methods
+*************************
+
+Elementary effect (EE) SA methods provide a solution to the local nature of the derivative-based methods by exploring the entire parametric range of each input parameter :cite:p:`iooss2015review`. However, EE methods still use OAT sampling and do not vary all input parameters simultaneously while exploring the parametric space. The OAT nature of EEs methods therefore prevents them from properly capturing the interactions between uncertain factors. EEs methods are computationally efficient compared to their All-At-a-Time (AAT) counterparts, making them more suitable when computational capacity is a limiting factor, while still allowing for some inferences regarding factor interactions.
+The most popular EE method is the Method of Morris :cite:p:`morris1991factorial`. Following the notation by :cite:p:`pianosi2016sensitivity`, this method calculates global sensitivity using the mean of the EEs (finite differences) of each parameter at different locations:
+
+.. math::
+
+  S_i = \mu_i^* = \frac{1}{r}\sum_{j=1}^r EE^j_i = \frac{1}{r}\sum_{j=1}^r \frac{g(\bar{x}_1,...\bar{x}_i+\Delta_i,...\bar{x}_N)-g(\bar{x}_1,...\bar{x}_i,...\bar{x}_N)}{\Delta_i}c_i
+
+
+with :math:`r` representing the number of sample repetitions (also refered to as trajectories) in the input space, usually set between 4 and 10 :cite:`saltelli2004sensitivity`. Each :math:`x_j` represents the points of each trajectory, with :math:`j=1,…, r`, selected as described in the sampling strategy for this method, found above. This method also produces the standard deviation of the EEs:
+
+.. math::
+
+  \sigma_i = \sqrt{\frac{1}{r}\sum_{j=1}^r(EE_i^j-\frac{1}{r}\sum_{j=1}^r EE^j_i)^2}
+
+
+which is a measure of parametric interactions. Higher values of :math:`\sigma_i` suggest model responses at different levels of factor :math:`x_i` are significantly different, which indicates considerable interactions between that and other uncertain factors. The values of :math:`\mu_i^*` and :math:`\sigma_i` for each factor allow us to draw several different conclusions, illustrated in :numref:`Figure_3_4`, following the example by :cite:`iooss2015review`. In this example, factors :math:`x_1`, :math:`x_2`, :math:`x_4`, and :math:`x_5` can be said to have an influence on the model outputs, with :math:`x_1`, :math:`x_4`, and :math:`x_5` having some interactive or non-linear effects. Depending on the orders of magnitude of :math:`\mu_i^*` and :math:`\sigma_i` one can indirectly deduce whether the factors have strong interactive effects, for example if a factor :math:`\sigma_i << \mu_i^*` then the relationship between that factor and the output can be assumed to be largely linear (note that this is still an OAT method and assumptions on factor interactions should be strongly caveated). Extensions of the Method of Morris have also been developed specifically for the purposes of factor fixing and explorations of parametric interactions (e.g., :cite:p:`borgonovo2010sensitivity,campolongo1999use,cropp2002new`).
+
+
+.. _Figure_3_4:
+.. figure:: _static/figure3_4_morris_method.png
+    :alt: Figure 3.4
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Illustrative results of the Morris Method. Factors :math:`x_1`, :math:`x_2`, :math:`x_4`, and :math:`x_5` have an influence on the model outputs, with :math:`x_1`, :math:`x_4`, and :math:`x_5` having interactive or non-linear effects. Whether or not a factor should be considered influential to the output depends on the output selected and is specific to the research context and purpose of the analysis, as discussed in :numref:`why_SA`.
diff --git a/docs/html/_sources/3.4.3_regression_based_methods.rst b/docs/html/_sources/3.4.3_regression_based_methods.rst
new file mode 100644
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+++ b/docs/html/_sources/3.4.3_regression_based_methods.rst
@@ -0,0 +1,27 @@
+Regression-based Methods
+************************
+Regression analysis is one of the oldest ways of investigating parametric importance and sensitivity :cite:p:`saltelli2004sensitivity`. Here, we describe some of the most popular regression-based sensitivity indices. One of the main sensitivity indices of this category is the standardized regression coefficient (SRC). To calculate SRC, a linear regression relationship needs to be fitted between the input vector, :math:`x`, and the model output of interest by using a least-square minimizing method:
+
+.. math::
+  y = b_0 + \sum_{i=1}^N b_ix_i
+
+where :math:`b_0` and :math:`b_i` (corresponding to the *i*-th model input) are regression coefficients. The following relationship can then be used to calculate the SRCs for different input values:
+
+.. math::
+  S_i=SRC_i=b_i\frac{\sigma_i}{\sigma_y}
+
+where :math:`\sigma_i` and :math:`\sigma_y` are standard deviations of *i*-th model input and output, respectively.
+
+Several other regression-based indices explore the correlation between input and output parameters as a proxy to model parametric sensitivity :cite:`helton1993uncertainty,iooss2015review,manache2008identification`. The Pearson correlation coefficient (PCC) can be used when a linear relationship exists between an uncertain factor, :math:`x_i`, and the output :math:`y`:
+
+.. math::
+  S_i=PCC=\frac{cov(x_i,y)}{\sigma_i\sigma_y}
+
+In cases when there are outliers in the data or the relationship between the uncertain factors and the output is not linear, rank-based correlation coefficients are preferred, for example, Spearman’s rank correlation coefficient (SRCC):
+
+.. math::
+  S_i=SRCC=\frac{cov(rx_i,ryi)}{\sigma_{ri}\sigma_{ry}}
+
+where the raw values of :math:`x_i` and :math:`y` and converted to ranks :math:`rx_i` and :math:`ry` respectively, which instead represent a measurement of the strength of the monotonic relationship, rather than linear relationship, between the input and output. Other regression-based metrics include the partial correlations coefficient, the partial rank correlations coefficient, and the Nash-Sutcliffe coefficient, more discussion on which can be found in :cite:p:`iooss2015review,borgonovo2016sensitivity`.
+
+Tree-based regression techniques have also been used for sensitivity analysis in an effort to address the challenges faced with nonlinear models :cite:p:`pappenberger2006ignorance`. Examples of these methods include the Patient Rule Induction Method (PRIM; :cite:p:`friedman1999bump`) and Classification And Regression Trees (CART; :cite:p:`breiman1984classification`). CART-based approaches also include boosting and bagging extensions :cite:p:`freund1999short,breiman1996bagging`. These methods are particularly useful when sensitivity analysis is used for factor mapping (i.e., when trying to identify which uncertain model factors produce a certain model behavior). :numref:`consequential_scenarios` elaborates on the use of these methods. Regression-based sensitivity analysis methods are global by nature and can explore the entire space of variables. However, the true level of comprehensiveness depends on the design of experiments and the number of simulations providing data to establish the regression relationships. Although they are usually computationally efficient, they do not produce significant information about parametric interactions :cite:p:`borgonovo2016sensitivity,saltelli2004sensitivity`.
diff --git a/docs/html/_sources/3.4.4_regional_sensitivity_analysis.rst b/docs/html/_sources/3.4.4_regional_sensitivity_analysis.rst
new file mode 100644
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+++ b/docs/html/_sources/3.4.4_regional_sensitivity_analysis.rst
@@ -0,0 +1,10 @@
+Regional Sensitivity Analysis
+*****************************
+Another method primarily applied for basic factor mapping applications is Regional Sensitivity Analysis (RSA; :cite:p:`hornberger1981approach`). RSA is a global sensitivity analysis method that is typically implemented using standard sampling methods such as latin hypercube sampling. It is performed by specifying a condition on the output space (e.g., an upper threshold) and classifying outputs that meet the condition as behavioral and the ones that fail it as non-behavioral (illustrated in :numref:`Figure_3_2` (b)). Note that the specified threshold depends on the nature of the problem, model, and the research question. It can reflect model-performance metrics (such as errors) or consequential decision-relevant metrics (such as unacceptable system outcomes). The behavioral and non-behavioral outputs are then traced back to their originating sampled factors, where differences between the distributions of samples can be used to determine their significance in producing each part of the output. The Kolmogorov-Smirnov divergence is commonly used to quantify the difference between the distribution of behavioral and non-behavioral parameters :cite:p:`pianosi2016sensitivity`:
+
+.. math::
+  S_i=|F_{x_i|y_b} (y \in Y_b)-F_{x_i|y_{nb}} (y \in Y_{nb})|
+
+where :math:`Y_b` represents the set of behavioral outputs, and :math:`F_{x_i|y_b}` is the empirical cumulative distribution function of the values of :math:`x_i` associated with values of :math:`y` that belong in the behavioral set. The :math:`nb` notation indicates the equivalent elements related to the non-behavioral set. Large differences between the two distributions indicate stronger effects by the parameters on the respective part of the output space.
+
+Used in a factor mapping setting, RSA can be applied for scenario discovery :cite:p:`lempert2006general,groves2007new`, the Generalized Likelihood Uncertainty Estimation method (GLUE; :cite:p:`beven_future_1992,beven2014glue,blasone2008generalized`) and other hybrid sensitivity analysis methods (e.g., :cite:p:`cryer1999regional,wei2013monte`). The fundamental shortcomings of RSA are that, in some cases, it could be hard to interpret the difference between behavioral and non-behavioral sample sets, and that insights about parametric correlations and interactions cannot always be uncovered :cite:p:`saltelli2004sensitivity`. For more elaborate discussions and illustrations of the RSA method, readers are directed to :cite:t:`saltelli2008global,tang2007comparing,young1999data` and references therein.
diff --git a/docs/html/_sources/3.4.5_variance_based_methods.rst b/docs/html/_sources/3.4.5_variance_based_methods.rst
new file mode 100644
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+++ b/docs/html/_sources/3.4.5_variance_based_methods.rst
@@ -0,0 +1,32 @@
+.. _variance-based_methods:
+
+Variance-based Methods
+**********************
+Variance-based sensitivity analysis methods hypothesize that various specified model factors contribute differently to the variation of model outputs; therefore, decomposition and analysis of output variance can determine a model’s sensitivity to input parameters :cite:p:`sobol2001global,saltelli2004sensitivity`. The most popular variance-based method is the Sobol method, which is a global sensitivity analysis method that takes into account complex and nonlinear factor interaction when calculating sensitivity indices, and employs more sophisticated sampling methods (e.g., the Sobol sampling method). The Sobol method is able to calculate three types of sensitivity indices that provide different types of information about model sensitivities. These indices include first-order, higher-order (e.g., second-, third-, etc. orders), and total-order sensitivities.
+
+The first-order sensitivity index indicates the percent of model output variance contributed by a factor individually (i.e., the effect of varying :math:`x_i` alone) and is obtained using the following :cite:p:`sobol2001global,saltelli2002making`:
+
+.. math::
+  S_i^1=\frac{V_{x_i}[E_{x_{\sim i}}(x_i)]}{V(y)}
+
+with :math:`E` and :math:`V` denoting the expected value and the variance, respectively. :math:`x_{\sim i}` denotes all factors expect for :math:`x_i`. The first-order sensitivity index (:math:`S_i^1`) can therefore also be thought of as the portion of total output variance (:math:`V_y`) that can be reduced if the uncertainty in factor :math:`x_i` is eliminated :cite:p:`saltelli2002sensitivity`. First-order sensitivity indices are usually used to understand the independent effect of a factor and to distinguish its individual versus interactive influence. It would be expected for linearly independent factors that they would only have first order indices (no interactions) that should correspond well with sensitivities obtained from simpler methods using OAT sampling.
+
+Higher-order sensitivity indices explore the interaction between two or more parameters that contribute to model output variations. For example, a second-order index indicates how interactions between a pair of factors can lead to change in model output variance and is calculated using the following relationship:
+
+.. math::
+  S_{ij}^2=\frac{V_{x_{i,j}}[E_{x_{\sim i,j}}(x_i,x_j)]}{V(y)}
+
+with :math:`i \ne j`. Higher order indices can be calculated by similar extensions (i.e., fixing additional operators together), but it is usually computationally expensive in practice.
+
+The total sensitivity analysis index represents the entire influence of an input factor on model outputs including all of its interactions with other factors :cite:p:`homma1996importance`. In other words, total-order indices include first-order and all higher-order interactions associated with each factor and can be estimated calculated using the following:
+
+.. math::
+  S_i^T= \frac{E_{x_{\sim i}}[V_{x_i}(x_{\sim i})]}{V(y)} = 1 - \frac{V_{x_{\sim i}}[E_{x_{i}}(x_{\sim i})]}{V(y)}
+
+This index reveals the expected portion of variance that remains if uncertainty is eliminated in all factors but :math:`x_i` :cite:p:`saltelli2002sensitivity`. The total sensitivity index is the overall best measure of sensitivity as it captures the full individual and interactive effects of model factors.
+
+Besides the Sobol method, there are some other variance-based sensitivity analysis methods, such as the Fourier amplitude sensitivity test (FAST; :cite:`cukier1973study, mcrae1982development`) and extended-FAST :cite:`saltelli1998alternative,vazquez2014global`, that have been used by the scientific community. However, Sobol remains by far the most common method of this class. Variance-based techniques have been widely used and have proved to be powerful in a variety of applications. Despite their popularity, some authors have expressed concerns about the methods’ appropriateness in some settings. Specifically, the presence of heavy-tailed distributions or outliers, or when model outputs are multimodal can bias the sensitivity indices produced by these methods :cite:p:`auder2008global,zadeh2017comparison,pianosi2015simple`. Moment-independent measures, discussed below, attempt to overcome these challenges.
+
+.. note::
+
+    Put this into practice! Click the following link to try out an interactive tutorial which demonstrates the application of a Sobol sensitivity analysis:  `Sobol SA using SALib Jupyter Notebook <nb_saltelli_sobol_>`_
diff --git a/docs/html/_sources/3.4.6_analysis_of_variance_anova.rst b/docs/html/_sources/3.4.6_analysis_of_variance_anova.rst
new file mode 100644
index 0000000..b359cd8
--- /dev/null
+++ b/docs/html/_sources/3.4.6_analysis_of_variance_anova.rst
@@ -0,0 +1,3 @@
+Analysis of Variance (ANOVA)
+****************************
+Analysis of Variance (ANOVA) was first introduced by :cite:t:`fisher1934statistical` and has since become a popular factor analysis method in physical experiments. ANOVA can be used as a sensitivity analysis method in computational experiments with a factorial design of experiment (referred to as factorial ANOVA). Note that Sobol can also be categorized as an ANOVA sensitivity analysis method, and that is why Sobol is sometimes referred to as a functional ANOVA :cite:p:`brevault2013comparison`. Factorial ANOVA methods are particularly suited for models and problems that have discrete input spaces, significantly reducing the computational time. More information about these methods can be found in :cite:p:`archer1997sensitivity,brevault2013comparison,owen2013variance`.
\ No newline at end of file
diff --git a/docs/html/_sources/3.4.7_moment_independent_density_based_methods.rst b/docs/html/_sources/3.4.7_moment_independent_density_based_methods.rst
new file mode 100644
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+++ b/docs/html/_sources/3.4.7_moment_independent_density_based_methods.rst
@@ -0,0 +1,8 @@
+Moment-Independent (Density-Based) Methods
+******************************************
+These methods typically compare the entire distribution (i.e., not just the variance) of input and output parameters in order to determine the sensitivity of the output to a particular input variable. Several moment-independent sensitivity analysis methods have been proposed in recent years. The delta (:math:`\delta`) moment-independent method calculates the difference between unconditional and conditional cumulative distribution functions of the output. The method was first introduced by :cite:`borgonovo2006measuring,borgonovo2007new` and has become widely used in various disciplines. The :math:`\delta` sensitivity index is defined as follows:
+
+.. math::
+  S_i=\delta_i=\frac{1}{2}E_{x_i}|f_y(y)-f_{y|x_i}(y)|dy
+
+where :math:`f_y(y)` is the probability density function of the entire model output :math:`y`, and :math:`f_{y|x_i}(y)` is the conditional density of :math:`y`, given that factor :math:`x_i` assumes a fixed value. The :math:`\delta_i` sensitivity indicator therefore represents the normalized expected shift in the distribution of :math:`y` provoked by :math:`x_i`. Moment-independent methods are advantageous in cases where we are concerned about the entire distribution of events, such as when uncertain factors lead to more extreme events in a system :cite:p:`hadjimichael_advancing_2020`. Further, they can be used with a pre-existing sample of data, without requiring a specific sampling scheme, unlike the previously reviewed methods :cite:p:`plischke2013global`. The :math:`\delta` sensitivity index does not include interactions between factors and it is therefore akin to the first order index produced by the Sobol method. Interactions between factors can still be estimated using this method, by conditioning the calculation on more than one uncertain factor being fixed :cite:p:`borgonovo2007new`.
diff --git a/docs/html/_sources/3.4_sensitivity_analysis_methods.rst b/docs/html/_sources/3.4_sensitivity_analysis_methods.rst
new file mode 100644
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--- /dev/null
+++ b/docs/html/_sources/3.4_sensitivity_analysis_methods.rst
@@ -0,0 +1,20 @@
+.. _sensitivity_analysis_methods:
+
+Sensitivity Analysis Methods
+############################
+
+In this section, we describe some of the most widely applied sensitivity analysis methods along with their mathematical definitions. We also provide a detailed discussion on applying each method, as well as a comparison of and their features and limitations.
+
+.. include:: 3.4.1_derivative_based_methods.rst
+
+.. include:: 3.4.2_elementary_effect_methods.rst
+
+.. include:: 3.4.3_regression_based_methods.rst
+
+.. include:: 3.4.4_regional_sensitivity_analysis.rst
+
+.. include:: 3.4.5_variance_based_methods.rst
+
+.. include:: 3.4.6_analysis_of_variance_anova.rst
+
+.. include:: 3.4.7_moment_independent_density_based_methods.rst
diff --git a/docs/html/_sources/3.5_how_to_choose_a_sensitivity_analysis_method_model_traits_and_dimensionality.rst b/docs/html/_sources/3.5_how_to_choose_a_sensitivity_analysis_method_model_traits_and_dimensionality.rst
new file mode 100644
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--- /dev/null
+++ b/docs/html/_sources/3.5_how_to_choose_a_sensitivity_analysis_method_model_traits_and_dimensionality.rst
@@ -0,0 +1,17 @@
+How To Choose A Sensitivity Analysis Method: Model Traits And Dimensionality
+############################################################################
+
+:numref:`Figure_3_5`, synthesized from variants found in :cite:p:`iooss2015review, pianosi2016sensitivity`, presents a graphical synthesis of the methods overviewed in this section, with regards to their appropriateness of application based on the complexity of the model at hand and the computational limits on the number of model evaluations afforded. The bars below each method also indicate the sensitivity analysis purposes they are most appropriate to address, which are in turn a reflection of the motivations and research questions the sensitivity analysis is called to address. Computational intensity is measured as a multiple of the number of model factors that are considered uncertain (:math:`d`). Increasing model complexity mandates that more advanced sensitivity analysis methods are applied to address potential nonlinearities, factor interactions, and discontinuities. Such methods can only be performed at increasing computational expense. For example, computationally cheap linear regression should not be used to assess factors’ importance if the model cannot be proven linear and the factors independent, because important relationships will invariably be missed (recall the example in :numref:`Figure_3_5`). When computational limits do constrain applications to make simplified assumptions and sensitivity techniques, any conclusions in such cases should be delivered with clear statements of the appropriate caveats.
+
+.. _Figure_3_5:
+.. figure:: _static/figure3_5classificationofmethods.png
+    :alt: Figure 3_5
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Classification of the sensitivity analysis methods overviewed in this section, with regards to their computational cost (horizontal axis), their appropriateness to model complexity (vertical axis), and the purpose they can be used for (colored bars). d: number of uncertain factors considered; ANOVA: Analysis of Variance; FAST: Fourier Amplitude Sensitivity Test; PRIM: Patient Rule Induction Method; CART: Classification and Regression Trees; SRCC: Spearman’s rank correlation coefficient: NSE: Nash–Sutcliffe efficiency; SRC: standardized regression coefficient; PCC: Pearson correlation coefficient. This figure is synthesized from variants found in :cite:p:`iooss2015review, pianosi2016sensitivity`.
+
+The reader should also be aware that the estimates of computational intensity that are given here are indicative of magnitude and would vary depending on the sampling technique, model complexity and the level of information being asked. For example, a Sobol sensitivity analysis typically requires a sample of size :math:`n * d+2` to produce first- and total-order indices, where :math:`d` is the number of uncertain factors and :math:`n` is a scaling factor, selected ad hoc, depending on model complexity :cite:p:`saltelli2002relative`. The scaling factor :math:`n` is typically set to at least 1000, but it should most appropriately be set on the basis of index convergence. In other words, a prudent analyst would perform the analysis several times with increasing :math:`n` and observe at what level the indices converge to stable values :cite:p:`NOSSENT20111515`. The level should be the minimum sample size used in subsequent sensitivity analyses of the same system. Furthermore, if the analyst would like to better understand the degrees of interaction between factors, requiring second-order indices, the sample size would have to increase to :math:`n * 2d+2` :cite:p:`saltelli2002relative`.
+
+Another important consideration is that methods that do not require specific sampling schemes can be performed in conjunction with others without requiring additional model evaluations. None of the regression-based methods, for example, require samples of specific structures or sizes, and can be combined with other methods for complementary purposes. For instance, one could complement a Sobol analysis with an application of CART, using the same data, but to address questions relating to factor mapping (e.g., we know factor :math:`x_i`  is important for a model output, but we would like to also know which of its values specifically push the output to undesirable states). Lastly, comparing results from different methods performed together can be especially useful in model diagnostic settings. For example, :cite:p:`hadjimichael_advancing_2020` used :math:`\delta` indices, first-order Sobol indices, and :math:`R^2`  values from linear regression, all performed on the same factors, to derive insights about the effects on factors on different moments of the output distribution and about the linearity of their relationship.
diff --git a/docs/html/_sources/3.6_software_toolkits.rst b/docs/html/_sources/3.6_software_toolkits.rst
new file mode 100644
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--- /dev/null
+++ b/docs/html/_sources/3.6_software_toolkits.rst
@@ -0,0 +1,19 @@
+.. _software_toolkits:
+
+Software Toolkits
+#################
+
+This section presents available open source sensitivity analysis software tools, based on the programming language they use and the methods they support :numref:`Figure_3_6`. Our review covers five widely used programming languages: R, MATLAB, Julia, Python, and C++, as well as one tool that provides a graphical user interface (GUI). Each available SA tool was assessed on the number of SA methods and design of experiments methods it supports. For example, the *sensobol* package in R only supports the variance-based Sobol method. However, it is the only package we came across that calculates third-order interactions among parameters. On the other side of the spectrum, there are SA software packages that contain several popular SA methods. For example, *SALib* in Python :cite:`herman2017salib` supports seven different SA methods. The *DifferentialEquations* package is a comprehensive package developed for Julia, and *GlobalSensitivityAnalysis* is another Julia package that has mostly adapted SALib methods. :numref:`Figure_3_6` also identifies the SA packages that have been updated since 2018, indicating active support and development.
+
+
+.. _Figure_3_6:
+.. figure:: _static/figure3_6_softwaretoolkits.png
+    :alt: Figure 3_6
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Sensitivity analysis packages available in different programming language platforms (R, Python, Julia, MATLAB, and C++), with the number of methods they support. Packages supporting more than five methods are indicated in pink. Packages updated since 2018 are indicated with asterisks. 
+
+
+
diff --git a/docs/html/_sources/3_sensitivity_analysis_the_basics.rst b/docs/html/_sources/3_sensitivity_analysis_the_basics.rst
new file mode 100644
index 0000000..8fee574
--- /dev/null
+++ b/docs/html/_sources/3_sensitivity_analysis_the_basics.rst
@@ -0,0 +1,30 @@
+.. _3_sensitivity_analysis_the_basics:
+
+********************************
+Sensitivity Analysis: The Basics
+********************************
+
+.. include:: 3.1_global_versus_local_sensitivity.rst
+
+.. include:: 3.2_why_perform_sensitivity_analysis.rst
+
+.. include:: 3.3_design_of_experiments.rst
+
+.. include:: 3.4_sensitivity_analysis_methods.rst
+
+.. include:: 3.5_how_to_choose_a_sensitivity_analysis_method_model_traits_and_dimensionality.rst
+
+.. include:: 3.6_software_toolkits.rst
+
+.. note::
+    The following articles are suggested as fundamental reading for the information presented in this section:
+
+    * Wagener, T., Pianosi, F., 2019. What has Global Sensitivity Analysis ever done for us? A systematic review to support scientific advancement and to inform policy-making in earth system modelling. *Earth-Science Reviews* 194, 1–18. https://doi.org/10.1016/j.earscirev.2019.04.006
+    * Pianosi, F., Beven, K., Freer, J., Hall, J.W., Rougier, J., Stephenson, D.B., Wagener, T., 2016. Sensitivity analysis of environmental models: A systematic review with practical workflow. Environmental Modelling & Software 79, 214–232. https://doi.org/10.1016/j.envsoft.2016.02.008
+
+    The following articles can be used as supplemental reading:
+
+    * Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S., 2008. Global Sensitivity Analysis: The Primer, 1st edition. ed. Wiley-Interscience, Chichester, England ; Hoboken, NJ.
+    * Montgomery, D.C., 2017. Design and analysis of experiments. John Wiley & Sons.
+    * Iooss, B., Lemaître, P., 2015. A Review on Global Sensitivity Analysis Methods, in: Dellino, G., Meloni, C. (Eds.), Uncertainty Management in Simulation-Optimization of Complex Systems: Algorithms and Applications, Operations Research/Computer Science Interfaces Series. Springer US, Boston, MA, pp. 101–122. https://doi.org/10.1007/978-1-4899-7547-8_5
+
diff --git a/docs/html/_sources/4.1_understanding_errors_what_is_controlling_model_performance.rst b/docs/html/_sources/4.1_understanding_errors_what_is_controlling_model_performance.rst
new file mode 100644
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--- /dev/null
+++ b/docs/html/_sources/4.1_understanding_errors_what_is_controlling_model_performance.rst
@@ -0,0 +1,25 @@
+.. _4_1_understanding_errors:
+
+Understanding Errors: What Is Controlling Model Performance?
+############################################################
+
+Sensitivity analysis is a diagnostic tool when reconciling model outputs with observed data. It is helpful for clarifying how and under what conditions modeling choices (structure, parameterization, data inputs, etc.) propagate through model components and manifest in their effects on model outputs. This exploration is performed through carefully designed sampling of multiple combinations of input parameters and subsequent evaluation of the model structures that are emerging as controlling factors. Model structure and parameterization are two of the most commonly explored aspects of models that have been a central focus when evaluating their performance relative to available observations :cite:`beven1993prophecy`. Addressing these issues plays an important role in establishing credibility in model predictions, particularly in the positivist natural sciences literature. Traditional model evaluations compare the model with observed data, and then rely on expert judgements of its acceptability based on the closeness between simulation and observation with one or a small number of selected metrics. This approach can be myopic, as it is often impossible to use one metric to attribute a certain error and to link that with different parts of the model and its parameters :cite:`gupta2008reconciling`. This means that, even when the error or fitting measure between the model estimates and observations is very small, it is not guaranteed that all the components in the model accurately represent the conceptual reality that the model is abstracting: propagated errors in different parts of the model might cancel each other out, or multiple parameterized implementations of the model can yield similar performance (i.e., equifinality :cite:`beven1993prophecy`).
+
+The inherent complexity of a system hinders accepting or rejecting a model based on one performance measure and different types of measures can aid in evaluating the various components of a model as essentially a multiobjective problem :cite:`gupta2012towards,beven2018hypothesis`. In addition, natural systems mimicked by the models contain various interacting components that might act differently across spatial and temporal domains :cite:`herman2013time,pianosi2016sensitivity`. This heterogeneity is lost when a single performance measure is used, as a highly dimensional and interactive system becomes aggregated through the averaging of spatial or temporal output errors :cite:`beven2002towards`. Therefore, diagnostic error analyses should consider multiple error signatures across different scales and states of concern when seeking to understand how model performance relates to observed data (:numref:`Figure_4_1`). Diverse error signatures can be used to measure the consistency of underlying processes and behaviors of the model and to evaluate the dynamics of model controls under changing temporal and spatial conditions :cite:`beven2001equifinality`. Within this framework, even minimal information extracted from the data can be beneficial as it helps us unearth structural inadequacies in the model. In this context, proper selection of measures of model performance and the number of measures could play consequential roles in our understanding of the model and its predictions :cite:`mcphail2018robustness`.
+
+As discussed earlier, instead of the traditional focus on using deterministic prediction that results in a single error measure, many plausible states and spaces could be searched for making different inferences and quantifying uncertainties. This process also requires estimates of prior probability distributions of all the important parameters and quantification of model behavior across input space. One strategy to reduce the search space is filtering of some model alternatives that are not consistent with observations and known system behaviors. Those implausible parts of the search space can be referred to as non-physical or non-behavioral alternatives :cite:`edwards2011precalibrating, beven2014glue`. This step is conducted before the Bayesian calibration exercise (see :numref:`A1_uncertainty_quantification`).
+
+A comprehensive model diagnostic workflow typically entails the components demonstrated in :numref:`Figure_4_1`. The workflow begins with the selection of model input parameters and their plausible ranges. After the parameter selection, we need to specify the design of experiment (:numref:`design_of_experiments`) and the sensitivity analysis method (:numref:`sensitivity_analysis_methods`) to be used. As previously discussed, these methods require different numbers of model simulations, and each method provides a different insights into the direct effects and interactions of the uncertain factors. In addition, the simulation time of the model and the available computational resources are two of the primary considerations that influence these decisions. After identifying the appropriate methods, we generate a matrix of input parameters, where each set of input parameters will be used to conduct a model simulation. The model can include one or more output variables that fluctuate in time and space. The next step is to analyze model performance by comparing model outputs with observations. As discussed earlier, the positivist model evaluation paradigm focuses on a single model performance metric (error), leading to a loss of information about model parameters and the suitability of the model's structure. However, a thorough investigation of the temporal and spatial signatures of model outputs using various performance metrics or time- and space-varying sensitivity analyses can shed more light on the fitness of each parameter set and the model’s internal structure. This analysis provides diagnostic feedback on the importance and range of model parameters and can guide further improvement of the model algorithm.
+
+.. _Figure_4_1:
+.. figure:: _static/figure4_1_diagnostic_workflow.png
+    :alt: Figure 4.1
+    :width: 500px
+    :figclass: margin-caption
+    :align: center
+
+    Diagnostic evaluation of model fidelity using sensitivity analysis methods.
+
+.. note::
+
+    Put this into practice! Click the following badge to try out an interactive tutorial on implementing a time-varying sensitivity analysis of HYMOD model parameters:  `HYMOD Jupyter Notebook <nb_hymod_>`_
diff --git a/docs/html/_sources/4.2_consequential_dynamics_what_is_controlling_model_behaviors_of_interest.rst b/docs/html/_sources/4.2_consequential_dynamics_what_is_controlling_model_behaviors_of_interest.rst
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+Consequential Dynamics: What is Controlling Model Behaviors of Interest?
+########################################################################
+
+Consequential changes in dynamic systems can take many forms, but most dynamic behavior can be categorized in a few basic patterns. Feedback structures inherent in a system, be they positive or negative, generate these patterns which can be grouped into three groups for the simplest of systems: exponential growth, goal seeking, and oscillation (:numref:`Figure_4_2`). A positive (or self-reinforcing) feedback gives rise to exponential growth, a negative (or self-correcting) feedback gives rise to a goal seeking mode, and negative feedbacks with time delays give rise to oscillatory behavior. Nonlinear interactions between the system’s feedback structures can give rise to more complex dynamic behavior modes, examples of which are also shown in :numref:`Figure_4_2`, adapted from :cite:t:`sterman2001system`.
+
+.. _Figure_4_2:
+.. figure:: _static/figure4_2_behavior_modes.png
+    :alt: Figure 4.2
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Common modes of behavior in dynamic systems, occurring based on the presence of positive and negative feedback relationships, and linear and non-linear interactions. Adapted from :cite:t:`sterman2001system`.
+
+The nature of feedback processes in a dynamic system shapes its fundamental behavior: positive feedbacks generate their own growth, negative feedbacks self-limit, seeking balance and equilibrium. In this manner, feedback processes give rise to different regimes, multiples of which could be present in each mode of behavior. Consider a population of mammals growing exponentially until it reaches the carrying capacity of its environment (referred to as S-shaped growth). When the population is exponentially growing, the regime is dominated by positive feedback relationships that reinforce its growth. As the population approaches its carrying capacity limit, negative feedback structures begin to dominate, counteracting the growth and establishing a stable equilibrium. Shifts between regimes can be thought of as tipping points, mathematically defined as unstable equilibria, where the presence of positive feedbacks amplifies disturbances and moves the system to a new equilibrium point. In the case of stable equilibria, the presence of negative feedbacks dampens any small disturbance and maintains the system at a stable state. As different feedback relationships govern each regime, different factors (those making up each feedback mechanism) are activated and shape the states the system is found in, as well as define the points of equilibria.
+
+For simple stylized models with a small number of states, system dynamics analysis can analytically derive these equilibria, the conditions for their stability and the factors determining them. The ability for this to be performed is, however, significantly challenged when it comes to systems that attempt to more closely resemble real complex systems. We argue this is the case for several reasons. First, besides generally exhibiting complex nonlinear dynamics, real world systems are also made up from larger numbers of interacting elements, which often makes the analytic derivation of system characteristics intractable :cite:p:`anderies2019knowledge,muneepeerakul2020emergence`. Second, human-natural systems temporally evolve and transform when human state-aware action is present. Consider, for instance, humans recreationally hunting the aforementioned population of mammals. Humans act based on the mammal population levels by enforcing hunting quotas or establishing protected territories or eliminating other predators. The mammal population reacts in a response, giving birth to ever changing state-action-consequence feedbacks, the path dependencies of which become difficult to diagnose and understand (e.g., :cite:p:`hadjimichael2020navigating`). Trying to simulate the combination of these two challenges (large numbers of state-aware agents interacting with a natural resource and with each other) produces intractable models that require advanced heuristics to analyze their properties and establish useful inferences.
+
+Sensitivity analysis paired with exploratory modeling methods offers a promising set of tools to address these challenges. We present a simple demonstrative application based on :cite:t:`quinn2017direct`. This stylistic example was first developed by :cite:t:`carpenter_management_1999` and represents a town that must balance its agricultural and industrial productivity with the pollution it creates in a downstream lake. Increased productivity allows for increased profits, which the town aims to maximize, but it also produces more pollution for the lake. Too much phosphorus pollution can cause irreversible eutrophication, a process known as “tipping” the lake. The model of phosphorus in the lake :math:`X_t` at time :math:`t` is governed by:
+
+.. math::
+
+  X_{t+1}= X_{t}+a_{t}+\frac{X_{t}^q} {1+X_{t}^q}-bX_t+\varepsilon
+
+where :math:`a_t \in [0,0.1]` is the town’s pollution release at each timestep, :math:`b` is the natural decay rate of phosphorus in the lake, :math:`q` defines the lake’s recycling rate (primarily through sediments), and :math:`\varepsilon` represents uncontrollable natural inflows of pollution modeled as a log-normal distribution with a given mean, :math:`\mu`, and standard deviation :math:`\sigma`.
+
+Panels (a-c) in :numref:`Figure_4_3` plot the fluxes of phosphorus into the lake versus the mass accumulation of phosphorus in the lake. The red line corresponds to the phosphorus sinks in the lake (natural decay), given by :math:`bX_t`. The grey shaded area represents the lake’s phosphorus recycling flux, given by :math:`\frac{X_{t}^q} {1+X_{t}^q}`. The points of intersection indicate the system’s equilibria, two of which are stable, and one is unstable (also known as the tipping point). The stable equilibrium in the bottom left of the figure reflects an oligotrophic lake, whereas the stable equilibrium in the top right represents a eutrophic lake. With increasing phosphorus values, the tipping point can be crossed, and the lake will experience irreversible eutrophication, as the recycling rate would exceed the removal rate even if the town’s pollution became zero. In the absence of anthropogenic and natural inflows of pollution in the lake (:math:`a_t` and :math:`\varepsilon` respectively), the area between the bottom-left black point and the white point in the middle can be considered as the safe operating space, before emission levels cross the tipping point.
+
+.. _Figure_4_3:
+.. figure:: _static/figure4_3_lake_problem_fluxes.png
+    :alt: Figure 4.3
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Fluxes of phosphorus with regards to mass of phosphorus in the lake and sensitivity analysis results, assuming :math:`b=0.42` and :math:`q=2`. (a) Fluxes of phosphorus assuming no emmisions policy and no natural inflows. (b-c) Fluxes phosphorus when applying two different emissions policies. The "Best economic policy" and the "Most reliable policy" have been identified by :cite:t:`quinn2017direct` and can be found at :cite:t:`quinn_julianneqlake_problem_dps_2017`. (d) Results of a sensitivity analysis on the parameters of the model most consequential to the reliability of the "Most reliable policy". The code to replicate the sensitivity analysis can be found at :cite:t:`hadka_project-platypusrhodium_2017`. Panels (a-c) are used courtesy of Julianne Quinn, University of Virginia.
+
+The town has identified two potential policies that can be used to manage this lake, one that maximizes its economic profits (“best economic policy”) and one that maximizes the time below the tipping point (“most reliable policy”). Panels (b-c) in :numref:`Figure_4_3` add the emissions from these policies to the recycling flux and show how the equilibria points shift as a result. In both cases the stable oligotrophic equilibrium increases and the tipping point decreases, narrowing the safe operating space :cite:`anderies2019knowledge, carpenter_allowing_2015`. The best economic policy results in a much narrower space of action, with the tipping point very close to the oligotrophic equilibrium. The performance of both policies depends significantly on the system parameters. For example, a higher value of :math:`b`, the natural decay rate, would shift the red line upward, moving the equilibria points and widening the safe operating space. Inversely, a higher value of :math:`q`, the lake’s recycling rate, would shift the recycling line upward, moving the tipping point lower and decreasing the safe operating space. The assumptions under which these policies were identified are therefore critical to their performance and any potential uncertainty in the parameter values could be detrimental to the system’s objectives being met.
+
+Sensitivity analysis can be used to clarify the role these parameters play on policy performance. :numref:`Figure_4_3` (d) shows the results of a Sobol sensitivity analysis on the reliability of the “most reliable” policy in a radial convergence diagram. The significance of each parameter is indicated by the size of circles corresponding to it. The size of the interior dark circle indicates the parameter’s first-order effects and the size of the exterior circle indicates the parameter’s total-order effects. The thickness of the lines between two parameters indicated the extent of their interaction (second-order effects). In this case, parameters :math:`b` and :math:`q` appear to have the most significant importance on the system, followed by the mean, :math:`\mu`, of the natural inflows. All these parameters function in a manner that shifts the location of the three equilibria and therefore policies that are identified ignoring this parametric uncertainty might fail to meet their intended goals.
+
+It is worth mentioning that current sensitivity analysis methods are somewhat challenged in addressing several system dynamics analysis questions. The fundamental reason is that sensitivity analysis methods and tools have been developed to gauge numerical sensitivity of model output to changes in factor values. This is natural, as most simulation studies (e.g., all aforementioned examples) have been traditionally concerned with this type of sensitivity. In system dynamics modeling, however, a more important and pertinent concern is changes between regimes or between behavior modes (also known as bifurcations) as a result of changes in model factors :cite:p:`hekimouglu2016sensitivity, sterman2001system`. This poses two new challenges. First, identifying a change in regime depends on several characteristics besides a change in output value, like the rate and direction of change. Second, behavior mode changes are qualitative and discontinuous, as equilibria change in stability but also move in and out of existence.
+
+Despite these challenges, recent advanced sensitivity analysis methods can help illuminate which factors in a system are most important in shaping boundary conditions (tipping points) between different regimes and determining changes in behavior modes. Reviewing such methods is outside the scope of this text, but the reader is directed to the examples of :cite:t:`eker2018practice` and :cite:t:`hadjimichael2020navigating`, who apply parameterised perturbation on the functional relationships of a system to study the effects of model structural uncertainty on model outputs and bifurcations, and :cite:t:`hekimouglu2016sensitivity` and :cite:t:`steinmann2020behavior` who, following wide sampling of uncertain inputs, cluster the resulting time series in modes of behavior and identify most important factors for each.
+
+.. note::
+
+    Put this into practice! Click the following badge to try out an interactive tutorial on performing a sensitivity analysis to discover consequential dynamics:  `Factor Discovery Jupyter Notebook <nb_fishery_dynamics_>`_
diff --git a/docs/html/_sources/4.3_consequential_scenarios_what_is_controlling_consequential_outcomes.rst b/docs/html/_sources/4.3_consequential_scenarios_what_is_controlling_consequential_outcomes.rst
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+.. _consequential_scenarios:
+
+Consequential Scenarios: What is Controlling Consequential Outcomes?
+####################################################################
+
+As overviewed in :numref:`perspectives`, most models are abstractions of systems in the real world. When sufficient confidence has been established in a model, it can then act as a surrogate for the actual system, in that the consequences of potential stressors, proposed actions or other changes can be evaluated by computer model simulations :cite:`bankes_exploratory_1993`. A model simulation then represents a computational experiment, which can be used to assess how the modeled system would behave should the various changes come to be. Steven Bankes coined the term exploratory modeling to describe the use of large sets of such computational experiments to investigate their implications on the system. :numref:`Figure_4_4` presents a typical workflow of an exploratory modeling application. Exploratory modeling approaches typically use sampling designs to generate large ensembles of states that represent combinations of changes happening together, spanning the entire range of potential values a factor might take (indicated in :numref:`Figure_4_4` by numbers 2-5). This perspective on modeling is particularly relevant to studies making long term projections into the future.
+
+.. _Figure_4_4:
+.. figure:: _static/figure4_4_exploratory_workflow.png
+    :alt: Figure 4_4
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    A typical exploratory modeling workflow
+
+
+In the long-term policy analysis literature, exploratory modeling has prominently placed itself as an alternative to traditional narrative scenario or assumptions-based planning approaches, in what can be summarized in the following two-pronged critique :cite:`bankes_exploratory_1993, bankes2001computer, lempert2003shaping`. The most prevalent criticism sees that the future and how it might evolve is both highly complex and deeply uncertain. Despite its benefits for interpretation and intuitive appeal, a small number of scenarios invariably misses many other potential futures that did not get selected as sufficiently representative of the future. This is especially the case for aggregate, narrative scenarios that describe simultaneous changes in multiple sectors together (e.g., "increased energy demand, combined with high agricultural land use and large economic growth"), such as the emission scenarios produced by the Intergovernmental Panel on Climate Change :cite:`lamontagne2018large`. The bias introduced by this reduced set of potential changes can skew inferences drawn from the model, particularly when the original narrative scenarios are focused on a single or narrow set of measures of system behavior.
+
+The second main criticism of traditional narrative scenario-based planning methods is that they provide no systematic way to distinguish which of the constituent factors lead to the undesirable consequences produced by a scenario. Narrative scenarios (e.g., the scenario matrix framework of RCPs-SSPs-SPAs; :cite:`o2014new`) encompass multiple changes happening together selected to span the range of potential changes but are not typically generated in a systematic factorial manner that considers the multiple ways the factors can be combined. This has two critical limitations. It obfuscates the role each component factor plays in the system, both in isolation and in combination with others (e.g., "is it the increased energy demand or the high agricultural land use that cause unbearable water stress?"). It also renders the delineation of how much change in a factor is critical near impossible. Consider, for example, narrative scenario A with a 5% increase in energy demand, and scenario B with a 30% increase in energy demand, which would have dire consequences. At which point between 5% and 30% do the dire consequences actually begin to occur? Such questions cannot be answered without a wide exploration of the space of potential changes. It should be noted that for some levels of model complexity and computational demands (e.g., global-scale models) there is little feasible recourse beyond the use of narrative scenarios.
+
+Exploratory modeling is typically paired with scenario discovery methods (indicated by number 9 in :numref:`Figure_4_4`) that identify which of the scenarios (also known as states of the world) generated indeed have consequences of interest for stakeholders and policy makers, in an approach referred to as ensemble-based scenario-discovery :cite:`bryant2010thinking, groves2007new, lempert2006general`. This approach therefore flips the planning analysis from one that attempts to predict future system conditions to one that attempts to discover the (un)desirable future conditions. Ensemble-based scenario discovery can thus inform what modeling choices yield the most consequential behavioral changes or outcomes, especially when considering deeply uncertain, scenario-informed projections :cite:`walker2013adapt,moallemi2020exploratory`. The relative likelihoods and relevance of the discovered scenarios can be subsequently evaluated by the practitioners a posteriori, within a richer context of knowing the wider set of potential consequences :cite:`dessai2009climate`. This can include changing how an analysis is framed (number 10 in :numref:`Figure_4_4`). For instance, one could initially focus on ensemble modeling of vulnerability using a single uncertain factor that is assumed to be well characterized by historical observations (e.g., streamflow; this step is represented by numbers 2-3 in :numref:`Figure_4_4`). The analysis can then shift to include projections of more factors treated as deeply uncertain (e.g., urbanization, population demands, temperature, and snow-melt) to yield a far wider space of challenging projected futures. UC experiments contrasting these two framings can be highly valuable for tracing how vulnerability inferences change as the modeled space of futures expands from the historical baseline :cite:`quinn2017direct`.
+
+An important nuance to be clarified here is that the focus or purpose of a modeling exercise plays a major role in whether a given factor of interest is considered well-characterized or deeply uncertain. Take the example context of characterizing temperature or streamflow extremes, where for each state variable of interest for a given location of focus there is a century of historical observations. Clearly, the observation technologies will have evolved over time uniquely for temperature and streamflow measurements and they likely lack replicate experiments (data uncertainty). A century of record will be insufficient to capture very high impact and rare extreme events (i.e., increasingly poor structural/parametric inference for the distributions of specific extreme single or compound events). The mechanistic processes as well as their evolving variability will be interdependent but uniquely different for each of these state variables. A large body of statistical literature exists focusing on the topics of synthetic weather :cite:p:`wilks1999weather,herrera2017review` or streamflow :cite:p:`lamontagne2018generating,medda2019comparison` generation that provides a rich suite of approaches for developing history-informed, well-characterized stochastic process models to better estimate rare individual or compound extremes. These history-focused approaches can be viewed as providing well-characterized quantifications of streamflow or temperature distributions; however, they do not capture how coupled natural-human processes can fundamentally change their dynamics when transitioning to projections of longer-term futures (e.g., streamflow and temperature in 2055). Consequently, changing the focus of the modeling to making long term projections of future streamflow or temperature now makes these processes deeply uncertain.
+
+Scenario discovery methods (number 9 in :numref:`Figure_4_4`) can be qualitative or quantitative and they generally attempt to distinguish futures in which a system or proposed policies to manage the system meet or miss their goals :cite:`groves2007new`. The emphasis placed by exploratory modeling on model outputs that have decision relevant consequences represents a shift toward a broader class of metrics that are reflective of the stakeholders’ concerns, agency and preferences (also discussed in :numref:`perspectives`). As a result, sensitivity analysis and scenario discovery methods in this context are therefore applied to performance metrics that go beyond model error but are rather focused on broader metrics such as the resilience of a sector, the reliability of a process, or the vulnerability of a population in the face of uncertainty. In exploratory modeling literature, this metric is most typically—but not always—a measure of robustness (number 8 in Fig. 13). Robustness is a property of a system or a design choice capturing its insensitivity to uncertainty and can be measured via a variety of means, most recently reviewed by :cite:t:`herman2015should` and :cite:t:`mcphail2018robustness`.
+
+Scenario discovery is typically performed through the use of algorithms applied on large databases of model runs, generated through exploratory modeling, with each model run representing the performance of the system in one potential state of the world. The algorithms seek to identify the combinations of factor values (e.g., future conditions) that best distinguish the cases in which the system does or does not meet its objectives. The most widely known classification algorithms are the Patient Rule Induction Method (PRIM; :cite:p:`friedman1999bump`) and Classification and Regression Trees (CART; :cite:p:`breiman1984classification`). These factor mapping algorithms create orthogonal boundaries (multi-dimensional hypercubes) between states of the world that are successful or unsuccessful in meeting the system’s goals :cite:p:`dalal2008low`. The algorithms attempt to strike a balance between simplicity of classification (and as a result, interpretability) and accuracy :cite:p:`groves2007new, bryant2010thinking, lempert2019robust`.
+
+Even though these approaches have been shown to yield interpretable and relevant scenarios :cite:p:`kwakkel2019supporting`, several authors have pointed out the limitations of these methods with regards to their division of space in orthogonal behavioral and non-behavioral regions :cite:p:`trindade2019deeply`. Due to their reliance on boundaries orthogonal to the uncertainty axes, PRIM and CART cannot capture interactions between the various uncertain factors considered, which can often be significant :cite:p:`quinn2018exploring`. More advanced methods have been proposed to address this drawback, with logistic regression being perhaps the most prominent :cite:p:`gold2019identifying, lamontagne2019robust, quinn2018exploring`. Logistic regression can produce boundaries that are not necessarily orthogonal to each uncertainty axis, nor necessarily linear, if interactive terms between two parameters are used to build the regression model. It also describes the probability that a state of the world belongs to the scenarios that lead to failure. This feature allows users to define regions of success based on a gradient of estimated probability of success in those worlds, unlike PRIM which only classifies states of the world in two regions :cite:p:`hadjimichael_defining_2020, quinn2018exploring`.
+
+Another more advanced factor mapping method is boosted trees :cite:p:`drucker1996boosting, freund1999short`. Boosted trees can avoid two limitations inherent to the application of logistic regression: i) to build a nonlinear classification model the interactive term between two uncertainties needs to be pre-specified and cannot be discovered (e.g., we need to know a-priori whether factor :math:`x_1` interacts with :math:`x_2` in a relationship that looks like :math:`x_1`·:math:`x_2` or :math:`x_1^{x_2}`); and ii) the subspaces defined are always convex. The application of such a factor mapping algorithm is limited in the presence of threshold-based rules with discrete actions in a modeled system (e.g., “if network capacity is low, build new infrastructure”), which results in failure regions that are nonlinear and non-convex :cite:p:`trindade2019deeply`. Boosting works by creating an ensemble of classifiers and forcing some of them to focus on the hard-to-learn parts of the problem, and others to focus on the easy-to-learn parts. Boosting applied to CART trees can avoid the aforementioned challenges faced by other scenario discovery methods, while resisting overfitting :cite:p:`murphy2012machine`, assuring the identified success and failure regions are still easy to interpret.
+
+Below we provide an example application of two scenario discovery methods, PRIM and logistic regression, using the lake problem introduced in the previous section. From the sensitivity analysis results presented in :numref:`Figure_4_3` (d), we can already infer that parameters :math:`b` and :math:`q` have important effects on model outputs (i.e., we have performed factor prioritization). Scenario discovery (i.e., factor mapping) complements this analysis by further identifying the specific values of b and q that can lead to consequential and undesirable outcomes. For the purposes of demonstration, we can assume the undesirable outcome in this case is defined as the management policy failing to achieve 90% reliability in a state of the world.
+
+.. _Figure_4_5:
+.. figure:: _static/figure14lake_problem_SD.png
+    :alt: Figure 14.
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Scenario discovery for the lake problem, using (a) PRIM and (b) logistic regression.
+
+:numref:`Figure_4_5` shows the results of scenario discovery, performed through (a) PRIM and (b) logistic regression. Each point in the two panels indicates a potential state of the world, generated through Latin Hypercube Sampling. Each point is colored by whether the policy meets the above performance criterion, with blue indicating success and red indicating failure. PRIM identifies several orthogonal areas of interest, one of which is shown in panel (a). As discussed above, this necessary orthogonality limits how PRIM identifies areas of success (the area within the box). As factors  :math:`b` and :math:`q` interact in this system, the transition boundary between the regions of success and failure is not orthogonal to any of the axes. As a result, a large number of points in the bottom right and the top left of the figure are left outside of the identified region. Logistic regression can overcome this limitation by identifying a diagonal boundary between the two regions, seen in panel (b). This method also produces a gradient of estimated probability of success across these regions.
+
+.. note::
+
+    Put this into practice! Click the following link to try out an interactive tutorial on performing factor mapping using logistic regression:  `Logistic Regression Jupyter Notebook <nb_logistic_regression_>`_
diff --git a/docs/html/_sources/4_sensitivity_analysis_diagnostic_and_exploratory_modeling.rst b/docs/html/_sources/4_sensitivity_analysis_diagnostic_and_exploratory_modeling.rst
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+.. _4_sensitivity_analysis:
+
+*******************************************************
+Sensitivity Analysis: Diagnostic & Exploratory Modeling
+*******************************************************
+
+.. include:: 4.1_understanding_errors_what_is_controlling_model_performance.rst
+
+.. include:: 4.2_consequential_dynamics_what_is_controlling_model_behaviors_of_interest.rst
+
+.. include:: 4.3_consequential_scenarios_what_is_controlling_consequential_outcomes.rst
+
+.. note::
+    The following articles are suggested as fundamental reading for the information presented in this section:
+
+    * Gupta, H.V., Wagener, T., Liu, Y., 2008. Reconciling theory with observations: elements of a diagnostic approach to model evaluation. *Hydrological Processes: An International Journal* 22, 3802–3813.
+    * Bankes, S., 1993. Exploratory Modeling for Policy Analysis. *Operations Research* 41, 435–449. https://doi.org/10.1287/opre.41.3.435
+    * Groves, D.G., Lempert, R.J., 2007. A new analytic method for finding policy-relevant scenarios. *Global Environmental Change* 17, 73–85. https://doi.org/10.1016/j.gloenvcha.2006.11.006
+
+
+    The following articles can be used as supplemental reading:
+
+    * Marchau, V.A.W.J., Walker, W.E., Bloemen, P.J.T.M., Popper, S.W. (Eds.), 2019. Decision Making under Deep Uncertainty: From Theory to Practice. Springer International Publishing. https://doi.org/10.1007/978-3-030-05252-2
+    * Sterman, J.D., 1994. Learning in and about complex systems. System Dynamics Review 10, 291–330. https://doi.org/10.1002/sdr.4260100214
diff --git a/docs/html/_sources/5_conclusion.rst b/docs/html/_sources/5_conclusion.rst
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+.. _5_conclusion:
+
+**********
+Conclusion
+**********
+
+As noted in the Introduction (:numref:`introduction`), the computational and conceptual challenges of the multi-model, transdisciplinary workflows that characterize ambitious projects such as IM3 have limited UC and UQ analyses. Moreover, the very nature and purpose of modeling and diagnostic model evaluation can have very diverse philosophical framings depending on the disciplines involved (see :numref:`Figure_1_1` and :numref:`perspectives`). The guidance provided in this text can be used to frame consistent and rigorous experimental designs for better understanding the consequences and insights from our modeling choices when seeking to capture complex human-natural systems. The progression of sections of this text provide a thorough introduction of the concepts and definitions of diagnostic model evaluation, sensitivity analysis and UC. In addition, we comprehensively discuss how specific modeling objectives and applications should guide the selection of appropriate techniques; broadly, these can include model diagnostics, in-depth analysis of the behavior of the abstracted system, and projections under conditions of deep uncertainty. This text also contains a detailed presentation of the main sensitivity analysis methods and a discussion of their features and main limitations. Readers are also provided with an overview of computer tools and platforms that have been developed and could be considered in addressing IM3 scientific questions. The appendices of this text include an overview of UQ methods, a terminology glossary of the key concepts as well as example test cases and scripts to showcase various UC related capabilities.
+
+Although we distinguish the UC and UQ model diagnostics, the reader should note that we suggest an overall consistent approach to both in this text by emphasizing “exploratory modeling” (see review by :cite:t:`moallemi2020exploratory`). Although data support, model complexity, and computational limits strongly distinguish the feasibility and appropriateness of various UC diagnostic tools (e.g., see :numref:`Figure_3_5`), we overall recommend that modelers view their work through the lens of cycles of learning. Iterative and deliberative exploration of model-based hypotheses and inferences for transdisciplinary teams is non-trivial and ultimately critical for mapping where innovations or insights are most consequential. Overall, we recommend approaching modeling with an openness to the diverse disciplinary perspectives such as those mirrored by the IM3 family of models in a progression from evaluating models relative to observed history to advanced formalized analyses to make inferences on multi-sector, multi-scale vulnerabilities and resilience. Exploratory modeling approaches can help fashion experiments with large numbers of alternative hypotheses on the co-evolutionary dynamics of influences, stressors, as well as path-dependent changes in the form and function of coupled human-natural systems :cite:`weaver_improving_2013`. This text guides the reader through the use of sensitivity analysis and uncertainty methods across the diverse perspectives that have shaped modern diagnostic and exploratory modeling.
diff --git a/docs/html/_sources/6_glossary.rst b/docs/html/_sources/6_glossary.rst
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+.. _glossary:
+
+========
+Glossary
+========
+
+**Design of experiment**: Provides a framework for the extraction of all plausible information about the impact of each factor on the output of the numerical model
+
+**Exploratory modeling**: Use of large ensembles of uncertain conditions to discover decision-relevant combinations of uncertain factors 
+
+**Factor**: Any model component that can affect model outputs: inputs, resolution levels, coupling relationships, model relationships and parameters. In models with acceptable model fidelity these factors may represent elements of the real-world system under study. 
+
+**Factor mapping**: A technique to identify which uncertain model factors lead to certain model behavior
+
+**Factor prioritization**: A technique to identify the uncertain factors which, when fixed to their true value, would lead to the greatest reduction in output variability
+
+**Factor screening**: A technique to identify model components that have a negligible effect or make no significant contributions to the variability of the outputs or metrics of interest 
+
+**First-, second-, total-order effects**: First-order effects indicate the percent of model output variance contributed by a factor individually. Second-order effects capture how interactions between a pair of parameter input variables can lead to change in model output. Total-order effects consider all the effects a factor has, individually and in interaction with other factors. 
+
+**Hindcasting**: A type of predictive check that uses the model to estimate output for past events to see how well the output matches the known results. 
+
+**Pre-calibration**: A hybrid uncertainty assessment method that involves identifying a plausible set of parameters using some prespecified screening criterion, such as the distance from the model results to the observations. 
+
+**Prior**: The best assessment of the probability of an event based on existing knowledge before a new experiment is conducted
+
+**Posterior**: The revised or updated probability of an event after taking into account new information 
+
+**Probabilistic inversion**: Uses additional information, for instance, a probabilistic expert assessment or survey result, to update an existing prior distribution
+
+**Return level**: A value that is expected to be equaled or exceeded on average once every interval of time (T) (with a probability of 1/T)
+
+**Return period**: The estimated time interval between events of a similar size or intensity/ 
+
+**Sampling**: The process of selecting model parameters or inputs that characterize the model uncertainty space. 
+
+**Scenario discovery**: Use of large ensembles of uncertain conditions to discover decision-relevant combinations of uncertain factors 
+
+**Sensitivity analysis**: Conducted to understand the factors and processes that most (or least) control a model’s outputs
+
+	*Local sensitivity analysis*: Model evaluation performed by varying uncertain factors around specific reference values
+
+	*Global sensitivity analysis*: Model evaluation performed by varying uncertain factors throughout their entire feasible value space
+
+**Uncertainty**
+
+*Deep uncertainty*: Refers to situations where expert opinions consulted on a decision do not know or cannot agree on system boundaries, or the outcomes of interest and their relative importance, or the prior probability Distribution for the various uncertain factors present
+
+*Epistemic uncertainty*: Systematic uncertainty that comes about due to the lack of knowledge or data to choose the best model
+
+*Ontological uncertainty*: Uncertainties due to processes, interactions, or futures, that are not contained within current conceptual models
+
+*Aleatory uncertainty*: Uncertainty due to natural randomness in processes 
+
+*Uncertainty characterization*: Model evaluation under alternative factor hypotheses to explore their implications for model output uncertainty
+
+*Uncertainty quantification*: Representation of model output uncertainty using probability distributions
+
+**Variance decomposition**: A technique to partition how much of the variability in a model’s output is due to different explanatory variables.  
diff --git a/docs/html/_sources/A1.1_UQ_Introduction.rst b/docs/html/_sources/A1.1_UQ_Introduction.rst
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+Introduction
+############
+
+As defined in :numref:`introduction`, uncertainty quantification (UQ) refers to the formal focus on the full specification of likelihoods as well as distributional forms necessary to infer the joint  probabilistic response across all modeled factors of interest :cite:`cooke1991experts`. This is in contrast to UC (the primary focus of the main document of this book), which is instead aimed at identifying which modeling choices yield the most consequential changes or outcomes and exploring alternative hypotheses related to the form and function of modeled systems :cite:`moallemi2020exploratory, walker2003defining`.
+
+UQ is important for quantifying the relative merits of hypotheses for at least three main reasons. First, identifying model parameters that are consistent with observations is an important part of model development. Due to several effects, including correlations between parameters, simplified or incomplete model structures (relative to the full real-world dynamics), and uncertainty in the observations, many different combinations of parameter values can be consistent with the model structure and the observations to varying extents. Accounting for this uncertainty is conceptually preferable to selecting a single “best fit” parameter vector, particularly as consistency with historical or present observations does not necessarily  guarantee skillful future projections.
+
+The act of quantification requires specific assumptions about distributional forms and likelihoods, which may be more or less justified depending on prior information about the system or model behavior. As a result, UQ is well-suited for studies accounting for or addressing hypotheses related to systems with a relatively large amount of available data and models which are computationally inexpensive, particularly when the emphasis is on prediction. As shown in :numref:`Figure_A1_1`, there is a fundamental tradeoff between the available number of model evaluations (for a fixed computational budget) and the number of parameters treated as uncertain. Sensitivity analyses are therefore part of a typical UQ workflow to identify which factors can be fixed and which ought to be prioritized in the UQ.
+
+.. _Figure_A1_1:
+.. figure:: _static/figureA1_1_UQ_approaches.png
+    :alt: Figure A1.1
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Overview of selected existing approaches for uncertainty quantification and their appropriateness given the number of uncertain model parameters and the number of available model simulations. Green shading denotes regions suitable for uncertainty quantification and red shading indicates regions more appropriate for uncertainty characterization.
+
+The choice of a particular UQ method depends on both the desired level of quantification and the ability to navigate the tradeoff between computational expense and the number of uncertain parameters (:numref:`Figure_A1_1`). For example, Markov chain Monte Carlo with a full system model can provide an improved representation of uncertainty compared to the coarser pre-calibration approach :cite:`ruckert_assessing_2017`, but requires many more model evaluations. The use of a surrogate model to approximate the full system model can reduce the number of needed model evaluations by several orders of magnitude, but the uncertainty quantification can only accommodate a limited number of parameters.
+
+The remainder of this appendix will focus on introducing workflows for particular UQ methods, including a brief discussion of advantages and limitations.
\ No newline at end of file
diff --git a/docs/html/_sources/A1.2_Parametric_Bootstrap.rst b/docs/html/_sources/A1.2_Parametric_Bootstrap.rst
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+Parametric Bootstrap
+####################
+
+The parametric bootstrap :cite:p:`efron_bootstrap_1986` refers to a process of model recalibration to alternate realizations of the data. The bootstrap was originally developed to estimate standard errors and confidence intervals without ascertaining key assumptions that might not hold given the available data. In a setting where observations can be viewed as independent realizations of an underlying stochastic process, a sufficiently rich dataset can be treated as a population representing the data distribution. New datasets are then generated by resampling from the data with replacement, and the model can be refit to each new dataset using maximum-likelihood estimation. The resulting distribution of estimates can then be viewed as a representation of parametric uncertainty.
+
+A typical workflow for the parametric bootstrap is shown in :numref:`Figure_A1_2`. After identifying outputs of interest and preparing the data, the parametric model is fit by some procedure such as minimizing root-mean-square-error or maximizing the likelihood. Alternate datasets are constructed by resampling from the population or by generating new samples from the fitted data-generating process. It is important at this step that the resampled quantities are independent of one another. For example, in the context of temporally- or spatially-correlated data, such as time series, the raw observations cannot be treated as independent realizations. However, the residuals resulting from fitting the model to the data could be (depending on their structure). For example, if the residuals are treated as independent, they can then be resampled with replacement, and these residuals added to the original model fit to create new realizations. If the residuals are assumed to be the result of an autoregressive process, this process could be fit to the original residual series and new residuals be created using this model :cite:`sriver_characterizing_2018`. The model is then refit to each new realization.
+
+.. _Figure_A1_2:
+.. figure:: _static/figureA1_2_bootstrap_workflow.png
+    :alt: Figure A1.2
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Workflow for the parametric bootstrap.
+
+The bootstrap is computationally convenient, particularly as the process of fitting the model to each realization can be easily parallelized. This approach also requires minimal prior assumptions. However, due to the assumption that the available data are representative of the underlying data distribution, the bootstrap can neglect key uncertainties which might influence the results. For example, when using an autoregressive process to generate new residuals, uncertainty in the autocorrelation parameter and innovation variance is neglected, which may bias estimates of, for example, low-probability but high-impact events :cite:`ruckert_effects_2017`.
diff --git a/docs/html/_sources/A1.3_Precalibration.rst b/docs/html/_sources/A1.3_Precalibration.rst
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+Pre-Calibration
+###############
+
+Pre-calibration :cite:`beven_future_1992, edwards_precalibrating_2011, boukouvalas_bayesian_2014` involves the identification of a plausible set of parameters using some prespecified screening criterion, such as the distance from the model results to the observations (based on an appropriate metric for the desired matching features, such as root-mean-squared error). A typical workflow is shown in :numref:`Figure_A1_3`. Parameter values are obtained by systematically sampling the input space (see :numref:`design_of_experiments`). After the model is evaluated at the samples, only those passing the distance criterion are retained. This selects a subset of the parameter space as “plausible” based on the screening criterion, though there is no assignment of probabilities within this plausible region.
+
+.. _Figure_A1_3:
+.. figure:: _static/figureA1_3_precal_workflow.png
+    :alt: Figure A1.3
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Workflow for pre-calibration.
+
+Pre-calibration can be useful for models which are inexpensive enough that a reasonable
+number of samples can be used to represent the parameter space, but which are too expensive to facilitate full uncertainty quantification. High-dimensional parameter spaces, which can be problematic for the uncertainty quantification methods below, may also be explored using pre-calibration. One key prerequisite to using this method is the ability to place a meaningful distance metric on the output space.
+
+However, pre-calibration results in a very coarse characterization of uncertainty, especially when considering a large number of parameters, as more samples are needed to fully characterize the parameter space. Due to the inability to evaluate the relative probability of regions of the parameter space beyond the binary plausible-and-implausible characterization, pre-calibration can also result in degraded hindcast and projection skills and parameter estimates :cite:`makowski_using_2002, shafii_uncertainty-based_2014, ruckert_assessing_2017`.
+
+A related method, widely used in hydrological studies, is generalized likelihood uncertainty estimation, or GLUE :cite:`beven_future_1992`. Unlike pre-calibration, the underlying argument for GLUE relies on the concept of equifinality :cite:`beven_equifinality_2001`, which posits that it is impossible to find a uniquely well-performing parameter vector for models of abstract environmental systems :cite:`beven_equifinality_2001, vrugt_embracing_2018`. In other words, there exist multiple parameter vectors which perform equally or similarly well. As with pre-calibration, GLUE uses a goodness-of-fit measure (though this is called a “likelihood” in the GLUE literature, as opposed to a statistical likelihood function :cite:`stedinger_appraisal_2008`) to evaluate samples. After setting a threshold of acceptable performance with respect to that measure, samples are evaluated and classified into “behavioral” or “non-behavioral” according to the threshold.
+
+.. note::
+
+    Put this into practice! Click the following badge to try out an interactive tutorial on utilizing Pre-Calibration and GLUE for HYMOD model calibration:  `Pre-Calibration Jupyter Notebook <nb_hymod_>`_
diff --git a/docs/html/_sources/A1.4_Markov_Chain_Monte_Carlo.rst b/docs/html/_sources/A1.4_Markov_Chain_Monte_Carlo.rst
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+Markov Chain Monte Carlo
+########################
+
+Markov chain Monte Carlo (MCMC) is a “gold standard” approach to full uncertainty quantification. MCMC refers to a category of algorithms which systematically sample from a target distribution (in this case, the posterior distribution) by constructing a Markov chain. A Markov chain is a probabilistic structure consisting of a state space, an initial probability distribution over the states, and a transition distribution between states. If a Markov chain satisfies certain properties :cite:p:`robert_monte_2013, robert_metropolishastings_2015`, the probability of being in each state will eventually converge to a stable, or stationary, distribution, regardless of the initial probabilities.
+
+MCMC algorithms construct a Markov chain of samples from a parameter space (the combination of model and statistical parameters). This Markov chain is constructed so that the stationary distribution is a target distribution, in this case the (Bayesian) posterior distribution. As a result, after the transient period, the resulting samples can be viewed as a set of dependent samples from the posterior (the dependence is due to the autocorrelation between samples resulting from the Markov chain transitions). Expected values can be computed from these samples (for example, using batch-means estimators :cite:p:`flegal_markov_2008`), or the chain can be sub-sampled or thinned and the resulting samples used as independent Monte Carlo samples due to the reduced or eliminated autocorrelation.
+
+A general workflow for MCMC is shown in :numref:`Figure_A1_4`. The first decision is whether to use the full model or a surrogate model (or emulator). Typical surrogates include Gaussian process emulation :cite:p:`currin_bayesian_1991, sacks_design_1989`, polynomial chaos expansions :cite:p:`ghanem_spectral_1991, xiu_wiener--askey_2002`, support vector machines :cite:p:`ciccazzo_svm_2016, pruett_creation_2016`, and neural networks :cite:p:`eason_adaptive_2014, gorissen_sequential_2009`. Surrogate modeling can be faster, but requires a sufficient number of model evaluations for the surrogate to accurately represent the model’s response surface, and this typically limits the number of parameters which can be included in the analysis.
+
+.. _Figure_A1_4:
+.. figure:: _static/figureA1_4_mcmc_workflow.png
+    :alt: Figure A1.4
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Workflow for Markov chain Monte Carlo.
+
+After selecting the variables which will be treated as uncertain, the next step is to specify the likelihood based on the selected surrogate model or the structure of the data-model residuals. For example, it may not always be appropriate to treat the residuals as independent and identically distributed (as is commonly done in linear regression). A mis-specification of the residual structure can result in biases and over- or under-confident inferences and projections :cite:p:`brynjarsdottir_learning_2014`.
+
+After specifying the prior distributions (see :numref:`critical_first_step`), the selected MCMC algorithm should be used to draw samples from the posterior distribution. There are many MCMC algorithms, all of which have advantages and disadvantages for a particular problem. These include the Metropolis-Hastings algorithm :cite:p:`robert_metropolishastings_2015` and Hamiltonian Monte Carlo :cite:p:`betancourt_conceptual_2017, neal_mcmc_2011`. Software packages typically implement one MCMC method, sometimes designed for a particular problem setting or likelihood specification. For example, R’s *adaptMCMC* implements an adaptive Metropolis-Hastings algorithm :cite:p:`vihola_robust_2012`, while *NIMBLE* :cite:p:`valpine_programming_2017, nimble_development_team_nimble_2021` uses a user-customizable Metropolis-Hastings implementation, as well as functionality for Gibbs sampling (which is a special case of Metropolis-Hastings where the prior distribution has a convenient mathematical form). Some recent implementations, such as *Stan* :cite:p:`stan_development_team_stan_2021`, *pyMC3* :cite:p:`salvatier_probabilistic_2016`, and *Turing* :cite:p:`ge_turing_2018` allow different algorithms to be used.
+
+A main consideration when using MCMC algorithms is testing for convergence to the target distribution. As convergence is guaranteed only for a sufficiently large number of transitions, it is impossible to conclude for certain that a chain has converged for a fixed number of iterations. However, several heuristics have been developed :cite:p:`flegal_markov_2008, gelman_inference_1992` to increase evidence that convergence has occurred.
diff --git a/docs/html/_sources/A1.5_Other_methods.rst b/docs/html/_sources/A1.5_Other_methods.rst
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+Other Methods
+#############
+
+Other common methods for UQ exist. These include sequential Monte Carlo, otherwise known as particle filtering :cite:p:`del_moral_sequential_2006, doucet_sequential_2000, liu_combined_2001`, where a number of particles are used to evaluate samples. An advantage of sequential Monte Carlo is that the vast majority of the computation can be parallelized, unlike with standard MCMC. A major weakness is the potential for degeneracy :cite:p:`doucet_sequential_2000`, where many particles have extremely small weights, resulting in the effective use of only a few samples.
+
+Another method is approximate Bayesian computation (ABC) :cite:p:`cabras_approximate_2015, lintusaari_fundamentals_2017, sunnaker_approximate_2013`. ABC is a likelihood-free approach that compares model output to a set of summary statistics. ABC is therefore well-suited for models and residual structures which do not lend themselves to a computationally-tractable likelihood, but the resulting inferences are known to be biased if the set of summary statistics is not sufficient, which can be difficult to know a-priori.
diff --git a/docs/html/_sources/A1.6_Critical_first_step.rst b/docs/html/_sources/A1.6_Critical_first_step.rst
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+.. _critical_first_step:
+
+The Critical First Step: How to Choose a Prior Distribution
+###########################################################
+
+Prior distributions play an important role in Bayesian uncertainty quantification, particularly when data is limited relative to the dimension of the model. Bayesian updating can be thought of as an information filter, where each additional datum is added to the information contained in the prior; eventually, the prior makes relatively little impact. In real world problems, it can be extremely difficult to assess how much data is required for the choice of prior to become less relevant. The choice of prior can also be influential when conducting SA prior to or without UQ. This is because a prior distribution for a sensitivity analysis for a given parameter which is much wider than the region where the model response surface is sensitive to the parameter value might cause the sensitivity calculation to underestimate the response in that potentially critical region. Similarly, a prior which is too narrow may miss regions where the model responds to the parameter altogether.
+
+Ideally, prior distributions are constructed independently of any analysis of the new data considered. This is because using data to inform the prior as well as to compute the likelihood reuses information in a potentially inappropriate way, which can lead to overconfident inferences. Following :cite:t:`jaynes_probability_1996, gelman_prior_2017` refers to the ideal prior as one which encodes all available information about the model. For practical reasons (difficulty of construction or computational inconvenience), most priors fail to achieve this ideal. These compromises mean that priors should be transparently articulated and justified, so that the impact of the choice of prior can be fully understood. When there is ambiguity about an appropriate prior, such as how fat the tails should be, an analyst should examine how sensitive the UQ results are to the choice of prior.
+
+Priors can also be classified in terms of the information encoded by them, demonstrated in :numref:`Figure_A1_5`. Non-informative priors (illustrated in :numref:`Figure_A1_5` (a)) allegedly correspond to (and are frequently justified by) a position of ignorance. A classic example is the use of a uniform distribution. A uniform prior can, however, be problematic, as it can lead to improper inferences by giving extremely large values the same prior probability as values which may seem more likely :cite:p:`gelman_prior_2017, robert_bayesian_2007`, and therefore does not really reflect a state of complete ignorance. In the extreme case of a uniform prior over the entire real line, every particular region has effectively a prior weight of zero, even though not all regions are a priori unlikely :cite:p:`robert_bayesian_2007`. Moreover, a uniform prior which excludes possible parameter values is not actually noninformative, as it assigns zero probability to those values while jumping to a nonzero probability as soon as the boundary is crossed. While a uniform prior can be problematic for the task of uncertainty quantification, it may be useful for an initial sensitivity analysis to identify the boundary of any regions where the model is sensitive to the parameter.
+
+.. _Figure_A1_5:
+.. figure:: _static/figureA1_5_priors_posteriors.png
+    :alt: Figure A1.5
+    :width: 700px
+    :figclass: margin-caption
+    :align: center
+
+    Impact of priors on posterior inferences. These plots show the results of inference for a linear regression model with 15 data points. The true value of the parameter is equal to -3. All priors have mean 0. In panel (a), a non-informative prior allows the tails of the posterior to extend freely, which may result in unreasonably large parameter values. In panel (b), a weakly informative prior constrains the tails more, but allows them to extend without too much restriction. In panel (c), an informative prior strongly constrains the tails of the posterior and biases the inference closer towards the prior mean (the posterior mean is -0.89 in this case, and closer to -3 in the other two cases).
+
+Informative priors strongly bound the range of probable values (illustrated in :numref:`Figure_A1_5` (c)). One example is a Gaussian distribution with a relatively small standard deviation, so that large values are assigned a close to null prior probability. Another example is the jump from zero to non-zero probability occurring at the truncation point of a truncated Gaussian, which could be justified based on information that the parameter cannot take on values beyond this point. Without this type of justification, however, priors may be too informative, failing to allow the information contained in the available data to update them.
+
+Finally, weakly informative priors (illustrated in :numref:`Figure_A1_5` (b)) fall in between :cite:p:`gelman_prior_2017`. They regularize better than non-informative priors, but allow for more inference flexibility than fully informative priors. An example might be a Gaussian distribution with a moderate standard deviation, which still assigns negligible probability for values far away from the mean, but is less constrained than a narrow Gaussian for a reasonably large area. A key note is that it is not necessarily better to be more informative if this cannot be justified by the available information.
\ No newline at end of file
diff --git a/docs/html/_sources/A1.7_Critical_final_step.rst b/docs/html/_sources/A1.7_Critical_final_step.rst
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+The Critical Final Step: Predictive Checks
+##########################################
+
+Every UQ workflow requires a number of choices, potentially including selecting prior distributions, the likelihood specification, and any used numerical models. Checking the appropriateness of these choices is an essential step for sound inferences, as misspecification can produce biased results :cite:p:`brynjarsdottir_learning_2014`. Model checking in this fashion is part of an iterative UQ process, as the results can reveal adjustments to the statistical model or the need to select a different numerical model :cite:p:`gelman_posterior_1996, gelman_bayesian_2020, gelman_philosophy_2013`.
+
+A classic example is the need to check the structure of residuals for correlations. Many standard statistical models, such as linear regression, assume that the residuals are independent and identically distributed from the error distribution. The presence of correlations, including temporal autocorrelations and spatial correlations, indicates a structural mismatch between the likelihood and the data. In these cases, the likelihood should be adjusted to account for these correlations.
+
+Checking residuals in this fashion is one example of a predictive check (or a posterior predictive check in the Bayesian setting). One way to view UQ is as a means to recover data-generating processes (associated with each parameter vector) consistent with the observations. Predictive checks compare the inferred data-generating process to the observations to determine whether the model is capable of appropriately capturing uncertainty. After conducting the UQ analysis, alternatively realized datasets are simulated from sampled parameters. These alternative datasets, or their summary statistics, can be tested against the observations to determine adequacy of the fit. Predictive checks are therefore a way of probing various model components to identify shortcomings that might result in biased inferences or poor projections, depending on the goal of the analysis.
+
+One example of a graphical predictive check for time series models is hindcasting, where predictive intervals are constructed from the alternative datasets and plotted along with the data. Hindcasts demonstrate how well the model is capable of capturing the broader dynamics of the data, as well as whether the parameter distributions produce appropriate levels of output uncertainty. A related quantitative check is the surprise index, which calculates the percentage of data points located within a fixed predictive interval. For example, the 90% predictive interval should contain approximately 90% of the data. More uncertainty than this reflects underconfidence, while less uncertainty reflects overconfidence. This could be the result of priors that are not appropriately informative, or a likelihood that does not account for correlations between data points appropriately. It could also be the result of a numerical model that isn’t sufficiently sensitive to the parameters that are treated as uncertain.
diff --git a/docs/html/_sources/A1.8_Key_take_home.rst b/docs/html/_sources/A1.8_Key_take_home.rst
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+Key Take-Home Points
+####################
+
+When appropriate, UQ is an important component of the exploratory modeling workflow. While a number of parameter sets could be consistent with observations, they may result in divergent model outputs when exposed to different future conditions. This can result in identifying risks which are not visible when selecting a “best fit” parameterization. Quantifying uncertainties also allows us to quantify the support for hypotheses, which is an essential part of the scientific process.
+
+Due to the scale and complexity of the experiments taking place in IM3, UQ has not been extensively used. The tradeoff between the available number of function evaluations and the number of uncertain parameters illustrated in :numref:`Figure_A1_1` is particularly challenging due to the increasing complexity of state-of-the-art models and the movement towards coupled, multisector models. This tradeoff can be addressed somewhat through the use of emulators and parallelizable methods. In particular, when attempting to navigate this tradeoff by limiting the number of uncertain parameters, it is important to carefully iterate with sensitivity analyses to ensure that critical parameters are identified.
+
+Specifying prior distributions and likelihoods is an ongoing challenge. Prior distributions, in particular, should be treated as deeply uncertain when appropriate. One key advantage of the methods described in this chapter is that they have the potential for increased transparency. When it is not possible to conduct a sensitivity analysis on a number of critical priors due to limited computational budgets, fully specifying and providing a justification for the utilized distributions allows other researchers to identify key assumptions and build on existing work. The same is true for the specification of likelihoods---while likelihood-free methods avoid the need to specify a likelihood function, they require other assumptions or choices, which should be described and justified as transparently as possible.
+
+We conclude this appendix with some key recommendations:
+1. UQ analysis does not require full confidence in priors and likelihoods. Rather, UQ should be treated as part of an exploratory modeling workflow, where hypotheses related to model structures, prior distributions, and likelihoods can be tested.
+2. For complex multisectoral models, UQ will typically require the use of a reduced set of parameters, either through emulation or by fixing the others to their best-fit values. These parameters should be selected through a careful sensitivity analysis.
+3. Avoid the use of supposedly “non-informative” priors, such as uniform priors, whenever possible. In the absence of strong information about parameter values, the use of weakly informative priors, such as diffuse normals, is preferable.
+4. Be cognizant of the limitations of conclusions that can be drawn by using each method. The bootstrap, for example, may result in overconfidence if the dataset is limited and is not truly representative of the underlying stochastic process.
+5. When using MCMC, Markov chains can not be shown to have converged to the target distribution, but rather evidence can be collected to demonstrate that it is likely that they have.
+6. Conduct predictive checks based on the assumptions underlying the choices made in the analysis, and iteratively update those choices if the assumptions prove to be ill-suited for the problem at hand.
diff --git a/docs/html/_sources/A1_Uncertainty_Quantification.rst b/docs/html/_sources/A1_Uncertainty_Quantification.rst
new file mode 100644
index 0000000..68718d4
--- /dev/null
+++ b/docs/html/_sources/A1_Uncertainty_Quantification.rst
@@ -0,0 +1,18 @@
+.. raw:: latex
+
+    \appendix
+
+.. _A1_uncertainty_quantification:
+
+**************************
+Uncertainty Quantification
+**************************
+
+.. include:: A1.1_UQ_Introduction.rst
+.. include:: A1.2_Parametric_Bootstrap.rst
+.. include:: A1.3_Precalibration.rst
+.. include:: A1.4_Markov_Chain_Monte_Carlo.rst
+.. include:: A1.5_Other_methods.rst
+.. include:: A1.6_Critical_first_step.rst
+.. include:: A1.7_Critical_final_step.rst
+.. include:: A1.8_Key_take_home.rst
diff --git a/docs/html/_sources/A2.1_fishgame.rst b/docs/html/_sources/A2.1_fishgame.rst
new file mode 100644
index 0000000..783399b
--- /dev/null
+++ b/docs/html/_sources/A2.1_fishgame.rst
@@ -0,0 +1,416 @@
+Fishery Dynamics Tutorial
+*************************
+
+.. note::
+    
+    | Run the tutorial interactively:  `Fishery Dynamics Notebook <nb_fishery_dynamics_>`_.
+    | Please be aware that notebooks can take a couple minutes to launch.
+    | To run the notebooks yourself, download the files `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks>`_ and use these `requirements <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt>`_.
+
+
+Tutorial: Sensitivity Analysis (SA) to discover factors shaping consequential dynamics
+======================================================================================
+
+This notebook demonstrates the application of sensitivity analysis to
+discover factors that shape the behavior modes of a socio-ecological
+system with dynamic human action.
+
+The system of differential equations below represent a system of prey
+(defined in the equation below as x) and predator (defined as y) fish,
+with a human actor harvesting the prey fish. You can read more about
+this system at `Hadjimichael et
+al. (2020) <https://doi.org/10.1155/2020/4170453>`__.
+
+
+.. image:: _static/eqn2.png
+
+The table below defines the parameters in the system and also denotes
+the baseline and ranges associated with each uncertain parameter.
+
+.. image:: _static/table1.png
+
+The system is simple but very rich in the dynamic behaviors it exhibits.
+This complexity is accompanied by the presence of several equilibria
+that come in and out of existence with different parameter values. The
+equilibria also change in their stability according to different
+parameter values, giving rise to different behavior modes as shown by
+the diverse predator and prey abundace trajectories in the figure below.
+
+.. image:: _static/Figure_1.png
+
+In the unharvested system (without the human actor) the stability of
+several of these equilibria can be derived analytically. The task
+becomes significantly more difficult when the adaptive human actor is
+introduced, deciding to harvest the system at different rates according
+to their objectives and preferences.
+
+Sensitivity analysis methods can help us identify the factors that most
+control these dynamics by exploring the space of parameter values and
+seeing how system outputs change as a result.
+
+Through previously conducted optimization, there already exists a set of
+potential harvesting strategies that were identified in pursuit of five
+objectives:
+
+-  Maximize Harvesting Discounted Profits (Net Present Value)
+-  Minimize Prey Population Deficit
+-  Minimize Longest Duration of Consecutive Low Harvest
+-  Maximize Worst Harvest Instance
+-  Minimize Harvest Variance
+
+The identified harvesting strategies also meet the necessary constraint
+of not causing inadvertent predator collapse.
+
+We will be examining the effects of parametric uncertainty on these
+identified strategies, particularly focusing on two strategies: one
+selected to maximize harvesting profits and one identified through
+previous analysis to perform ‘well enough’ for all objectives across a
+wide range of states of the world (referred to as the ‘robust’
+harvesting policy).
+
+Let’s get started!
+------------------
+
+In this tutorial, we will be loading in data that has been produced in
+Hadjimichael et al. (2020). Before we start our analysis, we’ll load the
+relevant Python libraries. **NOTE**: To step through the notebook,
+execute each gray (code) box by typing “Shift+Enter”.
+
+.. code:: ipython3
+
+    #Import necessary libraries
+    
+    import msdbook
+    import numpy as np
+    import matplotlib.pyplot as plt
+    from SALib.sample import saltelli
+    from SALib.analyze import sobol
+    from matplotlib import patheffects as pe
+    
+    # load example data
+    msdbook.install_package_data()
+    
+    %matplotlib inline
+    %config InlineBackend.print_figure_kwargs = {'bbox_inches':None}
+
+.. parsed-literal::
+
+    Downloading example data for msdbook version 0.1.5...
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/uncertain_params_bounds.txt
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_metric_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_vary_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_mth_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/solutions.resultfile
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/3704614_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/LHsamples_original_1000.txt
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/3704614_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/param_values.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_yr_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_mth_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7000550_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/collapse_days.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/hymod_params_256samples.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_vary_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7000550_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7200799_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_yr_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7200799_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/LeafCatch.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/hymod_simulations_256samples.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/Robustness.txt
+
+Step 1: Load identified solutions and explore performance
+---------------------------------------------------------
+
+Here we load in the solution set obtained in Hadjimichael et al. (2020).
+The solution set contains the decision variables and objectives
+associated with a variety of harvesting policies. For this tutorial, we
+focus on comparing two policies: harvesting profits and one that
+performs robustly across all objectives. Below, we are reading in the
+decision variables and objectives from an external file that can be
+found within the msdbook package data.
+
+.. code:: ipython3
+
+    robustness = msdbook.load_robustness_data()
+    results = msdbook.load_profit_maximization_data()
+    
+    robust_solution = np.argmax(robustness[:,-1]) #pick robust solution
+    profit_solution = np.argmin(results[:,6]) #pick profitable solution
+    objective_performance = -results[:,6:] #Retain objective values 
+    
+    # Get decision variables for each of the policies
+    highprofitpolicy = results[profit_solution,0:6]
+    mostrobustpolicy = results[robust_solution,0:6]
+
+
+Next we plot the identified solutions with regards to their objective
+performance in a parallel axis plot
+
+.. tip:: View the source code used to create this plot here: `plot_objective_performance <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-objective-performance>`_
+
+.. code:: ipython3
+
+    ax, ax1 = msdbook.plot_objective_performance(objective_performance, profit_solution, robust_solution)
+
+
+.. image:: _static/fishery_output_6_0.png
+
+
+The solution set from the optimization in Hadjimichael et al. (2020) are
+presented in a parallel axis plot where each of the five objectives (and
+one constraint) are represented as an axis. Each solution on the Pareto
+front is represented as a line where the color of the line indicates the
+value of the NPV objective. The preference for objective values is in
+the upward direction. Therefore, the ideal solution would be a line
+straight across the top of the plot that satisfies every objective.
+However, no such line exists because there are tradeoffs when sets of
+objectives are prioritized over the others. When lines cross in between
+axes, this indicates a tradeoff between objectives (as seen in the first
+two axes).The solution that is most robust in the NPV objective has the
+highest value on the first axis and is outlined in dark gold. The
+solution that is most robust across all objectives is outlined in a
+brighter yellow. A parallel axis is an effective visual to characterize
+high-dimensional tradeoffs in the system and visualize differences in
+performance across policies.
+
+Step 2: Use SALib to generate a sample for a Sobol sensitivity analysis
+-----------------------------------------------------------------------
+
+In Step 1, we showed how the optimized harvesting policies performed in
+the objective space, which utilized the baseline parameters outlined in
+the table above. Now, we are interested in understanding how sensitive
+our two policies are to alternative states of the world that may be
+characterized by different parameter values. To do so, we first need to
+define the problem dictionary that allows us to generate these
+alternative states of the world.
+
+.. code:: ipython3
+
+    # Set up SALib problem
+    problem = {
+      'num_vars': 9,
+      'names': ['a', 'b', 'c', 'd', 'h', 'K', 'm', 'sigmaX', 'sigmaY'],
+      'bounds': [[0.002, 2], [0.005, 1], [0.2, 1], [0.05, 0.2], [0.001, 1],
+                 [100, 5000], [0.1, 1.5], [0.001, 0.01], [0.001, 0.01]]
+    }
+
+
+Then we use the following command to generate a Saltelli sample from
+these defined ranges:
+
+.. code:: python
+
+   param_values = saltelli.sample(problem, 1024, calc_second_order=False)
+
+Generally, it is a good idea to save the result of the sample since it
+is often reused and regenerating it produces a different sample set. For
+this reason, we will load one from file that was previously generated.
+
+.. code:: ipython3
+
+    # load previously generated Saltelli sample from our msdbook package data
+    param_values = msdbook.load_saltelli_param_values()
+
+
+Step 3: Evaluate the system over all generated states of the world
+------------------------------------------------------------------
+
+Now we re-evaluate how well the policies do in the new states of the
+world. In order to characterize failure of a policy, we identify the
+states where the predator population collapses, as an inadvertent
+consequence of applying the harvesting strategy under a state of the
+world different from the one originally assumed. Due to how long this
+step takes to execute within the tutorial, we will read in the solutions
+from an external file. However, the block of code below shows how
+evaluation can be implemented.
+
+.. code:: python
+
+   # create array to store collapse values under both policies
+   collapse_days = np.zeros([len(param_values), 2])
+
+   # evaluate performance under every state
+   for i in range(len(param_values)):
+       
+       additional_inputs = np.append(['Previous_Prey'],
+                                     [param_values[i,0],
+                                      param_values[i,1],
+                                      param_values[i,2],
+                                      param_values[i,3],
+                                      param_values[i,4],
+                                      param_values[i,5],
+                                      param_values[i,6],
+                                      param_values[i,7],
+                                      param_values[i,8]])
+       
+       collapse_days[i,0]=fish_game(highprofitpolicy, additional_inputs)[1][0]
+       collapse_days[i,1]=fish_game(mostrobustpolicy, additional_inputs)[1][0]
+
+.. code:: ipython3
+
+    # load the simulation data from our msdbook package data
+    collapse_days = msdbook.load_collapse_data()
+
+
+Step 4: Calculate sensitivity indices
+-------------------------------------
+
+Now we use a Sobol sensitivity analysis to calculate first-order,
+second-order, and total-order sensitivity indices for each parameter and
+for each of the two policies. These indicies help determine which
+factors explain the most variability in the number of days of predator
+population collapse.
+
+.. code:: ipython3
+
+    #Perform the Sobol SA for the profit-maximizing solution
+    Si_profit = sobol.analyze(problem, collapse_days[:, 0], 
+                              calc_second_order=False, 
+                              conf_level=0.95, 
+                              print_to_console=True)
+
+
+.. code:: ipython3
+
+    #Perform the Sobol SA for the robust solution
+    Si_robustness = sobol.analyze(problem, 
+                                  collapse_days[:, 1], 
+                                  calc_second_order=False, 
+                                  conf_level=0.95, 
+                                  print_to_console=True)
+
+
+.. parsed-literal::
+
+                  ST   ST_conf
+    a       0.226402  0.036146
+    b       0.066819  0.013347
+    c       0.004395  0.004023
+    d       0.024509  0.006993
+    h       0.009765  0.005488
+    K       0.020625  0.009494
+    m       0.897971  0.066470
+    sigmaX  0.000136  0.000149
+    sigmaY  0.000739  0.001040
+                  S1   S1_conf
+    a       0.087936  0.044236
+    b       0.000554  0.021474
+    c      -0.002970  0.004590
+    d       0.001206  0.015881
+    h       0.004554  0.007998
+    K       0.003843  0.012661
+    m       0.751301  0.071862
+    sigmaX -0.000325  0.001245
+    sigmaY -0.001887  0.002768
+
+
+Looking at the total-order indices, (ST) factors :math:`m`, :math:`a`,
+:math:`b`, :math:`d` and :math:`K` explain a non-negligible amount of
+variance therefore have an effect on the stability of this system.
+Looking at the first-order indices (S1), we also see that besides
+factors :math:`m` and :math:`a`, all other factors are important in this
+system through their interactions, which make up the difference between
+their S1 and ST indices. This shows the danger of limiting sensitivity
+analyses to first order effects, as factor importance might be
+significantly misjudged.
+
+These findings are supported by the analytical condition of equilibrium
+stability in this system:
+
+.. image:: _static/eqn4.png
+
+In an unharvested system, this condition is both necessary and
+sufficient for the equilibrium of the two species coexisting to be
+stable.
+
+When adaptive human action is introduced however, this condition is
+still necessary, but no longer sufficient, as harvesting reduces the
+numbers of prey fish and as a result reduces the resources for the
+predator fish. Since this harvesting value is not constant, but can
+dynamically adapt according to the harvester’s objectives, it cannot be
+introduced into this simple equation.
+
+Step 5: Explore relationship between uncertain factors and performance
+----------------------------------------------------------------------
+
+In the following steps, we will use the results of our sensitivity
+analysis to investigate the relationships between parametric
+uncertainty, equilibrium stability and the performance of the two
+policies.
+
+We can use the top three factors identified (:math:`m`, :math:`a`, and
+:math:`b`) to visualize the performance of our policies in this
+three-dimensional parametric space.
+
+We first define the stability condition, as a function of :math:`b` and
+:math:`m`, and calculate the corresponding values of :math:`a`.
+
+.. code:: ipython3
+
+    def inequality(b, m, h, K):
+        return ((b**m)/(h*K)**(1-m))
+    
+    # boundary interval that separates successful and failed states of the world
+    b = np.linspace(start=0.005, stop=1, num=1000)
+    m = np.linspace(start=0.1, stop=1.5, num=1000)
+    h = np.linspace(start=0.001, stop=1, num=1000)
+    K = np.linspace(start=100, stop=2000, num=1000)
+    b, m = np.meshgrid(b, m)
+    a = inequality(b, m, h, K)
+    a = a.clip(0,2)
+
+
+.. tip:: View the source code used to create this plot here: `plot_factor_performance <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-factor-performance>`_
+
+.. code:: ipython3
+
+    # generate plot
+    ax1, ax2 = msdbook.plot_factor_performance(param_values, collapse_days, b, m, a)
+
+    
+.. image:: _static/fishery_output_22_0.png
+
+
+These figures show the combinations of factors that lead to success or
+failure in different states of the world for the profit-maximizing and
+robust policies. Each point is a state of the world, characterized by
+specific values of the parameters, and ideally, we would like the color
+of the point to be blue, to represent that there are a low number of
+days with a predator collapse in that world. The gray curve denotes the
+highly non-linear nature of the boundary, defined by the stability
+condition, that separates successful and failed states of the world. The
+figures demonstrate the following key points:
+
+First, as asserted above, the policies interact with the system in
+different and complex ways. In the presence of human action, the
+stability condition is not sufficient in determining whether the policy
+will succeed, even though it clearly shapes the system in a fundamental
+manner.
+
+Secondly, the robust policy manages to avoid collapse in many more of the sampled states of the world, indicated by the number of blue points. The robust policy avoids collapse in 31% of worlds versus 14% in the profit-maximizing policy.This presents a clear tradeoff between profit-maximizing performance androbustness against uncertainty.
+
+Tips to Apply Sobol SA and Scenario Discovery to your Problem
+-------------------------------------------------------------
+
+In this tutorial, we demonstrated a Sobol SA to identify the most
+important factors driving the behavior of a system (i.e. the number of
+the collapse days). In order to apply this methodology to your problem,
+you will need to have a set of optimized policies for your system that
+you are interested in analyzing. The general workflow is as follows:
+
+1. Choose sampling bounds for your parameters and set up the problem
+   dictionary as in Step 2 above.
+2. Generate samples, or alternative states of the world using the
+   ``saltelli.sample`` function.
+3. Evaluate your policies on the alternative states of the world. For
+   your application, you will also need to develop a rule for
+   determining success or failure of your policy in a new SOW. In this
+   tutorial, success was denoted by a small number of collapse days.
+   Ultimately, the rule will be specific to your application and can
+   include various satisficing criteria.
+4. Calculate the Sobol indices and discover the most important
+   parameters driving success and failure.
+5. Finally, use a similar plotting procedure as in step 5 to identify
+   the combination of parameter values that lead to success and failure
+   in the system.
diff --git a/docs/html/_sources/A2.2_saltelli.rst b/docs/html/_sources/A2.2_saltelli.rst
new file mode 100644
index 0000000..409681a
--- /dev/null
+++ b/docs/html/_sources/A2.2_saltelli.rst
@@ -0,0 +1,302 @@
+Sobol SA Tutorial
+*************************
+
+.. note::
+    
+    | Run the tutorial interactively:  `Sobol SA Tutorial <nb_saltelli_sobol_>`_.
+    | Please be aware that notebooks can take a couple minutes to launch.
+    | To run the notebooks yourself, download the files `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks>`_ and use these `requirements <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt>`_.
+
+Tutorial: Sensitivity Analysis (SA) using the Saltelli sampling scheme with Sobol SA
+====================================================================================
+
+In this tutorial, we will set up a workflow to investigate how sensitive
+the output of a function is to its inputs. Why might you want to do
+this? Imagine that this function represents a complex system, such as
+the rainfall-runoff process of a watershed model, and that you, the
+researcher, want to investigate how your choice of input parameter
+values are affecting the model’s characterization of runoff in the
+watershed. Your parameter values are likely uncertain and can take on
+any value in a pre-defined range. Using a Sobol SA will allow you to
+sample different values of your parameters and calculate how sensitive
+your output of interest is to certain parameters. Below, we demonstrate
+Sobol SA for a simple function to illustrate the method, but the
+workflow can be applied to your own problem of interest!
+
+In order to conduct this analysis, we will use the popular Python
+Sensitivity Analysis Library
+(`SALib <https://salib.readthedocs.io/en/latest/index.html>`__) to:
+
+#. Generate a problem set as a dictionary for our Ishigami function that has three inputs
+#. Generate 2048 samples for our problem set using the Saltelli [1]_ [2]_ sampling scheme
+#. Execute the Ishigami function for each of our samples and gather the outputs
+#. Compute the sensitivity analysis to generate first-order and total-order sensitivity indices using the Sobol [3]_ method
+#. Interpret the meaning of our results
+
+Let’s get started!
+------------------
+
+**NOTE**: Content from this tutorial is taken directly from the SALib
+`“Basics” <https://salib.readthedocs.io/en/latest/basics.html>`__
+walkthrough. To step through the notebook, execute each gray (code) box
+by typing “Shift+Enter”.
+
+.. code:: ipython3
+
+    #Import relevant libraries
+    import numpy as np
+    import matplotlib.pyplot as plt 
+    from mpl_toolkits import mplot3d
+    
+    from SALib.sample import saltelli
+    from SALib.analyze import sobol
+    from SALib.test_functions import Ishigami
+
+Step 1: Generate the problem dictionary
+---------------------------------------
+
+The Ishigami function is of the form:
+
+.. math:: f(x_1,x_2,x_3) = sin(x_1)+asin^2(x_2)+bx_3^4sin(x_1)
+
+The function has three inputs, 𝑥1, 𝑥2, 𝑥3 where 𝑥𝑖 ∈ [−𝜋, 𝜋]. The
+constants :math:`a` and :math:`b` are defined as 7.0 and 0.1
+respectively.
+
+.. code:: ipython3
+
+    #Create a problem dictionary. Here we supply the number of variables, the names of each variable, and the bounds of the variables.
+    problem = {
+        'num_vars': 3,
+        'names': ['x1', 'x2', 'x3'],
+        'bounds': [[-3.14159265359, 3.14159265359],
+                   [-3.14159265359, 3.14159265359],
+                   [-3.14159265359, 3.14159265359]]
+    }
+
+Step 2: Generate samples using the Saltelli sampling scheme
+-----------------------------------------------------------
+
+Sobol SA requires the use of the Saltelli sampling scheme. The output of
+the ``saltelli.sample`` function is a NumPy array that is of shape 2048
+by 3. The sampler generates 𝑁∗(2𝐷+2) samples, where in this example, N
+is 256 (the argument we supplied) and D is 3 (the number of model
+inputs), yielding 2048 samples. The keyword argument
+``calc_second_order=False`` will exclude second-order indices, resulting
+in a smaller sample matrix with 𝑁∗(𝐷+2) rows instead. Below, we plot the
+resulting Saltelli sample.
+
+.. code:: ipython3
+
+    #Generate parmeter values using the saltelli.sample function
+    param_values = saltelli.sample(problem, 256)
+    
+    print(f"`param_values` shape:  {param_values.shape}")
+
+
+
+.. parsed-literal::
+
+    `param_values` shape:  (2048, 3)
+
+
+.. code:: ipython3
+
+    #Plot the 2048 samples of the parameters 
+    
+    fig = plt.figure(figsize = (7, 5))
+    ax = plt.axes(projection ="3d")
+    ax.scatter3D(param_values[:,0], param_values[:,1], param_values[:,2])
+    ax.set_xlabel('X1 Parameter')
+    ax.set_ylabel('X2 Parameter')
+    ax.set_zlabel('X3 Parameter')
+    plt.title("Saltelli Sample of Parameter Values")
+    
+    plt.show()
+
+
+
+.. image:: _static/output_7_0.png
+
+
+Step 3: Execute the Ishigami function over our sample set
+---------------------------------------------------------
+
+SALib provides a nice wrapper to the Ishigami function that allows the
+user to directly pass the ``param_values`` array we just generated into
+the function directly.
+
+.. code:: ipython3
+
+    Y = Ishigami.evaluate(param_values)
+
+Step 4: Compute first-, second-, and total-order sensitivity indices using the Sobol method
+-------------------------------------------------------------------------------------------
+
+The ``sobol.analyze`` function will use our problem dictionary and the
+result of the Ishigami runs (``Y``) to compute first-, second-, and
+total-order indicies.
+
+.. code:: ipython3
+
+    Si = sobol.analyze(problem, Y)
+
+``Si`` is a Python dict with the keys “S1”, “S2”, “ST”, “S1_conf”,
+“S2_conf”, and “ST_conf”. The ``_conf`` keys store the corresponding
+confidence intervals, typically with a confidence level of 95%. Use the
+keyword argument ``print_to_console=True`` to print all indices. Or, we
+can print the individual values from ``Si`` as shown in the next step.
+
+Step 5: Interpret our results
+-----------------------------
+
+We execute the following code and take a look at our first-order indices
+(``S1``) for each of our three inputs. These indicies can be interpreted
+as the fraction of variance in the output that is explained by each
+input individually.
+
+.. code:: ipython3
+
+    first_order = Si['S1']
+    
+    print('First-order:')
+    print(f"x1: {first_order[0]}, x2: {first_order[1]}, x3: {first_order[2]}")
+
+
+.. parsed-literal::
+
+    First-order:
+    x1: 0.3184242969763115, x2: 0.4303808201623416, x3: 0.022687722804980225
+
+
+If we were to rank the importance of the inputs in how much they
+individually explain the variance in the output, we would rank them from
+greatest to least importance as follows: 𝑥2, 𝑥1 and then 𝑥3. Since 𝑥3
+only explains 2% of the output variance, it does not explain output
+variability meaningfully. Thus, this indicates that there is
+contribution to the output variance by 𝑥2 and 𝑥1 independently, whereas
+𝑥3 does not contribute to the output variance. Determining what inputs
+are most important or what index value is meaningful is a common
+question, but one for which there is no general rule or threshold. This
+question is problem and context-dependent, but procedures have been
+identified to rank order influential inputs and which can be used to
+identify the least influential factors. These factors can be fixed to
+simplify the model [4]_ [5]_ [6]_.
+
+Next, we evaluate the total-order indices, which measure the
+contribution to the output variance caused by varying the model input,
+including both its first-order effects (the input varying alone) and all
+higher-order interactions across the input parameters.
+
+.. code:: ipython3
+
+    total_order = Si['ST']
+    
+    print('Total-order:')
+    print(f"x1: {total_order[0]}, x2: {total_order[1]}, x3: {total_order[2]}")
+
+
+.. parsed-literal::
+
+    Total-order:
+    x1: 0.5184119098161343, x2: 0.41021260250026054, x3: 0.2299058431439953
+
+
+The magnitude of the total order indices are substantially larger than
+the first-order indices, which reveals that higher-order interactions
+are occurring, i.e. that the interactions across inputs are also
+explaining some of the total variance in the output. Note that 𝑥3 has
+non-negligible total-order indices, which indicates that it is not a
+consequential parameter when considered in isolation, but becomes
+consequential and explains 25% of variance in the output through its
+interactions with 𝑥1 and 𝑥2.
+
+Finally, we can investigate these higher order interactions by viewing
+the second-order indices. The second-order indicies measure the
+contribution to the output variance caused by the interaction between
+any two model inputs. Some computing error can appear in these
+sensitivity indices, such as negative values. Typically, these computing
+errors shrink as the number of samples increases.
+
+.. code:: ipython3
+
+    second_order = Si['S2']
+    
+    print("Second-order:")
+    print(f"x1-x2:  {second_order[0,1]}")
+    print(f"x1-x3:  {second_order[0,2]}")
+    print(f"x2-x3:  {second_order[1,2]}")
+
+
+
+.. parsed-literal::
+
+    Second-order:
+    x1-x2:  -0.043237389723234154
+    x1-x3:  0.17506452088709862
+    x2-x3:  -0.03430682392607577
+
+
+We can see that there are strong interactions between 𝑥1 and 𝑥3. Note
+that in the Ishigami function, these two variables are multiplied in the
+last term of the function, which leads to interactive effects. If we
+were considering first order indices alone, we would erroneously assume
+that 𝑥3 explains no variance in the output, but the second-order and
+total order indices reveal that this is not the case. It’s easy to
+understand where we might see interactive effects in the case of the
+simple Ishigami function. However, it’s important to remember that in
+more complex systems, there may be many higher-order interactions that
+are not apparent, but could be extremely consequential in explaining the
+variance of the output.
+
+Tips to Apply Sobol SA to Your Own Problem
+------------------------------------------
+
+In this tutorial, we demonstrated how to apply an SA analysis to a
+simple mathematical test function. In order to apply a Sobol SA to your
+own problem, you will follow the same general workflow that we defined
+above. You will need to:
+
+1. Choose sampling bounds for your parameters and set up the problem
+   dictionary as in Step 1 above.
+2. Generate samples using the ``saltelli.sample`` function. This step is
+   problem-dependent and note that the Sobol method can be
+   computationally intensive depending on the model being analyzed. For
+   example, for a simple rainfall-runoff model such as HYMOD, it has
+   been recommended to run a sample size of at least N = 10,000 (which
+   translates to 60,000 model runs). More complex models will be slower
+   to run and will also require more samples to calculate accurate
+   estimates of Sobol indices. Once you complete this process, pay
+   attention to the confidence bounds on your sensitivity indices to see
+   whether you need to run more samples.
+3. Run the parameter sets through your model. In the example above, the
+   Ishigami function could be evaluated through SALib since it is a
+   built in function. For your application, you will need to run these
+   parameter sets through the problem externally and save the output.
+   The output file should contain one row of output values for each
+   model run.
+4. Calculate the Sobol indices. Now, the Y will be a numpy array with
+   your external model output and you will need to include the parameter
+   samples as an additional argument.
+5. Finally, we interpet the results. If the confidence intervals of your
+   dominant indices are larger than roughly 10% of the value itself, you
+   may want to consider increasing your sample size as computation
+   permits. You should additionally read the references noted in Step 5
+   above to understand more about identifying important factors.
+
+
+.. only:: html
+    
+    **References**
+
+.. [1] Saltelli, A. (2002). "Making best use of model evaluations to compute sensitivity indices." Computer Physics Communications, 145(2):280-297, doi:`10.1016/S0010-4655(02)00280-1 <https://doi.org/10.1016/S0010-4655(02)00280-1>`_.
+
+.. [2] Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and S. Tarantola (2010). "Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index." Computer Physics Communications, 181(2):259-270, doi:`10.1016/j.cpc.2009.09.018 <https://doi.org/10.1016/j.cpc.2009.09.018>`_.
+
+.. [3] Sobol, I. M. (2001). "Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates." Mathematics and Computers in Simulation, 55(1-3):271-280, doi:`10.1016/S0378-4754(00)00270-6 <https://doi.org/10.1016/S0378-4754(00)00270-6>`_.
+
+.. [4] T.\  H.\  Andres, "Sampling methods and sensitivity analysis for large parameter sets," Journal of Statistical Computation and Simulation, vol. 57, no. 1–4, pp. 77–110, Apr. 1997, doi:`10.1080/00949659708811804 <https://doi.org/10.1080/00949659708811804>`_.
+
+.. [5] Y.\  Tang, P.\  Reed, T.\  Wagener, and K.\  van Werkhoven, "Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation," Hydrology and Earth System Sciences, vol. 11, no. 2, pp. 793–817, Feb. 2007, doi:`10.5194/hess-11-793-2007 <https://doi.org/10.5194/hess-11-793-2007>`_.
+
+.. [6] J.\  Nossent, P.\  Elsen, and W.\  Bauwens, "Sobol' sensitivity analysis of a complex environmental model," Environmental Modelling & Software, vol. 26, no. 12, pp. 1515–1525, Dec. 2011, doi:`10.1016/j.envsoft.2011.08.010 <https://doi.org/10.1016/j.envsoft.2011.08.010>`_.
diff --git a/docs/html/_sources/A2.3_logistic.rst b/docs/html/_sources/A2.3_logistic.rst
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+Logistic Regression Tutorial
+****************************
+
+.. note::
+
+    | Run the tutorial interactively:  `Logistic Regression Tutorial <nb_logistic_regression_>`_.
+    | Please be aware that notebooks can take a couple minutes to launch.
+    | To run the notebooks yourself, download the files `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks>`_ and use these `requirements <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt>`_.
+
+Tutorial: Logistic Regression for Factor Mapping
+================================================
+
+This tutorial replicates a scenario discovery analysis performed in
+`Hadjimichael et
+al. (2020) <https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020EF001503>`__.
+
+Background
+----------
+
+Planners in the the Upper Colorado River Basin (UCRB, shown in the
+figure below) are seeking to understand the vulnerability of water users
+to uncertainties stemming from climate change, population growth and
+water policy changes. The UCRB spans 25,682 km2 in western Colorado and
+is home to approximately 300,000 residents and 1,012 km2 of irrigated
+land. Several thousand irrigation ditches divert water from the main
+river and its tributaties for irrigation (shown as small black dots in
+the figure). Transmountain diversions of approximately 567,400,000 m3
+per year are exported for irrigation, industrial and municipal uses in
+northern and eastern Colorado, serving the major population centers of
+Denver and Colorado Springs. These diversions are carried through
+tunnels, shown as large black dots in the figure.
+
+.. image:: _static/basin_map.png
+
+An important planning consideration is the water rights of each user,
+defined by seniority across all water uses (irrigation diversions,
+transboundary diversions, power plants etc.) in the basin. To assess the
+vulnerability of users with varying degrees of water rights seniority,
+planners simulate the system across an ensemble of scenarios using the
+state of Colorado’s StateMod platform. The model simulates streamflow,
+diversions, instream demands, and reservoir operations.
+
+Hadjimichael et al. (2020) employ an exploratory analysis by simulating
+a large ensemble of plausible scenarios using StateMod and then
+identifying consequential decision-relevant combinations of uncertain
+factors, termed scenario discovery. Focusing on decision-relevant
+metrics (metrics that are important to the user, the scenario discovery
+is applied to the water shortages experienced by each individual user
+(i.e., not on a single basin-wide or sector-wide metric). For this
+training example, we’ll be performing scenario discovery for three
+different water users: two irrigation users and one municipal user.
+
+Let’s get started!
+------------------
+
+In this tutorial, we will be loading in data that has been produced in
+Hadjimichael et al. (2020). Before we start our analysis, we’ll load the
+relevant Python libraries, example data, and information for the three
+users.
+
+.. code:: ipython3
+
+    #import libraries 
+    import msdbook
+    import numpy as np
+    import pandas as pd
+    import matplotlib as mpl
+    import matplotlib.pyplot as plt
+    
+    # load example data from Hadjimichael et al. (2020)
+    msdbook.install_package_data()
+    
+    # Select the IDs for the three users that we will perform the analysis for
+    all_IDs = ['7000550','7200799','3704614'] 
+    usernames = ['Medium seniority irrigation',
+                 'Low seniority irrigation',
+                 'Transbasin municipal diversion']
+    nStructures = len(all_IDs)
+
+
+
+.. parsed-literal::
+
+    Downloading example data for msdbook version 0.1.5...
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/uncertain_params_bounds.txt
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_metric_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_vary_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_mth_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/solutions.resultfile
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/3704614_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/LHsamples_original_1000.txt
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/3704614_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/param_values.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_yr_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_mth_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7000550_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/collapse_days.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/hymod_params_256samples.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_vary_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7000550_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7200799_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_yr_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7200799_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/LeafCatch.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/hymod_simulations_256samples.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/Robustness.txt
+
+
+Step 1: Load Latin hypercube sample and set up problem
+------------------------------------------------------
+
+To examine regional vulnerability, we generate an ensemble of plausible
+future states of the world (SOWs) using Latin Hypercube Sampling. For
+this tutorial, we’ll load a file containing 1,000 samples across 14
+parameters. The sampled parameters encompass plausible changes to the
+future state of the basin, including changes to hydrology, water demands
+(irrigation, municipal & industry, transbasin), and institutional and
+environmental factors (environmental flows, reservoir storage, operation
+of the Shoshone Power Plant). These samples are taken from ranges
+identified in ``param_bounds``. Below we load in the 1000 samples, the
+range of values that the samples can take for each parameter, and the
+parameter names. More information on what each parameter constitutes can
+be found in Table 1 of Hadjimichael et al., 2020.
+
+.. code:: ipython3
+
+    #Identify the bounds for each of the 14 parameters 
+    param_bounds = msdbook.load_basin_param_bounds()
+    
+    #Load in the parameter samples 
+    LHsamples = msdbook.load_lhs_basin_sample()
+    
+    #Create an array of the parameter names
+    param_names=['Irrigation demand multiplier','Reservoir loss','Transbasin demand multiplier',
+                 'Municipal & industrial multiplier', 'Shoshone','Environmental flows',
+                 'Evaporation change','Mean dry flow','Dry flow variance',
+                 'Mean wet flow','Wet flow variance','Dry-dry probability',
+                 'Wet-wet probability', 'Snowmelt shift']
+
+
+Step 2: Define decision-relevant metrics for illustration
+---------------------------------------------------------
+
+Scenario discovery attempts to identify parametric regions that lead to
+‘success’ and ‘failure’. For this demonstration we’ll be defining
+‘success’ as states of the world where a shortage level doesn’t exceed
+its historical frequency.
+
+Step 3: Run the logistic regression
+-----------------------------------
+
+Logistic regression estimates the probability that a future SOW will be
+classified as a success or failure given a set of performance criteria.
+A logistic regression model is defined by:
+
+.. math:: ln \bigg (\frac{p_i}{1-p_i} \bigg ) = X^T_i \beta
+
+where :math:`p_i` is the probability the performance in the
+:math:`i^{th}` SOW will be classified as a success, :math:`X_i` is the
+vector of covariates describing the :math:`i^{th}` SOW, and
+:math:`\beta` is the vector of coefficients describing the relationship
+between the covariates and the response, which here will be estimated
+using maximum likelihood estimation.
+
+A logistic regression model was fit to the ensemble of SOWs using the
+performance criteria defined in step 2. Logistic regression modeling was
+conducted using the `Statsmodel
+Python <https://www.statsmodels.org/stable/index.html>`__ package. The
+data required for the full analysis is too large to include in this
+tutorial, but results can be found in the data file loaded below.
+
+The results files contain the occurence of different shortage frequency
+and magnitude combinations under the experiment, in increments of 10,
+between 0 and 100. These combinations (100 for each user) are
+alternative decision-relevant metrics that can be used for scenario
+discovery.
+
+.. code:: ipython3
+
+    # Set arrays for shortage frequencies and magnitudes
+    frequencies = np.arange(10, 110, 10)
+    magnitudes = np.arange(10, 110, 10)
+    realizations = 10
+    
+    # Load performance and pseudo r scores for each of the users
+    results = [msdbook.load_user_heatmap_array(all_IDs[i]) / 100 for i in range(len(all_IDs))]
+
+
+Step 4: Factor ranking
+----------------------
+
+To rank the importance of each uncertain factor, we utilize McFadden’s
+psuedo-R2, a measure that quantifies the improvement of the model when
+utilizing each given predictor as compared to prediction using the mean
+of the data set:
+
+.. math:: R^2_{McFadden}=1-\frac{ln \hat{L}(M_{full})}{ln \hat{L}(M_{intercept})}
+
+Here :math:`ln \hat{L}(M_{full})` is the log likelihood of the full
+model (including the predictor) and :math:`ln \hat{L}(M_{intercept})` is
+the log likelihood of the intercept model (which predicts the mean
+probability of success across all SOWs).
+
+Higher values of McFadden’s psuedo-R2 indicate higher factor importance
+(when the likelihood of the full model approaches one, the ratio of the
+likelihood of the full model compared to the intercept model will get
+very small).
+
+.. code:: ipython3
+
+    #Load the pseudo-R^2 scores
+    scores = [msdbook.load_user_pseudo_scores(all_IDs[i]) for i in range(len(all_IDs))]
+    
+    # Select indices of frequency and magnitudes that will be used for the visualization
+    freq = [1,0,0]
+    mag = [7,3,7]
+
+
+Step 5: Draw factor maps
+------------------------
+
+The McFadden’s psuedo-R2 scores files contain preliminary logistic
+regression results on parameter importance for each of these
+combinations. Using these psuedo-R2 scores we will identify the two most
+important factors for each metric which we’ll use to generate the final
+scenario discovery maps (note: there may be more than two important
+metrics for each user, but here we will demonstrate by mapping two).
+
+.. code:: ipython3
+
+    # setup figure
+    fig, axes = plt.subplots(3,1, figsize=(6,18), tight_layout=True)
+    fig.patch.set_facecolor('white')
+    
+    for i in range(len(axes.flat)):
+        
+        ax = axes.flat[i]
+        
+        allSOWsperformance = results[i]
+        all_pseudo_r_scores = scores[i]
+        
+        # construct dataframe
+        dta = pd.DataFrame(data=np.repeat(LHsamples, realizations, axis = 0), columns=param_names)
+        dta['Success'] = allSOWsperformance[freq[i],mag[i],:]
+        
+        pseudo_r_scores = all_pseudo_r_scores[str(frequencies[freq[i]])+'yrs_'+str(magnitudes[mag[i]])+'prc'].values
+        top_predictors = np.argsort(pseudo_r_scores)[::-1][:2] #Sort scores and pick top 2 predictors
+        
+        # define color map for dots representing SOWs in which the policy
+        # succeeds (light blue) and fails (dark red)
+        dot_cmap = mpl.colors.ListedColormap(np.array([[227,26,28],[166,206,227]])/255.0)
+        
+        # define color map for probability contours
+        contour_cmap = mpl.cm.get_cmap('RdBu')
+        
+        # define probability contours
+        contour_levels = np.arange(0.0, 1.05,0.1)
+        
+        # define base values of the predictors
+        SOW_values = np.array([1,1,1,1,0,0,1,1,1,1,1,0,0,0]) # default parameter values for base SOW
+        base = SOW_values[top_predictors]
+        ranges = param_bounds[top_predictors]
+        
+        # define grid of x (1st predictor), and y (2nd predictor) dimensions
+        # to plot contour map over
+        xgrid = np.arange(param_bounds[top_predictors[0]][0],
+                          param_bounds[top_predictors[0]][1], np.around((ranges[0][1]-ranges[0][0])/500,decimals=4))
+        ygrid = np.arange(param_bounds[top_predictors[1]][0],
+                          param_bounds[top_predictors[1]][1], np.around((ranges[1][1]-ranges[1][0])/500,decimals=4))
+        all_predictors = [ dta.columns.tolist()[i] for i in top_predictors]
+        dta['Interaction'] = dta[all_predictors[0]]*dta[all_predictors[1]]
+        
+        # logistic regression here
+        predictor_list = [all_predictors[i] for i in [0,1]]
+        result = msdbook.fit_logit(dta, predictor_list)
+        
+        # plot contour map
+        contourset = msdbook.plot_contour_map(ax, result, dta, contour_cmap,
+                                              dot_cmap, contour_levels, xgrid,
+                                              ygrid, all_predictors[0], all_predictors[1], base)
+        
+        ax.set_title(usernames[i])
+        
+    # set up colorbar
+    cbar_ax = fig.add_axes([0.98, 0.15, 0.05, 0.7])
+    cbar = fig.colorbar(contourset, cax=cbar_ax)
+    cbar_ax.set_ylabel('Probability of Success', fontsize=16)
+    cbar_ax.tick_params(axis='y', which='major', labelsize=12)
+
+
+
+.. parsed-literal::
+
+    /srv/conda/envs/notebook/lib/python3.7/site-packages/statsmodels/base/model.py:127: ValueWarning: unknown kwargs ['disp']
+      warnings.warn(msg, ValueWarning)
+
+
+.. parsed-literal::
+
+    Optimization terminated successfully.
+             Current function value: 0.378619
+             Iterations 8
+    Optimization terminated successfully.
+             Current function value: 0.397285
+             Iterations 8
+    Optimization terminated successfully.
+             Current function value: 0.377323
+             Iterations 8
+
+
+
+.. image:: _static/notebook_logistic_output_11_1.png
+
+
+The figure above demonstrates how different combinations of the
+uncertain factors lead to success or failure in different states of the
+world, which are denoted by the blue and red dots respectively. The
+probability of success and failure are further denoted by the contours
+in the figure. Several insights can be drawn from this figure.
+
+First, using metrics chosen to be decision-relevant (specific to each
+user) causes different factors to be identified as most important by
+this scenario-discovery exercise (the x- and y-axes for each of the
+subplots). In other words, depending on what the decision makers of this
+system want to prioritize they might choose to monitor different
+uncertain factors to track performance.
+
+Second, in the top panel, the two identified factors appear to also have
+an interactive effect on the metric used (shortages of a certain level
+and frequency in this example). In terms of scenario discovery, the
+Patient Rule Induction Method (PRIM) or Classification And Regression
+Trees (CART) would not be able to delineate this non-linear space and
+would therefore misclassify parameter combinations as ‘desirable’ when
+they were in fact undesirable, and vice versa.
+
+Lastly, logistic regression also produces contours of probability of
+success, i.e. different factor-value combinations are assigned different
+probabilities that a shortage level will be exceeded. This allows the
+decision makers to evaluate these insights while considering their risk
+aversion.
+
+Tips to Apply Scenario Discovery to Your Own Problem
+----------------------------------------------------
+
+In this tutorial, we demonstrated how to perform a scenario discovery
+analysis for three different users in the UCRB. The analysis allowed us
+to determine which parameters the users would be most affected by and to
+visualize how different ranges of these parameters lead to success and
+failure for different users. This framework can be applicable to any
+other application where it is of interest to characterize success and
+failure based on uncertain parameter ranges. In order to apply the same
+framework to your own problem:
+
+1. Choose sampling bounds for your parameters of interest, which will
+   represent uncertainties that characterize your system.
+2. Generate samples for these parameters (this can be done using the
+   ``saltelli.sample`` function or externally).
+3. Define what constitutes success and failure in your problem. In this
+   tutorial, success was defined based on not surpassing the historical
+   drought frequency. Choose a metric that is relevant to your problem
+   and decision-makers that might be involved. If your model involves an
+   optimization, you can also define metrics based on meeting certain
+   values of these objectives.
+4. Run the parameter sets through your model and calculate success and
+   failure based on your metrics and across different users if
+   applicable. This step will allow you to create the scatter plot part
+   of the final figure.
+5. If it is of interest, the contours on the figure can be created by
+   fitting the logistic regression model in a similiar manner as denoted
+   in Steps 3 and 5 of the tutorial.
+
+
diff --git a/docs/html/_sources/A2.4_hymod.rst b/docs/html/_sources/A2.4_hymod.rst
new file mode 100644
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+HYMOD Dynamics Tutorial
+*************************
+
+.. note::
+    
+    | Run the tutorial interactively:  `HYMOD Notebook <nb_hymod_>`_.
+    | Please be aware that notebooks can take a couple minutes to launch.
+    | To run the notebooks yourself, download the files `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks>`_ and use these `requirements <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt>`_.
+
+Tutorial: Sensitivity Analysis of the HYMOD Model
+=================================================
+
+The purpose of this tutorial is to demonstrate the global sensitivity
+analysis concepts and tools established in the Section 3.1 of the main
+text of this eBook. This demonstration will highlight the central role
+of design of experiments (Section 3.3), when implementing global
+sensitivity analysis tools described in Section 3.4.
+
+We’ll explore these tools and concepts using the HYdrological MODel
+(HYMOD), a rainfall-runoff model developed and used for river flow
+forecasting. HYMOD was chosen for demonstration because its purpose is
+to abstract highly complex and non-linear systems. The methods
+demonstrated in thistutorial can be applied to numerical models that
+simulate other complex non-linear systems.
+
+This tutorial will first introduce HYMOD and use it to simulate
+streamflows in a river basin. Next, we’ll employ sensitivity analysis
+concepets described in Section 3 of the main text to examine how values
+of HYMOD’s parameters impact streamflow predictions. We’ll then explore
+how the effects of these parameters may change over time using
+time-varying sensitivtiy analysis. Finally, we’ll demonstrate concepts
+presented in Chapter 7 through two ensemble-based methods of uncertainty
+quantification - Generalized Likelihood Uncertainty Estimation (GLUE)
+and Pre-Calibration.
+
+The tutorial includes the following steps:
+
+1. Introduction to HYMOD
+^^^^^^^^^^^^^^^^^^^^^^^^
+
+`1.1 - Introduction to a simple hydrologic model (HYMOD) <#hymod>`__
+
+`1.2 - Input Data <#inputs>`__
+
+`1.3 - Running a basic simulation <#baseline>`__
+
+`1.4 - Model outputs <#outputs>`__
+
+2. Global Sensitivity Analysis
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+`2.1 - Design of Experiments <#sensitivity>`__
+
+`2.2 - Sensitivity analysis for one output <#sametrics>`__
+
+`2.3 - Sensitivity analysis across multiple outputs <#diffperformance>`__
+
+`2.4 - Time-varying sensitivity analysis <#time>`__
+
+1 - Introduction to HYMOD
+=========================
+
+.. _hymod:
+
+1.1 Overview
+
+HYMOD is a hydrologic model (rainfall-runoff model) that simulates key
+hydrologic fluxes such as infiltration, streamflow and
+evapotranspiration. The model was originally developed and used for
+river flow forecasting, but it has also been been used to explore
+different sensitivity analysis (e.g., `Herman et al.,
+2013 <https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/wrcr.20124>`__),
+uncertainty quantification (e.g., `Smith et al.,
+2008 <https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2006WR005205>`__),
+and optimization (e.g., `Ye et al.,
+2014 <https://www.sciencedirect.com/science/article/pii/S0022169414006362?casa_token=IRqE19Hkfa8AAAAA:_fXOqfwpzxMpchvu8_0njCe0Ok9H29Gyw2F46l9PzG9UVODDTUg6wIOSiyp6uybGevNVnZ7N>`__)
+concepts.
+
+HYMOD accepts two inputs - daily precepitation and daily potential
+evapotranspiration (PET)- and generates predicitons of daily streamflow.
+HYMOD abstracts the highly non-linear process of runoff routing by
+dividing the flow into two components: quick flow, representing
+precipitation that quickly runs off the surface of the watershed into
+the stream, and slow flow, that moves through the soil and takes much
+longer to arrive at the stream.
+
+To generate streamflow predictions, HYMOD first models vertical
+processes within the watershed to determine how much water infiltrates
+and evaporates from the soil at a given time step. It then determines
+how much water should be partitioned into quick flow and slow flow
+processes. Within each process it abstracts residence time (the time it
+takes a unit volume of water to move through the watershed and into the
+stream) using a series of “reservoirs” each with a calibrated residence
+time.
+
+HYMOD’s representation of hydrologic processes are shown Figure 1 below
+and controlled by the following parameters:
+
+:math:`H_{uz}`: the maximum water storage capacity of the soil (mm)
+
+:math:`B_{exp}`: parameters describing the degree of spatial variability
+within the basin between 0 and Huz
+
+:math:`Alp`: Fraction of runoff contributing to quick flow
+
+:math:`K_q`: Quick flow residence time of linear infinite reservoir (the
+Kq values of all three linear reservoirs are the same)
+
+:math:`K_s`: Slow flow residence time of linear infinite reservoir
+
+
+.. image:: _static/hymod_schematic-DAVE.png
+
+Vertical Processes
+~~~~~~~~~~~~~~~~~~
+
+HYMOD models the fraction of water that is stored in the soil
+:math:`(F(XH_{uz}))` using the following relationship:
+
+.. math:: F(XH_{uz}) = 1 - (1 - \frac{XH_{uz}}{H_{uz}})^{B}
+
+where :math:`XH_{uz}` is the water storage capacity of the soil;
+:math:`H_{uz}` is the parameter describing basin maximum water
+storage capacity (mm); and :math:`B` is the parameter describing the
+degree of spatial variability within the basin.
+
+The portion of precipitation that exceeds the water storage capacity is
+treated as runoff.
+
+Horizontal Processes
+~~~~~~~~~~~~~~~~~~~~
+
+To route runoff to streamflow, the excess runoff from the vertical
+processes is split into quick flow and slow flow. The proportion of
+runoff partitioned into quick flow and slow flow is determined by a
+parameter :math:`Alp`, which ranges between 0 and 1. Quick flow is
+routed through :math:`N` identical quick flow tanks :math:`Q1, Q2... QN`
+(shown above as :math:`N=3`). The rate at which runoff moves through the
+quick flow system is described by the residence time of the quick frlow
+tanks, :math:`Kq` (day). Slow flow is routed through a parallel slow
+flow tank and the rate at which slow flow is routed is described by the
+slow flow residences time, :math:`Ks` (day).
+
+Citation: Wagener, T., Boyle, D. P., Lees, M. J., Wheater, H. S., Gupta,
+H. V., & Sorooshian, S. (2001). A framework for development and
+application of hydrological models. Hydrology and Earth System Sciences,
+5(1), 13-26.
+
+.. _inputs:
+
+1.2 Input data
+
+The HYMOD model only requires precipitation and potential
+evapotranspiration as inputs. For this example, we’ll run HYMOD using
+data from the Leaf River, a humid catchment located north of Collins
+Mississippi that has been widely used to explore HYMOD. The dataset also
+includes daily observed runoff that we later use to evaluate the
+performace of each sensitvity analysis sample set.
+
+In the following section of code, we’ll load the necessary python
+libraries and read in the input file. For this exercise we’ll only use
+the first eleven years of data. The first five rows of the input dataset
+are printed to show what they look like:
+
+.. code:: ipython3
+
+    import msdbook
+
+    import numpy as np
+    import pandas as pd
+    import seaborn as sns
+
+    from sklearn import metrics
+    from matplotlib import pyplot as plt
+
+    # load example data
+    msdbook.install_package_data()
+
+    # load the Leaf River HYMOD input file
+    leaf_data = msdbook.load_hymod_input_file()
+
+    # extract the first eleven years of data
+    leaf_data = leaf_data.iloc[0:4015].copy()
+
+    print('Leaf River Data structure:')
+
+    # There are only three columns in the file including precipitation, potential evapotranspiration and  streamflow
+    leaf_data.head()
+
+
+.. parsed-literal::
+
+    Downloading example data for msdbook version 0.1.5...
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/uncertain_params_bounds.txt
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_metric_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_vary_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_mth_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/solutions.resultfile
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/3704614_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/LHsamples_original_1000.txt
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/3704614_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/param_values.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_yr_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_mth_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7000550_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/collapse_days.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/hymod_params_256samples.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_vary_s1.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7000550_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7200799_heatmap.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/sa_by_yr_delta.npy
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/7200799_pseudo_r_scores.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/LeafCatch.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/hymod_simulations_256samples.csv
+    Unzipped: /srv/conda/envs/notebook/lib/python3.7/site-packages/msdbook/data/Robustness.txt
+    Leaf River Data structure:
+
+
+
+
+.. raw:: html
+
+    <div>
+    <style scoped>
+        .dataframe tbody tr th:only-of-type {
+            vertical-align: middle;
+        }
+
+        .dataframe tbody tr th {
+            vertical-align: top;
+        }
+
+        .dataframe thead th {
+            text-align: right;
+        }
+    </style>
+    <table border="1" class="dataframe">
+      <thead>
+        <tr style="text-align: right;">
+          <th></th>
+          <th>Precip</th>
+          <th>Pot_ET</th>
+          <th>Strmflw</th>
+        </tr>
+      </thead>
+      <tbody>
+        <tr>
+          <th>0</th>
+          <td>0.0</td>
+          <td>4.60</td>
+          <td>0.29</td>
+        </tr>
+        <tr>
+          <th>1</th>
+          <td>0.0</td>
+          <td>4.31</td>
+          <td>0.24</td>
+        </tr>
+        <tr>
+          <th>2</th>
+          <td>0.0</td>
+          <td>4.33</td>
+          <td>0.21</td>
+        </tr>
+        <tr>
+          <th>3</th>
+          <td>0.0</td>
+          <td>4.78</td>
+          <td>0.19</td>
+        </tr>
+        <tr>
+          <th>4</th>
+          <td>0.0</td>
+          <td>2.91</td>
+          <td>0.18</td>
+        </tr>
+      </tbody>
+    </table>
+    </div>
+
+
+
+To visualize catchment hydrology, streamflow and precipitation data are
+usually plotted together as a combined hydrograph (streamflow ) and
+hyetograph (rainfall, from Greek.hyetos, “rain”). Streamflow is plotted
+as a time series, while rainfall is shown as an inverted bar plot along
+the top of the graph. Streamflow labels are shown on the left y-axis,
+while rainfall labels are shown on the right y-axis.
+
+.. code:: ipython3
+
+    # make an axis for the hydrograph
+    fig, strmflw_ax = plt.subplots(figsize=[12,6])
+    strmflw_ax.set_ylim([0, 50])
+
+    #make a second y-axis for the hyetograph
+    precip_ax = strmflw_ax.twinx()
+    precip_ax.set_ylim([0, 200])
+    precip_ax.invert_yaxis()
+
+    precip = leaf_data['Precip']
+    strmflw_ax.plot(range(0, len(leaf_data['Precip'])), leaf_data['Strmflw'], color='lightcoral')
+    strmflw_ax.set_ylabel('Streamflow (mm/day)')
+
+    precip_ax.bar(range(0, len(leaf_data['Precip'])), leaf_data['Precip'], width=2)
+    precip_ax.set_ylabel('Rainfall (mm/day)')
+    precip_ax.legend(['Precip'], loc='center right')
+    strmflw_ax.legend(['Streamflow'],bbox_to_anchor=(1, 0.48))
+
+
+
+
+
+.. parsed-literal::
+
+    <matplotlib.legend.Legend at 0x7f53b95c6850>
+
+
+
+
+.. image:: _static/hymod1.png
+
+.. _baseline:
+
+1.3 Running a Baseline Model Simulation
+
+We’ll start our experiment by running HYMOD using its default
+parameters.
+
+.. code:: ipython3
+
+    # assign input parameters to generate a baseline simulated streamflow
+    Nq = 3  # Number of quickflow routing tanks
+    Kq = 0.5 # Quickflow routing tanks' rate parameter
+    Ks =  0.001 # Slowflow routing tank's rate parameter
+    Alp = 0.5 # Quick/slow split parameter
+    Huz = 100 # Maximum height of soil moisture accounting tank
+    B = 1.0 # Scaled distribution function shape parameter
+
+    # Note that the number of years is 11. One year of model warm-up and ten years are used for actual simulation
+    model = msdbook.hymod(Nq, Kq, Ks, Alp, Huz, B, leaf_data, ndays=4015)
+
+
+.. _outputs:
+
+1.4 Model Outputs
+
+Model outputs include actual evapotranspiration, quick and fast
+streamflow, and combined runoff. In this tutorial we focus on the total
+daily runoff, QQ (:math:`m^3/s`). We can use the following script to
+plot simulated streamflow against observed streamflow.
+
+.. tip:: View the source code used to create this plot here: `plot_observed_vs_simulated_streamflow <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-observed-vs-simulated-streamflow>`_
+
+.. code:: ipython3
+
+    ax = msdbook.plot_observed_vs_simulated_streamflow(df=leaf_data, hymod_dict=model)
+
+
+
+.. image:: _static/hymod2.png
+
+
+
+So how does our model perform? We can investigate model performance
+across several metrics:
+
+1: Mean Absolute Error (MAE); MAE conveys how the model performs on
+average across the 10 year simulation period, with smaller values
+indicating better performance. The absolute value is taken so that
+positive and negative errors do not cancel each other out.
+
+.. math:: MAE = \frac{1}{N}\sum_{t=0}^N\left\lvert Q_{sim,t}-Q_{obs,t}\right\rvert
+
+2: Root Mean Square Error (RMSE); RMSE is sum of square errors across
+the 10 year simulation period. RMSE is sensitive to large errors between
+the historical record and the simulated flows, and thus is useful for
+highlighting the model’s ability to capture of extreme flood events.
+
+.. math:: RMSE = \sqrt{\frac{1}{N}\sum_{t=1}^{N}(Q_{sim,t}-Q_{obs,t})^2}
+
+3: Log-Root Mean Square Error (Log(RMSE)) LOG(RMSE) focuses on model
+performance during low-flow events.
+
+.. math:: LOG(RMSE) = log(RMSE)
+
+.. code:: ipython3
+
+    mae = np.mean(abs(leaf_data['Strmflw'] - model['Q']))
+    mse = metrics.mean_squared_error(model['Q'], leaf_data['Strmflw'])
+    rmse = mse**(1/2)
+
+    print('MAE: ' + str(mae) + '\nRMSE: ' + str(mse) + '\nLOG(RMSE): ' + str(rmse))
+
+
+.. parsed-literal::
+
+    MAE: 1.0787471470460999
+    RMSE: 4.375695937555197
+    LOG(RMSE): 2.0918164206151544
+
+
+The error metrics show that HYMOD performs reasonably well, the MAE is
+around 1 :math:`m^3/s`, the RMSE is on the order of 10% of the largest
+observed streamflow and the LOG(RMSE) is fairly low.
+
+2- Global Sensitivity Analysis
+==============================
+
+.. _sensitivity:
+
+2.1 Experimental Design and Setup
+
+Now we’ll examine how sensitive streamflow simulations generated by
+HYMOD are to the model’s input parameters. We’ll perform global
+sensitivity analysis (see Section 3.1 of the main text) using the SALib
+Python library.
+
+.. code:: ipython3
+
+    from SALib.sample import saltelli
+    from SALib.analyze import sobol
+    from SALib.analyze import delta
+
+A first and critical step when conducting sensitivity analysis is
+determining the experimental design (see Design of Experiments, Section
+3.4 of the main text). Our experimental design involves defining the
+uncertainties that we’ll be examining, the output of interest, the
+ranges of each uncertainty that will be explored and the strategy for
+sampling the uncertainty space.
+
+For this experiment we’ll explore the five parameters highlighted in
+Figure 1. We’ll draw their ranges from existing literature on the model
+(note Jon H. paper). We’ll use a Sobol sampling an a quasi-random
+sampling with low sequences approach to sample the uncertainty space
+(Section 3.3.4).
+
+In this demonstration we’ll utilize Sobol Sensitivity Analysis, a
+variance based method (Section 3.4.5).
+
+To explore HYMOD’s behavoir, we’ll examine the sensitivity of four model
+ouputs to input parameters: 1) predicted flow, 2) Mean Absolute Error
+(compared with the calibaration data set), 3) Root Mean Square Error and
+4) Log Root Mean Square Error.
+
+This analysis will employ SALib, a Python implementation also utilized
+in the other SA tutorial (make this more formal).
+
+To start our analysis, we’ll create a dictionary that describes our
+model uncertainties and their ranges, this dictionary is named
+“problem_hymod” (SALib refers to these dictionaries as “problems”).
+
+.. code:: ipython3
+
+    problem_hymod = {
+        'num_vars': 5,
+        'names': ['Kq', 'Ks', 'Alp', 'Huz', 'B'],
+        'bounds': [[0.1, 1],  # Kq
+                   [0, 0.1],  # Ks
+                   [0, 1],    # Alp
+                   [0.1, 500],  # Huz
+                   [0, 1.9]]  # B
+    }
+
+After defining our uncertainites and ranges, we’ll use SALib to sample
+the uncertainty space and run the model for each of the sample sets. We
+will load a sample that has already been created ``param_values_hymod``
+for demonstration purposes. For HYMOD, literature recommends running at
+least N = 10,000 samples, to keep this demonstration easy to run
+however, we utilize only 256 sobol samples of uncertainties. To generate
+accurate approximations of second order sensitivity indicies SALib
+generates N*(2k+2) sets of samples, where N=256 and k=5 (number of
+uncertainties). For the math behind why this is needed, see (Saltelli,
+A., 2002. Making best use of model evaluations to compute sensitivity
+indices. Computer Physics Communications 145, 280–297.
+https://doi.org/10.1016/S0010-4655(02)00280-1).
+
+The actual model simulation takes an extended period, so we also load
+the simulation data from a previous run. The following demonstrates how
+to conduct this analysis:
+
+.. code:: python
+
+
+   # generate 256 samples.
+   param_values_hymod = saltelli.sample(problem_hymod, 256)
+
+   # dictionary to store outputs in
+   d_outputs = {}
+
+   # run simulation for each parameter sample
+   for i in range(0, len(param_values_hymod)):
+
+       # run model for each sensitivity analysis parameter sets
+       hymod_output = msdbook.hymod(Nq,
+                                    param_values_hymod[i, 0],
+                                    param_values_hymod[i, 1],
+                                    param_values_hymod[i, 2],
+                                    param_values_hymod[i, 3],
+                                    param_values_hymod[i, 4],
+                                    leaf_data,
+                                    ndays=4015)
+
+       # store the simulated total flow discharge
+       d_outputs[f"Q{i}"] = hymod_output["Q"]
+
+
+   Q_df_bw = pd.DataFrame(d_outputs)
+
+.. code:: ipython3
+
+    # load previously generated parameter values
+    param_values_hymod = msdbook.load_hymod_params()
+
+    # number of samples
+    n_samples = len(param_values_hymod)
+
+    # load previously generated hymod simulated outputs
+    Q_df_bw = msdbook.load_hymod_simulation()
+
+    # column names of each sample simulation number
+    sample_column_names = [i for i in Q_df_bw.columns if i[0] == 'Q']
+
+
+Running HYMOD - Model Warm-up
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+A hydrological model such as HYMOD usually includes ordinary
+differential equations that are sensitive to their initial condition.
+They also have components in their underlying formulation that have long
+memory such that prior time steps can affect their current simulations.
+For example, soil moisture or groundwater can hold water for a long time
+and therefore they are often considered to exhibit a long memory. This
+can affect the partitioning of water to runoff and infiltration, while
+also controlling the generation of base flow. Therefore, it is important
+to have a reasonable initial value for them. To achieve this,
+hydrologists usually extend their simulation period and after the
+simulations, they remove that extended time period that has unreasonable
+groundwater or surface water values. This time period is called the
+warm-up time period.
+
+Here we extended our simulation for one year (from 10 years to 11 years)
+and we removed the first year of simulation, therefore our warm-up
+period is one year.
+
+.. code:: ipython3
+
+    # exclude the first year of simulation from the simulations and reset the index
+    Q_df = Q_df_bw.iloc[365:4015].copy().reset_index(drop=True)
+
+    # exclude the first year of the input data and reset the index
+    leaf_data = leaf_data.iloc[365:4015].copy().reset_index(drop=True)
+
+
+Now that HYMOD has been warmed up, we’ll examine how HYMOD’s streamflow
+outputs vary under different sample sets, and compare them with the
+observed streamflow.
+
+.. code:: ipython3
+
+    # add date columns to our simulation data frame; for this data our start date is 1/1/2000
+    date_ts = pd.date_range(start='1/1/2000', periods=3650, freq='D')
+    Q_df['date'] = date_ts
+    Q_df['year'] = date_ts.year
+    Q_df['month'] = date_ts.month
+    Q_df['day'] = date_ts.day
+
+    # aggregate the simulated observed streamflow to monthly mean
+    df_sim_mth_mean = Q_df.groupby(['year', 'month'])[sample_column_names].mean()
+
+    # do the same for the observed data
+    date_ts = pd.date_range(start='1/1/2000', periods=len(leaf_data), freq='D')
+    leaf_data['date'] = date_ts
+    leaf_data['year'] = date_ts.year
+    leaf_data['month'] = date_ts.month
+    leaf_data['day'] = date_ts.day
+
+    # aggregate the daily observed streamflow to monthly mean
+    df_obs_mth_mean = leaf_data.groupby(['year', 'month']).mean()
+
+
+.. tip:: View the source code used to create this plot here: `plot_observed_vs_sensitivity_streamflow <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-observed-vs-sensitivity-streamflow>`_
+
+.. code:: ipython3
+
+    ax = msdbook.plot_observed_vs_sensitivity_streamflow(df_obs=df_obs_mth_mean,
+                                                         df_sim=df_sim_mth_mean)
+
+
+
+
+.. image:: _static/hymod3.png
+
+
+
+.. _sametrics:
+
+2.2 Sensitivity of streamflows to model parameters
+
+Now we’ll examine how each of HYMOD’s parameters impact the variance of
+simulated streamflows. Using SALib we’ll calculate the first order and
+total order sensitivity indicies of each model parameter. The first
+order sensitivity index measure’s the individual impact that a given
+parameter has on the variance of the simulated streamflows. The total
+order index measures the impact of a given parameter along with all
+interactions that other parameters have with the given parameter on
+simulated streamflows.
+
+We’ll start with an matrix, Y, which contains our simulated streamflows
+for every uncertainty sample. We’ll then use the sobol.analyze function
+from SALib to calculate the sensitivity indicies (Si). The arguments for
+this function are the problem dictionary defined in part 2.2 of this
+tutorial, and the matrix of simulated streamflows, Y.
+
+.. code:: ipython3
+
+    # overall aggregated indices
+    Y = Q_df[sample_column_names].mean().to_numpy()
+
+    # Perform analysis
+    Si = sobol.analyze(problem_hymod, Y)
+
+Now we can examine our results, we’ll print the first order and total
+order Si’s for each parameter, then visualize the results with bar plots
+
+.. code:: ipython3
+
+    print('First order indices = ', Si['S1'])
+
+    print('Total order indicies = ', Si['ST'])
+
+    sns.set_style('white')
+    fig = plt.figure(figsize=(8,4))
+    ax1 = fig.add_subplot(121)
+    ax1.bar(np.arange(5), Si['S1'])
+    ax1.set_xticklabels(['','Kq', 'Ks', 'Alp', 'Huz', 'B'])
+    ax1.set_ylabel('First order Si')
+    ax1.set_ylim([0,1])
+
+    ax2 = fig.add_subplot(122)
+    ax2.bar(np.arange(5), Si['ST'])
+    ax2.set_xticklabels(['','Kq', 'Ks', 'Alp', 'Huz', 'B'])
+    ax2.set_ylabel('Total order Si')
+    ax2.set_ylim([0,1])
+
+
+
+.. parsed-literal::
+
+    First order indices =  [9.55550001e-05 7.49249463e-04 5.62386413e-04 7.03327551e-01
+     2.53701895e-01]
+    Total order indicies =  [1.76174200e-06 1.63288175e-03 3.41378460e-04 6.88983864e-01
+     2.53922146e-01]
+
+
+.. parsed-literal::
+
+    /srv/conda/envs/notebook/lib/python3.7/site-packages/ipykernel_launcher.py:9: UserWarning: FixedFormatter should only be used together with FixedLocator
+      if __name__ == '__main__':
+    /srv/conda/envs/notebook/lib/python3.7/site-packages/ipykernel_launcher.py:15: UserWarning: FixedFormatter should only be used together with FixedLocator
+      from ipykernel import kernelapp as app
+
+
+
+
+.. parsed-literal::
+
+    (0.0, 1.0)
+
+
+
+
+.. image:: _static/hymod4.png
+
+
+
+Our findings indicate that in this instance, the streamflow estimate
+from HYMOD is highly sensitive to soil moisture parameters Huz and B and
+hardly affected by the routing parameters. Notably, there is very little
+interactions between parameters causing the total order indicies to be
+nearly identical to the first order indicies.
+
+
+.. _diffperformance:
+
+2.3 How do different performance metrics affect the results of our
+sensitivity analysis?
+
+Streamflow has many different properties. In this section, we discuss
+how the selection of metrics can lead to fundamentally different
+sensitivity analysis results. For example, one can only focus on
+aggregated streamflow metrics such as mean (what has been presented so
+far), or only on extreme events such as drought or floods.
+
+Here we compare three different metrics: 1- Mean error (ME) 2- Root Mean
+Square Error (RMSE) 3- Log-Root Mean Square Error (Log(RMSE))
+
+Each of these metrics focuses on a specific attribute of streamflow. For
+example, RMSE highlights the impacts of extreme flood events, while
+LOG(RMSE) focuses on model performance during low-flow events.
+
+.. code:: ipython3
+
+    # calculate error metrics
+    mae = Q_df[sample_column_names].apply(lambda x: abs(x-leaf_data["Strmflw"]), axis=0)
+    mse = Q_df[sample_column_names].apply(lambda x: metrics.mean_squared_error(x, leaf_data["Strmflw"]), axis=0)
+    rmse = mse**(1/2)
+
+    # add error metrics to a dictionary
+    d_metrics = {'MAE': mae.mean().values,
+                 'RMSE': rmse.values,
+                 'LOG[RMSE]': np.log10(rmse.values)}
+
+    # convert to a dataframe
+    df_metrics_SA = pd.DataFrame(d_metrics)
+
+
+We can use the following to calculate the SA indices for each metric and
+visualize it.
+
+.. code:: ipython3
+
+    df_metric_s1_result = pd.DataFrame(np.zeros((3, 5)), columns=['Kq', 'Ks', 'Alp', 'Huz', 'B'])
+    df_metric_sT_result = pd.DataFrame(np.zeros((3, 5)), columns=['Kq', 'Ks', 'Alp', 'Huz', 'B'])
+
+    # conduct sensitivity analysis for each metric
+    for index, i in enumerate(d_metrics.keys()):
+
+        # get the data as a numpy array for the target metric
+        Y = d_metrics[i]
+
+        # use the metric to conduct SA
+        Si = sobol.analyze(problem_hymod, Y, print_to_console=False)
+
+        # add the sensitivity indices to the output data frame
+        df_metric_s1_result.iloc[index, :] = Si['S1']
+        df_metric_sT_result.iloc[index, :] = Si['ST']
+
+.. code:: ipython3
+
+    # create seaborn heatmap with required labels
+    fig = plt.figure(figsize=(12,4))
+    ax1 = fig.add_subplot(121)
+    # labels for y-axis
+    y_axis_labels = ['Mean Absoulte Error', 'RSME', 'Log(RMSE)']
+
+    # plot heatmap
+    ax1 = sns.heatmap(df_metric_s1_result, yticklabels=y_axis_labels, annot=True,  cmap='inferno_r', cbar_kws={'label': 'Si'}, cbar=False)
+    ax1.figure.axes[-1].yaxis.label.set_size(14)
+    ax1.set_title('First Order Sensitivity')
+
+    ax2 = fig.add_subplot(122)
+    ax2 = sns.heatmap(df_metric_sT_result, yticklabels=y_axis_labels, annot=True,  cmap='inferno_r', cbar_kws={'label': 'Si'})
+    ax2.figure.axes[-1].yaxis.label.set_size(14)
+    ax2.set_title('Total Order Sensitivity')
+
+
+
+
+
+.. parsed-literal::
+
+    Text(0.5, 1.0, 'Total Order Sensitivity')
+
+
+
+
+.. image:: _static/hymod5.png
+
+
+
+The first order sensitivity indicies indicate that HYMOD’s sensitivity
+to its parameters is different depending on how its output is measured.
+Unsurprisingly, the mean absolute error is highly sensitive to the soil
+moisture accounting parameters Huz and B, just like the overall
+streamflow predictions above. However, when we examine RMSE and
+log(RMSE), the routing parameters Alp become sensitive, and the
+sensitivity to parameter B is reduced. As described above, RMSE and
+LOG(RMSE) respond to model performance in high-flow and low flow periods
+respectively. Our results indicate that for these flow regimes Alp, the
+parameter that governs the split between quick and slow flow is an
+important factor. While still the parameter with the highest most effect
+on all three measures, Huz is much less influential for RMSE and
+LOG(RMSE) than it is for MAE.
+
+The total order sensitivity indicies review a different, more complex
+story. While the MAE sensitivity is relatively governed by first order
+effects (like the streamflow predictions above), the RMSE and LOG(RMSE)
+error metrics show significant interactions. Alp has the highest total
+order sensitivity for RMSE and is eqal to Huz for Log(RMSE). Kq, which
+has a relatively low first order sensitivity index, shows strong
+contribution to variance when interactions are taken into account.
+
+Radial convergence plots are a helpful way to visualize the interactions
+between parameters. These plots array the model parameters in a circle
+and plot the first order, total order and second order Sobol sensitivity
+indices for each parameter. The first order sensitivity is shown as the
+size of a closed circle, the total order as the size of a larger open
+circle and the second order as the thickness of a line connecting two
+parameters. Below is an example of a radial convergence plot for the
+LOG(RMSE) measure. The plot indicates strong interactions between the
+Huz and Alp parameters, as well as Alp and Kq. There is also an
+interaction between Alp and Ks.
+
+.. code:: ipython3
+
+    import numpy as np
+    import itertools
+    import seaborn as sns
+    import math
+    sns.set_style('whitegrid', {'axes_linewidth': 0, 'axes.edgecolor': 'white'})
+
+    def is_significant(value, confidence_interval, threshold="conf"):
+        if threshold == "conf":
+            return value - abs(confidence_interval) > 0
+        else:
+            return value - abs(float(threshold)) > 0
+
+    def grouped_radial(SAresults, parameters, radSc=2.0, scaling=1, widthSc=0.5, STthick=1, varNameMult=1.3, colors=None, groups=None, gpNameMult=1.5, threshold="conf"):
+        # Derived from https://github.com/calvinwhealton/SensitivityAnalysisPlots
+        fig, ax = plt.subplots(1, 1)
+        color_map = {}
+
+        # initialize parameters and colors
+        if groups is None:
+
+            if colors is None:
+                colors = ["k"]
+
+            for i, parameter in enumerate(parameters):
+                color_map[parameter] = colors[i % len(colors)]
+        else:
+            if colors is None:
+                colors = sns.color_palette("deep", max(3, len(groups)))
+
+            for i, key in enumerate(groups.keys()):
+                #parameters.extend(groups[key])
+
+                for parameter in groups[key]:
+                    color_map[parameter] = colors[i % len(colors)]
+
+        n = len(parameters)
+        angles = radSc*math.pi*np.arange(0, n)/n
+        x = radSc*np.cos(angles)
+        y = radSc*np.sin(angles)
+
+        # plot second-order indices
+        for i, j in itertools.combinations(range(n), 2):
+            #key1 = parameters[i]
+            #key2 = parameters[j]
+
+            if is_significant(SAresults["S2"][i][j], SAresults["S2_conf"][i][j], threshold):
+                angle = math.atan((y[j]-y[i])/(x[j]-x[i]))
+
+                if y[j]-y[i] < 0:
+                    angle += math.pi
+
+                line_hw = scaling*(max(0, SAresults["S2"][i][j])**widthSc)/2
+
+                coords = np.empty((4, 2))
+                coords[0, 0] = x[i] - line_hw*math.sin(angle)
+                coords[1, 0] = x[i] + line_hw*math.sin(angle)
+                coords[2, 0] = x[j] + line_hw*math.sin(angle)
+                coords[3, 0] = x[j] - line_hw*math.sin(angle)
+                coords[0, 1] = y[i] + line_hw*math.cos(angle)
+                coords[1, 1] = y[i] - line_hw*math.cos(angle)
+                coords[2, 1] = y[j] - line_hw*math.cos(angle)
+                coords[3, 1] = y[j] + line_hw*math.cos(angle)
+
+                ax.add_artist(plt.Polygon(coords, color="0.75"))
+
+        # plot total order indices
+        for i, key in enumerate(parameters):
+            if is_significant(SAresults["ST"][i], SAresults["ST_conf"][i], threshold):
+                ax.add_artist(plt.Circle((x[i], y[i]), scaling*(SAresults["ST"][i]**widthSc)/2, color='w'))
+                ax.add_artist(plt.Circle((x[i], y[i]), scaling*(SAresults["ST"][i]**widthSc)/2, lw=STthick, color='0.4', fill=False))
+
+        # plot first-order indices
+        for i, key in enumerate(parameters):
+            if is_significant(SAresults["S1"][i], SAresults["S1_conf"][i], threshold):
+                ax.add_artist(plt.Circle((x[i], y[i]), scaling*(SAresults["S1"][i]**widthSc)/2, color='0.4'))
+
+        # add labels
+        for i, key in enumerate(parameters):
+            ax.text(varNameMult*x[i], varNameMult*y[i], key, ha='center', va='center',
+                    rotation=angles[i]*360/(2*math.pi) - 90,
+                    color=color_map[key])
+
+        if groups is not None:
+            for i, group in enumerate(groups.keys()):
+                print(group)
+                group_angle = np.mean([angles[j] for j in range(n) if parameters[j] in groups[group]])
+
+                ax.text(gpNameMult*radSc*math.cos(group_angle), gpNameMult*radSc*math.sin(group_angle), group, ha='center', va='center',
+                    rotation=group_angle*360/(2*math.pi) - 90,
+                    color=colors[i % len(colors)])
+
+        ax.set_facecolor('white')
+        ax.set_xticks([])
+        ax.set_yticks([])
+        plt.axis('equal')
+        plt.axis([-2*radSc, 2*radSc, -2*radSc, 2*radSc])
+        #plt.show()
+
+
+        return fig
+
+    # define groups for parameter uncertainties
+    groups={"Soil Moisture" : ["Huz", "B"],
+            "Routing" : ["Alp", "Kq", "Ks"]}
+
+
+    fig = grouped_radial(Si, ['Kq', 'Ks', 'Alp', 'Huz', 'B'], groups=groups, threshold=0.025)
+
+
+.. parsed-literal::
+
+    Soil Moisture
+    Routing
+
+
+
+.. image:: _static/hymod6.png
+
+
+2.4 Time-Varying Sensitivity Analysis
+
+.. _time:
+
+In section 2.5 we saw how performing sensitivity analysis on different
+measurements of model output can yeild in different results on the
+importance of each uncertain input. In this section we’ll examine how
+performing this analysis over time can yeild additional insight into the
+performance of HYMOD. We’ll first examine how model sensitivities vary
+by month, then examine how they change across each year of the
+simulation.
+
+For this demonstration, we’ll focus only on the monthly streamflow
+predictions generated by HYMOD.
+
+2.4.1 Sensitivity analysis indices for each month
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+.. code:: ipython3
+
+    # aggregate simulated streamflow data to monthly time series
+    df_sim_by_mth_mean = Q_df.groupby('month')[sample_column_names].mean()
+
+    # aggregate observed streamflow data to monthly time series
+    df_obs_by_mth_mean = leaf_data.groupby('month').mean()
+
+
+We can use the following to calculate the SA indices for each month and
+visualize it. Results are pre-loaded for efficiency.
+
+.. code:: python
+
+   # set up dataframes to store outputs
+   df_mth_s1 = pd.DataFrame(np.zeros((12,5)), columns=['Kq', 'Ks', 'Alp', 'Huz', 'B'])
+   df_mth_delta = df_mth_s1.copy()
+
+   # iterate through each month
+   for i in range(0, 12):
+
+       # generate the simulation data
+       Y = df_sim_by_mth_mean.iloc[i, :].to_numpy()
+
+       # run SA
+       Si = delta.analyze(problem_hymod, param_values_hymod, Y, print_to_console=False)
+
+       # add to output dataframes
+       df_mth_s1.iloc[i, :] = np.maximum(Si['S1'], 0)
+       df_mth_delta.iloc[i, :] = np.maximum(Si['delta'], 0)
+
+   # convert to arrays
+   arr_mth_s1 = df_mth_s1.values
+   arr_mth_delta = df_mth_delta.values
+
+First-order Indices
+^^^^^^^^^^^^^^^^^^^
+
+The following can be used to visualize the time-varying first-order
+indices. The first order represents the direct impacts of a specific
+parameter on model outputs.
+
+.. tip:: View the source code used to create this plot here: `plot_monthly_heatmap <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-monthly-heatmap>`_
+
+.. code:: ipython3
+
+    # load previously ran data
+    arr_mth_delta, arr_mth_s1 = msdbook.load_hymod_monthly_simulations()
+
+    # plot figure
+    ax, ax2 = msdbook.plot_monthly_heatmap(arr_sim=arr_mth_s1.T,
+                                           df_obs=df_obs_by_mth_mean,
+                                           title='First Order - Mean Monthly SA')
+
+
+
+
+.. image:: _static/hymod7.png
+
+
+
+This figure demonstrates the first order sensitivity indices when the
+streamflow data are aggregated by month. The purple line represents the
+observed monthly discharge. The figure indicates that the first order
+indices are highest for B and Huz across all months and lowest for Alp,
+Ks, and Kq. Note that in the months with the highest flow, Ks becomes an
+influential parameter.
+
+Total-order indices
+^^^^^^^^^^^^^^^^^^^
+
+We can also focus on the total order sensitivity index that includes
+first-order SA indices and interactions between parameters
+
+.. code:: ipython3
+
+    # plot figure
+    ax, ax2 = msdbook.plot_monthly_heatmap(arr_sim=arr_mth_delta.T,
+                                           df_obs=df_obs_by_mth_mean,
+                                           title='Total Order - Mean monthly SA')
+
+
+
+
+.. image:: _static/hymod8.png
+
+
+
+Notably, the total order sensitivity results are different than the
+first order sensitivity results, which indicates that interactions
+between the parameters (particularly in regards to routing parameters
+:math:`Kq`, :math:`Ks`, and :math:`Alp`) contribute to changes in HYMOD
+output.
+
+2.4.2 Annual sensitivity analysis indices
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+.. code:: ipython3
+
+    # group by year and get mean
+    df_sim_by_yr_mean = Q_df.groupby(['year'])[sample_column_names].mean()
+
+    # group input data and get mean
+    df_obs_by_yr_mean = leaf_data.groupby(['year']).mean()
+
+
+We can also calculate the sensitivity analysis indices for each
+individual year. This will allow us to understand if model control
+changes during different years. The following code first aggregates the
+outputs to annual time steps, and then calculates the SA indices.
+
+.. code:: python
+
+   # set up dataframes to store outputs
+   df_yr_s1 = pd.DataFrame(np.zeros((10, 5)), columns=['Kq', 'Ks', 'Alp', 'Huz', 'B'])
+   df_yr_delta = df_yr_s1.copy()
+
+   # iterate through each year
+   for i in range(0, 10):
+
+       # generate the simulation data
+       Y = df_sim_by_yr_mean.iloc[i, :].to_numpy()
+
+       # run SA
+       Si = delta.analyze(problem_hymod, param_values_hymod, Y, print_to_console=False)
+
+       # add to output dataframes
+       df_yr_s1.iloc[i, :] = np.maximum(Si['S1'], 0)
+       df_yr_delta.iloc[i, :] = np.maximum(Si['delta'], 0)
+
+   # convert to arrays
+   arr_yr_s1 = df_mth_s1.values
+   arr_yr_delta = df_mth_delta.values
+
+First-order indices
+^^^^^^^^^^^^^^^^^^^
+
+.. tip:: View the source code used to create this plot here: `plot_annual_heatmap <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-annual-heatmap>`_
+
+.. code:: ipython3
+
+    # load previously ran data
+    arr_yr_delta, arr_yr_s1 = msdbook.load_hymod_annual_simulations()
+
+    # plot figure
+    ax, ax2 = msdbook.plot_annual_heatmap(arr_sim=arr_yr_s1.T,
+                                          df_obs=df_obs_by_yr_mean,
+                                          title='First Order - Mean Annual SA')
+
+
+
+
+.. image:: _static/hymod9.png
+
+
+
+The first order sensitivities at the annual scale are not unlike the
+first order monthly sensitivities. Once again, sensitivities vary across
+year and Huz and B are the most consequential parameters.
+
+Total-order indices
+^^^^^^^^^^^^^^^^^^^
+
+.. code:: ipython3
+
+    # plot figure
+    ax, ax2 = msdbook.plot_annual_heatmap(arr_sim=arr_yr_delta.T,
+                                          df_obs=df_obs_by_yr_mean,
+                                          title='Total Order - Mean Annual SA and Observed flow')
+
+
+
+.. image:: _static/hymod10.png
+
+
+
+Our results indicate that sensitivity analysis indices vary in different
+years and now that interactions are included, the Kq, Ks, and Alp
+variables impact the sensitivity of the streamflow output.
+
+2.4.3 Monthly time-varying sensitivity analysis
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+Although time-varying sensitivity analysis (TVSA) at average monthly and
+average annual temporal resolutions is informative, TVSA is susceptible
+to the aggregation issue that we discussed earlier in section 3-2. To
+avoid that we can further discretize our time domain to zoom into
+individual months. This will provide us with even more information about
+model behavior and the sensitivity of different parameters in different
+states of the system. The block of code demonstrates how to implement
+the monthly TVSA.
+
+.. code:: python
+
+   # set up dataframes to store outputs
+   df_vary_s1 = pd.DataFrame(np.zeros((df_obs_mth_mean.shape[0], 5)),
+                             columns=['Kq', 'Ks', 'Alp', 'Huz', 'B'])
+
+   df_vary_delta = df_vary_s1.copy()
+
+   # iterate through each month
+   for i in range(0, df_obs_mth_mean.shape[0]):
+
+       # generate the simulation data
+       Y = df_sim_mth_mean.iloc[i, :].to_numpy()
+
+       # run SA
+       Si = delta.analyze(problem_hymod, param_values_hymod, Y, print_to_console=False)
+
+       # add to output dataframes
+       df_vary_s1.iloc[i, :] = np.maximum(Si['S1'], 0)
+       df_vary_delta.iloc[i, :] = np.maximum(Si['delta'], 0)
+
+   # convert to arrays
+   arr_vary_s1 = df_vary_s1.values
+   arr_vary_delta = df_vary_delta.values
+
+First-order indices
+^^^^^^^^^^^^^^^^^^^
+
+.. tip:: View the source code used to create this plot here: `plot_varying_heatmap <https://uc-ebook.org/docs/html/A3_plotting_code.html#plot-varying-heatmap>`_
+
+.. code:: ipython3
+
+    # load in previously ran data
+    arr_vary_delta, arr_vary_s1 = msdbook.load_hymod_varying_simulations()
+
+    # plot figure
+    ax, ax2 = msdbook.plot_varying_heatmap(arr_sim=arr_vary_s1.T,
+                                          df_obs=df_obs_mth_mean,
+                                          title='First Order - Time-Varying SA')
+
+
+
+.. image:: _static/hymod11.png
+
+
+
+Compared to the TVSA when streamflow was aggregated, this figure
+suggests that Kq is indeed a relevant parameter for influencing
+streamflow output when individual months are considered.
+
+Total order - time varying sensitivity analysis
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+.. code:: ipython3
+
+    # plot figure
+    ax, ax2 = msdbook.plot_varying_heatmap(arr_sim=arr_vary_delta.T,
+                                          df_obs=df_obs_mth_mean,
+                                          title='Total Order - Time-Varying SA')
+
+
+
+
+.. image:: _static/hymod12.png
+
+
+
+As above, the total order sensitivities further indicate the importance
+of Kq that is not apparent if aggregation is utilized.
+
+Tips to Apply this methodology to your own problem
+==================================================
+
+In this tutorial, we demonstrated how to use global sensitivtiy analysis
+to explore a complex, non-linear model. We showed how measuring
+sensitivity across multiple measures of model performance and temporal
+aggregations yeilding differing results about model
+sensitivity/behavoir. While these results may seem contraditory, they
+provide useful insight into the behavoir of HYMOD. Would we expect the
+same parameters to control high flow and low flow regimes within the
+model? Maybe, depending on the system, but also, maybe not. This
+analysis can provide insight into how the model responds to its input
+parameters, allowing us to compare the results to our expectaions. This
+may allow us to find problems with our intial assumptions, or call
+attention to model features that can be improved or expanded upon.
+Depending on the model and context, it may also yield insight into the
+workings of the underlying system.
+
+To run this tutorial on your own model you will need to:
+
+1. Design your experiment by choosing sampling bounds for your
+   parameters and setting up the problem dictionary as in step 2-2
+2. Choose the parameters of interest
+3. Generate samples using the saltelli.sample function. This step is
+   problem-dependent and note that the Sobol method can be
+   computationally intensive depending on the model being analyzed. More
+   complex models will be slower to run and will also require more
+   samples to calculate accurate estimates of Sobol indices. Once you
+   complete this process, pay attention to the confidence bounds on your
+   sensitivity indices to see whether you need to run more samples.
+4. Run the parameter sets through your model and record each of the
+   desired model outputs.
+5. Calculate the Sobol indices for each performance criteria. Now, the Y
+   will be a numpy array with your external model output and you will
+   need to include the parameter samples as an additional argument.
+6. Follow the procedure in step 2.6 to disaggregate performance across
+   time
diff --git a/docs/html/_sources/A2.5_discovery.rst b/docs/html/_sources/A2.5_discovery.rst
new file mode 100644
index 0000000..4487f45
--- /dev/null
+++ b/docs/html/_sources/A2.5_discovery.rst
@@ -0,0 +1,384 @@
+Time-evolving scenario discovery for infrastructure pathways
+****************************************************************************************************
+
+.. note::
+
+    | Run the tutorial interactively:  `Scenario Discovery Notebook <nb_discovery_>`_.
+    | Please be aware that notebooks can take a couple minutes to launch.
+    | To run the notebooks yourself, download the files `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks>`_ and use these `requirements <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt>`_.
+
+
+
+Time-evolving scenario discovery for infrastructure pathways
+============================================================
+
+The purpose of this tutorial is to explore time-evolving vulnerability for systems that dynamically adapt to changing conditions. Using an example from water supply planning, we'll first examine how performance of a dynamic infrastructure pathway policy changes over time, then use factor mapping (main text Chapter 4.3) to understand which combinations of uncertaities generate vulnerability for two water utilities. Next, we'll perform factor prioritization (main text Chapter 4.3) to determine which uncertainties have the most influence on water supply performance. Finally, we'll provide an open platform to explore vulnerability across multiple measures of performance and different combinations of uncertainties.
+
+Background
+----------
+
+The Bedford-Greene metropolitan area (Figure 1) is a stylized water resources test case where two urban water utilities seek to develop an infrastructure and investment and management strategy to confront growing demands and changing climate. The utilities have agreed to jointly construct a new water treatment plant on Lake Classon, a large regional resource. Both utilities have also identified a set of individual infrastructure options to construct if necessary.
+
+.. figure:: _static/Map_small.png
+   :alt: Figure 1
+
+   Figure 1
+
+The utilities are formulating a cooperative and adaptive regional management strategy that uses a risk-of-failure (ROF) metric to trigger both short term drought mitigation actions (water use restrictions and treated transfers between utilities) and long-term infrastructure investment decisions (shown in Figure 2a). Both utilities have specified a set of risk triggers and developed a construction order for available infrastructure options.
+
+The utilities have run a Monte Carlo simulation to evaluate how these policies respond to a wide array of future States Of the World (SOWs). Each SOW represents a different combinations of thirteen uncertain system inputs including demand growth rates, changes to streamflows and financial variables. In this context, a fully specified SOW is composed of one sample of uncertain model inputs (e.g. one projection of demand growth rate coupled with one future streamflow scenario and one projection of future financial conditions). The water utilities used Latin Hypercube sampling (Chapter 3.3 of the main text) to develop an ensemble of 1,000 plausible future SOWs. The Monte Carlo simulation evaluates each candidate water supply infrastructure investment and management policy across all 1,000 SOWs, as shown in Figure 2b. For more details on the Monte Carlo sampling for this type  of analysis, see Trindade et al., (2019).
+
+The ROF-based policies respond to each SOW by generating a unique infrastructure pathway - a sequence of infrastructure investment decisions over time. Infrastructure pathways over a set of 1,000 future SOWs are shown in Figure 2c. Infrastructure options are arrayed along the vertical axis and the sequence of infrastructure investments triggered by the policy is plotted as pathways over time.  Since the adaptive rule system generates a unique infrastructure sequence for each scenario, Figure 2c summarizes the ensemble of pathways by clustering SOWs according to infrastructure intensity. Dark green lines represent SOWs where the utilities heavily invest in new infrastructure, light green lines represent SOWs with low infrastructure investment and medium shaded lines represent moderate investment. The shading behind each pathway represents the frequency that each infrastructure option was triggered over time across sampled scenarios
+
+.. figure:: _static/Policy_MonteCarlo_Pathways_small.png
+   :alt: Figure 2
+
+   Figure 2
+
+Evaluating Robustness over time
+-------------------------------
+
+The two water utilities are interested in mainting both supply
+reliability and financial stability across the broadest set of plausible
+future SOWs. To measure the performance of the infrastructure pathway
+policy, they’ve defined five critical performance criteria:
+
+-  Reliability > 99%
+-  Restriction Frequency < 20%
+-  Peak Financial Cost < 80% of annual revenue (a measure of debt
+   service spending)
+-  Worst-case drought management cost < 10% of annual revenue (a measure
+   of unexpected drought costs)
+-  Unit Cost of Expansion < 5 dollars/kgal
+
+To assess the robustness of the infrastructure pathway policy, the two
+utilities apply a satisficing metric, which measures the percentage of
+sampled SOWs where the pathway policy meets the peformance criteria:
+
+:math:`R =\frac{1}{N}\sum_{j=1}^{N}\Lambda_{\theta,j}`
+
+Where, :math:`\Lambda\_{\theta,j}=`
+
+.. math::
+
+   \begin{cases}
+   1, \quad \textrm{if}\ F(\theta)_{j}\leq \Phi_j \\
+   0, \quad \textrm{otherwise}
+   \end{cases}
+
+And :math:`\Phi` is a vector of performance criteria for utility
+:math:`j`, :math:`\theta` is the portfolio and :math:`N` is the total
+number of sampled SOWs.
+
+Below, we'll visualize how robustness for the two utilities evolves over the 45-year planning horizon. We'll assess robustness across three time periods, near-term (first 10 years), mid-term (22 years) and long term (45 years).
+
+We start by loading robustness values for the both utilities. These values are calculated by applying the robustness metric above across 2,000 simulated SOWs. To make this exercise computationally tractable, we've precomputed these values, which can be found in the files "short_term_robustness.csv", "mid_term_robustness.csv" and "long_term_robustness.csv". These values are calculated using the function "check_rdm_meet_criteria" within the helper functions.
+
+.. code:: ipython3
+
+    import numpy as np
+    from matplotlib import pyplot as plt
+    from functions.eBook_SD_helpers import check_rdm_meet_criteria, create_sd_input, plot_selected_tree_maps, get_factor_importances, open_exploration
+    import seaborn as sns
+
+    # load Deeply uncertain factors
+    rdm_factors = np.loadtxt('data/DU_Factors.csv', delimiter= ',')
+
+    sns.set()
+    short_term_robustness = np.loadtxt('data/short_term_robustness.csv', delimiter= ',')
+    mid_term_robustness = np.loadtxt('data/mid_term_robustness.csv', delimiter = ',')
+    long_term_robustness = np.loadtxt('data/long_term_robustness.csv', delimiter = ',')
+
+    # plot robustness over time
+    fig =plt.figure(figsize=(9,3))
+    plt.plot([10,22,45], [short_term_robustness[5]*100, mid_term_robustness[5]*100,long_term_robustness[5]*100], c='#B19CD9')
+    plt.plot([10, 22, 45], [short_term_robustness[11]*100, mid_term_robustness[11]*100, long_term_robustness[11]*100], c= '#43b284')
+    plt.scatter([10,22,45], [short_term_robustness[5]*100, mid_term_robustness[5]*100,long_term_robustness[5]*100], s=100, c='#B19CD9')
+    plt.scatter([10, 22, 45], [short_term_robustness[11]*100, mid_term_robustness[11]*100, long_term_robustness[11]*100], s=100, c='#43b284')
+    plt.xlabel('Time Horizon (yrs)')
+    plt.ylabel('Robustness (% SOWs)')
+    plt.legend(['Bedford', 'Greene'])
+    plt.title('Robustness Over Time')
+    plt.ylim([0, 107])
+
+
+
+
+
+.. parsed-literal::
+
+    (0.0, 107.0)
+
+
+
+
+.. image:: _static/discovery_4_1.png
+
+
+Exploring performance evolution
+-------------------------------
+
+The figure above reveals that the robustness of both water utilities degrades over time, with Bedford's robustness declining further than Greene. This suggests that the proposed pathway policy is likely insufficient to meet the long-term needs of the two utilities. But how is the current policy insufficient? To answer that question we examine the performance measures that fail to meet performance criteria for each utility across the three planning horizons.
+
+.. code:: ipython3
+
+    # Plot the type of vulnerability over time
+
+    ### Bedford ###
+    plot_robustness_1 = np.zeros([3,5])
+    # Determine the percentage of failure SOWs that violate each criterion (note some SOWS fail multiple criteria, so this may some to >1)
+    criteria = ['Reliability', 'Restriction Frequency', 'Peak Financial Cost', 'Worst-case drought\nManagement Cost', 'Stranded Assets']
+    plot_robustness_1[0,:] = (1 - short_term_robustness[0:5])/(1-short_term_robustness[5])
+    plot_robustness_1[1,:] = (1 - mid_term_robustness[0:5])/(1-mid_term_robustness[5])
+    plot_robustness_1[2,:] = (1 - long_term_robustness[0:5])/(1-long_term_robustness[5])
+
+    # Plot over time
+    fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(9,4))
+    axes[0].bar(np.arange(5), plot_robustness_1[0,:], color='#B19CD9')
+    axes[0].set_xticks(np.arange(5))
+    axes[0].set_xticklabels(criteria, rotation='vertical')
+    axes[0].set_ylim([0,1])
+    axes[0].set_title('10-year Horizon')
+    axes[0].set_ylabel('Fraction of failure SOWs')
+    axes[1].bar(np.arange(5), plot_robustness_1[1,:], color='#B19CD9')
+    axes[1].set_xticks(np.arange(5))
+    axes[1].set_xticklabels(criteria, rotation='vertical')
+    axes[1].set_ylim([0,1])
+    axes[1].set_title('22-year Horizon')
+    axes[2].bar(np.arange(5), plot_robustness_1[2,:], color='#B19CD9')
+    axes[2].set_xticks(np.arange(5))
+    axes[2].set_xticklabels(criteria, rotation='vertical')
+    axes[2].set_title('45-year Horizon')
+    axes[2].set_ylim([0,1])
+    fig.suptitle('Bedford')
+    plt.tight_layout()
+
+    ### Greene ###
+    # Determine the percentage of failure SOWs that violate each criterion (note some SOWS fail multiple criteria, so this may some to >1)
+    plot_robustness_2 = np.zeros([3, 5])
+    plot_robustness_2[0, :] = (1 - short_term_robustness[6:11]) / (1 - short_term_robustness[11])
+    plot_robustness_2[1, :] = (1 - mid_term_robustness[6:11]) / (1 - mid_term_robustness[11])
+    plot_robustness_2[2, :] = (1 - long_term_robustness[6:11]) / (1 - long_term_robustness[11])
+
+    fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(9, 4))
+    axes[0].bar(np.arange(5), plot_robustness_2[0, :], color='#43b284')
+    axes[0].set_xticks(np.arange(5))
+    axes[0].set_xticklabels(criteria, rotation='vertical')
+    axes[0].set_title('10-year Horizon')
+    axes[0].set_ylim([0,1])
+    axes[0].set_ylabel('Fraction of failure SOWs')
+    axes[1].bar(np.arange(5), plot_robustness_2[1, :], color='#43b284')
+    axes[1].set_xticks(np.arange(5))
+    axes[1].set_xticklabels(criteria, rotation='vertical')
+    axes[1].set_title('22-year Horizon')
+    axes[1].set_ylim([0,1])
+    axes[2].bar(np.arange(5), plot_robustness_2[2, :], color='#43b284')
+    axes[2].set_xticks(np.arange(5))
+    axes[2].set_xticklabels(criteria, rotation='vertical')
+    axes[2].set_title('45-year Horizon')
+    axes[2].set_ylim([0,1])
+    fig.suptitle('Greene')
+    plt.tight_layout()
+
+
+
+
+.. image:: _static/discovery_6_0.png
+
+
+
+.. image:: _static/discovery_6_1.png
+
+
+In the figures above, we observe that the vulnerability of both utilities changes in different ways. Early in the simulation period, Bedford is vulnerable to failures in reliability (though the robustness figure created in step B5.2 reveals that these failures are very rare). As the simulation period progresses, Bedford's vulnerability expands to include failures in restriction frequency and worst-case cost. These failures indicate that the utility has an overall inability to manage drought conditions and future conditions progress.
+
+Greene shows a very different evolution in vulnerability. Early in the simulation period, failures manifest in the restriction frequency objective, suggesting that the utility must rely on water use restrictions to maintain supply reliability. As the simulation progresses however, the vulnerability evolves. When evaluated across the 45-year planning horizon, a new failure modes emerges - financial failure manifesting in peak financial cost and stranded assets. This suggests that the proposed pathway policy may be over-investing in new infrastructure, straining the utility's budget with large debt payments that are unnecessary to maintain supply reliability.
+
+How do deep uncertainties generate vulnerability
+------------------------------------------------
+
+While the evolution of robustness provides insight into how the system
+evolves over time, it does not reveal *why* each utility is vulnerable.
+To examine how deep uncertainties generate vulnerability over time for
+the two utilities, we perform scenario discovery (factor mapping,
+Chapter 4.3). Here we’ll utilize gradient boosted trees to identify
+regions of the uncertainty space that cause the utilities to fail to
+meet performance criteria.
+
+.. code:: ipython3
+
+    # import the performance data across 2000 SOWs for three time periods
+    short_term_performance = np.loadtxt('data/short_term_performance.csv', delimiter= ',')
+    mid_term_performance = np.loadtxt('data/mid_term_performance.csv', delimiter = ',')
+    long_term_performance = np.loadtxt('data/long_term_performance.csv', delimiter = ',')
+
+    satisficing_criteria = [.98, .2, .8, .1, 5]
+
+    # transform into scenario discovery input
+    short_term_SD_input = create_sd_input(short_term_performance, satisficing_criteria)
+    mid_term_SD_input = create_sd_input(mid_term_performance, satisficing_criteria)
+    long_term_SD_input = create_sd_input(long_term_performance, satisficing_criteria)
+
+    # factor mapping Bedford
+    fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(9,3))
+    plot_selected_tree_maps(5, 'short_term', 0, 6, satisficing_criteria, 0, axes[0])
+    axes[0].set_title('10-year Horizon')
+    plot_selected_tree_maps(5, 'mid_term', 0, 6, satisficing_criteria, 0, axes[1])
+    axes[1].set_title('22-year Horizon')
+    plot_selected_tree_maps(5, 'long_term', 0, 1, satisficing_criteria, 0, axes[2])
+    axes[2].set_title('45-year Horizon')
+    fig.suptitle('Bedford Factor Maps')
+    plt.tight_layout()
+
+    # factor mapping Greene
+    fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(9,3))
+    plot_selected_tree_maps(11, 'short_term', 0, 8, satisficing_criteria, 0, axes[0])
+    axes[0].set_title('10-year Horizon')
+    plot_selected_tree_maps(11, 'mid_term', 0, 6, satisficing_criteria, 0, axes[1])
+    axes[1].set_title('22-year Horizon')
+    plot_selected_tree_maps(11, 'long_term', 0, 1, satisficing_criteria, 0, axes[2])
+    axes[2].set_title('45-year Horizon')
+    fig.suptitle('Greene Factor Maps')
+    plt.tight_layout()
+
+
+
+.. parsed-literal::
+
+    Factor map for Bedford
+    Factor map for Bedford
+    Factor map for Bedford
+    Factor map for Greene
+    Factor map for Greene
+    Factor map for Greene
+
+
+
+.. image:: _static/discovery_9_1.png
+
+
+
+.. image:: _static/discovery_9_2.png
+
+
+In the figures above, we learn more about how the vulnerability of the two utilities evolves over time. Bedford begins with very few possible failures but appears vulnerable to high demand growth scenarios under future scenarios with high demands. When evaluated across a 22-year planning horizon, Bedford is vulnerable when the near-term demand growth is high and water use restrictions are less effective than predicted. Under the full 45-year planning horizon, Bedford is vulnerable to sustained high levels of demand growth, failing if either near-term or mid-term demand growth exceeds expected levels.
+
+Greene's vulnerability evolves differently. It begins with vulnerability to high demand growth, but as the simulation progresses (and infrastructure is constructed), the utility becomes vulnerable to low-demand growth futures which cause the failures in financial criteria shown in section B.5.3. This indicates that the pathway policy over-builds in many SOWs, and becomes financially unstable if demand does not grow sufficiently to provide revenue to cover debt service payments.
+
+Which uncertainties have the most influence on time-evolving performance?
+-------------------------------------------------------------------------
+
+The factor maps generated in B.5.4 present the vulnerability generated by the two most important deep uncertainties as determined by Gradient Boosted Trees. Yet the factor prioritization shows that more than two uncertainties are influential to regional performance. Further, we can observe that individual uncertainties have different impacts on each performance obejctive, and these impacts may change over time. In the cells below, explore the impact of deep uncertainty by generating factor maps for different combinations of deep uncertain factors, objectives and time horizons.
+
+.. code:: ipython3
+
+    sns.set_style('white')
+    uncertainties = ['D1', 'D2', 'D3', 'BT', 'BM', 'DR', 'RE', 'EV', 'PM', 'CT', 'IA', 'IF', 'IP']
+    uncertainties = ['Near-term demand', 'Mid-term demand', 'Long-term demand', 'Bond Term', 'Bond Rate', 'Discount Rate', 'Restriction Effectiveness', 'Evaporation Rate', 'Permitting time', 'Construction time', 'Inflow Amplitude', 'Inflow Frequency', 'Inflow Period']
+
+    u1_st_FI = get_factor_importances(short_term_SD_input, rdm_factors, 250, 4, 5)
+    u1_mt_FI = get_factor_importances(mid_term_SD_input, rdm_factors, 250, 4, 5)
+    u1_lt_FI = get_factor_importances(long_term_SD_input, rdm_factors, 250, 4, 5)
+
+    u1_all = np.vstack([u1_st_FI,u1_mt_FI, u1_lt_FI])
+    u1_all = np.transpose(u1_all)
+
+    # factor ranking -- utility 2
+    u2_st_FI = get_factor_importances(short_term_SD_input, rdm_factors, 250, 4, 11)
+    u2_mt_FI = get_factor_importances(mid_term_SD_input, rdm_factors, 250, 4, 11)
+    u2_lt_FI = get_factor_importances(long_term_SD_input, rdm_factors, 250, 4, 11)
+    u2_all = np.vstack([u2_st_FI,u2_mt_FI, u2_lt_FI])
+    u2_all = np.transpose(u2_all)
+
+    fig, (ax, ax2, cax) = plt.subplots(ncols=3,figsize=(5,5),
+                      gridspec_kw={"width_ratios":[1,1, 0.1]})
+    fig.subplots_adjust(wspace=0.3)
+    im = ax.imshow(u1_all, cmap='Reds', vmin=0, vmax=.3)
+    ax.set_yticks(np.arange(13))
+    ax.set_yticklabels(uncertainties)
+    ax.set_xticks(np.arange(3))
+    ax.set_xlabel('Time Horizon')
+    ax.set_title('Bedford')
+
+    im1 = ax2.imshow(u2_all, cmap='Reds', vmin=0, vmax=.3)
+    ax2.set_yticks(np.arange(13))
+    ax2.set_yticklabels([])
+    ax2.set_xticks(np.arange(3))
+    ax2.set_xlabel('Time Horizon')
+    ax2.set_title('Greene')
+    fig.colorbar(im, cax=cax, label='Factor Importance')
+    plt.tight_layout()
+
+
+
+
+.. image:: _static/discovery_12_0.png
+
+
+
+
+The Figure above shows the factor importance as determined by gradient boosted trees for both utilities across the three planning horizons. While near-term demand growth is important for both utilities under all three planning horizons, the importance of other factors evolves over time. For example, restriction effectiveness plays an important role for Greene under the 22-year planning horizon but disappears under the 45-year planning horizon. In contrast, the bond interest rate is important for predicting success over the 45-year planning horizon, but does not appear important over the 10- or 22-year planning horizons. These findings highlight how assumptions about the planning period can have a large impact on modeling outcomes.
+
+Open exploration
+----------------
+
+In the cell below, use the function to explore how factor maps change
+for the two utilities based upon the uncertainties plotted, the
+objectives of interest and the time horizon.
+
+.. code:: ipython3
+
+    # specify the utility ("Bedford" or "Greene")
+    utility = "Bedford"
+
+    # specify which performance objectives to investigate (note that not all performance objectives have failures, which may result in a blank factor map)
+    # set this to one of the following: "Reliability", "Restriction Frequency", "Peak Financial Cost", "Worst Case Cost" or "Unit Cost"
+    objective = "Reliability"
+
+    # select uncertainties from the following list: 'D1', 'D2', 'D3', 'BT', 'BM', 'DR', 'RE', 'EV', 'PM', 'CT', 'IA', 'IF', 'IP'
+    uncertainty_1 = 'D1'
+    uncertainty_2 = 'D2'
+
+    # The code below will plot factor maps over the three planning horizons for the information above
+    fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(9,3))
+    open_exploration(utility, objective, 'short_term', uncertainty_1, uncertainty_2, axes[0])
+    open_exploration(utility, objective, 'mid_term', uncertainty_1, uncertainty_2, axes[1])
+    open_exploration(utility, objective, 'long_term', uncertainty_1, uncertainty_2, axes[2])
+    plt.tight_layout()
+
+
+
+.. parsed-literal::
+
+    Factor map for Bedford, reliability
+    Factor map for Bedford, reliability
+    Factor map for Bedford, reliability
+
+
+
+.. image:: _static/discovery_16_1.png
+
+
+
+
+Tips to apply this methodology to your own problem
+--------------------------------------------------
+
+In this tutorial, we demonstrated time-evolving scenario discovery for a
+cooperative water supply system. To apply this workflow to your own
+problem:
+
+1. Choose sampling bounds for your parameters of interest, which will
+   represent uncertainties that characterize your system.
+2. Generate samples for these parameters (this can be done using the
+   saltelli.sample function as in B.2 or done with another package).
+3. Define performance criteria for your problem
+4. Evaluate parameter sets through your model, and save performance
+   measures across multiple time horizons
+5. Draw from the supporting code for this tutorial to perform scneario
+   discovery and visualize results
+
+
+References
+--------------------------------------------------
+Trindade, B. C., Reed, P. M., & Characklis, G. W. (2019). Deeply uncertain pathways: Integrated multi-city regional water supply infrastructure investment and portfolio management. Advances in Water Resources, 134, 103442.
diff --git a/docs/html/_sources/A2.6_hmm.rst b/docs/html/_sources/A2.6_hmm.rst
new file mode 100644
index 0000000..781567d
--- /dev/null
+++ b/docs/html/_sources/A2.6_hmm.rst
@@ -0,0 +1,975 @@
+A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios Tutorial
+****************************************************************************************************
+
+.. note::
+
+    | Run the tutorial interactively:  `HMM Notebook <nb_hmm_>`_.
+    | Please be aware that notebooks can take a couple minutes to launch.
+    | To run the notebooks yourself, download the files `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/tree/main/notebooks>`_ and use these `requirements <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/requirements.txt>`_.
+
+
+A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios
+============================================================================
+
+In this notebook, we will be covering the basics of fitting a Hidden
+Markov Model-based synthetic streamflow generator for a single site in
+the Upper Colorado River Basin. First, we will characterize the observed
+historical flow in the basin from 1909-2013. Then, we will fit a
+synthetic streamflow generator to the observed flows in the basin in
+order to create stationary synthetic flows. Finally, we will create a
+non-stationary version of the generator to create flows that could be
+representative of plausible future climate in the region. We ultimately
+show how to place the synthetically generated flows in the context of
+physically-informed CMIP5 projections to compare the two methods.
+
+Background
+----------
+
+In the Western United States (US), and particularly the Colorado River
+Basin, a recent study used tree-ring reconstructions to suggest that the
+megadrought that has been occurring in the Southwest over the past 22
+years is the region’s worst drought since about 800 AD (Williams et al.,
+2022). The study’s lead author, UCLA climatologist Park Williams,
+suggested that had the sequence of wet-dry years occurred without
+anthropogenic forcing, the 2000s would have likely still been dry, but
+not on the same level as the worst of the last millennium’s
+megadroughts.
+
+The recent trend of warming and reduced soil moisture in the Southwest
+US is highly challenging from a water systems planning and management
+perspective for the Colorado River Basin. Given the wide recognition
+that the river is over-allocated, the most recent drought highlights the
+difficulty of sustaining the flow requirements as dictated by the
+Colorado Compact. Thus, there has been an increasing focus in
+exploratory modeling efforts to clarify how vulnerable water systems in
+this region are to plausible drought streamflow scenarios for the
+future. In this tutorial, we’ll discuss how to create these scenarios
+using a Hidden Markov Model (HMM)- based streamflow synthetic generator.
+As discussed in `Section
+2.1 <https://uc-ebook.org/docs/html/2_diagnostic_modeling_overview_and_perspectives.html#overview-of-model-diagnostics>`__
+and
+`4.2 <https://uc-ebook.org/docs/html/4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#consequential-dynamics-what-is-controlling-model-behaviors-of-interest>`__
+of the eBook, future climate conditions in the basin represent a deep
+uncertainty that can lead to highly consequential water scarcity
+outcomes. It is advantageous to create a model such as the HMM-based
+generator in order to facilitate the creation of many ensembles of
+streamflow that can ultimately be used to force regional water systems
+models to understand how variability and drought extremes affect
+regional water shortages, operations, and policies.
+
+.. raw:: html
+
+  <figure>
+  <center><img src="_static/1-lede_pix_-_usbr-powell-8517152024_aa66437c2e_o-cropped.jpg" style="width:50%"></center>
+  <figcaption style="text-align:center"> Lake Powell shows persistent effects from drought (Source: U.S. Bureau of Reclamation) </figcaption>
+  </figure>
+
+Let’s Get Started!
+==================
+
+Observed Record
+---------------
+
+First, let’s take a look at the observed data from 1909-2013 for a
+specific site. In this example, we use the outlet gauge of the Upper
+Colorado River (USGS Gauge 09163500 at the Colorado-Utah state line).
+Below, we create a plot of the annual streamflow.
+
+.. code:: ipython3
+
+    # Import libraries
+    import matplotlib as mpl
+    import matplotlib.pyplot as plt
+    import matplotlib.patches as patches
+    import numpy as np
+    import pandas as pd
+    from random import random
+    from SALib.sample import latin
+    from scipy import stats as ss
+    import statistics 
+    import statsmodels.api as sm
+    
+    
+    # Import helper functions from local package 
+    from functions import fitmodel
+    from functions import plotstates
+    from functions import plotdist
+
+
+.. code:: ipython3
+
+    # Read in annual historical data
+    AnnualQ = pd.read_csv('data/uc_historical.csv')
+    AnnualQ['Year'] = list(range(1909, 2014))
+    
+    # Plot a line graph
+    fig, ax = plt.subplots(figsize=(12, 8))
+    ax.plot(AnnualQ.iloc[:, 1],
+            AnnualQ.iloc[:, 0],
+            color='#005F73',
+            label='Annual')
+    
+    # Add labels and title
+    ax.set_title("Upper Colorado Annual Flow")
+    ax.set_xlabel("Year", fontsize=16)
+    ax.set_ylabel("Annual Flow (cubic feet per year)", fontsize=16)
+    plt.xticks(fontsize=12)
+    plt.yticks(fontsize=12)
+    mpl.rc('legend', fontsize=16)
+    legend = plt.legend(loc="upper right") 
+    plt.show()
+    plt.close()
+
+
+
+
+.. image:: _static/hmm_9_0.png
+
+
+Let’s calculate an 11-year rolling mean of the same data to get a sense
+of long-term trends.
+
+.. code:: ipython3
+
+    fig, ax = plt.subplots(figsize=(12, 8))
+    
+    # Plot the original line graph
+    plt.plot(AnnualQ.iloc[:,1],
+             AnnualQ.iloc[:,0],
+             color='#005F73',
+             label='Annual')
+    
+    # Plot an 11-year rolling mean 
+    plt.plot(AnnualQ.iloc[:, 1].rolling(11).mean(),
+             AnnualQ.iloc[:, 0].rolling(11).mean(),
+             color='#183A2E',
+             label='11-Year Rolling Mean')
+    
+    # Add labels and title
+    plt.title("Upper Colorado Annual Flow")
+    ax.set_xlabel("Year",fontsize=16)
+    ax.set_ylabel("Annual Flow (cubic feet per year)", fontsize=16)
+    plt.xticks(fontsize=12)
+    plt.yticks(fontsize=12)
+    mpl.rc('legend', fontsize=16)
+    legend = plt.legend() 
+    plt.show()
+    plt.close()
+
+
+
+
+.. image:: _static/hmm_11_0.png
+
+
+The Colorado Compact, which prescribes flows between the Upper and Lower
+Colorado Basins, was negotiated using data prior to 1922, a time period
+revealed by the above figure to be one of the consistently wetter
+periods on record. It’s clear today that since the 1980s, the Southwest
+US has been experiencing aridification (Overpeck et al., 2020) and that
+this observed record alone isn’t an accurate representation of what
+future climate might look like in this region.
+
+Let’s get a little more specific and formally quantify decadal droughts
+that have occurred in the observed period. We use a metric proposed in
+Ault et al. (2014). The authors define a decadal drought as when the
+11-year rolling mean falls below a threshold that is 1/2 a standard
+deviation below the overall mean of the record. We can then highlight
+the block of years that fall in a decadal drought using yellow
+rectangles below.
+
+.. code:: ipython3
+
+    # Define drought threshold 
+    std = statistics.stdev(AnnualQ.iloc[:, 0])
+    threshold = np.mean(AnnualQ.iloc[:, 0] - (0.5 * std))
+    
+    # Find where the rolling mean dip below the threshold?
+    drought_instances = [i for i, v in enumerate(AnnualQ.iloc[:,0].rolling(11).mean()) if v < threshold]
+    drought_years = AnnualQ.iloc[:, 1].rolling(11).mean()[drought_instances]
+    
+    # Add labels and title
+    fig, ax = plt.subplots(figsize=(12, 8))
+    ax.plot(AnnualQ.iloc[:,1],
+            AnnualQ.iloc[:,0],
+            color='#005F73',
+            label='Annual') 
+    
+    ax.plot(AnnualQ.iloc[:,1].rolling(11,center=True).mean(),
+            AnnualQ.iloc[:,0].rolling(11,center=True).mean(),
+            color='#183A2E',
+            label='11-Year Rolling Mean')
+    
+    ax.axhline(y=threshold, 
+               color='black', 
+               linestyle='--',
+               label='Drought Threshold')
+    
+    # Visualize the drought periods as yellow rectangles
+    for i in drought_years:
+        
+        # Plot a box centered around those values and with 5 years on either side.
+        rect = patches.Rectangle((i-5,0), 11,2e7, linewidth=1, edgecolor='#EFE2BE', facecolor='#EFE2BE')
+        
+        # Add the patch to the Axes
+        ax.add_patch(rect)
+        
+        
+    plt.title("Upper Colorado Annual Flow")
+    ax.set_xlabel("Year", fontsize=16)
+    ax.set_ylabel("Annual Flow (cubic feet per year)", fontsize=16)
+    plt.xticks(fontsize=12)
+    plt.yticks(fontsize=12)
+    mpl.rc('legend', fontsize=16)
+    legend = plt.legend() 
+    plt.show()
+    plt.close()
+
+
+
+
+.. image:: _static/hmm_14_0.png
+
+
+By this metric, the Upper Colorado Basin region has experienced two
+decadal droughts over the last century.
+
+Synthetic Stationary Generator to Better Quantify Natural Variability
+---------------------------------------------------------------------
+
+It is important to remember that the streamflow that we have observed in
+the region over the last century is only one instance of the hydrology
+that could occur since the atmosphere is an inherently stochastic
+system. Thus, we require a tool that will allow us to see multiple
+plausible realizations of the streamflow record to understand the
+internal variability that characterizes the historical period. One
+observed realization of historical streamflow is limited in its ability
+to capture rare extremes; plausible (but not observed) alternative
+instances of streamflow records can help to fill this gap. The tool that
+we use to develop synthetic flows for the region is a Gaussian Hidden
+Markov Model (HMM). If a system follows a Markov process, it switches
+between a number of “hidden states” dictated by a transition matrix.
+Each state has its own Gaussian probability distribution (defined by a
+mean and standard deviation) and one can draw from this distribution to
+create synthetic flows that fit the properties of the historical
+distribution. HMMs are an attractive choice for this region because they
+can simulate persistence (i.e., long duration droughts), which is a
+characteristic of the region’s hydro-climatology. The figure below shows
+an example of a 2-state Gaussian HMM that we will be fitting for this
+example.
+
+.. raw:: html
+
+  <figure>
+      <center><img src="_static/HMM_example.png" style="width:75%"></center>
+  <figcaption style="text-align:center"> Two-state Gaussian HMM with mean and standard deviation parameters</figcaption>
+  </figure>
+
+Below is the code that fits the HMM model to the last 2/3 of the
+historical record of log annual flows at the CO-UT stateline gauge and
+creates an alternative trace of 105 years. A subset of the dataset is
+chosen in order to minimize overfitting and to retain a set of data for
+validation of the model. When we fit our model, we utilize the
+Baum-Welch algorithm (a special version of the expectation-maximization
+algorithm) to find the optimal parameters that maximize the likelihood
+of seeing the observed flows. Ultimately, the algorithm will return a
+mean and standard deviation associated with each state (mus and sigmas
+defined below) and a 2x2 transition probability matrix that captures the
+likelihood of transitioning between states (P). We can also retrieve the
+annual hidden states across the observed series, also known as the
+Viterbi sequence of states, which classifies each year in a “wet” or
+“dry” state.
+
+.. code:: ipython3
+
+    # Number of years for alternative trace
+    n_years = 105
+    
+    # Import historical data that it used to fit HMM model 
+    AnnualQ_h = pd.read_csv('data/uc_historical.csv')
+    
+    # Fit the model and pull out relevant parameters and samples 
+    logQ = np.log(AnnualQ_h)
+    hidden_states, mus, sigmas, P, logProb, samples, model = fitmodel.fitHMM(logQ, n_years)
+
+
+We’ve fit our HMM, but what does the model look like? Let’s plot the
+annual time series of hidden states, or the Viterbi sequence. In the
+code, above, we have defined that the drier state is always represented
+by state 0. Thus, we know that hidden_states = 0 corresponds to the dry
+state and hidden_states = 1 to the wet state.
+
+.. code:: ipython3
+
+    # Plot Vitebi sequence 
+    plotstates.plotTimeSeries(np.log(AnnualQ.iloc[:,0]), hidden_states, 'Annual Flow (cube feet per year)')
+
+
+
+
+.. image:: _static/hmm_21_0.png
+
+
+In the figure above, we see that the years with the higher log flows
+tend to be classified in a “wet” state and the opposite is true of the
+“dry” state. We can also print the transition matrix, which shows the
+likelihood of transitioning between states. Note that the system has a
+high likelihood of persisting in the same state.
+
+.. code:: ipython3
+
+    print(model.transmat_)
+
+
+.. parsed-literal::
+
+    [[0.65095026 0.34904974]
+     [0.3205531  0.6794469 ]]
+
+
+Let’s also plot the distribution of log annual flows associated with the
+wet and dry states.
+
+.. code:: ipython3
+
+    # Plot wet and dry state distributions
+    plotdist.plotDistribution(logQ, mus, sigmas, P)
+
+
+
+
+.. image:: _static/hmm_25_0.png
+
+
+The wet state distribution is characterized by a greater mean flow, but
+note that there is significant overlap in the tails of the distributions
+below which demonstrates why years with similiar flows can be classified
+in different states.
+
+Now let’s see what the drought dynamics look like in the synthetic
+scenario that we created using the same definition that we had used for
+the historical period.
+
+.. code:: ipython3
+
+    # Retrieve samples and back-transform out of log space
+    AnnualQ_s = np.exp(samples[0])
+    AnnualQ_s = pd.DataFrame(AnnualQ_s)
+    AnnualQ_s['Year'] = list(range(1909, 2014))
+    
+    # Define drought threshold 
+    std=statistics.stdev(AnnualQ_s.iloc[:, 0])
+    threshold=np.mean(AnnualQ_s.iloc[:, 0] - (0.5 * std))
+    
+    # Where does the rolling mean dip below the threshold
+    drought_instances = [i for i,v in enumerate(AnnualQ_s.iloc[:, 0].rolling(11).mean()) if v < threshold]
+    drought_years = AnnualQ_s.iloc[:, 1].rolling(11).mean()[drought_instances]
+    
+    #Visualize the streamflow scenario
+    fig, ax = plt.subplots(figsize=(12, 8))
+    
+    #Plot the original line graph
+    ax.plot(AnnualQ_s.iloc[:,1],
+            AnnualQ_s.iloc[:,0],
+            color='#005F73',
+            label='Annual')
+    
+    #Plot a 11-year rolling mean 
+    ax.plot(AnnualQ_s.iloc[:,1],
+            AnnualQ_s.iloc[:,0].rolling(11, center=True).mean(),
+            color='#183A2E',
+            label='11-Year Rolling Mean')
+    
+    # Add labels and title
+    ax.axhline(y=threshold, 
+               color='black', 
+               linestyle='--',
+               label='Drought Threshold')
+    
+    
+    for i in drought_years:
+        
+        #Plot a box centered around those values and with 5 years on either side.
+        rect = patches.Rectangle((i - 5,
+                                  0), 
+                                  11,
+                                  2e7, 
+                                  linewidth=1, 
+                                  edgecolor='#EFE2BE', 
+                                  facecolor='#EFE2BE')
+        
+        # Add the patch to the Axes
+        ax.add_patch(rect)
+        
+        
+    plt.title("Upper Colorado Annual Flow (Synthetic Stationary)",fontsize=16)
+    plt.xlabel("Year", fontsize=16)
+    plt.ylabel("Annual Flow (cubic feet per year)", fontsize=16)
+    mpl.rc('legend', fontsize=16)
+    plt.legend() 
+    plt.xticks(fontsize=12)
+    plt.yticks(fontsize=12)
+    plt.show()
+    plt.close()
+
+
+
+
+.. image:: _static/hmm_28_0.png
+
+
+You can sample from the model and create more 105-year traces and note
+how the location and number of decadal droughts changes. This
+demonstrates how different the historical record can look just within
+the range of natural variability. It’s also important to remember that
+when droughts occur can also define the ultimate effect of the drought
+(i.e. is it a time when there is a large population growth or a time
+when humans can adapt by conserving or building more infrastructure?). A
+hydrologic drought need not manifest into an agricultural or operational
+drought of the same magnitude if stored surface water is available.
+
+We externally run the HMM many times to create a dataset of 100
+instances of the 105-year traces and 1000 instances of the 105-year
+traces that are available in the package
+(“synthetic_stationary_small_sample_100.csv”,“synthetic_stationary_large_sample_1000”).
+The shaded green lines correspond to the flow duration curves (FDCs) for
+the generated streamflow traces in comparison with the FDC of the
+historical record in beige.
+
+.. raw:: html
+
+  <figure>
+      <center><img src="_static/stationary_synthetic_FDC.png" style="width:75%"></center>
+  <figcaption style="text-align:center">Generated streamflow traces in comparison with the FDC of the historical record.</figcaption>
+  </figure>
+
+As expected, the stationary synthetic FDCs envelope the historical FDC
+and particularly, the synthetic traces offer many more instances of low
+flow conditions that could lead to more extreme drought conditions than
+what has been observed historically. It is also useful to check for
+convergence of samples and to determine how many samples are needed to
+fully represent internal variability. Above we see that the extension to
+1000 instances of 105-year traces fills out regions of the FDC,
+including creating some more extreme drought conditions, but that
+additional samples will likely not fill out the FDC substantially more.
+
+Non-Stationary Synthetic Generator to Impose Climate Changes
+------------------------------------------------------------
+
+Now, we create flows under non-stationary conditions to get a better
+understanding of what flows can look like under climate changes. In
+order to create flows under non-stationary conditions, we can toggle the
+parameters of the HMM model in order to create systematic changes to the
+model that can represent a changing climate. The HMM has 6 parameters
+that define it. When we fit the historical model, the parameters that
+are fit represent a baseline parameter value. In this non-stationary
+generator, we define a range to sample these parameters from.
+
++-----------------------+---------------+-------------+-------------+
+| Parameter             | Current Value | Lower Bound | Upper Bound |
++=======================+===============+=============+=============+
+| Log-Space Wet State   | 1.00          | 0.98        | 1.02        |
+| Mean Multiplier       |               |             |             |
++-----------------------+---------------+-------------+-------------+
+| Log-Space Dry State   | 1.00          | 0.98        | 1.02        |
+| Mean Multiplier       |               |             |             |
++-----------------------+---------------+-------------+-------------+
+| Log-Space Wet State   | 1.00          | 0.75        | 1.25        |
+| Standard Deviation    |               |             |             |
+| Multiplier            |               |             |             |
++-----------------------+---------------+-------------+-------------+
+| Log-Space Dry State   | 1.00          | 0.75        | 1.25        |
+| Standard Deviation    |               |             |             |
+| Multiplier            |               |             |             |
++-----------------------+---------------+-------------+-------------+
+| Change in Dry-Dry     | 0.00          | -0.30       | +0.30       |
+| Transition            |               |             |             |
+| Probability           |               |             |             |
++-----------------------+---------------+-------------+-------------+
+| Change in Wet-Wet     | 0.00          | -0.30       | +0.30       |
+| Transition            |               |             |             |
+| Probability           |               |             |             |
++-----------------------+---------------+-------------+-------------+
+
+Now let’s sample 1000 times from these bounds to create 1000 new
+parameterizations of the model. Here we use SALib and the Latin
+Hypercube sample function.
+
+.. code:: ipython3
+
+    # Create problem structure with parameters that we want to sample 
+    problem = {
+        'num_vars': 6,
+        'names': ['wet_mu', 'dry_mu', 'wet_std','dry_std','dry_tp',"wet_tp"],
+        'bounds': [[0.98, 1.02],
+                   [0.98, 1.02],
+                   [0.75,1.25],
+                   [0.75,1.25],
+                   [-0.3,0.3],
+                   [-0.3,0.3]]
+    }
+    
+    # generate 1000 parameterizations
+    n_samples = 1000
+    
+    # set random seed for reproducibility
+    seed_value = 123
+    
+    # Generate our samples 
+    LHsamples = latin.sample(problem, n_samples, seed_value)
+
+
+Now let’s look at what some of the traces look like in our
+non-stationary generator. Let’s choose a random instance from the
+1000-member space and adjust the parameters accordingly.
+
+.. code:: ipython3
+
+    # Define static parameters 
+    n_years = 105
+    
+    # Sample parameter; Adjust to any sample number from 0-999 
+    sample = 215 
+    
+    # Create empty arrays to store the new Gaussian HMM parameters for each SOW
+    Pnew = np.empty([2,2])
+    piNew = np.empty([2])
+    musNew_HMM = np.empty([2])
+    sigmasNew_HMM = np.empty([2])
+    logAnnualQ_s = np.empty([n_years]) 
+    
+    # Calculate new transition matrix and stationary distribution of SOW at last node as well as new means and standard deviations
+    Pnew[0, 0] = max(0.0, min(1.0, P[0, 0] + LHsamples[sample][4]))
+    Pnew[1, 1] = max(0.0, min(1.0, P[1, 1] + LHsamples[sample][5]))
+    Pnew[0, 1] = 1 - Pnew[0, 0]
+    Pnew[1, 0] = 1 - Pnew[1, 1]
+    eigenvals, eigenvecs = np.linalg.eig(np.transpose(Pnew))
+    one_eigval = np.argmin(np.abs(eigenvals - 1))
+    piNew = np.divide(np.dot(np.transpose(Pnew), eigenvecs[:, one_eigval]), 
+                      np.sum(np.dot(np.transpose(Pnew), eigenvecs[:,one_eigval])))
+    
+    musNew_HMM[0] = mus[0] * LHsamples[sample][1]
+    musNew_HMM[1] = mus[1] * LHsamples[sample][0]
+    sigmasNew_HMM[0] = sigmas[0] * LHsamples[sample][3]
+    sigmasNew_HMM[1] = sigmas[1] * LHsamples[sample][2]
+    
+    # Generate first state and log-space annual flow at last node
+    states = np.empty([n_years])
+    if random() <= piNew[0]:
+        states[0] = 0
+        logAnnualQ_s[0] = ss.norm.rvs(musNew_HMM[0], sigmasNew_HMM[0])
+    else:
+        states[0] = 1
+        logAnnualQ_s[0] = ss.norm.rvs(musNew_HMM[1], sigmasNew_HMM[1])
+    
+    # Generate remaining state trajectory and log space flows at last node
+    for j in range(1, n_years):
+        if random() <= Pnew[int(states[j-1]), int(states[j-1])]:
+            states[j] = states[j-1]
+        else:
+            states[j] = 1 - states[j-1]
+    
+        if states[j] == 0:
+            logAnnualQ_s[j] = ss.norm.rvs(musNew_HMM[0], sigmasNew_HMM[0])
+        else:
+            logAnnualQ_s[j] = ss.norm.rvs(musNew_HMM[1], sigmasNew_HMM[1])
+    
+    # Convert log-space flows to real-space flows
+    AnnualQ_s = np.exp(logAnnualQ_s)-1
+
+
+Now let’s see what this synthetic trace looks like.
+
+.. code:: ipython3
+
+    # Retrieve samples and back-transform out of log space
+    AnnualQ_s = pd.DataFrame(AnnualQ_s)
+    AnnualQ_s['Year'] = list(range(1909, 2014))
+    
+    # Define drought threshold 
+    std = statistics.stdev(AnnualQ_s.iloc[:, 0])
+    threshold = np.mean(AnnualQ_s.iloc[:, 0] - (0.5 * std))
+    
+    # Where does the rolling mean dip below the threshold
+    drought_instances = [i for i, v in enumerate(AnnualQ_s.iloc[:, 0].rolling(11).mean()) if v < threshold]
+    drought_years = AnnualQ_s.iloc[:, 1].rolling(11).mean()[drought_instances]
+    
+    # Visualize the streamflow scenario
+    fig, ax = plt.subplots(figsize=(12, 8))
+    
+    # Plot the original line graph
+    ax.plot(AnnualQ_s.iloc[:,1],
+            AnnualQ_s.iloc[:,0],
+            color='#005F73',
+            label='Annual')
+    
+    # Plot a 11-year rolling mean 
+    ax.plot(AnnualQ_s.iloc[:, 1],
+            AnnualQ_s.iloc[:, 0].rolling(11, center=True).mean(),
+            color='#183A2E',
+            label='11-Year Rolling Mean')
+    
+    # Add labels and title
+    ax.axhline(y=threshold, 
+               color='black', 
+               linestyle='--',
+               label='Drought Threshold')
+    
+    
+    for i in drought_years:
+        
+        # Plot a box centered around those values and with 5 years on either side.
+        rect = patches.Rectangle((i - 5,0), 
+                                 11,
+                                 2e7, 
+                                 linewidth=1, 
+                                 edgecolor='#EFE2BE', 
+                                 facecolor='#EFE2BE')
+        
+        # Add the patch to the Axes
+        ax.add_patch(rect)
+        
+        
+    plt.title("Annual Flow (Synthetic Non-Stationary)", fontsize=16)
+    plt.xlabel("Year", fontsize=16)
+    plt.ylabel("Annual Flow (cubic feet per year)", fontsize=16)
+    plt.xticks(fontsize=12)
+    plt.yticks(fontsize=12)
+    mpl.rc('legend', fontsize=16)
+    legend = plt.legend(loc="upper right") 
+    plt.show()
+    plt.close()
+    
+
+
+
+
+.. image:: _static/hmm_40_0.png
+
+
+Above is the example trace from the new non-stationary model. You may
+see fewer or more decadal drought instances. We can further summarize
+overall decadal drought characteristics across the samples. Let’s plot a
+histogram of the total number of times we go below the drought threshold
+across these realizations.
+
+.. code:: ipython3
+
+    decadal_drought_occurence=np.empty([1000]) 
+    
+    for y in range(1000):
+        
+        # Create empty arrays to store the new Gaussian HMM parameters for each SOW
+        Pnew = np.empty([2, 2])
+        piNew = np.empty([2])
+        musNew_HMM = np.empty([2])
+        sigmasNew_HMM = np.empty([2])
+        logAnnualQ_s = np.empty([n_years]) 
+        
+        # Calculate new transition matrix and stationary distribution of SOW at last node
+        # as well as new means and standard deviations
+    
+        Pnew[0, 0] = max(0.0,min(1.0, P[0, 0] + LHsamples[y][4]))
+        Pnew[1, 1] = max(0.0,min(1.0, P[1, 1] + LHsamples[y][5]))
+        Pnew[0, 1] = 1 - Pnew[0, 0]
+        Pnew[1, 0] = 1 - Pnew[1, 1]
+        eigenvals, eigenvecs = np.linalg.eig(np.transpose(Pnew))
+        one_eigval = np.argmin(np.abs(eigenvals - 1))
+        piNew = np.divide(np.dot(np.transpose(Pnew), eigenvecs[:, one_eigval]),
+                          np.sum(np.dot(np.transpose(Pnew), eigenvecs[:, one_eigval])))
+    
+        musNew_HMM[0] = mus[0] * LHsamples[y][1]
+        musNew_HMM[1] = mus[1] * LHsamples[y][0]
+        sigmasNew_HMM[0] = sigmas[0] * LHsamples[y][3]
+        sigmasNew_HMM[1] = sigmas[1] * LHsamples[y][2]
+    
+        # Generate first state and log-space annual flow at last node
+        states = np.empty([n_years])
+        if random() <= piNew[0]:
+            states[0] = 0
+            logAnnualQ_s[0] = ss.norm.rvs(musNew_HMM[0], sigmasNew_HMM[0])
+        else:
+            states[0] = 1
+            logAnnualQ_s[0] = ss.norm.rvs(musNew_HMM[1], sigmasNew_HMM[1])
+    
+        # generate remaining state trajectory and log space flows at last node
+        for j in range(1, n_years):
+            if random() <= Pnew[int(states[j-1]), int(states[j-1])]:
+                states[j] = states[j-1]
+            else:
+                states[j] = 1 - states[j-1]
+    
+            if states[j] == 0:
+                logAnnualQ_s[j] = ss.norm.rvs(musNew_HMM[0], sigmasNew_HMM[0])
+            else:
+                logAnnualQ_s[j] = ss.norm.rvs(musNew_HMM[1], sigmasNew_HMM[1])
+    
+        # Convert log-space flows to real-space flows
+        AnnualQ_s = np.exp(logAnnualQ_s) - 1
+        AnnualQ_s = pd.DataFrame(AnnualQ_s)
+        AnnualQ_s['Year'] = list(range(1909, 2014))
+    
+        # Define drought threshold 
+        std = statistics.stdev(AnnualQ_s.iloc[:, 0])
+        threshold = np.mean(AnnualQ_s.iloc[:, 0] - (0.5 * std))
+    
+        # Where does the rolling mean dip below the threshold
+        drought_instances = [i for i, v in enumerate(AnnualQ_s.iloc[:, 0].rolling(11).mean()) if v < threshold]
+        decadal_drought_occurence[y] = len(drought_instances)
+        
+
+.. code:: ipython3
+
+    fig, ax = plt.subplots(figsize=(12, 8))
+    ax.hist(decadal_drought_occurence,label='Non-Stationary generator',color="#005F73")
+    ax.set_xlabel('Number of Instances of Decadal Drought',fontsize=16)
+    ax.set_ylabel('Frequency',fontsize=16)
+    ax.axvline(x=2, color='r', linestyle='-',label='Observed')
+    mpl.rc('legend', fontsize = 16)
+    plt.xticks(fontsize = 12)
+    plt.yticks(fontsize = 12)
+    plt.show()
+    plt.close()
+
+
+
+
+.. image:: _static/hmm_43_0.png
+
+
+Note how many more instances of the decadal droughts we are creating
+with the non-stationary generator than our observed 105-year trace which
+creates a rich space in which we can test our models. Just as we did
+with the stationary generator, we can externally run the non-stationary
+generator to create 10,000 instances of the 105-year traces that are
+available in the package
+(“synthetic_nonstationary_large_sample_10000.csv”). The shaded green and
+blue lines correspond to the FDCs for the stationary and non-stationary
+generated streamflow traces in comparison with the FDC of the historical
+record in beige. Note how the non-stationary generator produces even
+more drought extremes than the stationary non-synthetic traces.
+
+.. raw:: html
+
+  <figure>
+      <center><img src="_static/nonstationary_synthetic_FDC.png" style="width:75%"></center>
+  <figcaption style="text-align:center">Generated streamflow traces in comparison with the FDC of the historical record.</figcaption>
+  </figure>
+
+Placing CMIP5 Projections in the Context of Non-Stationary Flows
+----------------------------------------------------------------
+
+We have broadened the drought conditions that we are creating which that
+can be very useful to understand how our water systems model performs
+under potentially extreme scenarios. However, it’s useful to compare our
+bottom-up synthetically generated flows in the context of global
+physically-driven CMIP5 projections to get a better understanding of how
+the two approaches compare. We first aquire 97 CMIP5 projections from
+the Colorado River Water Availability Study (CWCB, 2012). In each of
+these projections, monthly precipitation factor changes and temperature
+delta changes were computed between mean projected 2035–2065 climate
+statistics and mean historical climate statistics from 1950–2013. These
+97 different combinations of 12 monthly precipitation multipliers and 12
+monthly temperature delta shifts were applied to historical
+precipitation and temperature time series from 1950–2013. The resulting
+climate time series were run through a Variable Infiltration Capacity
+(VIC) model of the UCRB, resulting in 97 time series of projected future
+streamflows at the Colorado‐Utah state line.
+
+We fit an HMM to each trace of projected streamflow and get a set of
+corresponding HMM parameters. Then we take the ratio between these
+parameters and the baseline HMM parameters that we calculated earlier in
+the notebook in order to calculate the multipliers associated with each
+CMIP5 projection. This is all done externally, so we import the
+resulting multipliers in the next line.
+
+.. code:: ipython3
+
+    # Read in CMIP5 and paleo multipliers
+    CMIP5_multipliers = pd.read_csv('data/CMIP5_SOWs.txt', header=None, sep=" ")
+
+
+Let’s plot a response surface that will allow us to see how combinations
+of HMM parameters tend to influence decadal drought. In order to get a
+continuous surface, we’ll fit a non-linear regression to the parameter
+values and then predict the decadal drought over a set of grid points.
+We fit the response surface for two parameters that should have an
+affect on decadal drought: the dry distribution mean and the dry-dry
+transition probabilites.
+
+.. code:: ipython3
+
+    # Choose two parameters to fit the response surface for
+    mu_dry=[i[1] for i in LHsamples]
+    tp_dry=[i[4] for i in LHsamples]
+    
+    # Create an interpolation grid
+    xgrid = np.arange(np.min(mu_dry),
+                      np.max(mu_dry), 
+                      (np.max(mu_dry) - np.min(mu_dry)) / 100)
+    
+    ygrid = np.arange(np.min(tp_dry),
+                      np.max(tp_dry), 
+                      (np.max(tp_dry) - np.min(tp_dry)) / 100)
+    
+    # Fit regression 
+    d = {'Dry_Tp': tp_dry, 
+         'Dry_Mu': mu_dry,
+         'Drought_Occurrence':decadal_drought_occurence}
+    
+    df = pd.DataFrame(d)
+    df['Intercept'] = np.ones(np.shape(df)[0])
+    df['Interaction'] = df['Dry_Tp'] * df['Dry_Mu']
+    cols = ['Intercept'] + ['Dry_Mu'] + ['Dry_Tp'] + ['Interaction']
+    ols = sm.OLS(df['Drought_Occurrence'], df[cols])
+    result = ols.fit()
+    
+    # Calculate drought occurence for each grid point
+    X, Y = np.meshgrid(xgrid, ygrid)
+    x = X.flatten()
+    y = Y.flatten()
+    grid = np.column_stack([np.ones(len(x)), x, y, x * y])
+    z = result.predict(grid)
+    z[z < 0.0] = 0.0 # replace negative shortage predictions with 0
+
+
+Let’s plot our results:
+
+.. code:: ipython3
+
+    # Set color gradient for response surface 
+    drought_map = mpl.cm.get_cmap('RdBu_r')
+    
+    # Reshape our predicted drought occurrence and define bounds of colors 
+    Z = np.reshape(z, np.shape(X))
+    vmin = np.min([np.min(z), np.min(df['Drought_Occurrence'].values)])
+    vmax = 15
+    norm = mpl.colors.Normalize(vmin, vmax)
+    
+    # Plot response surface and CMIP5 projections
+    fig, ax = plt.subplots(figsize=(12, 8))
+    ax.contourf(X, Y, Z, cmap=drought_map, norm=norm)
+    ax.scatter(CMIP5_multipliers.iloc[:,7],
+               CMIP5_multipliers.iloc[:,12], 
+               c='#ffffb3', 
+               edgecolor='none', 
+               s=30)
+    cbar = ax.figure.colorbar(mpl.cm.ScalarMappable(norm=norm, cmap=drought_map), ax=ax)
+    ax.set_xlim(np.nanmin(X), np.nanmax(X))
+    ax.set_ylim(np.nanmin(Y), np.nanmax(Y))
+    ax.set_xlabel('Dry State Mu', fontsize=14)
+    ax.set_ylabel('Dry-Dry Transition Probability', fontsize=14)
+    ax.tick_params(axis='both', labelsize=14)
+    cbar.ax.set_ylabel('Decadal Drought Occurrence', rotation=-90, fontsize=14, labelpad=15)
+    cbar.ax.tick_params(axis='y',labelsize=14)
+    plt.show()
+    plt.close()
+    
+
+
+
+
+.. image:: _static/hmm_53_0.png
+
+
+We see the influence of the dry state mean and dry-dry transition
+parameters. We’re likely to see more decadal droughts when we (1)
+increase the dry-dry transition probability, which inherently will
+increase persistence of the dry state, and (2) when we make the dry
+state log mean drier. Note that the CMIP5 scenarios tend to span the
+extent of the dry mean sample space, but are less representative of the
+dry transition probability sample space, which suggests that the types
+of hydrological droughts represented in the projections tend to only be
+wetter to slightly drier than our baseline. Both methods of producing
+these scenarios are valid, though studies have suggested that
+globally-resolved GCMs may be inappropriate to represent regional
+extremes. Ultimately, if your goal is to produce a variety of ensembles
+that are characterized by many different drought characteristics, you
+will likely find that a generator approach will serve this purpose
+better.
+
+Tips to Create an HMM-Based Generator for your System
+-----------------------------------------------------
+
+In this tutorial, we demonstrated how to fit an HMM-based generator for
+a single gauge located in the Upper Colorado River Basin. In order to
+apply this methodology to your problem, you will need to first ask:
+
+(1) Is this model appropriate for my location of interest? We have
+    applied this style of generator to locations where persistent wet
+    and dry states are characteristic, which tends to be in the Western
+    US. Ultimately the best way to judge if an HMM is useful for your
+    application is to fit the model and explore the resulting
+    distributions. Are there two (or more) distinct states that emerge?
+    If not, then your location may not exhibit the type of persistence
+    that an HMM-based generator is useful for. You can consider
+    exploring other styles of generators such as the Kirsch-Nowak
+    generator (Kirsch et al., 2013).
+
+(2) Do I have the right datasets? We use annual data for our location of
+    interest. In this notebook, the HMM is fit to log annual flows.
+    Ultimately, it can be disaggregated to daily flows (using a
+    reference historical daily dataset) to be useful in water resources
+    operational applications. You could also disaggregate to a finer
+    resolution than daily if the historical dataset exists.
+
+If you meet these requirements, feel free to proceed through fitting the
+model using the code available in the notebook. Be sure to consider the
+appropirate number of samples to generate (both in a stationary and
+non-stationary case). Make sure that you test multiple sample sizes and
+continue to increase your sample size until you converge to a consistent
+representation of extremes. What is the appropriate number of LHS
+samples of the parameters to use? In this experiment we used 1,000
+samples of parameters due to extensive stability tests described in
+Quinn et al. (2020).
+
+Finally, to learn more about this test case refer to Hadmichael et
+al. (2020a) and Hadmichael et al. (2020b). For another study on
+synthetic drought generation to support vulnerability assessments in the
+Research Triangle region of North Carolina, please refer to Herman et
+al. (2016)
+
+References
+----------
+
+Ault, T. R., Cole, J. E., Overpeck, J. T., Pederson, G. T., & Meko, D.
+M. (2014). Assessing the risk of persistent drought using climate model
+simulations and paleoclimate data. Journal of Climate, 27(20),
+7529-7549.
+
+CWCB (2012).Colorado River Water Availability Study Phase I Report.
+Colorado Water Conservation Board
+
+Hadjimichael, A., Quinn, J., Wilson, E., Reed, P., Basdekas, L., Yates,
+D., & Garrison, M. (2020a). Defining robustness, vulnerabilities, and
+consequential scenarios for diverse stakeholder interests in
+institutionally complex river basins. Earth’s Future, 8(7),
+e2020EF001503.
+
+Hadjimichael, A., Quinn, J., & Reed, P. (2020). Advancing diagnostic
+model evaluation to better understand water shortage mechanisms in
+institutionally complex river basins. Water Resources Research, 56(10),
+e2020WR028079.
+
+Herman, J. D., Zeff, H. B., Lamontagne, J. R., Reed, P. M., &
+Characklis, G. W. (2016). Synthetic drought scenario generation to
+support bottom-up water supply vulnerability assessments. Journal of
+Water Resources Planning and Management, (11), 04016050.
+
+Kirsch, B. R., Characklis, G. W., & Zeff, H. B. (2013). Evaluating the
+impact of alternative hydro-climate scenarios on transfer agreements:
+Practical improvement for generating synthetic streamflows. Journal of
+Water Resources Planning and Management, 139(4), 396-406.
+
+Overpeck, J.T. & Udall, B. (2020) “Climate change and the aridification
+of North America.” Proceedings of the national academy of sciences
+117.22 11856-11858.
+
+Quinn, J. D., Hadjimichael, A., Reed,P. M., & Steinschneider, S. (2020).
+Canexploratory modeling of water scarcity vulnerabilities and robustness
+bescenario neutral?Earth’s Future,8,e2020EF001650.
+https://doi.org/10.1029/2020EF001650Received
+
+Williams, A. P., Cook, B. I., & Smerdon, J. E. (2022). Rapid
+intensification of the emerging southwestern North American megadrought
+in 2020–2021. Nature Climate Change, 12(3), 232-234.
diff --git a/docs/html/_sources/A2_Jupyter_Notebooks.rst b/docs/html/_sources/A2_Jupyter_Notebooks.rst
new file mode 100644
index 0000000..cfa77c0
--- /dev/null
+++ b/docs/html/_sources/A2_Jupyter_Notebooks.rst
@@ -0,0 +1,12 @@
+.. _A2_Jupyter_Notebooks:
+
+**************************
+Jupyter Notebook Tutorials
+**************************
+
+.. include:: A2.1_fishgame.rst
+.. include:: A2.2_saltelli.rst
+.. include:: A2.3_logistic.rst
+.. include:: A2.4_hymod.rst
+.. include:: A2.5_discovery.rst
+.. include:: A2.6_hmm.rst
diff --git a/docs/html/_sources/A3_plotting_code.rst b/docs/html/_sources/A3_plotting_code.rst
new file mode 100644
index 0000000..f3b3499
--- /dev/null
+++ b/docs/html/_sources/A3_plotting_code.rst
@@ -0,0 +1,549 @@
+Plotting Code Samples
+*********************
+
+hymod.ipynb
+---------------------------------------
+
+The following are the plotting functions as described in the ``hymod.ipynb`` Jupyter notebook tutorial.
+
+The following are the necessary package imports to run these functions:
+
+.. code-block:: python
+
+    import numpy as np
+    import seaborn as sns
+    import matplotlib.pyplot as plt
+
+    from matplotlib.lines import Line2D
+
+
+**plot_observed_vs_simulated_streamflow()**
+___________________________________________
+
+.. code-block:: python
+
+    def plot_observed_vs_simulated_streamflow(df, hymod_dict, figsize=[12, 6]):
+        """Plot observed versus simulated streamflow.
+
+        :param df:              Dataframe of hymod input data including columns for precip, potential evapotranspiration,
+                                and streamflow
+
+        :param hymod_dict:      A dictionary of hymod outputs
+        :type hymod_dict:       dict
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        # set plot style
+        plt.style.use('seaborn-white')
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # plot observed streamflow
+        ax.plot(range(0, len(df['Strmflw'])), df['Strmflw'], color='pink')
+
+        # plot simulated streamflow
+        ax.plot(range(0, len(df['Strmflw'])), hymod_dict['Q'], color='black')
+
+        # set axis labels
+        ax.set_ylabel('Streamflow($m^3/s$)')
+        ax.set_xlabel('Days')
+
+        # set plot title
+        plt.title('Observed vs. Simulated Streamflow')
+
+        return ax
+
+
+**plot_observed_vs_sensitivity_streamflow()**
+_____________________________________________
+
+.. code-block:: python
+
+    def plot_observed_vs_sensitivity_streamflow(df_obs, df_sim, figsize=[10, 4]):
+        """Plot observed streamflow versus simulations generated from sensitivity analysis.
+
+        :param df_obs:          Dataframe of mean monthly hymod input data including columns for precip,
+                                potential evapotranspiration, and streamflow
+
+        :param df_sim:          Dataframe of mean monthly simulation data from sensitivity analysis
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        month_list = range(len(df_sim))
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # set labels
+        ax.set_xlabel('Days')
+        ax.set_ylabel('Flow Discharge (m^3/s)')
+
+        # plots all simulated streamflow cases under different sample sets
+        for i in df_sim.columns:
+            plt.plot(month_list, df_sim[i], color="pink", alpha=0.2)
+
+        # plot observed streamflow
+        plt.plot(month_list, df_obs['Strmflw'], color="black")
+
+        plt.title('Observed vs. Sensitivity Analysis Outputs')
+
+        return ax
+
+
+**plot_monthly_heatmap()**
+__________________________
+
+.. code-block:: python
+
+    def plot_monthly_heatmap(arr_sim, df_obs, title='', figsize=[14, 6]):
+        """Plot a sensitivity metric overlain by observed flow.
+
+        :param arr_sim:         Numpy array of simulated metrics
+
+        :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis
+
+        :param title:           Title of plot
+        :type title:            str
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # plot heatmap
+        sns.heatmap(arr_sim,
+                    ax=ax,
+                    yticklabels=['Kq', 'Ks', 'Alp', 'Huz', 'B'],
+                    cmap=sns.color_palette("ch:s=-.2,r=.6"))
+
+        # setup overlay axis
+        ax2 = ax.twinx()
+
+        # plot line
+        ax2.plot(np.arange(0.5, 12.5), df_obs['Strmflw'], color='slateblue')
+
+        # plot points on line
+        ax2.plot(np.arange(0.5, 12.5), df_obs['Strmflw'], color='slateblue', marker='o')
+
+        # set axis limits and labels
+        ax.set_ylim(0, 5)
+        ax.set_xlim(0, 12)
+        ax.set_xticklabels(['jan', 'feb', 'mar', 'apr', 'may', 'jun', 'jul', 'aug', 'sep', 'oct', 'nov', 'dec'])
+        ax2.set_ylabel('Flow Discharge($m^3/s$)')
+
+        plt.title(title)
+
+        plt.show()
+
+        return ax, ax2
+
+
+**plot_annual_heatmap()**
+__________________________
+
+.. code-block:: python
+
+    def plot_annual_heatmap(arr_sim, df_obs, title='', figsize=[14,5]):
+        """Plot a sensitivity metric overlain by observed flow..
+
+        :param arr_sim:         Numpy array of simulated metrics
+
+        :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis
+
+        :param title:           Title of plot
+        :type title:            str
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # plot heatmap
+        sns.heatmap(arr_sim, ax=ax, cmap=sns.color_palette("YlOrBr"))
+
+        # setup overlay axis
+        ax2 = ax.twinx()
+
+        # plot line
+        ax2.plot(np.arange(0.5, 10.5), df_obs['Strmflw'], color='slateblue')
+
+        # plot points on line
+        ax2.plot(np.arange(0.5, 10.5), df_obs['Strmflw'], color='slateblue', marker='o')
+
+        # set up axis lables and limits
+        ax.set_ylim(0, 5)
+        ax.set_xlim(0, 10)
+        ax.set_yticklabels(['Kq', 'Ks', 'Alp', 'Huz', 'B'])
+        ax.set_xticklabels(range(2000, 2010))
+        ax2.set_ylabel('Flow Discharge($m^3/s$)')
+
+        plt.title(title)
+
+        return ax, ax2
+
+
+**plot_varying_heatmap()**
+___________________________
+
+.. code-block:: python
+
+    def plot_varying_heatmap(arr_sim, df_obs, title='', figsize=[14,5]):
+        """Plot a sensitivity metric overlain by observed flow..
+
+        :param arr_sim:         Numpy array of simulated metrics
+
+        :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis
+
+        :param title:           Title of plot
+        :type title:            str
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # plot heatmap
+        sns.heatmap(arr_sim,
+                    ax=ax,
+                    yticklabels=['Kq', 'Ks', 'Alp', 'Huz', 'B'],
+                    cmap=sns.light_palette("seagreen", as_cmap=True))
+
+        n_years = df_obs.shape[0]
+
+        # setup overlay axis
+        ax2 = ax.twinx()
+
+        # plot line
+        ax2.plot(range(0, n_years), df_obs['Strmflw'], color='slateblue')
+
+        # plot points on line
+        ax2.plot(range(0, n_years), df_obs['Strmflw'], color='slateblue', marker='o')
+
+        # set up axis lables and limits
+        ax.set_ylim(0, 5)
+        ax.set_xlim(-0.5, 119.5)
+        ax2.set_ylabel('Flow Discharge')
+        ax.set_xlabel('Number of Months')
+
+        plt.title(title)
+
+        return ax, ax2
+
+
+**plot_precalibration_flow()**
+_______________________________
+
+.. code-block:: python
+
+    def plot_precalibration_flow(df_sim, df_obs, figsize=[10, 4]):
+        """Plot flow discharge provided by the ensemble of parameters sets from Pre-Calibration versus the observed
+        flow data.
+
+        :param df_sim:          Dataframe of simulated metrics
+
+        :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # set axis labels
+        ax.set_xlabel('Days')
+        ax.set_ylabel('Flow Discharge')
+
+        # plot pre-calibration results
+        for i in range(df_sim.shape[1]):
+            plt.plot(range(len(df_sim)), df_sim.iloc[:, i],  color="lightgreen", alpha=0.2)
+
+        # plot observed
+        plt.plot(range(len(df_sim)), df_obs['Strmflw'],  color="black")
+
+        plt.title('Observed vs. Pre-Calibration Outputs')
+
+        # customize legend
+        custom_lines = [Line2D([0], [0],  color="lightgreen", lw=4),
+                        Line2D([0], [0], color="black", lw=4)]
+        plt.legend(custom_lines, ['Pre-Calibration', 'Observed'])
+
+        return ax
+
+
+**plot_precalibration_glue()**
+_______________________________
+
+.. code-block:: python
+
+    def plot_precalibration_glue(df_precal, df_glue, df_obs, figsize=[10, 4]):
+        """Plot flow discharge provided by the ensemble of parameters sets from Pre-Calibration versus the observed
+        flow data.
+
+        :param df_sim:          Dataframe of simulated metrics
+
+        :param df_obs:          Dataframe of mean monthly observed data from sensitivity analysis
+
+        :param figsize:         Matplotlib figure size
+        :type figsize:          list
+
+        """
+
+        # set up figure
+        fig, ax = plt.subplots(figsize=figsize)
+
+        # set axis labels
+        ax.set_xlabel('Days')
+        ax.set_ylabel('Flow Discharge')
+
+        # plot pre-calibration results
+        for i in range(df_precal.shape[1]):
+            plt.plot(range(len(df_precal)), df_precal.iloc[:, i],  color="lightgreen", alpha=0.2)
+
+        # plot glue
+        for i in range(df_glue.shape[1]):
+            plt.plot(range(len(df_glue)), df_glue.iloc[:, i], color="lightblue", alpha=0.2)
+
+        # plot observed
+        plt.plot(range(len(df_precal)), df_obs['Strmflw'],  color="black")
+
+        plt.title('Observed vs. Sensitivity Analysis Outputs across GLUE/Pre-Calibration')
+
+        # customize legend
+        custom_lines = [Line2D([0], [0],  color="lightgreen", lw=4),
+                        Line2D([0], [0], color="lightblue", lw=4),
+                        Line2D([0], [0], color="black", lw=4)]
+        plt.legend(custom_lines, ['Pre-Calibration', 'GLUE', 'Observed'])
+
+        return ax
+
+
+fishery_dynamics.ipynb
+---------------------------------------
+
+The following are the plotting functions as described in the ``fishery_dynamics.ipynb`` Jupyter notebook tutorial.
+
+The following are the necessary package imports to run these functions:
+
+.. code-block:: python
+
+    import numpy as np
+    import matplotlib.pyplot as plt
+
+    from matplotlib import patheffects as pe
+
+
+**plot_objective_performance()**
+_________________________________
+
+.. code-block:: python
+
+    def plot_objective_performance(objective_performance, profit_solution, robust_solution, figsize=(18, 9)):
+        """Plot the identified solutions with regards to their objective performance
+        in a parallel axis plot
+
+        :param objective_performance:               Objective performance array
+        :param profit_solution:                     Profitable solutions array
+        :param robust_solution:                     Robust solutions array
+        :param figsize:                             Figure size
+        :type figsize:                              tuple
+
+        """
+
+        # create the figure object
+        fig = plt.figure(figsize=figsize)
+
+        # set up subplot axis object
+        ax = fig.add_subplot(1, 1, 1)
+
+        # labels where constraint is always 0
+        objs_labels = ['Net present\nvalue (NPV)',
+                       'Prey population deficit',
+                       'Longest duration\nof low harvest',
+                       'Worst harvest instance',
+                       'Variance of harvest',
+                       'Duration of predator\npopulation collapse']
+
+        # normalization across objectives
+        mins = objective_performance.min(axis=0)
+        maxs = objective_performance.max(axis=0)
+        norm_reference = objective_performance.copy()
+
+        for i in range(5):
+            mm = objective_performance[:, i].min()
+            mx = objective_performance[:, i].max()
+            if mm != mx:
+                norm_reference[:, i] = (objective_performance[:, i] - mm) / (mx - mm)
+            else:
+                norm_reference[:, i] = 1
+
+        # colormap from matplotlib
+        cmap = plt.cm.get_cmap("Blues")
+
+        # plot all solutions
+        for i in range(len(norm_reference[:, 0])):
+            ys = np.append(norm_reference[i, :], 1.0)
+            xs = range(len(ys))
+            ax.plot(xs, ys, c=cmap(ys[0]), linewidth=2)
+
+        # to highlight robust solutions
+        ys = np.append(norm_reference[profit_solution, :], 1.0)  # Most profitable
+        xs = range(len(ys))
+        l1 = ax.plot(xs[0:6],
+                     ys[0:6],
+                     c=cmap(ys[0]),
+                     linewidth=3,
+                     label='Most robust in NPV',
+                     path_effects=[pe.Stroke(linewidth=6, foreground='darkgoldenrod'), pe.Normal()])
+
+        ys = np.append(norm_reference[robust_solution, :], 1.0)  # Most robust in all criteria
+        xs = range(len(ys))
+        l2 = ax.plot(xs[0:6],
+                     ys[0:6],
+                     c=cmap(ys[0]),
+                     linewidth=3,
+                     label='Most robust across criteria',
+                     path_effects=[pe.Stroke(linewidth=6, foreground='gold'), pe.Normal()])
+
+        # build colorbar
+        sm = plt.cm.ScalarMappable(cmap=cmap)
+        sm.set_array([objective_performance[:, 0].min(), objective_performance[:, 0].max()])
+        cbar = fig.colorbar(sm)
+        cbar.ax.set_ylabel("\nNet present value (NPV)")
+
+        # tick values
+        minvalues = ["{0:.3f}".format(mins[0]),
+                     "{0:.3f}".format(-mins[1]),
+                     str(-mins[2]),
+                     "{0:.3f}".format(-mins[3]),
+                     "{0:.2f}".format(-mins[4]),
+                     str(0)]
+
+        maxvalues = ["{0:.2f}".format(maxs[0]),
+                     "{0:.3f}".format(-maxs[1]),
+                     str(-maxs[2]),
+                     "{0:.2f}".format(maxs[3]),
+                     "{0:.2f}".format(-maxs[4]),
+                     str(0)]
+
+        ax.set_ylabel("Preference ->", size=12)
+        ax.set_yticks([])
+        ax.set_xticks([0, 1, 2, 3, 4, 5])
+        ax.set_xticklabels([minvalues[i] + '\n' + objs_labels[i] for i in range(len(objs_labels))])
+
+        # make a twin axis for toplabels
+        ax1 = ax.twiny()
+        ax1.set_yticks([])
+        ax1.set_xticks([0, 1, 2, 3, 4, 5])
+        ax1.set_xticklabels([maxvalues[i] for i in range(len(maxs) + 1)])
+
+        return ax, ax1
+
+
+**plot_factor_performance()**
+_________________________________
+
+.. code-block:: python
+
+    def plot_factor_performance(param_values, collapse_days, b, m, a):
+        """Visualize the performance of our policies in three-dimensional
+        parametric space.
+
+        :param param_values:                Saltelli sample array
+        :param collapse_days:               Simulation array
+        :param b:                           b parameter boundary interval
+        :param m:                           m parameter boundary interval
+        :param a:                           a parameter boundary interval
+
+        """
+
+        # set colormap
+        cmap = plt.cm.get_cmap("RdBu_r")
+
+        # build figure object
+        fig = plt.figure(figsize=plt.figaspect(0.5), dpi=600, constrained_layout=True)
+
+        # set up scalable colormap
+        sm = plt.cm.ScalarMappable(cmap=cmap)
+
+        # set up subplot for profit maximizing policy
+        ax1 = fig.add_subplot(1, 2, 1, projection='3d')
+
+        # add point data for profit plot
+        sows = ax1.scatter(param_values[:,1],
+                           param_values[:,6],
+                           param_values[:,0],
+                           c=collapse_days[:,0],
+                           cmap=cmap,
+                           s=0.5)
+
+        # add surface data for boundary separating successful and failed states of the world
+        pts_ineq = ax1.plot_surface(b, m, a, color='black', alpha=0.25, zorder=1)
+
+        # add reference point to plot
+        pt_ref = ax1.scatter(0.5, 0.7, 0.005, c='black', s=50, zorder=0)
+
+        # set up plot aesthetics and labels
+        ax1.set_xlabel("b")
+        ax1.set_ylabel("m")
+        ax1.set_zlabel("a")
+        ax1.set_zlim([0.0, 2.0])
+        ax1.set_xlim([0.0, 1.0])
+        ax1.set_ylim([0.0, 1.5])
+        ax1.xaxis.set_view_interval(0,  0.5)
+        ax1.set_facecolor('white')
+        ax1.view_init(12, -17)
+        ax1.set_title('Profit maximizing policy')
+
+        # set up subplot for robust policy
+        ax2 = fig.add_subplot(1, 2, 2, projection='3d')
+
+        # add point data for robust plot
+        sows = ax2.scatter(param_values[:,1],
+                           param_values[:,6],
+                           param_values[:,0],
+                           c=collapse_days[:,1],
+                           cmap=cmap,
+                           s=0.5)
+
+        # add surface data for boundary separating successful and failed states of the world
+        pts_ineq = ax2.plot_surface(b, m, a, color='black', alpha=0.25, zorder=1)
+
+        # add reference point to plot
+        pt_ref = ax2.scatter(0.5, 0.7, 0.005, c='black', s=50, zorder=0)
+
+        # set up plot aesthetics and labels
+        ax2.set_xlabel("b")
+        ax2.set_ylabel("m")
+        ax2.set_zlabel("a")
+        ax2.set_zlim([0.0, 2.0])
+        ax2.set_xlim([0.0, 1.0])
+        ax2.set_ylim([0.0, 1.5])
+        ax2.xaxis.set_view_interval(0, 0.5)
+        ax2.set_facecolor('white')
+        ax2.view_init(12, -17)
+        ax2.set_title('Robust policy')
+
+        # set up colorbar
+        sm.set_array([collapse_days.min(), collapse_days.max()])
+        cbar = fig.colorbar(sm)
+        cbar.set_label('Days with predator collapse')
+
+        return ax1, ax2
diff --git a/docs/html/_sources/R.Bibliography.rst b/docs/html/_sources/R.Bibliography.rst
new file mode 100644
index 0000000..60cc211
--- /dev/null
+++ b/docs/html/_sources/R.Bibliography.rst
@@ -0,0 +1,6 @@
+============
+Bibliography
+============
+
+.. bibliography::
+    :style: unsrt
diff --git a/docs/html/_sources/acknowledgements.rst b/docs/html/_sources/acknowledgements.rst
new file mode 100644
index 0000000..1f34009
--- /dev/null
+++ b/docs/html/_sources/acknowledgements.rst
@@ -0,0 +1,25 @@
+
+Acknowledgements
+################
+
+This e-book was developed by the `Integrated Multisector, Multiscale Modeling (IM3) project <https://im3.pnnl.gov>`_, supported by the `U.S. Department of Energy <https://www.energy.gov/>`_, `Office of Science <https://www.energy.gov/science/office-science>`_, as part of research in the `MultiSector Dynamics <https://climatemodeling.science.energy.gov/program/multisector-dynamics>`_, Earth and Environmental System Modeling Program.
+
+The authors would like to thank Casey Burleyson (PNNL) and Isaac Thompson (PNNL) for their insightful feedback on drafts of the text and tutorials.
+
+Open source under license `CC BY-NC-ND-4.0 <https://creativecommons.org/licenses/by-nc-nd/4.0/>`_.
+
+Portions of cover image by `V.Cid7413 <https://commons.wikimedia.org/wiki/File:The_Amazing_Earth.jpg>`_, distributed under a `CC-BY-4.0 <https://creativecommons.org/licenses/by/4.0>`_ license.
+
+Open Source Disclaimer:
+
+    This material was prepared as an account of work sponsored by an agency of the United States Government.  Neither the United States Government nor the United States Department of Energy, nor Battelle, nor any of their employees, nor any jurisdiction or organization that has cooperated in the development of these materials, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness or any information, apparatus, product, software, or process disclosed, or represents that its use would not infringe privately owned rights.
+    Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or Battelle Memorial Institute. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
+
+| PACIFIC NORTHWEST NATIONAL LABORATORY
+| operated by
+| BATTELLE
+| for the
+| UNITED STATES DEPARTMENT OF ENERGY
+| under Contract DE-AC05-76RL01830
+
+Copyright (c) 2022-present; Battelle Memorial Institute
diff --git a/docs/html/_sources/citation.rst b/docs/html/_sources/citation.rst
new file mode 100644
index 0000000..63ee0ef
--- /dev/null
+++ b/docs/html/_sources/citation.rst
@@ -0,0 +1,29 @@
+Suggested Citation
+###################
+
+    Reed, P.M., Hadjimichael, A., Malek, K., Karimi, T., Vernon, C.R., Srikrishnan, V., Gupta, R.S., Gold, D.F., Lee, B., Keller, K., Thurber, T.B, & Rice, J.S. (2022). Addressing Uncertainty in Multisector Dynamics Research [Book]. Zenodo. https://doi.org/10.5281/zenodo.6110623
+
+.. only:: html
+
+    .. raw:: html
+
+        <div style="display:flex;flex-direction:row;align-items:center;justify-content:start;margin:1em 0;">
+            <a
+                class="btn btn-light"
+                style="margin-right:1em;"
+                role="button"
+                href="data:text/plain;base64,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"
+                download="uc-ebook.bib"
+            >
+                <i style="margin-right:0.5em;" class="fas fa-download"></i>Download citation (.bib)
+            </a>
+            <a
+                class="btn btn-light"
+                style="margin-right:1em;"
+                role="button"
+                href="data:text/plain;base64,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"
+                download="uc-ebook.ris"
+            >
+                <i style="margin-right:0.5em;" class="fas fa-download"></i>Download citation (.ris)
+            </a>
+        </div>
\ No newline at end of file
diff --git a/docs/html/_sources/code_of_conduct.rst b/docs/html/_sources/code_of_conduct.rst
new file mode 100644
index 0000000..56672b8
--- /dev/null
+++ b/docs/html/_sources/code_of_conduct.rst
@@ -0,0 +1,58 @@
+Code of Conduct
+===============
+
+Our Pledge
+----------
+
+We as members, contributors, and leaders pledge to make participation in our
+community a harassment-free experience for everyone, regardless of age, body
+size, visible or invisible disability, ethnicity, sex characteristics, gender
+identity and expression, level of experience, education, socio-economic status,
+nationality, personal appearance, race, religion, or sexual identity and
+orientation.
+
+We pledge to act and interact in ways that contribute to an open, welcoming,
+diverse, inclusive, and healthy community.
+
+Our Standards
+-------------
+
+Examples of behavior that contributes to a positive environment for our
+community include:
+
+- Demonstrating empathy and kindness toward other people
+- Being respectful of differing opinions, viewpoints, and experiences
+- Giving and gracefully accepting constructive feedback
+- Accepting responsibility and apologizing to those affected by our mistakes,
+  and learning from the experience
+- Focusing on what is best not just for us as individuals, but for the overall
+  community
+
+Examples of unacceptable behavior include:
+
+- The use of sexualized language or imagery, and sexual attention or
+  advances of any kind
+- Trolling, insulting or derogatory comments, and personal or political attacks
+- Public or private harassment
+- Publishing others' private information, such as a physical or email
+  address, without their explicit permission
+- Other conduct which could reasonably be considered inappropriate in a
+  professional setting
+
+Scope
+-----
+
+This Code of Conduct applies within all community spaces, and also applies
+when an individual is officially representing the community in public spaces.
+Examples of representing our community include using an official e-mail
+address, posting via an official social media account, or acting as an appointed
+representative at an online or offline event.
+
+Attribution
+-----------
+
+This Code of Conduct is adapted from the Contributor Covenant, version 2.0,
+available at https://www.contributor-covenant.org/version/2/0/code_of_conduct.html.
+
+For answers to common questions about this code of conduct, see the FAQ at
+https://www.contributor-covenant.org/faq. Translations are available at https://www.contributor-covenant.org/translations.
diff --git a/docs/html/_sources/contributing.rst b/docs/html/_sources/contributing.rst
new file mode 100644
index 0000000..917dda6
--- /dev/null
+++ b/docs/html/_sources/contributing.rst
@@ -0,0 +1,18 @@
+******************
+Contribution Guide
+******************
+
+Our eBook is a living product that we hope continues to grow over time to stay relevant with methodological and technological advancements in the field of uncertainty quantificaiton and MultiSector Dynamics at large.  We extend an invitation to you the reader to contribute to one of our interactive Jupyter notebook tutorials.  If you feel you have a contribution that would be relevant and generalizable to the MSD community, please submit a proposal idea `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/issues/new?assignees=thurber%2C+crvernon&labels=triage&projects=&template=contribution_proposal.yml&title=Contribution+Proposal>`_.  Your proposal will be reviewing by our team and feedback and/or a decision will be provided shortly.  Any contribution accepted will receive it's own DOI and citation so that you may provide an independent reference to your work as you see fit.
+
+Please consider the following requirements for contribution:
+
+- All elements of your contribution MUST be fully open-source and can be distributed with an `Open Source Initiative approved license <https://opensource.org/licenses/>`_ with an understanding that they may be used in community demonstrations and other activities.  Author citations will be present in the notebook for any contributed work to ensure the author(s) receive full credit for their contribution.
+- Any data or code reused in your submission must be correctly cited giving credit to the original authors. 
+- The notebook provided must be written in English and able to be a stand-alone product that needs no further explanation past what is written in the notebook to make use of it.
+- The provided work is not merely a regurgitation of an existing tutorial or demonstration but represents a novel contribution.
+- All contributions and communication thereof must abide by our `code of conduct <https://uc-ebook.org/docs/html/code_of_conduct.html>`_.
+
+
+If you feel your work meets the criteria above, please submit a proposal issue `here <https://github.com/IMMM-SFA/msd_uncertainty_ebook/issues/new?assignees=thurber%2C+crvernon&labels=triage&projects=&template=contribution_proposal.yml&title=Contribution+Proposal>`_.  If your proposal is approved, please submit your draft notebook for review as an attachment using our submission `template <https://github.com/IMMM-SFA/msd_uncertainty_ebook/blob/main/.github/PULL_REQUEST_TEMPLATE/contribution_checklist.md>`_.  We will then review your submission and provide feedback.  Once your contribution has been deemed ready to deploy, we will generate a DOI for your work, launch it to our MSD-LIVE set of interactive notebooks, and feature the contribution in the index of our eBook.
+
+Please feel free to reach out with any further questions.
diff --git a/docs/html/_sources/examples.rst b/docs/html/_sources/examples.rst
new file mode 100644
index 0000000..ca7125a
--- /dev/null
+++ b/docs/html/_sources/examples.rst
@@ -0,0 +1,44 @@
+========
+Examples
+========
+
+
+Using mathjax
+===================
+
+The **first-order sensitivity index** indicates the percent of model output variance contributed by a factor individually (i.e., the effect of varying *x*\ :sub:`i`\ alone) and is obtained using the following (Saltelli, 2002a; Sobol, 2001):
+
+.. math::
+
+  S_i^1 = \frac{V_{x_i} [E_{x\sim_i} (x_i)]}{V(y)}
+
+with *E* and *V* denoting the expected value and the variance, respectively.
+
+Building a codeblock
+=======================
+
+.. code-block:: python
+  :linenos:
+
+   import ebook
+
+   ebook.plot_experimental_design()
+
+Testing out a note
+==================
+
+.. note::
+  Keep track of the latest in programming solutions on the `Water Programming <https://waterprogramming.wordpress.com/>`_ blog!
+
+
+Insert a figure with a caption
+==============================
+
+Be sure to add your figure into 'msd_uncertainty_ebook/docs/source/_static'
+
+  .. figure:: _static/im3.png
+      :alt: IM3 logo
+      :width: 100px
+      :align: center
+
+      This is my caption.
diff --git a/docs/html/_sources/index.rst b/docs/html/_sources/index.rst
new file mode 100644
index 0000000..170c127
--- /dev/null
+++ b/docs/html/_sources/index.rst
@@ -0,0 +1,134 @@
+.. msd_uncertainty_ebook documentation master file, created by
+   sphinx-quickstart on Wed May 26 22:27:12 2021.
+   You can adapt this file completely to your liking, but it should at least
+   contain the root `toctree` directive.
+
+
+=======================================================
+Addressing Uncertainty in MultiSector Dynamics Research
+=======================================================
+
+.. only:: html
+
+    .. epigraph::
+
+        Patrick M. Reed, Antonia Hadjimichael, Keyvan Malek,
+        Tina Karimi, Chris R. Vernon, Vivek Srikrishnan, Rohini S. Gupta,
+        David F. Gold, Ben Lee, Klaus Keller, Travis B. Thurber, Jennie S. Rice
+
+    A practical guide to sensitivity analysis and diagnostic model evaluation techniques for confronting the computational and conceptual challenges of multi-model, transdisciplinary workflows.
+
+    **Topics in the Book**
+
+    * `Preface <preface.html>`_
+    * `Introduction <1_introduction.html>`_
+    * `An overview of diagnostic modeling and perspectives on model evaluation <2_diagnostic_modeling_overview_and_perspectives.html>`_
+    * `A framework for the basic methods and concepts used in sensitivity analysis <3_sensitivity_analysis_the_basics.html>`_
+    * `Technical applications supporting diagnostic model evaluation and exploration <4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html>`_
+
+    **Interactive Tutorials**
+
+    * `Factor Discovery <nb_fishery_dynamics_>`_
+    * `Model Calibration <nb_hymod_>`_
+    * `Sobol Sensitivity Analysis <nb_saltelli_sobol_>`_
+    * `Factor Mapping using Logistic Regression <nb_logistic_regression_>`_
+    * `Time-evolving scenario discovery for infrastructure pathways <nb_discovery_>`_
+    * `A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios <nb_hmm_>`_
+
+    .. tip::
+
+        .. raw:: html
+
+            <p>
+                Use the sidebar on the left to quickly navigate the eBook!
+                <br/>
+                Click or tap the <i style="padding: 0 0.25rem;" class="fas fa-bars"></i> icon to show and hide the sidebar.
+            </p>
+
+    .. admonition:: Info
+
+        .. raw:: html
+
+            <p>
+                Report a typo or just pass along something you like about the book by <a target="_blank" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/issues/new?assignees=thurber%2C+crvernon&labels=documentation%2C+triage&template=custom.md&title=Publication+Feedback">opening an issue</a> on GitHub!
+                <br/>
+                Click or tap the <i style="padding: 0 0.25rem;" class="fab fa-github"></i> icon to find this link again.
+            </p>
+
+.. raw:: latex
+
+    \frontmatter
+    \sphinxmaketitle
+
+.. toctree::
+    :hidden:
+    :glob:
+
+    preface
+    citation
+    acknowledgements
+    code_of_conduct
+    contributing
+
+.. raw:: latex
+
+    \sphinxtableofcontents
+
+.. raw:: latex
+
+    \mainmatter
+
+.. toctree::
+    :hidden:
+    :includehidden:
+    :numbered: 4
+    :maxdepth: 4
+    :caption: Contents
+    :name: mastertoc
+    :glob:
+
+    1_introduction
+    2_diagnostic_modeling_overview_and_perspectives
+    3_sensitivity_analysis_the_basics
+    4_sensitivity_analysis_diagnostic_and_exploratory_modeling
+    5_conclusion
+
+.. raw:: latex
+
+    \appendix
+
+.. appendix::
+    :hidden:
+    :numbered: 4
+    :maxdepth: 4
+    :caption: Appendices
+    :glob:
+
+    A1_Uncertainty_Quantification
+    A2_Jupyter_Notebooks
+    A3_plotting_code
+
+.. raw:: latex
+
+    \backmatter
+
+.. toctree::
+    :hidden:
+    :maxdepth: 1
+    :caption: Glossary
+    :glob:
+
+    6_glossary
+
+
+.. raw:: latex
+
+    \backmatter
+
+.. toctree::
+    :hidden:
+    :maxdepth: 1
+    :caption: References
+    :glob:
+
+    R.Bibliography
diff --git a/docs/html/_sources/modules.rst b/docs/html/_sources/modules.rst
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--- /dev/null
+++ b/docs/html/_sources/modules.rst
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+nanites
+=======
+
+.. toctree::
+   :maxdepth: 4
+
diff --git a/docs/html/_sources/preface.rst b/docs/html/_sources/preface.rst
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+
+.. _preface:
+
+*******
+Preface
+*******
+
+This online book is meant to provide an open science “living” resource on uncertainty characterization methods for the MultiSector Dynamics (MSD) community and other technical communities confronting sustainability, climate, and energy transition challenges. The last decade has seen rapid growth in science efforts seeking to address the interconnected nature of these challenges across scales, sectors, and systems. Accompanying these advances is the growing realization that the deep integration of research from many disciplinary fields is non-trivial and raises important questions. How and why models are developed seems to have an obvious answer (“to gain understanding”). But what does it actually mean to gain understanding? What if a small change in a model or its data fundamentally changes our perceptions of what we thought we understood? What controls the outcomes of our model(s)? How do we understand the implications of model coupling, such as when one model is on the receiving end of several other models that are considered “input data”? 
+
+The often quoted “All models are wrong, but some are useful.” (George Box) is a bit of a conflation trap, often used to excuse known weaknesses in complex models as just an unavoidable outcome of being a modeler. In fact, the quote actually refers to a specific class of small-scale statistical models within an application context that assures a much higher degree of understanding and data quality control than is typical for the coupled human-natural systems applications in the MSD area. Moreover, Box was actually warning readers to avoid overparameterization and emphasizing the need to better understand what underlying factors cause your model to be wrong :cite:p:`box_1976`. 
+
+So, in short, there is a tension when attaining better performance by means of increasing the complexity of a model or model-based workflow. Box highlights that a modeler requires a clear diagnostic understanding of this performance-complexity tradeoff. If we move from small-scale models simulating readily-observed phenomena to the MSD context, things get quite a bit more complicated. How can we provide robust insights for unseen futures that emerge across a myriad of human and natural systems? Sometimes even asking, “what is a model?” or “what is data?” is complicated (e.g., data assimilated weather products, satellite-based signals translated through retrieval algorithms, demographic changes, resource demands, etc.). This MSD guidance text seeks to help readers navigate these challenges. It is meant to serve as an evolving resource that helps the MSD community learn how to better address uncertainty while working with complex chains of models bridging sectors, scales, and systems. It is not intended to be an exhaustive resource, but instead should be seen as a guided tour through state-of-the-science methods in uncertainty characterization, including global sensitivity analysis and exploratory modeling, to provide insights into complex human-natural systems interactions. 
+
+To aid readers in navigating the text, the key goals for each chapter are summarized below.
+
+:numref:`introduction` uses the `Integrated Multisector Multiscale Modeling <https://im3.pnnl.gov>`_ project as a living lab to encapsulate the challenges that emerge in bridging disciplines to make consequential model-based insights while acknowledging the tremendous array of uncertainties that shape them.
+
+:numref:`2_diagnostic_modeling` helps the reader to better understand the importance of using diagnostic modeling to interrogate why uncertain model behaviors may emerge. The chapter also aids readers to better understand the diverse disciplinary perspectives that exist on how best to pursue consequential model-based discoveries.
+
+:numref:`3_sensitivity_analysis_the_basics` is a technical tools-focused primer for readers on the key elements of uncertainty characterization that includes ensemble-based design of experiments, quantitative methods for computing global sensitivities, and a summary of existing software packages.
+
+:numref:`4_sensitivity_analysis` narrates for readers how and why the tools from the previous chapter can be applied in a range of tasks from diagnosing model performance to formal exploratory modeling methods for making consequential model-based discoveries.
+
+The supplemental appendices provided in the text are also important resources for readers. They provide a glossary to help bridge terminology challenges, a brief summary of uncertainty quantification tools for more advanced readers, and a suite of Jupyter notebook tutorials that provide hands-on training tied to the contents of :numref:`3_sensitivity_analysis_the_basics` and :numref:`4_sensitivity_analysis`.
+
+This text was written with a number of different audiences in mind. 
+
+Technical experts in uncertainty may find this to be a helpful and unique resource bridging a number of perspectives that have not been combined in prior books (e.g., formal model diagnostics, global sensitivity analysis, and exploratory modeling under deep uncertainty). 
+
+Readers from different sector-specific and disciplinary-specific backgrounds can use this text to better understand potential differences and opportunities in how to make model-based insights.
+
+Academic or junior researchers can utilize this freely available text for training and teaching resources that include hands-on coding experiences.
+
+This text itself represents our strong commitment to open science and will evolve as a living resource as the communities of researchers provide feedback, innovations, and future tools.
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diff --git a/docs/html/_static/hymod_schematic-DAVE.png b/docs/html/_static/hymod_schematic-DAVE.png
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diff --git a/docs/html/_static/im3.png b/docs/html/_static/im3.png
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diff --git a/docs/html/_static/images/logo_binder.svg b/docs/html/_static/images/logo_binder.svg
new file mode 100644
index 0000000..45fecf7
--- /dev/null
+++ b/docs/html/_static/images/logo_binder.svg
@@ -0,0 +1,19 @@
+<?xml version="1.0" encoding="utf-8"?>
+<!-- Generator: Adobe Illustrator 23.0.1, SVG Export Plug-In . SVG Version: 6.00 Build 0)  -->
+<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px"
+	 viewBox="0 0 44.4 44.4" style="enable-background:new 0 0 44.4 44.4;" xml:space="preserve">
+<style type="text/css">
+	.st0{fill:none;stroke:#F5A252;stroke-width:5;stroke-miterlimit:10;}
+	.st1{fill:none;stroke:#579ACA;stroke-width:5;stroke-miterlimit:10;}
+	.st2{fill:none;stroke:#E66581;stroke-width:5;stroke-miterlimit:10;}
+</style>
+<title>logo</title>
+<g>
+	<path class="st0" d="M33.9,6.4c3.6,3.9,3.4,9.9-0.5,13.5s-9.9,3.4-13.5-0.5s-3.4-9.9,0.5-13.5l0,0C24.2,2.4,30.2,2.6,33.9,6.4z"/>
+	<path class="st1" d="M35.1,27.3c2.6,4.6,1.1,10.4-3.5,13c-4.6,2.6-10.4,1.1-13-3.5s-1.1-10.4,3.5-13l0,0
+		C26.6,21.2,32.4,22.7,35.1,27.3z"/>
+	<path class="st2" d="M25.9,17.8c2.6,4.6,1.1,10.4-3.5,13s-10.4,1.1-13-3.5s-1.1-10.4,3.5-13l0,0C17.5,11.7,23.3,13.2,25.9,17.8z"/>
+	<path class="st1" d="M19.2,26.4c3.1-4.3,9.1-5.2,13.3-2.1c1.1,0.8,2,1.8,2.7,3"/>
+	<path class="st0" d="M19.9,19.4c-3.6-3.9-3.4-9.9,0.5-13.5s9.9-3.4,13.5,0.5"/>
+</g>
+</svg>
diff --git a/docs/html/_static/images/logo_colab.png b/docs/html/_static/images/logo_colab.png
new file mode 100644
index 0000000..b7560ec
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diff --git a/docs/html/_static/images/logo_deepnote.svg b/docs/html/_static/images/logo_deepnote.svg
new file mode 100644
index 0000000..fa77ebf
--- /dev/null
+++ b/docs/html/_static/images/logo_deepnote.svg
@@ -0,0 +1 @@
+<svg viewBox="0 0 128 128" fill="none" xmlns="http://www.w3.org/2000/svg"><path fill-rule="evenodd" clip-rule="evenodd" d="M0 128h52.512l29.539-11.077-11.077-43.487-34.051 3.693L0 128Z" fill="#0076D4"/><path fill-rule="evenodd" clip-rule="evenodd" d="M52.513 128s16.6-8.759 19.673-24.277c3.072-15.517-12.091-26.594-35.263-26.594 0-.41 20.343-28.718 20.343-28.718l49.4 1.435L95.71 107.7l-20.452 15.978L52.513 128Z" fill="#002868"/><path fill-rule="evenodd" clip-rule="evenodd" d="M0 60.718 41.025.001s1.006.01 3.282 0c16.082-.068 81.23 3.12 81.23 60.368 0 65.352-73.025 67.631-73.025 67.631s30.495-5.839 30.495-34.816c0-28.978-27.541-32.466-45.264-32.466H0Z" fill="#00A9FF"/></svg>
diff --git a/docs/html/_static/images/logo_jupyterhub.svg b/docs/html/_static/images/logo_jupyterhub.svg
new file mode 100644
index 0000000..60cfe9f
--- /dev/null
+++ b/docs/html/_static/images/logo_jupyterhub.svg
@@ -0,0 +1 @@
+<svg id="Layer_1" data-name="Layer 1" xmlns="http://www.w3.org/2000/svg" width="38.73" height="50" viewBox="0 0 38.73 50"><defs><style>.cls-1{fill:#767677;}.cls-2{fill:#f37726;}.cls-3{fill:#9e9e9e;}.cls-4{fill:#616262;}.cls-5{font-size:17.07px;fill:#fff;font-family:Roboto-Regular, Roboto;}</style></defs><title>logo_jupyterhub</title><g id="Canvas"><path id="path7_fill" data-name="path7 fill" class="cls-1" d="M39.51,3.53a3,3,0,0,1-1.7,2.9A3,3,0,0,1,34.48,6a3,3,0,0,1-.82-3.26,3,3,0,0,1,1.05-1.41A3,3,0,0,1,37.52.86a2.88,2.88,0,0,1,1,.6,3,3,0,0,1,.7.93,3.18,3.18,0,0,1,.28,1.14Z" transform="translate(-1.87 -0.69)"/><path id="path8_fill" data-name="path8 fill" class="cls-2" d="M21.91,38.39c-8,0-15.06-2.87-18.7-7.12a19.93,19.93,0,0,0,37.39,0C37,35.52,30,38.39,21.91,38.39Z" transform="translate(-1.87 -0.69)"/><path id="path9_fill" data-name="path9 fill" class="cls-2" d="M21.91,10.78c8,0,15.05,2.87,18.69,7.12a19.93,19.93,0,0,0-37.39,0C6.85,13.64,13.86,10.78,21.91,10.78Z" transform="translate(-1.87 -0.69)"/><path id="path10_fill" data-name="path10 fill" class="cls-3" d="M10.88,46.66a3.86,3.86,0,0,1-.52,2.15,3.81,3.81,0,0,1-1.62,1.51,3.93,3.93,0,0,1-2.19.34,3.79,3.79,0,0,1-2-.94,3.73,3.73,0,0,1-1.14-1.9,3.79,3.79,0,0,1,.1-2.21,3.86,3.86,0,0,1,1.33-1.78,3.92,3.92,0,0,1,3.54-.53,3.85,3.85,0,0,1,2.14,1.93,3.74,3.74,0,0,1,.37,1.43Z" transform="translate(-1.87 -0.69)"/><path id="path11_fill" data-name="path11 fill" class="cls-4" d="M4.12,9.81A2.18,2.18,0,0,1,2.9,9.48a2.23,2.23,0,0,1-.84-1A2.26,2.26,0,0,1,1.9,7.26a2.13,2.13,0,0,1,.56-1.13,2.18,2.18,0,0,1,2.36-.56,2.13,2.13,0,0,1,1,.76,2.18,2.18,0,0,1,.42,1.2A2.22,2.22,0,0,1,4.12,9.81Z" transform="translate(-1.87 -0.69)"/></g><text class="cls-5" transform="translate(5.24 30.01)">Hub</text></svg>
diff --git a/docs/html/_static/language_data.js b/docs/html/_static/language_data.js
new file mode 100644
index 0000000..367b8ed
--- /dev/null
+++ b/docs/html/_static/language_data.js
@@ -0,0 +1,199 @@
+/*
+ * language_data.js
+ * ~~~~~~~~~~~~~~~~
+ *
+ * This script contains the language-specific data used by searchtools.js,
+ * namely the list of stopwords, stemmer, scorer and splitter.
+ *
+ * :copyright: Copyright 2007-2024 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+
+var stopwords = ["a", "and", "are", "as", "at", "be", "but", "by", "for", "if", "in", "into", "is", "it", "near", "no", "not", "of", "on", "or", "such", "that", "the", "their", "then", "there", "these", "they", "this", "to", "was", "will", "with"];
+
+
+/* Non-minified version is copied as a separate JS file, if available */
+
+/**
+ * Porter Stemmer
+ */
+var Stemmer = function() {
+
+  var step2list = {
+    ational: 'ate',
+    tional: 'tion',
+    enci: 'ence',
+    anci: 'ance',
+    izer: 'ize',
+    bli: 'ble',
+    alli: 'al',
+    entli: 'ent',
+    eli: 'e',
+    ousli: 'ous',
+    ization: 'ize',
+    ation: 'ate',
+    ator: 'ate',
+    alism: 'al',
+    iveness: 'ive',
+    fulness: 'ful',
+    ousness: 'ous',
+    aliti: 'al',
+    iviti: 'ive',
+    biliti: 'ble',
+    logi: 'log'
+  };
+
+  var step3list = {
+    icate: 'ic',
+    ative: '',
+    alize: 'al',
+    iciti: 'ic',
+    ical: 'ic',
+    ful: '',
+    ness: ''
+  };
+
+  var c = "[^aeiou]";          // consonant
+  var v = "[aeiouy]";          // vowel
+  var C = c + "[^aeiouy]*";    // consonant sequence
+  var V = v + "[aeiou]*";      // vowel sequence
+
+  var mgr0 = "^(" + C + ")?" + V + C;                      // [C]VC... is m>0
+  var meq1 = "^(" + C + ")?" + V + C + "(" + V + ")?$";    // [C]VC[V] is m=1
+  var mgr1 = "^(" + C + ")?" + V + C + V + C;              // [C]VCVC... is m>1
+  var s_v   = "^(" + C + ")?" + v;                         // vowel in stem
+
+  this.stemWord = function (w) {
+    var stem;
+    var suffix;
+    var firstch;
+    var origword = w;
+
+    if (w.length < 3)
+      return w;
+
+    var re;
+    var re2;
+    var re3;
+    var re4;
+
+    firstch = w.substr(0,1);
+    if (firstch == "y")
+      w = firstch.toUpperCase() + w.substr(1);
+
+    // Step 1a
+    re = /^(.+?)(ss|i)es$/;
+    re2 = /^(.+?)([^s])s$/;
+
+    if (re.test(w))
+      w = w.replace(re,"$1$2");
+    else if (re2.test(w))
+      w = w.replace(re2,"$1$2");
+
+    // Step 1b
+    re = /^(.+?)eed$/;
+    re2 = /^(.+?)(ed|ing)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      re = new RegExp(mgr0);
+      if (re.test(fp[1])) {
+        re = /.$/;
+        w = w.replace(re,"");
+      }
+    }
+    else if (re2.test(w)) {
+      var fp = re2.exec(w);
+      stem = fp[1];
+      re2 = new RegExp(s_v);
+      if (re2.test(stem)) {
+        w = stem;
+        re2 = /(at|bl|iz)$/;
+        re3 = new RegExp("([^aeiouylsz])\\1$");
+        re4 = new RegExp("^" + C + v + "[^aeiouwxy]$");
+        if (re2.test(w))
+          w = w + "e";
+        else if (re3.test(w)) {
+          re = /.$/;
+          w = w.replace(re,"");
+        }
+        else if (re4.test(w))
+          w = w + "e";
+      }
+    }
+
+    // Step 1c
+    re = /^(.+?)y$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(s_v);
+      if (re.test(stem))
+        w = stem + "i";
+    }
+
+    // Step 2
+    re = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      suffix = fp[2];
+      re = new RegExp(mgr0);
+      if (re.test(stem))
+        w = stem + step2list[suffix];
+    }
+
+    // Step 3
+    re = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      suffix = fp[2];
+      re = new RegExp(mgr0);
+      if (re.test(stem))
+        w = stem + step3list[suffix];
+    }
+
+    // Step 4
+    re = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/;
+    re2 = /^(.+?)(s|t)(ion)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(mgr1);
+      if (re.test(stem))
+        w = stem;
+    }
+    else if (re2.test(w)) {
+      var fp = re2.exec(w);
+      stem = fp[1] + fp[2];
+      re2 = new RegExp(mgr1);
+      if (re2.test(stem))
+        w = stem;
+    }
+
+    // Step 5
+    re = /^(.+?)e$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(mgr1);
+      re2 = new RegExp(meq1);
+      re3 = new RegExp("^" + C + v + "[^aeiouwxy]$");
+      if (re.test(stem) || (re2.test(stem) && !(re3.test(stem))))
+        w = stem;
+    }
+    re = /ll$/;
+    re2 = new RegExp(mgr1);
+    if (re.test(w) && re2.test(w)) {
+      re = /.$/;
+      w = w.replace(re,"");
+    }
+
+    // and turn initial Y back to y
+    if (firstch == "y")
+      w = firstch.toLowerCase() + w.substr(1);
+    return w;
+  }
+}
+
diff --git a/docs/html/_static/locales/ar/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/ar/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..15541a6
Binary files /dev/null and b/docs/html/_static/locales/ar/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/ar/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/ar/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..34d404c
--- /dev/null
+++ b/docs/html/_static/locales/ar/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ar\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "طباعة إلى PDF"
+
+msgid "Theme by the"
+msgstr "موضوع بواسطة"
+
+msgid "Download source file"
+msgstr "تنزيل ملف المصدر"
+
+msgid "open issue"
+msgstr "قضية مفتوحة"
+
+msgid "Contents"
+msgstr "محتويات"
+
+msgid "previous page"
+msgstr "الصفحة السابقة"
+
+msgid "Download notebook file"
+msgstr "تنزيل ملف دفتر الملاحظات"
+
+msgid "Copyright"
+msgstr "حقوق النشر"
+
+msgid "Download this page"
+msgstr "قم بتنزيل هذه الصفحة"
+
+msgid "Source repository"
+msgstr "مستودع المصدر"
+
+msgid "By"
+msgstr "بواسطة"
+
+msgid "repository"
+msgstr "مخزن"
+
+msgid "Last updated on"
+msgstr "آخر تحديث في"
+
+msgid "Toggle navigation"
+msgstr "تبديل التنقل"
+
+msgid "Sphinx Book Theme"
+msgstr "موضوع كتاب أبو الهول"
+
+msgid "suggest edit"
+msgstr "أقترح تحرير"
+
+msgid "Open an issue"
+msgstr "افتح قضية"
+
+msgid "Launch"
+msgstr "إطلاق"
+
+msgid "Fullscreen mode"
+msgstr "وضع ملء الشاشة"
+
+msgid "Edit this page"
+msgstr "قم بتحرير هذه الصفحة"
+
+msgid "By the"
+msgstr "بواسطة"
+
+msgid "next page"
+msgstr "الصفحة التالية"
diff --git a/docs/html/_static/locales/bg/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/bg/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..da95120
Binary files /dev/null and b/docs/html/_static/locales/bg/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/bg/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/bg/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..7420c19
--- /dev/null
+++ b/docs/html/_static/locales/bg/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: bg\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Печат в PDF"
+
+msgid "Theme by the"
+msgstr "Тема от"
+
+msgid "Download source file"
+msgstr "Изтеглете изходния файл"
+
+msgid "open issue"
+msgstr "отворен брой"
+
+msgid "Contents"
+msgstr "Съдържание"
+
+msgid "previous page"
+msgstr "предишна страница"
+
+msgid "Download notebook file"
+msgstr "Изтеглете файла на бележника"
+
+msgid "Copyright"
+msgstr "Авторско право"
+
+msgid "Download this page"
+msgstr "Изтеглете тази страница"
+
+msgid "Source repository"
+msgstr "Хранилище на източника"
+
+msgid "By"
+msgstr "От"
+
+msgid "repository"
+msgstr "хранилище"
+
+msgid "Last updated on"
+msgstr "Последна актуализация на"
+
+msgid "Toggle navigation"
+msgstr "Превключване на навигацията"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема на книгата Sphinx"
+
+msgid "suggest edit"
+msgstr "предложи редактиране"
+
+msgid "Open an issue"
+msgstr "Отворете проблем"
+
+msgid "Launch"
+msgstr "Стартиране"
+
+msgid "Fullscreen mode"
+msgstr "Режим на цял екран"
+
+msgid "Edit this page"
+msgstr "Редактирайте тази страница"
+
+msgid "By the"
+msgstr "По"
+
+msgid "next page"
+msgstr "Следваща страница"
diff --git a/docs/html/_static/locales/bn/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/bn/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..6b96639
Binary files /dev/null and b/docs/html/_static/locales/bn/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/bn/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/bn/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..63a07c3
--- /dev/null
+++ b/docs/html/_static/locales/bn/LC_MESSAGES/booktheme.po
@@ -0,0 +1,63 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: bn\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "পিডিএফ প্রিন্ট করুন"
+
+msgid "Theme by the"
+msgstr "থিম দ্বারা"
+
+msgid "Download source file"
+msgstr "উত্স ফাইল ডাউনলোড করুন"
+
+msgid "open issue"
+msgstr "খোলা সমস্যা"
+
+msgid "previous page"
+msgstr "আগের পৃষ্ঠা"
+
+msgid "Download notebook file"
+msgstr "নোটবুক ফাইল ডাউনলোড করুন"
+
+msgid "Copyright"
+msgstr "কপিরাইট"
+
+msgid "Download this page"
+msgstr "এই পৃষ্ঠাটি ডাউনলোড করুন"
+
+msgid "Source repository"
+msgstr "উত্স সংগ্রহস্থল"
+
+msgid "By"
+msgstr "দ্বারা"
+
+msgid "Last updated on"
+msgstr "সর্বশেষ আপডেট"
+
+msgid "Toggle navigation"
+msgstr "নেভিগেশন টগল করুন"
+
+msgid "Sphinx Book Theme"
+msgstr "স্পিনিক্স বুক থিম"
+
+msgid "Open an issue"
+msgstr "একটি সমস্যা খুলুন"
+
+msgid "Launch"
+msgstr "শুরু করা"
+
+msgid "Edit this page"
+msgstr "এই পৃষ্ঠাটি সম্পাদনা করুন"
+
+msgid "By the"
+msgstr "দ্বারা"
+
+msgid "next page"
+msgstr "পরবর্তী পৃষ্ঠা"
diff --git a/docs/html/_static/locales/ca/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/ca/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..a4dd30e
Binary files /dev/null and b/docs/html/_static/locales/ca/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/ca/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/ca/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..8fb358b
--- /dev/null
+++ b/docs/html/_static/locales/ca/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ca\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Imprimeix a PDF"
+
+msgid "Theme by the"
+msgstr "Tema del"
+
+msgid "Download source file"
+msgstr "Baixeu el fitxer font"
+
+msgid "open issue"
+msgstr "número obert"
+
+msgid "previous page"
+msgstr "Pàgina anterior"
+
+msgid "Download notebook file"
+msgstr "Descarregar fitxer de quadern"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Download this page"
+msgstr "Descarregueu aquesta pàgina"
+
+msgid "Source repository"
+msgstr "Dipòsit de fonts"
+
+msgid "By"
+msgstr "Per"
+
+msgid "Last updated on"
+msgstr "Darrera actualització el"
+
+msgid "Toggle navigation"
+msgstr "Commuta la navegació"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema del llibre Esfinx"
+
+msgid "suggest edit"
+msgstr "suggerir edició"
+
+msgid "Open an issue"
+msgstr "Obriu un número"
+
+msgid "Launch"
+msgstr "Llançament"
+
+msgid "Edit this page"
+msgstr "Editeu aquesta pàgina"
+
+msgid "By the"
+msgstr "Per la"
+
+msgid "next page"
+msgstr "pàgina següent"
diff --git a/docs/html/_static/locales/cs/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/cs/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..c39e01a
Binary files /dev/null and b/docs/html/_static/locales/cs/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/cs/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/cs/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..c6ef469
--- /dev/null
+++ b/docs/html/_static/locales/cs/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: cs\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Tisk do PDF"
+
+msgid "Theme by the"
+msgstr "Téma od"
+
+msgid "Download source file"
+msgstr "Stáhněte si zdrojový soubor"
+
+msgid "open issue"
+msgstr "otevřené číslo"
+
+msgid "Contents"
+msgstr "Obsah"
+
+msgid "previous page"
+msgstr "předchozí stránka"
+
+msgid "Download notebook file"
+msgstr "Stáhnout soubor poznámkového bloku"
+
+msgid "Copyright"
+msgstr "autorská práva"
+
+msgid "Download this page"
+msgstr "Stáhněte si tuto stránku"
+
+msgid "Source repository"
+msgstr "Zdrojové úložiště"
+
+msgid "By"
+msgstr "Podle"
+
+msgid "repository"
+msgstr "úložiště"
+
+msgid "Last updated on"
+msgstr "Naposledy aktualizováno"
+
+msgid "Toggle navigation"
+msgstr "Přepnout navigaci"
+
+msgid "Sphinx Book Theme"
+msgstr "Téma knihy Sfinga"
+
+msgid "suggest edit"
+msgstr "navrhnout úpravy"
+
+msgid "Open an issue"
+msgstr "Otevřete problém"
+
+msgid "Launch"
+msgstr "Zahájení"
+
+msgid "Fullscreen mode"
+msgstr "Režim celé obrazovky"
+
+msgid "Edit this page"
+msgstr "Upravit tuto stránku"
+
+msgid "By the"
+msgstr "Podle"
+
+msgid "next page"
+msgstr "další strana"
diff --git a/docs/html/_static/locales/da/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/da/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..f43157d
Binary files /dev/null and b/docs/html/_static/locales/da/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/da/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/da/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..306a38e
--- /dev/null
+++ b/docs/html/_static/locales/da/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: da\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Udskriv til PDF"
+
+msgid "Theme by the"
+msgstr "Tema af"
+
+msgid "Download source file"
+msgstr "Download kildefil"
+
+msgid "open issue"
+msgstr "åbent nummer"
+
+msgid "Contents"
+msgstr "Indhold"
+
+msgid "previous page"
+msgstr "forrige side"
+
+msgid "Download notebook file"
+msgstr "Download notesbog-fil"
+
+msgid "Copyright"
+msgstr "ophavsret"
+
+msgid "Download this page"
+msgstr "Download denne side"
+
+msgid "Source repository"
+msgstr "Kildelager"
+
+msgid "By"
+msgstr "Ved"
+
+msgid "repository"
+msgstr "lager"
+
+msgid "Last updated on"
+msgstr "Sidst opdateret den"
+
+msgid "Toggle navigation"
+msgstr "Skift navigation"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx bogtema"
+
+msgid "suggest edit"
+msgstr "foreslå redigering"
+
+msgid "Open an issue"
+msgstr "Åbn et problem"
+
+msgid "Launch"
+msgstr "Start"
+
+msgid "Fullscreen mode"
+msgstr "Fuldskærmstilstand"
+
+msgid "Edit this page"
+msgstr "Rediger denne side"
+
+msgid "By the"
+msgstr "Ved"
+
+msgid "next page"
+msgstr "Næste side"
diff --git a/docs/html/_static/locales/de/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/de/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..648b565
Binary files /dev/null and b/docs/html/_static/locales/de/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/de/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/de/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..4925360
--- /dev/null
+++ b/docs/html/_static/locales/de/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: de\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "In PDF drucken"
+
+msgid "Theme by the"
+msgstr "Thema von der"
+
+msgid "Download source file"
+msgstr "Quelldatei herunterladen"
+
+msgid "open issue"
+msgstr "offenes Thema"
+
+msgid "Contents"
+msgstr "Inhalt"
+
+msgid "previous page"
+msgstr "vorherige Seite"
+
+msgid "Download notebook file"
+msgstr "Notebook-Datei herunterladen"
+
+msgid "Copyright"
+msgstr "Urheberrechte ©"
+
+msgid "Download this page"
+msgstr "Laden Sie diese Seite herunter"
+
+msgid "Source repository"
+msgstr "Quell-Repository"
+
+msgid "By"
+msgstr "Durch"
+
+msgid "repository"
+msgstr "Repository"
+
+msgid "Last updated on"
+msgstr "Zuletzt aktualisiert am"
+
+msgid "Toggle navigation"
+msgstr "Navigation umschalten"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx-Buch-Thema"
+
+msgid "suggest edit"
+msgstr "vorschlagen zu bearbeiten"
+
+msgid "Open an issue"
+msgstr "Öffnen Sie ein Problem"
+
+msgid "Launch"
+msgstr "Starten"
+
+msgid "Fullscreen mode"
+msgstr "Vollbildmodus"
+
+msgid "Edit this page"
+msgstr "Bearbeite diese Seite"
+
+msgid "By the"
+msgstr "Bis zum"
+
+msgid "next page"
+msgstr "Nächste Seite"
diff --git a/docs/html/_static/locales/el/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/el/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..fca6e93
Binary files /dev/null and b/docs/html/_static/locales/el/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/el/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/el/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..3e01acb
--- /dev/null
+++ b/docs/html/_static/locales/el/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: el\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Εκτύπωση σε PDF"
+
+msgid "Theme by the"
+msgstr "Θέμα από το"
+
+msgid "Download source file"
+msgstr "Λήψη αρχείου προέλευσης"
+
+msgid "open issue"
+msgstr "ανοιχτό ζήτημα"
+
+msgid "Contents"
+msgstr "Περιεχόμενα"
+
+msgid "previous page"
+msgstr "προηγούμενη σελίδα"
+
+msgid "Download notebook file"
+msgstr "Λήψη αρχείου σημειωματάριου"
+
+msgid "Copyright"
+msgstr "Πνευματική ιδιοκτησία"
+
+msgid "Download this page"
+msgstr "Λήψη αυτής της σελίδας"
+
+msgid "Source repository"
+msgstr "Αποθήκη πηγής"
+
+msgid "By"
+msgstr "Με"
+
+msgid "repository"
+msgstr "αποθήκη"
+
+msgid "Last updated on"
+msgstr "Τελευταία ενημέρωση στις"
+
+msgid "Toggle navigation"
+msgstr "Εναλλαγή πλοήγησης"
+
+msgid "Sphinx Book Theme"
+msgstr "Θέμα βιβλίου Sphinx"
+
+msgid "suggest edit"
+msgstr "προτείνω επεξεργασία"
+
+msgid "Open an issue"
+msgstr "Ανοίξτε ένα ζήτημα"
+
+msgid "Launch"
+msgstr "Εκτόξευση"
+
+msgid "Fullscreen mode"
+msgstr "ΛΕΙΤΟΥΡΓΙΑ ΠΛΗΡΟΥΣ ΟΘΟΝΗΣ"
+
+msgid "Edit this page"
+msgstr "Επεξεργαστείτε αυτήν τη σελίδα"
+
+msgid "By the"
+msgstr "Από το"
+
+msgid "next page"
+msgstr "επόμενη σελίδα"
diff --git a/docs/html/_static/locales/eo/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/eo/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..d1072bb
Binary files /dev/null and b/docs/html/_static/locales/eo/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/eo/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/eo/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..f7ed226
--- /dev/null
+++ b/docs/html/_static/locales/eo/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: eo\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Presi al PDF"
+
+msgid "Theme by the"
+msgstr "Temo de la"
+
+msgid "Download source file"
+msgstr "Elŝutu fontodosieron"
+
+msgid "open issue"
+msgstr "malferma numero"
+
+msgid "Contents"
+msgstr "Enhavo"
+
+msgid "previous page"
+msgstr "antaŭa paĝo"
+
+msgid "Download notebook file"
+msgstr "Elŝutu kajeran dosieron"
+
+msgid "Copyright"
+msgstr "Kopirajto"
+
+msgid "Download this page"
+msgstr "Elŝutu ĉi tiun paĝon"
+
+msgid "Source repository"
+msgstr "Fonto-deponejo"
+
+msgid "By"
+msgstr "De"
+
+msgid "repository"
+msgstr "deponejo"
+
+msgid "Last updated on"
+msgstr "Laste ĝisdatigita la"
+
+msgid "Toggle navigation"
+msgstr "Ŝalti navigadon"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinksa Libro-Temo"
+
+msgid "suggest edit"
+msgstr "sugesti redaktadon"
+
+msgid "Open an issue"
+msgstr "Malfermu numeron"
+
+msgid "Launch"
+msgstr "Lanĉo"
+
+msgid "Fullscreen mode"
+msgstr "Plenekrana reĝimo"
+
+msgid "Edit this page"
+msgstr "Redaktu ĉi tiun paĝon"
+
+msgid "By the"
+msgstr "Per la"
+
+msgid "next page"
+msgstr "sekva paĝo"
diff --git a/docs/html/_static/locales/es/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/es/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..ba2ee4d
Binary files /dev/null and b/docs/html/_static/locales/es/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/es/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/es/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..5e0029e
--- /dev/null
+++ b/docs/html/_static/locales/es/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: es\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Imprimir en PDF"
+
+msgid "Theme by the"
+msgstr "Tema por el"
+
+msgid "Download source file"
+msgstr "Descargar archivo fuente"
+
+msgid "open issue"
+msgstr "Tema abierto"
+
+msgid "Contents"
+msgstr "Contenido"
+
+msgid "previous page"
+msgstr "pagina anterior"
+
+msgid "Download notebook file"
+msgstr "Descargar archivo de cuaderno"
+
+msgid "Copyright"
+msgstr "Derechos de autor"
+
+msgid "Download this page"
+msgstr "Descarga esta pagina"
+
+msgid "Source repository"
+msgstr "Repositorio de origen"
+
+msgid "By"
+msgstr "Por"
+
+msgid "repository"
+msgstr "repositorio"
+
+msgid "Last updated on"
+msgstr "Ultima actualización en"
+
+msgid "Toggle navigation"
+msgstr "Navegación de palanca"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema del libro de la esfinge"
+
+msgid "suggest edit"
+msgstr "sugerir editar"
+
+msgid "Open an issue"
+msgstr "Abrir un problema"
+
+msgid "Launch"
+msgstr "Lanzamiento"
+
+msgid "Fullscreen mode"
+msgstr "Modo de pantalla completa"
+
+msgid "Edit this page"
+msgstr "Edita esta página"
+
+msgid "By the"
+msgstr "Por el"
+
+msgid "next page"
+msgstr "siguiente página"
diff --git a/docs/html/_static/locales/et/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/et/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..983b823
Binary files /dev/null and b/docs/html/_static/locales/et/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/et/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/et/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..8680982
--- /dev/null
+++ b/docs/html/_static/locales/et/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: et\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Prindi PDF-i"
+
+msgid "Theme by the"
+msgstr "Teema"
+
+msgid "Download source file"
+msgstr "Laadige alla lähtefail"
+
+msgid "open issue"
+msgstr "avatud küsimus"
+
+msgid "Contents"
+msgstr "Sisu"
+
+msgid "previous page"
+msgstr "eelmine leht"
+
+msgid "Download notebook file"
+msgstr "Laadige sülearvuti fail alla"
+
+msgid "Copyright"
+msgstr "Autoriõigus"
+
+msgid "Download this page"
+msgstr "Laadige see leht alla"
+
+msgid "Source repository"
+msgstr "Allikahoidla"
+
+msgid "By"
+msgstr "Kõrval"
+
+msgid "repository"
+msgstr "hoidla"
+
+msgid "Last updated on"
+msgstr "Viimati uuendatud"
+
+msgid "Toggle navigation"
+msgstr "Lülita navigeerimine sisse"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinksiraamatu teema"
+
+msgid "suggest edit"
+msgstr "soovita muuta"
+
+msgid "Open an issue"
+msgstr "Avage probleem"
+
+msgid "Launch"
+msgstr "Käivitage"
+
+msgid "Fullscreen mode"
+msgstr "Täisekraanirežiim"
+
+msgid "Edit this page"
+msgstr "Muutke seda lehte"
+
+msgid "By the"
+msgstr "Autor"
+
+msgid "next page"
+msgstr "järgmine leht"
diff --git a/docs/html/_static/locales/fi/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/fi/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..d8ac054
Binary files /dev/null and b/docs/html/_static/locales/fi/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/fi/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/fi/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..34dac21
--- /dev/null
+++ b/docs/html/_static/locales/fi/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: fi\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Tulosta PDF-tiedostoon"
+
+msgid "Theme by the"
+msgstr "Teeman tekijä"
+
+msgid "Download source file"
+msgstr "Lataa lähdetiedosto"
+
+msgid "open issue"
+msgstr "avoin ongelma"
+
+msgid "Contents"
+msgstr "Sisällys"
+
+msgid "previous page"
+msgstr "Edellinen sivu"
+
+msgid "Download notebook file"
+msgstr "Lataa muistikirjatiedosto"
+
+msgid "Copyright"
+msgstr "Tekijänoikeus"
+
+msgid "Download this page"
+msgstr "Lataa tämä sivu"
+
+msgid "Source repository"
+msgstr "Lähteen arkisto"
+
+msgid "By"
+msgstr "Tekijä"
+
+msgid "repository"
+msgstr "arkisto"
+
+msgid "Last updated on"
+msgstr "Viimeksi päivitetty"
+
+msgid "Toggle navigation"
+msgstr "Vaihda navigointia"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx-kirjan teema"
+
+msgid "suggest edit"
+msgstr "ehdottaa muokkausta"
+
+msgid "Open an issue"
+msgstr "Avaa ongelma"
+
+msgid "Launch"
+msgstr "Tuoda markkinoille"
+
+msgid "Fullscreen mode"
+msgstr "Koko näytön tila"
+
+msgid "Edit this page"
+msgstr "Muokkaa tätä sivua"
+
+msgid "By the"
+msgstr "Mukaan"
+
+msgid "next page"
+msgstr "seuraava sivu"
diff --git a/docs/html/_static/locales/fr/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/fr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..f663d39
Binary files /dev/null and b/docs/html/_static/locales/fr/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/fr/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/fr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..8991a1b
--- /dev/null
+++ b/docs/html/_static/locales/fr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: fr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Imprimer au format PDF"
+
+msgid "Theme by the"
+msgstr "Thème par le"
+
+msgid "Download source file"
+msgstr "Télécharger le fichier source"
+
+msgid "open issue"
+msgstr "signaler un problème"
+
+msgid "Contents"
+msgstr "Contenu"
+
+msgid "previous page"
+msgstr "page précédente"
+
+msgid "Download notebook file"
+msgstr "Télécharger le fichier notebook"
+
+msgid "Copyright"
+msgstr "droits d'auteur"
+
+msgid "Download this page"
+msgstr "Téléchargez cette page"
+
+msgid "Source repository"
+msgstr "Dépôt source"
+
+msgid "By"
+msgstr "Par"
+
+msgid "repository"
+msgstr "dépôt"
+
+msgid "Last updated on"
+msgstr "Dernière mise à jour le"
+
+msgid "Toggle navigation"
+msgstr "Basculer la navigation"
+
+msgid "Sphinx Book Theme"
+msgstr "Thème du livre Sphinx"
+
+msgid "suggest edit"
+msgstr "suggestion de modification"
+
+msgid "Open an issue"
+msgstr "Ouvrez un problème"
+
+msgid "Launch"
+msgstr "lancement"
+
+msgid "Fullscreen mode"
+msgstr "Mode plein écran"
+
+msgid "Edit this page"
+msgstr "Modifier cette page"
+
+msgid "By the"
+msgstr "Par le"
+
+msgid "next page"
+msgstr "page suivante"
diff --git a/docs/html/_static/locales/hr/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/hr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..eca4a1a
Binary files /dev/null and b/docs/html/_static/locales/hr/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/hr/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/hr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..42c4233
--- /dev/null
+++ b/docs/html/_static/locales/hr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: hr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Ispis u PDF"
+
+msgid "Theme by the"
+msgstr "Tema autora"
+
+msgid "Download source file"
+msgstr "Preuzmi izvornu datoteku"
+
+msgid "open issue"
+msgstr "otvoreno izdanje"
+
+msgid "Contents"
+msgstr "Sadržaj"
+
+msgid "previous page"
+msgstr "Prethodna stranica"
+
+msgid "Download notebook file"
+msgstr "Preuzmi datoteku bilježnice"
+
+msgid "Copyright"
+msgstr "Autorska prava"
+
+msgid "Download this page"
+msgstr "Preuzmite ovu stranicu"
+
+msgid "Source repository"
+msgstr "Izvorno spremište"
+
+msgid "By"
+msgstr "Po"
+
+msgid "repository"
+msgstr "spremište"
+
+msgid "Last updated on"
+msgstr "Posljednje ažuriranje:"
+
+msgid "Toggle navigation"
+msgstr "Uključi / isključi navigaciju"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema knjige Sphinx"
+
+msgid "suggest edit"
+msgstr "predloži uređivanje"
+
+msgid "Open an issue"
+msgstr "Otvorite izdanje"
+
+msgid "Launch"
+msgstr "Pokrenite"
+
+msgid "Fullscreen mode"
+msgstr "Način preko cijelog zaslona"
+
+msgid "Edit this page"
+msgstr "Uredite ovu stranicu"
+
+msgid "By the"
+msgstr "Od strane"
+
+msgid "next page"
+msgstr "sljedeća stranica"
diff --git a/docs/html/_static/locales/id/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/id/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..d07a06a
Binary files /dev/null and b/docs/html/_static/locales/id/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/id/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/id/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..b8d8d89
--- /dev/null
+++ b/docs/html/_static/locales/id/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: id\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Cetak ke PDF"
+
+msgid "Theme by the"
+msgstr "Tema oleh"
+
+msgid "Download source file"
+msgstr "Unduh file sumber"
+
+msgid "open issue"
+msgstr "masalah terbuka"
+
+msgid "Contents"
+msgstr "Isi"
+
+msgid "previous page"
+msgstr "halaman sebelumnya"
+
+msgid "Download notebook file"
+msgstr "Unduh file notebook"
+
+msgid "Copyright"
+msgstr "hak cipta"
+
+msgid "Download this page"
+msgstr "Unduh halaman ini"
+
+msgid "Source repository"
+msgstr "Repositori sumber"
+
+msgid "By"
+msgstr "Oleh"
+
+msgid "repository"
+msgstr "gudang"
+
+msgid "Last updated on"
+msgstr "Terakhir diperbarui saat"
+
+msgid "Toggle navigation"
+msgstr "Alihkan navigasi"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema Buku Sphinx"
+
+msgid "suggest edit"
+msgstr "menyarankan edit"
+
+msgid "Open an issue"
+msgstr "Buka masalah"
+
+msgid "Launch"
+msgstr "Meluncurkan"
+
+msgid "Fullscreen mode"
+msgstr "Mode layar penuh"
+
+msgid "Edit this page"
+msgstr "Edit halaman ini"
+
+msgid "By the"
+msgstr "Oleh"
+
+msgid "next page"
+msgstr "halaman selanjutnya"
diff --git a/docs/html/_static/locales/it/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/it/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..53ba476
Binary files /dev/null and b/docs/html/_static/locales/it/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/it/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/it/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..36fca59
--- /dev/null
+++ b/docs/html/_static/locales/it/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: it\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Stampa in PDF"
+
+msgid "Theme by the"
+msgstr "Tema di"
+
+msgid "Download source file"
+msgstr "Scarica il file sorgente"
+
+msgid "open issue"
+msgstr "questione aperta"
+
+msgid "Contents"
+msgstr "Contenuti"
+
+msgid "previous page"
+msgstr "pagina precedente"
+
+msgid "Download notebook file"
+msgstr "Scarica il file del taccuino"
+
+msgid "Copyright"
+msgstr "Diritto d'autore"
+
+msgid "Download this page"
+msgstr "Scarica questa pagina"
+
+msgid "Source repository"
+msgstr "Repository di origine"
+
+msgid "By"
+msgstr "Di"
+
+msgid "repository"
+msgstr "repository"
+
+msgid "Last updated on"
+msgstr "Ultimo aggiornamento il"
+
+msgid "Toggle navigation"
+msgstr "Attiva / disattiva la navigazione"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema del libro della Sfinge"
+
+msgid "suggest edit"
+msgstr "suggerisci modifica"
+
+msgid "Open an issue"
+msgstr "Apri un problema"
+
+msgid "Launch"
+msgstr "Lanciare"
+
+msgid "Fullscreen mode"
+msgstr "Modalità schermo intero"
+
+msgid "Edit this page"
+msgstr "Modifica questa pagina"
+
+msgid "By the"
+msgstr "Dal"
+
+msgid "next page"
+msgstr "pagina successiva"
diff --git a/docs/html/_static/locales/iw/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/iw/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..a45c657
Binary files /dev/null and b/docs/html/_static/locales/iw/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/iw/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/iw/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..dede9cb
--- /dev/null
+++ b/docs/html/_static/locales/iw/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: iw\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "הדפס לקובץ PDF"
+
+msgid "Theme by the"
+msgstr "נושא מאת"
+
+msgid "Download source file"
+msgstr "הורד את קובץ המקור"
+
+msgid "open issue"
+msgstr "בעיה פתוחה"
+
+msgid "Contents"
+msgstr "תוכן"
+
+msgid "previous page"
+msgstr "עמוד קודם"
+
+msgid "Download notebook file"
+msgstr "הורד קובץ מחברת"
+
+msgid "Copyright"
+msgstr "זכויות יוצרים"
+
+msgid "Download this page"
+msgstr "הורד דף זה"
+
+msgid "Source repository"
+msgstr "מאגר המקורות"
+
+msgid "By"
+msgstr "על ידי"
+
+msgid "repository"
+msgstr "מאגר"
+
+msgid "Last updated on"
+msgstr "עודכן לאחרונה ב"
+
+msgid "Toggle navigation"
+msgstr "החלף ניווט"
+
+msgid "Sphinx Book Theme"
+msgstr "נושא ספר ספינקס"
+
+msgid "suggest edit"
+msgstr "מציע לערוך"
+
+msgid "Open an issue"
+msgstr "פתח גיליון"
+
+msgid "Launch"
+msgstr "לְהַשִׁיק"
+
+msgid "Fullscreen mode"
+msgstr "מצב מסך מלא"
+
+msgid "Edit this page"
+msgstr "ערוך דף זה"
+
+msgid "By the"
+msgstr "דרך"
+
+msgid "next page"
+msgstr "עמוד הבא"
diff --git a/docs/html/_static/locales/ja/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/ja/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..1cefd29
Binary files /dev/null and b/docs/html/_static/locales/ja/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/ja/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/ja/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..2615f0d
--- /dev/null
+++ b/docs/html/_static/locales/ja/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ja\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "PDFに印刷"
+
+msgid "Theme by the"
+msgstr "のテーマ"
+
+msgid "Download source file"
+msgstr "ソースファイルをダウンロード"
+
+msgid "open issue"
+msgstr "未解決の問題"
+
+msgid "Contents"
+msgstr "目次"
+
+msgid "previous page"
+msgstr "前のページ"
+
+msgid "Download notebook file"
+msgstr "ノートブックファイルをダウンロード"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Download this page"
+msgstr "このページをダウンロード"
+
+msgid "Source repository"
+msgstr "ソースリポジトリ"
+
+msgid "By"
+msgstr "著者"
+
+msgid "repository"
+msgstr "リポジトリ"
+
+msgid "Last updated on"
+msgstr "最終更新日"
+
+msgid "Toggle navigation"
+msgstr "ナビゲーションを切り替え"
+
+msgid "Sphinx Book Theme"
+msgstr "スフィンクスの本のテーマ"
+
+msgid "suggest edit"
+msgstr "編集を提案する"
+
+msgid "Open an issue"
+msgstr "問題を報告"
+
+msgid "Launch"
+msgstr "起動"
+
+msgid "Fullscreen mode"
+msgstr "全画面モード"
+
+msgid "Edit this page"
+msgstr "このページを編集"
+
+msgid "By the"
+msgstr "によって"
+
+msgid "next page"
+msgstr "次のページ"
diff --git a/docs/html/_static/locales/ko/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/ko/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..06c7ec9
Binary files /dev/null and b/docs/html/_static/locales/ko/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/ko/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/ko/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..c9e13a4
--- /dev/null
+++ b/docs/html/_static/locales/ko/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ko\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "PDF로 인쇄"
+
+msgid "Theme by the"
+msgstr "테마별"
+
+msgid "Download source file"
+msgstr "소스 파일 다운로드"
+
+msgid "open issue"
+msgstr "열린 문제"
+
+msgid "Contents"
+msgstr "내용"
+
+msgid "previous page"
+msgstr "이전 페이지"
+
+msgid "Download notebook file"
+msgstr "노트북 파일 다운로드"
+
+msgid "Copyright"
+msgstr "저작권"
+
+msgid "Download this page"
+msgstr "이 페이지 다운로드"
+
+msgid "Source repository"
+msgstr "소스 저장소"
+
+msgid "By"
+msgstr "으로"
+
+msgid "repository"
+msgstr "저장소"
+
+msgid "Last updated on"
+msgstr "마지막 업데이트"
+
+msgid "Toggle navigation"
+msgstr "탐색 전환"
+
+msgid "Sphinx Book Theme"
+msgstr "스핑크스 도서 테마"
+
+msgid "suggest edit"
+msgstr "편집 제안"
+
+msgid "Open an issue"
+msgstr "이슈 열기"
+
+msgid "Launch"
+msgstr "시작하다"
+
+msgid "Fullscreen mode"
+msgstr "전체 화면으로보기"
+
+msgid "Edit this page"
+msgstr "이 페이지 편집"
+
+msgid "By the"
+msgstr "에 의해"
+
+msgid "next page"
+msgstr "다음 페이지"
diff --git a/docs/html/_static/locales/lt/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/lt/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..4468ba0
Binary files /dev/null and b/docs/html/_static/locales/lt/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/lt/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/lt/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..35eabd9
--- /dev/null
+++ b/docs/html/_static/locales/lt/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: lt\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Spausdinti į PDF"
+
+msgid "Theme by the"
+msgstr "Tema"
+
+msgid "Download source file"
+msgstr "Atsisiųsti šaltinio failą"
+
+msgid "open issue"
+msgstr "atviras klausimas"
+
+msgid "Contents"
+msgstr "Turinys"
+
+msgid "previous page"
+msgstr "Ankstesnis puslapis"
+
+msgid "Download notebook file"
+msgstr "Atsisiųsti nešiojamojo kompiuterio failą"
+
+msgid "Copyright"
+msgstr "Autorių teisės"
+
+msgid "Download this page"
+msgstr "Atsisiųskite šį puslapį"
+
+msgid "Source repository"
+msgstr "Šaltinio saugykla"
+
+msgid "By"
+msgstr "Iki"
+
+msgid "repository"
+msgstr "saugykla"
+
+msgid "Last updated on"
+msgstr "Paskutinį kartą atnaujinta"
+
+msgid "Toggle navigation"
+msgstr "Perjungti naršymą"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinkso knygos tema"
+
+msgid "suggest edit"
+msgstr "pasiūlyti redaguoti"
+
+msgid "Open an issue"
+msgstr "Atidarykite problemą"
+
+msgid "Launch"
+msgstr "Paleiskite"
+
+msgid "Fullscreen mode"
+msgstr "Pilno ekrano režimas"
+
+msgid "Edit this page"
+msgstr "Redaguoti šį puslapį"
+
+msgid "By the"
+msgstr "Prie"
+
+msgid "next page"
+msgstr "Kitas puslapis"
diff --git a/docs/html/_static/locales/lv/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/lv/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..74aa4d8
Binary files /dev/null and b/docs/html/_static/locales/lv/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/lv/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/lv/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..ee1bd08
--- /dev/null
+++ b/docs/html/_static/locales/lv/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: lv\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Drukāt PDF formātā"
+
+msgid "Theme by the"
+msgstr "Autora tēma"
+
+msgid "Download source file"
+msgstr "Lejupielādēt avota failu"
+
+msgid "open issue"
+msgstr "atklāts jautājums"
+
+msgid "Contents"
+msgstr "Saturs"
+
+msgid "previous page"
+msgstr "iepriekšējā lapa"
+
+msgid "Download notebook file"
+msgstr "Lejupielādēt piezīmju grāmatiņu"
+
+msgid "Copyright"
+msgstr "Autortiesības"
+
+msgid "Download this page"
+msgstr "Lejupielādējiet šo lapu"
+
+msgid "Source repository"
+msgstr "Avota krātuve"
+
+msgid "By"
+msgstr "Autors"
+
+msgid "repository"
+msgstr "krātuve"
+
+msgid "Last updated on"
+msgstr "Pēdējoreiz atjaunināts"
+
+msgid "Toggle navigation"
+msgstr "Pārslēgt navigāciju"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinksa grāmatas tēma"
+
+msgid "suggest edit"
+msgstr "ieteikt rediģēt"
+
+msgid "Open an issue"
+msgstr "Atveriet problēmu"
+
+msgid "Launch"
+msgstr "Uzsākt"
+
+msgid "Fullscreen mode"
+msgstr "Pilnekrāna režīms"
+
+msgid "Edit this page"
+msgstr "Rediģēt šo lapu"
+
+msgid "By the"
+msgstr "Ar"
+
+msgid "next page"
+msgstr "nākamā lapaspuse"
diff --git a/docs/html/_static/locales/ml/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/ml/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..2736e8f
Binary files /dev/null and b/docs/html/_static/locales/ml/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/ml/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/ml/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..d471277
--- /dev/null
+++ b/docs/html/_static/locales/ml/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ml\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "PDF- ലേക്ക് പ്രിന്റുചെയ്യുക"
+
+msgid "Theme by the"
+msgstr "പ്രമേയം"
+
+msgid "Download source file"
+msgstr "ഉറവിട ഫയൽ ഡൗൺലോഡുചെയ്യുക"
+
+msgid "open issue"
+msgstr "തുറന്ന പ്രശ്നം"
+
+msgid "previous page"
+msgstr "മുൻപത്തെ താൾ"
+
+msgid "Download notebook file"
+msgstr "നോട്ട്ബുക്ക് ഫയൽ ഡൺലോഡ് ചെയ്യുക"
+
+msgid "Copyright"
+msgstr "പകർപ്പവകാശം"
+
+msgid "Download this page"
+msgstr "ഈ പേജ് ഡൗൺലോഡുചെയ്യുക"
+
+msgid "Source repository"
+msgstr "ഉറവിട ശേഖരം"
+
+msgid "By"
+msgstr "എഴുതിയത്"
+
+msgid "Last updated on"
+msgstr "അവസാനം അപ്‌ഡേറ്റുചെയ്‌തത്"
+
+msgid "Toggle navigation"
+msgstr "നാവിഗേഷൻ ടോഗിൾ ചെയ്യുക"
+
+msgid "Sphinx Book Theme"
+msgstr "സ്ഫിങ്ക്സ് പുസ്തക തീം"
+
+msgid "suggest edit"
+msgstr "എഡിറ്റുചെയ്യാൻ നിർദ്ദേശിക്കുക"
+
+msgid "Open an issue"
+msgstr "ഒരു പ്രശ്നം തുറക്കുക"
+
+msgid "Launch"
+msgstr "സമാരംഭിക്കുക"
+
+msgid "Edit this page"
+msgstr "ഈ പേജ് എഡിറ്റുചെയ്യുക"
+
+msgid "By the"
+msgstr "എഴുതിയത്"
+
+msgid "next page"
+msgstr "അടുത്ത പേജ്"
diff --git a/docs/html/_static/locales/mr/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/mr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..fe53010
Binary files /dev/null and b/docs/html/_static/locales/mr/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/mr/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/mr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..f3694ac
--- /dev/null
+++ b/docs/html/_static/locales/mr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: mr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "पीडीएफवर मुद्रित करा"
+
+msgid "Theme by the"
+msgstr "द्वारा थीम"
+
+msgid "Download source file"
+msgstr "स्त्रोत फाइल डाउनलोड करा"
+
+msgid "open issue"
+msgstr "खुला मुद्दा"
+
+msgid "previous page"
+msgstr "मागील पान"
+
+msgid "Download notebook file"
+msgstr "नोटबुक फाईल डाउनलोड करा"
+
+msgid "Copyright"
+msgstr "कॉपीराइट"
+
+msgid "Download this page"
+msgstr "हे पृष्ठ डाउनलोड करा"
+
+msgid "Source repository"
+msgstr "स्त्रोत भांडार"
+
+msgid "By"
+msgstr "द्वारा"
+
+msgid "Last updated on"
+msgstr "अखेरचे अद्यतनित"
+
+msgid "Toggle navigation"
+msgstr "नेव्हिगेशन टॉगल करा"
+
+msgid "Sphinx Book Theme"
+msgstr "स्फिंक्स बुक थीम"
+
+msgid "suggest edit"
+msgstr "संपादन सुचवा"
+
+msgid "Open an issue"
+msgstr "एक मुद्दा उघडा"
+
+msgid "Launch"
+msgstr "लाँच करा"
+
+msgid "Edit this page"
+msgstr "हे पृष्ठ संपादित करा"
+
+msgid "By the"
+msgstr "द्वारा"
+
+msgid "next page"
+msgstr "पुढील पृष्ठ"
diff --git a/docs/html/_static/locales/ms/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/ms/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..f02603f
Binary files /dev/null and b/docs/html/_static/locales/ms/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/ms/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/ms/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..65b7c60
--- /dev/null
+++ b/docs/html/_static/locales/ms/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ms\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Cetak ke PDF"
+
+msgid "Theme by the"
+msgstr "Tema oleh"
+
+msgid "Download source file"
+msgstr "Muat turun fail sumber"
+
+msgid "open issue"
+msgstr "isu terbuka"
+
+msgid "previous page"
+msgstr "halaman sebelumnya"
+
+msgid "Download notebook file"
+msgstr "Muat turun fail buku nota"
+
+msgid "Copyright"
+msgstr "hak cipta"
+
+msgid "Download this page"
+msgstr "Muat turun halaman ini"
+
+msgid "Source repository"
+msgstr "Repositori sumber"
+
+msgid "By"
+msgstr "Oleh"
+
+msgid "Last updated on"
+msgstr "Terakhir dikemas kini pada"
+
+msgid "Toggle navigation"
+msgstr "Togol navigasi"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema Buku Sphinx"
+
+msgid "suggest edit"
+msgstr "cadangkan edit"
+
+msgid "Open an issue"
+msgstr "Buka masalah"
+
+msgid "Launch"
+msgstr "Lancarkan"
+
+msgid "Edit this page"
+msgstr "Edit halaman ini"
+
+msgid "By the"
+msgstr "Oleh"
+
+msgid "next page"
+msgstr "muka surat seterusnya"
diff --git a/docs/html/_static/locales/nl/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/nl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..e59e7ec
Binary files /dev/null and b/docs/html/_static/locales/nl/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/nl/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/nl/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..71bd1cd
--- /dev/null
+++ b/docs/html/_static/locales/nl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: nl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Afdrukken naar pdf"
+
+msgid "Theme by the"
+msgstr "Thema door de"
+
+msgid "Download source file"
+msgstr "Download het bronbestand"
+
+msgid "open issue"
+msgstr "open probleem"
+
+msgid "Contents"
+msgstr "Inhoud"
+
+msgid "previous page"
+msgstr "vorige pagina"
+
+msgid "Download notebook file"
+msgstr "Download notebookbestand"
+
+msgid "Copyright"
+msgstr "auteursrechten"
+
+msgid "Download this page"
+msgstr "Download deze pagina"
+
+msgid "Source repository"
+msgstr "Bronopslagplaats"
+
+msgid "By"
+msgstr "Door"
+
+msgid "repository"
+msgstr "repository"
+
+msgid "Last updated on"
+msgstr "Laatst geupdate op"
+
+msgid "Toggle navigation"
+msgstr "Schakel navigatie"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx-boekthema"
+
+msgid "suggest edit"
+msgstr "suggereren bewerken"
+
+msgid "Open an issue"
+msgstr "Open een probleem"
+
+msgid "Launch"
+msgstr "Lancering"
+
+msgid "Fullscreen mode"
+msgstr "Volledig scherm"
+
+msgid "Edit this page"
+msgstr "bewerk deze pagina"
+
+msgid "By the"
+msgstr "Door de"
+
+msgid "next page"
+msgstr "volgende bladzijde"
diff --git a/docs/html/_static/locales/no/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/no/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..6cd15c8
Binary files /dev/null and b/docs/html/_static/locales/no/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/no/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/no/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..b21346a
--- /dev/null
+++ b/docs/html/_static/locales/no/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: no\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Skriv ut til PDF"
+
+msgid "Theme by the"
+msgstr "Tema av"
+
+msgid "Download source file"
+msgstr "Last ned kildefilen"
+
+msgid "open issue"
+msgstr "åpent nummer"
+
+msgid "Contents"
+msgstr "Innhold"
+
+msgid "previous page"
+msgstr "forrige side"
+
+msgid "Download notebook file"
+msgstr "Last ned notatbokfilen"
+
+msgid "Copyright"
+msgstr "opphavsrett"
+
+msgid "Download this page"
+msgstr "Last ned denne siden"
+
+msgid "Source repository"
+msgstr "Kildedepot"
+
+msgid "By"
+msgstr "Av"
+
+msgid "repository"
+msgstr "oppbevaringssted"
+
+msgid "Last updated on"
+msgstr "Sist oppdatert den"
+
+msgid "Toggle navigation"
+msgstr "Bytt navigasjon"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx boktema"
+
+msgid "suggest edit"
+msgstr "foreslå redigering"
+
+msgid "Open an issue"
+msgstr "Åpne et problem"
+
+msgid "Launch"
+msgstr "Start"
+
+msgid "Fullscreen mode"
+msgstr "Fullskjerm-modus"
+
+msgid "Edit this page"
+msgstr "Rediger denne siden"
+
+msgid "By the"
+msgstr "Ved"
+
+msgid "next page"
+msgstr "neste side"
diff --git a/docs/html/_static/locales/pl/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/pl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..9ebb584
Binary files /dev/null and b/docs/html/_static/locales/pl/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/pl/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/pl/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..1b7233f
--- /dev/null
+++ b/docs/html/_static/locales/pl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: pl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Drukuj do PDF"
+
+msgid "Theme by the"
+msgstr "Motyw autorstwa"
+
+msgid "Download source file"
+msgstr "Pobierz plik źródłowy"
+
+msgid "open issue"
+msgstr "otwarty problem"
+
+msgid "Contents"
+msgstr "Zawartość"
+
+msgid "previous page"
+msgstr "Poprzednia strona"
+
+msgid "Download notebook file"
+msgstr "Pobierz plik notatnika"
+
+msgid "Copyright"
+msgstr "prawa autorskie"
+
+msgid "Download this page"
+msgstr "Pobierz tę stronę"
+
+msgid "Source repository"
+msgstr "Repozytorium źródłowe"
+
+msgid "By"
+msgstr "Przez"
+
+msgid "repository"
+msgstr "magazyn"
+
+msgid "Last updated on"
+msgstr "Ostatnia aktualizacja"
+
+msgid "Toggle navigation"
+msgstr "Przełącz nawigację"
+
+msgid "Sphinx Book Theme"
+msgstr "Motyw książki Sphinx"
+
+msgid "suggest edit"
+msgstr "zaproponuj edycję"
+
+msgid "Open an issue"
+msgstr "Otwórz problem"
+
+msgid "Launch"
+msgstr "Uruchomić"
+
+msgid "Fullscreen mode"
+msgstr "Pełny ekran"
+
+msgid "Edit this page"
+msgstr "Edytuj tę strone"
+
+msgid "By the"
+msgstr "Przez"
+
+msgid "next page"
+msgstr "Następna strona"
diff --git a/docs/html/_static/locales/pt/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/pt/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..d0ddb87
Binary files /dev/null and b/docs/html/_static/locales/pt/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/pt/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/pt/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..1b27314
--- /dev/null
+++ b/docs/html/_static/locales/pt/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: pt\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Imprimir em PDF"
+
+msgid "Theme by the"
+msgstr "Tema por"
+
+msgid "Download source file"
+msgstr "Baixar arquivo fonte"
+
+msgid "open issue"
+msgstr "questão aberta"
+
+msgid "Contents"
+msgstr "Conteúdo"
+
+msgid "previous page"
+msgstr "página anterior"
+
+msgid "Download notebook file"
+msgstr "Baixar arquivo de notebook"
+
+msgid "Copyright"
+msgstr "direito autoral"
+
+msgid "Download this page"
+msgstr "Baixe esta página"
+
+msgid "Source repository"
+msgstr "Repositório fonte"
+
+msgid "By"
+msgstr "De"
+
+msgid "repository"
+msgstr "repositório"
+
+msgid "Last updated on"
+msgstr "Última atualização em"
+
+msgid "Toggle navigation"
+msgstr "Alternar de navegação"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema do livro Sphinx"
+
+msgid "suggest edit"
+msgstr "sugerir edição"
+
+msgid "Open an issue"
+msgstr "Abra um problema"
+
+msgid "Launch"
+msgstr "Lançamento"
+
+msgid "Fullscreen mode"
+msgstr "Modo tela cheia"
+
+msgid "Edit this page"
+msgstr "Edite essa página"
+
+msgid "By the"
+msgstr "Pelo"
+
+msgid "next page"
+msgstr "próxima página"
diff --git a/docs/html/_static/locales/ro/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/ro/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..3c36ab1
Binary files /dev/null and b/docs/html/_static/locales/ro/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/ro/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/ro/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..1783ad2
--- /dev/null
+++ b/docs/html/_static/locales/ro/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ro\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Imprimați în PDF"
+
+msgid "Theme by the"
+msgstr "Tema de"
+
+msgid "Download source file"
+msgstr "Descărcați fișierul sursă"
+
+msgid "open issue"
+msgstr "problema deschisă"
+
+msgid "Contents"
+msgstr "Cuprins"
+
+msgid "previous page"
+msgstr "pagina anterioară"
+
+msgid "Download notebook file"
+msgstr "Descărcați fișierul notebook"
+
+msgid "Copyright"
+msgstr "Drepturi de autor"
+
+msgid "Download this page"
+msgstr "Descarcă această pagină"
+
+msgid "Source repository"
+msgstr "Depozit sursă"
+
+msgid "By"
+msgstr "De"
+
+msgid "repository"
+msgstr "repertoriu"
+
+msgid "Last updated on"
+msgstr "Ultima actualizare la"
+
+msgid "Toggle navigation"
+msgstr "Comutare navigare"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema Sphinx Book"
+
+msgid "suggest edit"
+msgstr "sugerează editare"
+
+msgid "Open an issue"
+msgstr "Deschideți o problemă"
+
+msgid "Launch"
+msgstr "Lansa"
+
+msgid "Fullscreen mode"
+msgstr "Modul ecran întreg"
+
+msgid "Edit this page"
+msgstr "Editați această pagină"
+
+msgid "By the"
+msgstr "Langa"
+
+msgid "next page"
+msgstr "pagina următoare"
diff --git a/docs/html/_static/locales/ru/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/ru/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..6b8ca41
Binary files /dev/null and b/docs/html/_static/locales/ru/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/ru/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/ru/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..b1176b7
--- /dev/null
+++ b/docs/html/_static/locales/ru/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ru\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Распечатать в PDF"
+
+msgid "Theme by the"
+msgstr "Тема от"
+
+msgid "Download source file"
+msgstr "Скачать исходный файл"
+
+msgid "open issue"
+msgstr "открытый вопрос"
+
+msgid "Contents"
+msgstr "Содержание"
+
+msgid "previous page"
+msgstr "Предыдущая страница"
+
+msgid "Download notebook file"
+msgstr "Скачать файл записной книжки"
+
+msgid "Copyright"
+msgstr "авторское право"
+
+msgid "Download this page"
+msgstr "Загрузите эту страницу"
+
+msgid "Source repository"
+msgstr "Исходный репозиторий"
+
+msgid "By"
+msgstr "По"
+
+msgid "repository"
+msgstr "хранилище"
+
+msgid "Last updated on"
+msgstr "Последнее обновление"
+
+msgid "Toggle navigation"
+msgstr "Переключить навигацию"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема книги Сфинкс"
+
+msgid "suggest edit"
+msgstr "предложить редактировать"
+
+msgid "Open an issue"
+msgstr "Открыть вопрос"
+
+msgid "Launch"
+msgstr "Запуск"
+
+msgid "Fullscreen mode"
+msgstr "Полноэкранный режим"
+
+msgid "Edit this page"
+msgstr "Редактировать эту страницу"
+
+msgid "By the"
+msgstr "Посредством"
+
+msgid "next page"
+msgstr "Следующая страница"
diff --git a/docs/html/_static/locales/sk/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/sk/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..59bd0dd
Binary files /dev/null and b/docs/html/_static/locales/sk/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/sk/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/sk/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..6501288
--- /dev/null
+++ b/docs/html/_static/locales/sk/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sk\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Tlač do PDF"
+
+msgid "Theme by the"
+msgstr "Téma od"
+
+msgid "Download source file"
+msgstr "Stiahnite si zdrojový súbor"
+
+msgid "open issue"
+msgstr "otvorené vydanie"
+
+msgid "Contents"
+msgstr "Obsah"
+
+msgid "previous page"
+msgstr "predchádzajúca strana"
+
+msgid "Download notebook file"
+msgstr "Stiahnite si zošit"
+
+msgid "Copyright"
+msgstr "Autorské práva"
+
+msgid "Download this page"
+msgstr "Stiahnite si túto stránku"
+
+msgid "Source repository"
+msgstr "Zdrojové úložisko"
+
+msgid "By"
+msgstr "Autor:"
+
+msgid "repository"
+msgstr "Úložisko"
+
+msgid "Last updated on"
+msgstr "Posledná aktualizácia dňa"
+
+msgid "Toggle navigation"
+msgstr "Prepnúť navigáciu"
+
+msgid "Sphinx Book Theme"
+msgstr "Téma knihy Sfinga"
+
+msgid "suggest edit"
+msgstr "navrhnúť úpravu"
+
+msgid "Open an issue"
+msgstr "Otvorte problém"
+
+msgid "Launch"
+msgstr "Spustiť"
+
+msgid "Fullscreen mode"
+msgstr "Režim celej obrazovky"
+
+msgid "Edit this page"
+msgstr "Upraviť túto stránku"
+
+msgid "By the"
+msgstr "Podľa"
+
+msgid "next page"
+msgstr "ďalšia strana"
diff --git a/docs/html/_static/locales/sl/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/sl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..87bf26d
Binary files /dev/null and b/docs/html/_static/locales/sl/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/sl/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/sl/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..3c7e3a8
--- /dev/null
+++ b/docs/html/_static/locales/sl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Natisni v PDF"
+
+msgid "Theme by the"
+msgstr "Tema avtorja"
+
+msgid "Download source file"
+msgstr "Prenesite izvorno datoteko"
+
+msgid "open issue"
+msgstr "odprto vprašanje"
+
+msgid "Contents"
+msgstr "Vsebina"
+
+msgid "previous page"
+msgstr "Prejšnja stran"
+
+msgid "Download notebook file"
+msgstr "Prenesite datoteko zvezka"
+
+msgid "Copyright"
+msgstr "avtorske pravice"
+
+msgid "Download this page"
+msgstr "Prenesite to stran"
+
+msgid "Source repository"
+msgstr "Izvorno skladišče"
+
+msgid "By"
+msgstr "Avtor"
+
+msgid "repository"
+msgstr "odlagališče"
+
+msgid "Last updated on"
+msgstr "Nazadnje posodobljeno dne"
+
+msgid "Toggle navigation"
+msgstr "Preklopi navigacijo"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema knjige Sphinx"
+
+msgid "suggest edit"
+msgstr "predlagajte urejanje"
+
+msgid "Open an issue"
+msgstr "Odprite številko"
+
+msgid "Launch"
+msgstr "Kosilo"
+
+msgid "Fullscreen mode"
+msgstr "Celozaslonski način"
+
+msgid "Edit this page"
+msgstr "Uredite to stran"
+
+msgid "By the"
+msgstr "Avtor"
+
+msgid "next page"
+msgstr "Naslednja stran"
diff --git a/docs/html/_static/locales/sr/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/sr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..ec740f4
Binary files /dev/null and b/docs/html/_static/locales/sr/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/sr/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/sr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..773b8ad
--- /dev/null
+++ b/docs/html/_static/locales/sr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Испис у ПДФ"
+
+msgid "Theme by the"
+msgstr "Тхеме би"
+
+msgid "Download source file"
+msgstr "Преузми изворну датотеку"
+
+msgid "open issue"
+msgstr "отворено издање"
+
+msgid "Contents"
+msgstr "Садржај"
+
+msgid "previous page"
+msgstr "Претходна страница"
+
+msgid "Download notebook file"
+msgstr "Преузмите датотеку бележнице"
+
+msgid "Copyright"
+msgstr "Ауторско право"
+
+msgid "Download this page"
+msgstr "Преузмите ову страницу"
+
+msgid "Source repository"
+msgstr "Изворно спремиште"
+
+msgid "By"
+msgstr "Од стране"
+
+msgid "repository"
+msgstr "спремиште"
+
+msgid "Last updated on"
+msgstr "Последње ажурирање"
+
+msgid "Toggle navigation"
+msgstr "Укључи / искључи навигацију"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема књиге Спхинк"
+
+msgid "suggest edit"
+msgstr "предложи уређивање"
+
+msgid "Open an issue"
+msgstr "Отворите издање"
+
+msgid "Launch"
+msgstr "Лансирање"
+
+msgid "Fullscreen mode"
+msgstr "Режим целог екрана"
+
+msgid "Edit this page"
+msgstr "Уредите ову страницу"
+
+msgid "By the"
+msgstr "Од"
+
+msgid "next page"
+msgstr "Следећа страна"
diff --git a/docs/html/_static/locales/sv/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/sv/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..b07dc76
Binary files /dev/null and b/docs/html/_static/locales/sv/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/sv/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/sv/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..bcac54c
--- /dev/null
+++ b/docs/html/_static/locales/sv/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sv\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Skriv ut till PDF"
+
+msgid "Theme by the"
+msgstr "Tema av"
+
+msgid "Download source file"
+msgstr "Ladda ner källfil"
+
+msgid "open issue"
+msgstr "öppna problemrapport"
+
+msgid "Contents"
+msgstr "Innehåll"
+
+msgid "previous page"
+msgstr "föregående sida"
+
+msgid "Download notebook file"
+msgstr "Ladda ner notebook-fil"
+
+msgid "Copyright"
+msgstr "Upphovsrätt"
+
+msgid "Download this page"
+msgstr "Ladda ner den här sidan"
+
+msgid "Source repository"
+msgstr "Källkodsrepositorium"
+
+msgid "By"
+msgstr "Av"
+
+msgid "repository"
+msgstr "repositorium"
+
+msgid "Last updated on"
+msgstr "Senast uppdaterad den"
+
+msgid "Toggle navigation"
+msgstr "Växla navigering"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx Boktema"
+
+msgid "suggest edit"
+msgstr "föreslå ändring"
+
+msgid "Open an issue"
+msgstr "Öppna en problemrapport"
+
+msgid "Launch"
+msgstr "Öppna"
+
+msgid "Fullscreen mode"
+msgstr "Fullskärmsläge"
+
+msgid "Edit this page"
+msgstr "Redigera den här sidan"
+
+msgid "By the"
+msgstr "Av den"
+
+msgid "next page"
+msgstr "nästa sida"
diff --git a/docs/html/_static/locales/ta/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/ta/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..29f52e1
Binary files /dev/null and b/docs/html/_static/locales/ta/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/ta/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/ta/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..b48bdfa
--- /dev/null
+++ b/docs/html/_static/locales/ta/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ta\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "PDF இல் அச்சிடுக"
+
+msgid "Theme by the"
+msgstr "வழங்கிய தீம்"
+
+msgid "Download source file"
+msgstr "மூல கோப்பைப் பதிவிறக்குக"
+
+msgid "open issue"
+msgstr "திறந்த பிரச்சினை"
+
+msgid "previous page"
+msgstr "முந்தைய பக்கம்"
+
+msgid "Download notebook file"
+msgstr "நோட்புக் கோப்பைப் பதிவிறக்கவும்"
+
+msgid "Copyright"
+msgstr "பதிப்புரிமை"
+
+msgid "Download this page"
+msgstr "இந்தப் பக்கத்தைப் பதிவிறக்கவும்"
+
+msgid "Source repository"
+msgstr "மூல களஞ்சியம்"
+
+msgid "By"
+msgstr "வழங்கியவர்"
+
+msgid "Last updated on"
+msgstr "கடைசியாக புதுப்பிக்கப்பட்டது"
+
+msgid "Toggle navigation"
+msgstr "வழிசெலுத்தலை நிலைமாற்று"
+
+msgid "Sphinx Book Theme"
+msgstr "ஸ்பிங்க்ஸ் புத்தக தீம்"
+
+msgid "suggest edit"
+msgstr "திருத்த பரிந்துரைக்கவும்"
+
+msgid "Open an issue"
+msgstr "சிக்கலைத் திறக்கவும்"
+
+msgid "Launch"
+msgstr "தொடங்க"
+
+msgid "Edit this page"
+msgstr "இந்தப் பக்கத்தைத் திருத்தவும்"
+
+msgid "By the"
+msgstr "மூலம்"
+
+msgid "next page"
+msgstr "அடுத்த பக்கம்"
diff --git a/docs/html/_static/locales/te/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/te/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..0a5f4b4
Binary files /dev/null and b/docs/html/_static/locales/te/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/te/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/te/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..952278f
--- /dev/null
+++ b/docs/html/_static/locales/te/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: te\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "PDF కి ముద్రించండి"
+
+msgid "Theme by the"
+msgstr "ద్వారా థీమ్"
+
+msgid "Download source file"
+msgstr "మూల ఫైల్‌ను డౌన్‌లోడ్ చేయండి"
+
+msgid "open issue"
+msgstr "ఓపెన్ ఇష్యూ"
+
+msgid "previous page"
+msgstr "ముందు పేజి"
+
+msgid "Download notebook file"
+msgstr "నోట్బుక్ ఫైల్ను డౌన్లోడ్ చేయండి"
+
+msgid "Copyright"
+msgstr "కాపీరైట్"
+
+msgid "Download this page"
+msgstr "ఈ పేజీని డౌన్‌లోడ్ చేయండి"
+
+msgid "Source repository"
+msgstr "మూల రిపోజిటరీ"
+
+msgid "By"
+msgstr "ద్వారా"
+
+msgid "Last updated on"
+msgstr "చివరిగా నవీకరించబడింది"
+
+msgid "Toggle navigation"
+msgstr "నావిగేషన్‌ను టోగుల్ చేయండి"
+
+msgid "Sphinx Book Theme"
+msgstr "సింహిక పుస్తక థీమ్"
+
+msgid "suggest edit"
+msgstr "సవరించమని సూచించండి"
+
+msgid "Open an issue"
+msgstr "సమస్యను తెరవండి"
+
+msgid "Launch"
+msgstr "ప్రారంభించండి"
+
+msgid "Edit this page"
+msgstr "ఈ పేజీని సవరించండి"
+
+msgid "By the"
+msgstr "ద్వారా"
+
+msgid "next page"
+msgstr "తరువాతి పేజీ"
diff --git a/docs/html/_static/locales/tg/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/tg/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..b21c6c6
Binary files /dev/null and b/docs/html/_static/locales/tg/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/tg/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/tg/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..c33dc42
--- /dev/null
+++ b/docs/html/_static/locales/tg/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: tg\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Чоп ба PDF"
+
+msgid "Theme by the"
+msgstr "Мавзӯъи аз"
+
+msgid "Download source file"
+msgstr "Файли манбаъро зеркашӣ кунед"
+
+msgid "open issue"
+msgstr "барориши кушод"
+
+msgid "Contents"
+msgstr "Мундариҷа"
+
+msgid "previous page"
+msgstr "саҳифаи қаблӣ"
+
+msgid "Download notebook file"
+msgstr "Файли дафтарро зеркашӣ кунед"
+
+msgid "Copyright"
+msgstr "Ҳуқуқи муаллиф"
+
+msgid "Download this page"
+msgstr "Ин саҳифаро зеркашӣ кунед"
+
+msgid "Source repository"
+msgstr "Анбори манбаъ"
+
+msgid "By"
+msgstr "Бо"
+
+msgid "repository"
+msgstr "анбор"
+
+msgid "Last updated on"
+msgstr "Last навсозӣ дар"
+
+msgid "Toggle navigation"
+msgstr "Гузаришро иваз кунед"
+
+msgid "Sphinx Book Theme"
+msgstr "Сфинкс Мавзӯи китоб"
+
+msgid "suggest edit"
+msgstr "пешниҳод вироиш"
+
+msgid "Open an issue"
+msgstr "Масъаларо кушоед"
+
+msgid "Launch"
+msgstr "Оғоз"
+
+msgid "Fullscreen mode"
+msgstr "Ҳолати экрани пурра"
+
+msgid "Edit this page"
+msgstr "Ин саҳифаро таҳрир кунед"
+
+msgid "By the"
+msgstr "Бо"
+
+msgid "next page"
+msgstr "саҳифаи оянда"
diff --git a/docs/html/_static/locales/th/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/th/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..abede98
Binary files /dev/null and b/docs/html/_static/locales/th/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/th/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/th/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..9d24294
--- /dev/null
+++ b/docs/html/_static/locales/th/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: th\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "พิมพ์เป็น PDF"
+
+msgid "Theme by the"
+msgstr "ธีมโดย"
+
+msgid "Download source file"
+msgstr "ดาวน์โหลดไฟล์ต้นฉบับ"
+
+msgid "open issue"
+msgstr "เปิดปัญหา"
+
+msgid "Contents"
+msgstr "สารบัญ"
+
+msgid "previous page"
+msgstr "หน้าที่แล้ว"
+
+msgid "Download notebook file"
+msgstr "ดาวน์โหลดไฟล์สมุดบันทึก"
+
+msgid "Copyright"
+msgstr "ลิขสิทธิ์"
+
+msgid "Download this page"
+msgstr "ดาวน์โหลดหน้านี้"
+
+msgid "Source repository"
+msgstr "ที่เก็บซอร์ส"
+
+msgid "By"
+msgstr "โดย"
+
+msgid "repository"
+msgstr "ที่เก็บ"
+
+msgid "Last updated on"
+msgstr "ปรับปรุงล่าสุดเมื่อ"
+
+msgid "Toggle navigation"
+msgstr "ไม่ต้องสลับช่องทาง"
+
+msgid "Sphinx Book Theme"
+msgstr "ธีมหนังสือสฟิงซ์"
+
+msgid "suggest edit"
+msgstr "แนะนำแก้ไข"
+
+msgid "Open an issue"
+msgstr "เปิดปัญหา"
+
+msgid "Launch"
+msgstr "เปิด"
+
+msgid "Fullscreen mode"
+msgstr "โหมดเต็มหน้าจอ"
+
+msgid "Edit this page"
+msgstr "แก้ไขหน้านี้"
+
+msgid "By the"
+msgstr "โดย"
+
+msgid "next page"
+msgstr "หน้าต่อไป"
diff --git a/docs/html/_static/locales/tl/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/tl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..8df1b73
Binary files /dev/null and b/docs/html/_static/locales/tl/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/tl/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/tl/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..20e0d07
--- /dev/null
+++ b/docs/html/_static/locales/tl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: tl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "I-print sa PDF"
+
+msgid "Theme by the"
+msgstr "Tema ng"
+
+msgid "Download source file"
+msgstr "Mag-download ng file ng pinagmulan"
+
+msgid "open issue"
+msgstr "bukas na isyu"
+
+msgid "previous page"
+msgstr "Nakaraang pahina"
+
+msgid "Download notebook file"
+msgstr "Mag-download ng file ng notebook"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Download this page"
+msgstr "I-download ang pahinang ito"
+
+msgid "Source repository"
+msgstr "Pinagmulan ng imbakan"
+
+msgid "By"
+msgstr "Ni"
+
+msgid "Last updated on"
+msgstr "Huling na-update noong"
+
+msgid "Toggle navigation"
+msgstr "I-toggle ang pag-navigate"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema ng Sphinx Book"
+
+msgid "suggest edit"
+msgstr "iminumungkahi i-edit"
+
+msgid "Open an issue"
+msgstr "Magbukas ng isyu"
+
+msgid "Launch"
+msgstr "Ilunsad"
+
+msgid "Edit this page"
+msgstr "I-edit ang pahinang ito"
+
+msgid "By the"
+msgstr "Sa pamamagitan ng"
+
+msgid "next page"
+msgstr "Susunod na pahina"
diff --git a/docs/html/_static/locales/tr/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/tr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..029ae18
Binary files /dev/null and b/docs/html/_static/locales/tr/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/tr/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/tr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..a77eb02
--- /dev/null
+++ b/docs/html/_static/locales/tr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: tr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "PDF olarak yazdır"
+
+msgid "Theme by the"
+msgstr "Tarafından tema"
+
+msgid "Download source file"
+msgstr "Kaynak dosyayı indirin"
+
+msgid "open issue"
+msgstr "Açık konu"
+
+msgid "Contents"
+msgstr "İçindekiler"
+
+msgid "previous page"
+msgstr "önceki sayfa"
+
+msgid "Download notebook file"
+msgstr "Defter dosyasını indirin"
+
+msgid "Copyright"
+msgstr "Telif hakkı"
+
+msgid "Download this page"
+msgstr "Bu sayfayı indirin"
+
+msgid "Source repository"
+msgstr "Kaynak kod deposu"
+
+msgid "By"
+msgstr "Tarafından"
+
+msgid "repository"
+msgstr "depo"
+
+msgid "Last updated on"
+msgstr "Son güncelleme tarihi"
+
+msgid "Toggle navigation"
+msgstr "Gezinmeyi değiştir"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfenks Kitap Teması"
+
+msgid "suggest edit"
+msgstr "düzenleme öner"
+
+msgid "Open an issue"
+msgstr "Bir sorunu açın"
+
+msgid "Launch"
+msgstr "Başlatmak"
+
+msgid "Fullscreen mode"
+msgstr "Tam ekran modu"
+
+msgid "Edit this page"
+msgstr "Bu sayfayı düzenle"
+
+msgid "By the"
+msgstr "Tarafından"
+
+msgid "next page"
+msgstr "sonraki Sayfa"
diff --git a/docs/html/_static/locales/uk/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/uk/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..16ab789
Binary files /dev/null and b/docs/html/_static/locales/uk/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/uk/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/uk/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..993dd07
--- /dev/null
+++ b/docs/html/_static/locales/uk/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: uk\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "Друк у форматі PDF"
+
+msgid "Theme by the"
+msgstr "Тема від"
+
+msgid "Download source file"
+msgstr "Завантажити вихідний файл"
+
+msgid "open issue"
+msgstr "відкритий випуск"
+
+msgid "Contents"
+msgstr "Зміст"
+
+msgid "previous page"
+msgstr "Попередня сторінка"
+
+msgid "Download notebook file"
+msgstr "Завантажте файл блокнота"
+
+msgid "Copyright"
+msgstr "Авторське право"
+
+msgid "Download this page"
+msgstr "Завантажте цю сторінку"
+
+msgid "Source repository"
+msgstr "Джерело сховища"
+
+msgid "By"
+msgstr "Автор"
+
+msgid "repository"
+msgstr "сховище"
+
+msgid "Last updated on"
+msgstr "Останнє оновлення:"
+
+msgid "Toggle navigation"
+msgstr "Переключити навігацію"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема книги \"Сфінкс\""
+
+msgid "suggest edit"
+msgstr "запропонувати редагувати"
+
+msgid "Open an issue"
+msgstr "Відкрийте випуск"
+
+msgid "Launch"
+msgstr "Запуск"
+
+msgid "Fullscreen mode"
+msgstr "Повноекранний режим"
+
+msgid "Edit this page"
+msgstr "Редагувати цю сторінку"
+
+msgid "By the"
+msgstr "По"
+
+msgid "next page"
+msgstr "Наступна сторінка"
diff --git a/docs/html/_static/locales/ur/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/ur/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..de8c84b
Binary files /dev/null and b/docs/html/_static/locales/ur/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/ur/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/ur/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..2f77426
--- /dev/null
+++ b/docs/html/_static/locales/ur/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ur\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "پی ڈی ایف پرنٹ کریں"
+
+msgid "Theme by the"
+msgstr "کے ذریعہ تھیم"
+
+msgid "Download source file"
+msgstr "سورس فائل ڈاؤن لوڈ کریں"
+
+msgid "open issue"
+msgstr "کھلا مسئلہ"
+
+msgid "previous page"
+msgstr "سابقہ ​​صفحہ"
+
+msgid "Download notebook file"
+msgstr "نوٹ بک فائل ڈاؤن لوڈ کریں"
+
+msgid "Copyright"
+msgstr "کاپی رائٹ"
+
+msgid "Download this page"
+msgstr "اس صفحے کو ڈاؤن لوڈ کریں"
+
+msgid "Source repository"
+msgstr "ماخذ ذخیرہ"
+
+msgid "By"
+msgstr "بذریعہ"
+
+msgid "Last updated on"
+msgstr "آخری بار تازہ کاری ہوئی"
+
+msgid "Toggle navigation"
+msgstr "نیویگیشن ٹوگل کریں"
+
+msgid "Sphinx Book Theme"
+msgstr "سپنکس بک تھیم"
+
+msgid "suggest edit"
+msgstr "ترمیم کی تجویز کریں"
+
+msgid "Open an issue"
+msgstr "ایک مسئلہ کھولیں"
+
+msgid "Launch"
+msgstr "لانچ کریں"
+
+msgid "Edit this page"
+msgstr "اس صفحے میں ترمیم کریں"
+
+msgid "By the"
+msgstr "کی طرف"
+
+msgid "next page"
+msgstr "اگلا صفحہ"
diff --git a/docs/html/_static/locales/vi/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/vi/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..2bb3255
Binary files /dev/null and b/docs/html/_static/locales/vi/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/vi/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/vi/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..33159f3
--- /dev/null
+++ b/docs/html/_static/locales/vi/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: vi\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "In sang PDF"
+
+msgid "Theme by the"
+msgstr "Chủ đề của"
+
+msgid "Download source file"
+msgstr "Tải xuống tệp nguồn"
+
+msgid "open issue"
+msgstr "vấn đề mở"
+
+msgid "Contents"
+msgstr "Nội dung"
+
+msgid "previous page"
+msgstr "trang trước"
+
+msgid "Download notebook file"
+msgstr "Tải xuống tệp sổ tay"
+
+msgid "Copyright"
+msgstr "Bản quyền"
+
+msgid "Download this page"
+msgstr "Tải xuống trang này"
+
+msgid "Source repository"
+msgstr "Kho nguồn"
+
+msgid "By"
+msgstr "Bởi"
+
+msgid "repository"
+msgstr "kho"
+
+msgid "Last updated on"
+msgstr "Cập nhật lần cuối vào"
+
+msgid "Toggle navigation"
+msgstr "Chuyển đổi điều hướng thành"
+
+msgid "Sphinx Book Theme"
+msgstr "Chủ đề sách nhân sư"
+
+msgid "suggest edit"
+msgstr "đề nghị chỉnh sửa"
+
+msgid "Open an issue"
+msgstr "Mở một vấn đề"
+
+msgid "Launch"
+msgstr "Phóng"
+
+msgid "Fullscreen mode"
+msgstr "Chế độ toàn màn hình"
+
+msgid "Edit this page"
+msgstr "chỉnh sửa trang này"
+
+msgid "By the"
+msgstr "Bằng"
+
+msgid "next page"
+msgstr "Trang tiếp theo"
diff --git a/docs/html/_static/locales/zh_CN/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/zh_CN/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..0e3235d
Binary files /dev/null and b/docs/html/_static/locales/zh_CN/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/zh_CN/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/zh_CN/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..2e519ef
--- /dev/null
+++ b/docs/html/_static/locales/zh_CN/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: zh_CN\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "列印成 PDF"
+
+msgid "Theme by the"
+msgstr "主题作者："
+
+msgid "Download source file"
+msgstr "下载源文件"
+
+msgid "open issue"
+msgstr "创建议题"
+
+msgid "Contents"
+msgstr "目录"
+
+msgid "previous page"
+msgstr "上一页"
+
+msgid "Download notebook file"
+msgstr "下载笔记本文件"
+
+msgid "Copyright"
+msgstr "版权"
+
+msgid "Download this page"
+msgstr "下载此页面"
+
+msgid "Source repository"
+msgstr "源码库"
+
+msgid "By"
+msgstr "作者："
+
+msgid "repository"
+msgstr "仓库"
+
+msgid "Last updated on"
+msgstr "上次更新时间："
+
+msgid "Toggle navigation"
+msgstr "显示或隐藏导航栏"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx Book 主题"
+
+msgid "suggest edit"
+msgstr "提出修改建议"
+
+msgid "Open an issue"
+msgstr "创建议题"
+
+msgid "Launch"
+msgstr "启动"
+
+msgid "Fullscreen mode"
+msgstr "全屏模式"
+
+msgid "Edit this page"
+msgstr "编辑此页面"
+
+msgid "By the"
+msgstr "作者："
+
+msgid "next page"
+msgstr "下一页"
diff --git a/docs/html/_static/locales/zh_TW/LC_MESSAGES/booktheme.mo b/docs/html/_static/locales/zh_TW/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000..9116fa9
Binary files /dev/null and b/docs/html/_static/locales/zh_TW/LC_MESSAGES/booktheme.mo differ
diff --git a/docs/html/_static/locales/zh_TW/LC_MESSAGES/booktheme.po b/docs/html/_static/locales/zh_TW/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..beecb07
--- /dev/null
+++ b/docs/html/_static/locales/zh_TW/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: zh_TW\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Print to PDF"
+msgstr "列印成 PDF"
+
+msgid "Theme by the"
+msgstr "佈景主題作者："
+
+msgid "Download source file"
+msgstr "下載原始檔"
+
+msgid "open issue"
+msgstr "公開的問題"
+
+msgid "Contents"
+msgstr "目錄"
+
+msgid "previous page"
+msgstr "上一頁"
+
+msgid "Download notebook file"
+msgstr "下載 Notebook 檔案"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Download this page"
+msgstr "下載此頁面"
+
+msgid "Source repository"
+msgstr "來源儲存庫"
+
+msgid "By"
+msgstr "作者："
+
+msgid "repository"
+msgstr "儲存庫"
+
+msgid "Last updated on"
+msgstr "最後更新時間："
+
+msgid "Toggle navigation"
+msgstr "顯示或隱藏導覽列"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx Book 佈景主題"
+
+msgid "suggest edit"
+msgstr "提出修改建議"
+
+msgid "Open an issue"
+msgstr "開啟議題"
+
+msgid "Launch"
+msgstr "啟動"
+
+msgid "Fullscreen mode"
+msgstr "全螢幕模式"
+
+msgid "Edit this page"
+msgstr "編輯此頁面"
+
+msgid "By the"
+msgstr "作者："
+
+msgid "next page"
+msgstr "下一頁"
diff --git a/docs/html/_static/minus.png b/docs/html/_static/minus.png
new file mode 100644
index 0000000..d96755f
Binary files /dev/null and b/docs/html/_static/minus.png differ
diff --git a/docs/html/_static/nbsphinx-broken-thumbnail.svg b/docs/html/_static/nbsphinx-broken-thumbnail.svg
new file mode 100644
index 0000000..4919ca8
--- /dev/null
+++ b/docs/html/_static/nbsphinx-broken-thumbnail.svg
@@ -0,0 +1,9 @@
+<svg xmlns="http://www.w3.org/2000/svg" width="100" height="100">
+  <style>
+    svg { fill: lightcoral; }
+    @media (prefers-color-scheme: dark) {
+      svg { fill: crimson; }
+    }
+  </style>
+  <path d="m 80.856887,38.34474 h 2.903414 l 7.066397,13.332221 V 38.34474 h 2.092166 V 54.281494 H 90.01545 L 82.949053,40.949273 v 13.332221 h -2.092166 z m -13.812565,0 h 10.076555 v 1.814633 h -7.920343 v 4.718048 h 7.589439 v 1.814634 h -7.589439 v 5.774805 h 8.11248 v 1.814634 H 67.044322 Z m -14.335606,0 h 2.156212 v 6.735493 L 62.01672,38.34474 h 2.775323 l -7.909669,7.429324 8.475407,8.50743 h -2.839368 l -7.653485,-7.674833 v 7.674833 h -2.156212 z m -10.738365,1.462381 q -2.348349,0 -3.73601,1.750588 -1.376987,1.750588 -1.376987,4.771419 0,3.010158 1.376987,4.760746 1.387661,1.750587 3.73601,1.750587 2.34835,0 3.714663,-1.750587 1.376986,-1.750588 1.376986,-4.760746 0,-3.020831 -1.376986,-4.771419 -1.366313,-1.750588 -3.714663,-1.750588 z m 0,-1.750588 q 3.351736,0 5.358507,2.252281 2.006772,2.241606 2.006772,6.020314 0,3.768034 -2.006772,6.020315 -2.006771,2.241606 -5.358507,2.241606 -3.362409,0 -5.379855,-2.241606 -2.006772,-2.241607 -2.006772,-6.020315 0,-3.778708 2.006772,-6.020314 2.017446,-2.252281 5.379855,-2.252281 z m -14.100771,8.75294 q 0.693831,0.234835 1.344964,1.003385 0.661808,0.768551 1.323615,2.113515 l 2.188235,4.355121 H 30.410068 L 28.371273,50.193231 Q 27.581374,48.592083 26.834172,48.069042 26.097644,47.546 24.816726,47.546 h -2.34835 v 6.735494 H 20.312164 V 38.34474 h 4.867489 q 2.732625,0 4.077588,1.142152 1.344964,1.142151 1.344964,3.447804 0,1.505078 -0.704505,2.49779 -0.69383,0.992711 -2.02812,1.376987 z m -5.401204,-6.692797 v 5.657388 h 2.711277 q 1.55845,0 2.348349,-0.715179 0.800574,-0.725854 0.800574,-2.124189 0,-1.398336 -0.800574,-2.102841 -0.789899,-0.715179 -2.348349,-0.715179 z M 7.4709616,46.670707 v 5.838851 H 10.92944 q 1.739914,0 2.57251,-0.71518 0.843271,-0.725853 0.843271,-2.209583 0,-1.494404 -0.843271,-2.198909 -0.832596,-0.715179 -2.57251,-0.715179 z m 0,-6.554031 v 4.803443 h 3.1916204 q 1.579799,0 2.34835,-0.587088 0.779225,-0.597761 0.779225,-1.814634 0,-1.206197 -0.779225,-1.803959 -0.768551,-0.597762 -2.34835,-0.597762 z M 5.3147497,38.34474 h 5.5079473 q 2.465767,0 3.800057,1.024734 1.334289,1.024734 1.334289,2.914088 0,1.462382 -0.683156,2.327001 -0.683156,0.86462 -2.006772,1.078106 1.590474,0.341579 2.465768,1.430359 0.885968,1.078106 0.885968,2.700602 0,2.134863 -1.451707,3.298364 -1.451707,1.1635 -4.130961,1.1635 H 5.3147497 Z M 65.365328,74.550032 H 67.52154 V 88.67217 h 7.760238 v 1.814634 h -9.91645 z m -6.447298,0 h 2.156214 V 90.486804 H 58.91803 Z m -9.955303,2.124192 -2.924765,7.931025 h 5.860208 z m -1.216872,-2.124192 h 2.444423 l 6.073692,15.936772 h -2.241606 l -1.451709,-4.088268 h -7.183824 l -1.451707,4.088268 H 41.66149 Z m -21.264514,0 h 2.903417 L 36.45116,87.882271 V 74.550032 h 2.092171 V 90.486804 H 35.639914 L 28.573506,77.154569 v 13.332235 h -2.092165 z m 53.454813,-9.781663 v 5.838858 h 3.458479 q 1.739919,0 2.572513,-0.71518 0.843272,-0.725854 0.843272,-2.209586 0,-1.494406 -0.843272,-2.198912 -0.832594,-0.71518 -2.572513,-0.71518 z m 0,-6.554037 v 4.803447 h 3.191622 q 1.579803,0 2.348355,-0.587087 0.779225,-0.597763 0.779225,-1.814636 0,-1.206199 -0.779225,-1.803962 -0.768552,-0.597762 -2.348355,-0.597762 z m -2.156212,-1.771939 h 5.507949 q 2.465772,0 3.800065,1.024736 1.334288,1.024735 1.334288,2.914092 0,1.462383 -0.683157,2.327003 -0.683157,0.864621 -2.006771,1.078107 1.590471,0.341579 2.46577,1.43036 0.885967,1.078107 0.885967,2.700605 0,2.134866 -1.451708,3.298368 -1.451709,1.163502 -4.130963,1.163502 h -5.72144 z m -18.86154,0 h 3.212969 l 4.066921,10.845118 4.088269,-10.845118 H 73.49953 V 72.379166 H 71.396692 V 58.385121 L 67.287075,69.315634 H 65.120183 L 61.010567,58.385121 v 13.994045 h -2.092165 z m -16.246333,0 h 2.166892 v 9.681616 q 0,2.561839 0.928668,3.693318 0.928667,1.120804 3.010159,1.120804 2.070819,0 2.999486,-1.120804 0.928667,-1.131479 0.928667,-3.693318 v -9.681616 h 2.166886 v 9.948474 q 0,3.116904 -1.547776,4.70738 -1.537104,1.590474 -4.547263,1.590474 -3.020833,0 -4.568615,-1.590474 -1.537104,-1.590476 -1.537104,-4.70738 z m -16.192953,0 h 2.156213 v 6.532689 h 7.83496 v -6.532689 h 2.156213 v 15.936773 h -2.156213 v -7.589448 h -7.83496 v 7.589448 h -2.156213 z m -15.563168,0 h 13.481671 v 1.814636 H 18.740226 V 72.379166 H 16.573339 V 58.257029 h -5.657391 z" />
+</svg>
diff --git a/docs/html/_static/nbsphinx-code-cells.css b/docs/html/_static/nbsphinx-code-cells.css
new file mode 100644
index 0000000..a3fb27c
--- /dev/null
+++ b/docs/html/_static/nbsphinx-code-cells.css
@@ -0,0 +1,259 @@
+/* remove conflicting styling from Sphinx themes */
+div.nbinput.container div.prompt *,
+div.nboutput.container div.prompt *,
+div.nbinput.container div.input_area pre,
+div.nboutput.container div.output_area pre,
+div.nbinput.container div.input_area .highlight,
+div.nboutput.container div.output_area .highlight {
+    border: none;
+    padding: 0;
+    margin: 0;
+    box-shadow: none;
+}
+
+div.nbinput.container > div[class*=highlight],
+div.nboutput.container > div[class*=highlight] {
+    margin: 0;
+}
+
+div.nbinput.container div.prompt *,
+div.nboutput.container div.prompt * {
+    background: none;
+}
+
+div.nboutput.container div.output_area .highlight,
+div.nboutput.container div.output_area pre {
+    background: unset;
+}
+
+div.nboutput.container div.output_area div.highlight {
+    color: unset;  /* override Pygments text color */
+}
+
+/* avoid gaps between output lines */
+div.nboutput.container div[class*=highlight] pre {
+    line-height: normal;
+}
+
+/* input/output containers */
+div.nbinput.container,
+div.nboutput.container {
+    display: -webkit-flex;
+    display: flex;
+    align-items: flex-start;
+    margin: 0;
+    width: 100%;
+}
+@media (max-width: 540px) {
+    div.nbinput.container,
+    div.nboutput.container {
+        flex-direction: column;
+    }
+}
+
+/* input container */
+div.nbinput.container {
+    padding-top: 5px;
+}
+
+/* last container */
+div.nblast.container {
+    padding-bottom: 5px;
+}
+
+/* input prompt */
+div.nbinput.container div.prompt pre,
+/* for sphinx_immaterial theme: */
+div.nbinput.container div.prompt pre > code {
+    color: #307FC1;
+}
+
+/* output prompt */
+div.nboutput.container div.prompt pre,
+/* for sphinx_immaterial theme: */
+div.nboutput.container div.prompt pre > code {
+    color: #BF5B3D;
+}
+
+/* all prompts */
+div.nbinput.container div.prompt,
+div.nboutput.container div.prompt {
+    width: 4.5ex;
+    padding-top: 5px;
+    position: relative;
+    user-select: none;
+}
+
+div.nbinput.container div.prompt > div,
+div.nboutput.container div.prompt > div {
+    position: absolute;
+    right: 0;
+    margin-right: 0.3ex;
+}
+
+@media (max-width: 540px) {
+    div.nbinput.container div.prompt,
+    div.nboutput.container div.prompt {
+        width: unset;
+        text-align: left;
+        padding: 0.4em;
+    }
+    div.nboutput.container div.prompt.empty {
+        padding: 0;
+    }
+
+    div.nbinput.container div.prompt > div,
+    div.nboutput.container div.prompt > div {
+        position: unset;
+    }
+}
+
+/* disable scrollbars and line breaks on prompts */
+div.nbinput.container div.prompt pre,
+div.nboutput.container div.prompt pre {
+    overflow: hidden;
+    white-space: pre;
+}
+
+/* input/output area */
+div.nbinput.container div.input_area,
+div.nboutput.container div.output_area {
+    -webkit-flex: 1;
+    flex: 1;
+    overflow: auto;
+}
+@media (max-width: 540px) {
+    div.nbinput.container div.input_area,
+    div.nboutput.container div.output_area {
+        width: 100%;
+    }
+}
+
+/* input area */
+div.nbinput.container div.input_area {
+    border: 1px solid #e0e0e0;
+    border-radius: 2px;
+    /*background: #f5f5f5;*/
+}
+
+/* override MathJax center alignment in output cells */
+div.nboutput.container div[class*=MathJax] {
+    text-align: left !important;
+}
+
+/* override sphinx.ext.imgmath center alignment in output cells */
+div.nboutput.container div.math p {
+    text-align: left;
+}
+
+/* standard error */
+div.nboutput.container div.output_area.stderr {
+    background: #fdd;
+}
+
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+.ansi-blue-intense-fg { color: #0065CA; }
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+.ansi-white-intense-bg { background-color: #A1A6B2; }
+
+.ansi-default-inverse-fg { color: #FFFFFF; }
+.ansi-default-inverse-bg { background-color: #000000; }
+
+.ansi-bold { font-weight: bold; }
+.ansi-underline { text-decoration: underline; }
+
+
+div.nbinput.container div.input_area div[class*=highlight] > pre,
+div.nboutput.container div.output_area div[class*=highlight] > pre,
+div.nboutput.container div.output_area div[class*=highlight].math,
+div.nboutput.container div.output_area.rendered_html,
+div.nboutput.container div.output_area > div.output_javascript,
+div.nboutput.container div.output_area:not(.rendered_html) > img{
+    padding: 5px;
+    margin: 0;
+}
+
+/* fix copybtn overflow problem in chromium (needed for 'sphinx_copybutton') */
+div.nbinput.container div.input_area > div[class^='highlight'],
+div.nboutput.container div.output_area > div[class^='highlight']{
+    overflow-y: hidden;
+}
+
+/* hide copy button on prompts for 'sphinx_copybutton' extension ... */
+.prompt .copybtn,
+/* ... and 'sphinx_immaterial' theme */
+.prompt .md-clipboard.md-icon {
+    display: none;
+}
+
+/* Some additional styling taken form the Jupyter notebook CSS */
+.jp-RenderedHTMLCommon table,
+div.rendered_html table {
+  border: none;
+  border-collapse: collapse;
+  border-spacing: 0;
+  color: black;
+  font-size: 12px;
+  table-layout: fixed;
+}
+.jp-RenderedHTMLCommon thead,
+div.rendered_html thead {
+  border-bottom: 1px solid black;
+  vertical-align: bottom;
+}
+.jp-RenderedHTMLCommon tr,
+.jp-RenderedHTMLCommon th,
+.jp-RenderedHTMLCommon td,
+div.rendered_html tr,
+div.rendered_html th,
+div.rendered_html td {
+  text-align: right;
+  vertical-align: middle;
+  padding: 0.5em 0.5em;
+  line-height: normal;
+  white-space: normal;
+  max-width: none;
+  border: none;
+}
+.jp-RenderedHTMLCommon th,
+div.rendered_html th {
+  font-weight: bold;
+}
+.jp-RenderedHTMLCommon tbody tr:nth-child(odd),
+div.rendered_html tbody tr:nth-child(odd) {
+  background: #f5f5f5;
+}
+.jp-RenderedHTMLCommon tbody tr:hover,
+div.rendered_html tbody tr:hover {
+  background: rgba(66, 165, 245, 0.2);
+}
+
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new file mode 100644
index 0000000..365c27a
--- /dev/null
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+.nbsphinx-gallery {
+    display: grid;
+    grid-template-columns: repeat(auto-fill, minmax(160px, 1fr));
+    gap: 5px;
+    margin-top: 1em;
+    margin-bottom: 1em;
+}
+
+.nbsphinx-gallery > a {
+    padding: 5px;
+    border: 1px dotted currentColor;
+    border-radius: 2px;
+    text-align: center;
+}
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+.nbsphinx-gallery > a:hover {
+    border-style: solid;
+}
+
+.nbsphinx-gallery img {
+    max-width: 100%;
+    max-height: 100%;
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+
+.nbsphinx-gallery > a > div:first-child {
+    display: flex;
+    align-items: start;
+    justify-content: center;
+    height: 120px;
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new file mode 100644
index 0000000..9dca758
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+html[data-theme="light"] .highlight span.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+html[data-theme="light"] .highlight td.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+html[data-theme="light"] .highlight span.linenos.special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
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+html[data-theme="light"] .highlight { background: #fefefe; color: #080808 }
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+html[data-theme="light"] .highlight .mo { color: #7f4707 } /* Literal.Number.Oct */
+html[data-theme="light"] .highlight .sa { color: #00622f } /* Literal.String.Affix */
+html[data-theme="light"] .highlight .sb { color: #00622f } /* Literal.String.Backtick */
+html[data-theme="light"] .highlight .sc { color: #00622f } /* Literal.String.Char */
+html[data-theme="light"] .highlight .dl { color: #00622f } /* Literal.String.Delimiter */
+html[data-theme="light"] .highlight .sd { color: #00622f } /* Literal.String.Doc */
+html[data-theme="light"] .highlight .s2 { color: #00622f } /* Literal.String.Double */
+html[data-theme="light"] .highlight .se { color: #00622f } /* Literal.String.Escape */
+html[data-theme="light"] .highlight .sh { color: #00622f } /* Literal.String.Heredoc */
+html[data-theme="light"] .highlight .si { color: #00622f } /* Literal.String.Interpol */
+html[data-theme="light"] .highlight .sx { color: #00622f } /* Literal.String.Other */
+html[data-theme="light"] .highlight .sr { color: #a12236 } /* Literal.String.Regex */
+html[data-theme="light"] .highlight .s1 { color: #00622f } /* Literal.String.Single */
+html[data-theme="light"] .highlight .ss { color: #005b82 } /* Literal.String.Symbol */
+html[data-theme="light"] .highlight .bp { color: #7f4707 } /* Name.Builtin.Pseudo */
+html[data-theme="light"] .highlight .fm { color: #005b82 } /* Name.Function.Magic */
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+html[data-theme="light"] .highlight .vi { color: #a12236 } /* Name.Variable.Instance */
+html[data-theme="light"] .highlight .vm { color: #7f4707 } /* Name.Variable.Magic */
+html[data-theme="light"] .highlight .il { color: #7f4707 } /* Literal.Number.Integer.Long */
+html[data-theme="dark"] .highlight pre { line-height: 125%; }
+html[data-theme="dark"] .highlight td.linenos .normal { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+html[data-theme="dark"] .highlight span.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+html[data-theme="dark"] .highlight td.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+html[data-theme="dark"] .highlight span.linenos.special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+html[data-theme="dark"] .highlight .hll { background-color: #ffd9002e }
+html[data-theme="dark"] .highlight { background: #2b2b2b; color: #f8f8f2 }
+html[data-theme="dark"] .highlight .c { color: #ffd900 } /* Comment */
+html[data-theme="dark"] .highlight .err { color: #ffa07a } /* Error */
+html[data-theme="dark"] .highlight .k { color: #dcc6e0 } /* Keyword */
+html[data-theme="dark"] .highlight .l { color: #ffd900 } /* Literal */
+html[data-theme="dark"] .highlight .n { color: #f8f8f2 } /* Name */
+html[data-theme="dark"] .highlight .o { color: #abe338 } /* Operator */
+html[data-theme="dark"] .highlight .p { color: #f8f8f2 } /* Punctuation */
+html[data-theme="dark"] .highlight .ch { color: #ffd900 } /* Comment.Hashbang */
+html[data-theme="dark"] .highlight .cm { color: #ffd900 } /* Comment.Multiline */
+html[data-theme="dark"] .highlight .cp { color: #ffd900 } /* Comment.Preproc */
+html[data-theme="dark"] .highlight .cpf { color: #ffd900 } /* Comment.PreprocFile */
+html[data-theme="dark"] .highlight .c1 { color: #ffd900 } /* Comment.Single */
+html[data-theme="dark"] .highlight .cs { color: #ffd900 } /* Comment.Special */
+html[data-theme="dark"] .highlight .gd { color: #00e0e0 } /* Generic.Deleted */
+html[data-theme="dark"] .highlight .ge { font-style: italic } /* Generic.Emph */
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+html[data-theme="dark"] .highlight .na { color: #ffd900 } /* Name.Attribute */
+html[data-theme="dark"] .highlight .nb { color: #ffd900 } /* Name.Builtin */
+html[data-theme="dark"] .highlight .nc { color: #00e0e0 } /* Name.Class */
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+html[data-theme="dark"] .highlight .nd { color: #ffd900 } /* Name.Decorator */
+html[data-theme="dark"] .highlight .ni { color: #abe338 } /* Name.Entity */
+html[data-theme="dark"] .highlight .ne { color: #dcc6e0 } /* Name.Exception */
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+html[data-theme="dark"] .highlight .nl { color: #ffd900 } /* Name.Label */
+html[data-theme="dark"] .highlight .nn { color: #f8f8f2 } /* Name.Namespace */
+html[data-theme="dark"] .highlight .nx { color: #f8f8f2 } /* Name.Other */
+html[data-theme="dark"] .highlight .py { color: #00e0e0 } /* Name.Property */
+html[data-theme="dark"] .highlight .nt { color: #00e0e0 } /* Name.Tag */
+html[data-theme="dark"] .highlight .nv { color: #ffa07a } /* Name.Variable */
+html[data-theme="dark"] .highlight .ow { color: #dcc6e0 } /* Operator.Word */
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+html[data-theme="dark"] .highlight .w { color: #f8f8f2 } /* Text.Whitespace */
+html[data-theme="dark"] .highlight .mb { color: #ffd900 } /* Literal.Number.Bin */
+html[data-theme="dark"] .highlight .mf { color: #ffd900 } /* Literal.Number.Float */
+html[data-theme="dark"] .highlight .mh { color: #ffd900 } /* Literal.Number.Hex */
+html[data-theme="dark"] .highlight .mi { color: #ffd900 } /* Literal.Number.Integer */
+html[data-theme="dark"] .highlight .mo { color: #ffd900 } /* Literal.Number.Oct */
+html[data-theme="dark"] .highlight .sa { color: #abe338 } /* Literal.String.Affix */
+html[data-theme="dark"] .highlight .sb { color: #abe338 } /* Literal.String.Backtick */
+html[data-theme="dark"] .highlight .sc { color: #abe338 } /* Literal.String.Char */
+html[data-theme="dark"] .highlight .dl { color: #abe338 } /* Literal.String.Delimiter */
+html[data-theme="dark"] .highlight .sd { color: #abe338 } /* Literal.String.Doc */
+html[data-theme="dark"] .highlight .s2 { color: #abe338 } /* Literal.String.Double */
+html[data-theme="dark"] .highlight .se { color: #abe338 } /* Literal.String.Escape */
+html[data-theme="dark"] .highlight .sh { color: #abe338 } /* Literal.String.Heredoc */
+html[data-theme="dark"] .highlight .si { color: #abe338 } /* Literal.String.Interpol */
+html[data-theme="dark"] .highlight .sx { color: #abe338 } /* Literal.String.Other */
+html[data-theme="dark"] .highlight .sr { color: #ffa07a } /* Literal.String.Regex */
+html[data-theme="dark"] .highlight .s1 { color: #abe338 } /* Literal.String.Single */
+html[data-theme="dark"] .highlight .ss { color: #00e0e0 } /* Literal.String.Symbol */
+html[data-theme="dark"] .highlight .bp { color: #ffd900 } /* Name.Builtin.Pseudo */
+html[data-theme="dark"] .highlight .fm { color: #00e0e0 } /* Name.Function.Magic */
+html[data-theme="dark"] .highlight .vc { color: #ffa07a } /* Name.Variable.Class */
+html[data-theme="dark"] .highlight .vg { color: #ffa07a } /* Name.Variable.Global */
+html[data-theme="dark"] .highlight .vi { color: #ffa07a } /* Name.Variable.Instance */
+html[data-theme="dark"] .highlight .vm { color: #ffd900 } /* Name.Variable.Magic */
+html[data-theme="dark"] .highlight .il { color: #ffd900 } /* Literal.Number.Integer.Long */
\ No newline at end of file
diff --git a/docs/html/_static/sbt-webpack-macros.html b/docs/html/_static/sbt-webpack-macros.html
new file mode 100644
index 0000000..6cbf559
--- /dev/null
+++ b/docs/html/_static/sbt-webpack-macros.html
@@ -0,0 +1,11 @@
+<!--
+  All these macros are auto-generated and must **NOT** be edited by hand.
+  See the webpack.config.js file, to learn more about how this is generated.
+-->
+{% macro head_pre_bootstrap() %}
+  <link href="{{ pathto('_static/styles/sphinx-book-theme.css', 1) }}" rel="stylesheet">
+{% endmacro %}
+
+{% macro body_post() %}
+  <script src="{{ pathto('_static/scripts/sphinx-book-theme.js', 1) }}"></script>
+{% endmacro %}
diff --git a/docs/html/_static/scripts/bootstrap.js b/docs/html/_static/scripts/bootstrap.js
new file mode 100644
index 0000000..c8178de
--- /dev/null
+++ b/docs/html/_static/scripts/bootstrap.js
@@ -0,0 +1,3 @@
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t=this._mergeConfigObj(t),t=this._configAfterMerge(t),this._typeCheckConfig(t),t}_configAfterMerge(t){return t}_mergeConfigObj(t,e){const i=Ft(e)?_e.getDataAttribute(e,"config"):{};return{...this.constructor.Default,..."object"==typeof i?i:{},...Ft(e)?_e.getDataAttributes(e):{},..."object"==typeof t?t:{}}}_typeCheckConfig(t,e=this.constructor.DefaultType){for(const[n,s]of Object.entries(e)){const e=t[n],o=Ft(e)?"element":null==(i=e)?`${i}`:Object.prototype.toString.call(i).match(/\s([a-z]+)/i)[1].toLowerCase();if(!new RegExp(s).test(o))throw new TypeError(`${this.constructor.NAME.toUpperCase()}: Option "${n}" provided type "${o}" but expected type "${s}".`)}var i}}class ve extends be{constructor(t,e){super(),(t=Ht(t))&&(this._element=t,this._config=this._getConfig(e),Nt.set(this._element,this.constructor.DATA_KEY,this))}dispose(){Nt.remove(this._element,this.constructor.DATA_KEY),fe.off(this._element,this.constructor.EVENT_KEY);for(const t of 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i=this._getItemIndex(this._getActive());if(i===t)return;const n=t>i?ze:Re;this._slide(n,e[t])}dispose(){this._swipeHelper&&this._swipeHelper.dispose(),super.dispose()}_configAfterMerge(t){return t.defaultInterval=t.interval,t}_addEventListeners(){this._config.keyboard&&fe.on(this._element,Qe,(t=>this._keydown(t))),"hover"===this._config.pause&&(fe.on(this._element,Xe,(()=>this.pause())),fe.on(this._element,Ue,(()=>this._maybeEnableCycle()))),this._config.touch&&je.isSupported()&&this._addTouchEventListeners()}_addTouchEventListeners(){for(const t of we.find(".carousel-item img",this._element))fe.on(t,Ge,(t=>t.preventDefault()));const t={leftCallback:()=>this._slide(this._directionToOrder(qe)),rightCallback:()=>this._slide(this._directionToOrder(Ve)),endCallback:()=>{"hover"===this._config.pause&&(this.pause(),this.touchTimeout&&clearTimeout(this.touchTimeout),this.touchTimeout=setTimeout((()=>this._maybeEnableCycle()),500+this._config.interval))}};this._swipeHelper=new je(this._element,t)}_keydown(t){if(/input|textarea/i.test(t.target.tagName))return;const e=oi[t.key];e&&(t.preventDefault(),this._slide(this._directionToOrder(e)))}_getItemIndex(t){return this._getItems().indexOf(t)}_setActiveIndicatorElement(t){if(!this._indicatorsElement)return;const e=we.findOne(ii,this._indicatorsElement);e.classList.remove(ei),e.removeAttribute("aria-current");const i=we.findOne(`[data-bs-slide-to="${t}"]`,this._indicatorsElement);i&&(i.classList.add(ei),i.setAttribute("aria-current","true"))}_updateInterval(){const t=this._activeElement||this._getActive();if(!t)return;const e=Number.parseInt(t.getAttribute("data-bs-interval"),10);this._config.interval=e||this._config.defaultInterval}_slide(t,e=null){if(this._isSliding)return;const i=this._getActive(),n=t===ze,s=e||Gt(this._getItems(),i,n,this._config.wrap);if(s===i)return;const 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.collapse.collapsing").filter((t=>t!==this._element)).map((t=>Ei.getOrCreateInstance(t,{toggle:!1})))),t.length&&t[0]._isTransitioning)return;if(fe.trigger(this._element,hi).defaultPrevented)return;for(const e of t)e.hide();const e=this._getDimension();this._element.classList.remove(gi),this._element.classList.add(_i),this._element.style[e]=0,this._addAriaAndCollapsedClass(this._triggerArray,!0),this._isTransitioning=!0;const i=`scroll${e[0].toUpperCase()+e.slice(1)}`;this._queueCallback((()=>{this._isTransitioning=!1,this._element.classList.remove(_i),this._element.classList.add(gi,mi),this._element.style[e]="",fe.trigger(this._element,di)}),this._element,!0),this._element.style[e]=`${this._element[i]}px`}hide(){if(this._isTransitioning||!this._isShown())return;if(fe.trigger(this._element,ui).defaultPrevented)return;const t=this._getDimension();this._element.style[t]=`${this._element.getBoundingClientRect()[t]}px`,qt(this._element),this._element.classList.add(_i),this._element.classList.remove(gi,mi);for(const t of this._triggerArray){const e=we.getElementFromSelector(t);e&&!this._isShown(e)&&this._addAriaAndCollapsedClass([t],!1)}this._isTransitioning=!0,this._element.style[t]="",this._queueCallback((()=>{this._isTransitioning=!1,this._element.classList.remove(_i),this._element.classList.add(gi),fe.trigger(this._element,fi)}),this._element,!0)}_isShown(t=this._element){return t.classList.contains(mi)}_configAfterMerge(t){return t.toggle=Boolean(t.toggle),t.parent=Ht(t.parent),t}_getDimension(){return this._element.classList.contains("collapse-horizontal")?"width":"height"}_initializeChildren(){if(!this._config.parent)return;const t=this._getFirstLevelChildren(vi);for(const e of t){const t=we.getElementFromSelector(e);t&&this._addAriaAndCollapsedClass([e],this._isShown(t))}}_getFirstLevelChildren(t){const e=we.find(bi,this._config.parent);return we.find(t,this._config.parent).filter((t=>!e.includes(t)))}_addAriaAndCollapsedClass(t,e){if(t.length)for(const i of t)i.classList.toggle("collapsed",!e),i.setAttribute("aria-expanded",e)}static jQueryInterface(t){const e={};return"string"==typeof t&&/show|hide/.test(t)&&(e.toggle=!1),this.each((function(){const i=Ei.getOrCreateInstance(this,e);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t]()}}))}}fe.on(document,pi,vi,(function(t){("A"===t.target.tagName||t.delegateTarget&&"A"===t.delegateTarget.tagName)&&t.preventDefault();for(const t of we.getMultipleElementsFromSelector(this))Ei.getOrCreateInstance(t,{toggle:!1}).toggle()})),Qt(Ei);const Ai="dropdown",Ti=".bs.dropdown",Ci=".data-api",Oi="ArrowUp",xi="ArrowDown",ki=`hide${Ti}`,Li=`hidden${Ti}`,Si=`show${Ti}`,Di=`shown${Ti}`,$i=`click${Ti}${Ci}`,Ii=`keydown${Ti}${Ci}`,Ni=`keyup${Ti}${Ci}`,Pi="show",Mi='[data-bs-toggle="dropdown"]:not(.disabled):not(:disabled)',ji=`${Mi}.${Pi}`,Fi=".dropdown-menu",Hi=Kt()?"top-end":"top-start",Bi=Kt()?"top-start":"top-end",Wi=Kt()?"bottom-end":"bottom-start",zi=Kt()?"bottom-start":"bottom-end",Ri=Kt()?"left-start":"right-start",qi=Kt()?"right-start":"left-start",Vi={autoClose:!0,boundary:"clippingParents",display:"dynamic",offset:[0,2],popperConfig:null,reference:"toggle"},Yi={autoClose:"(boolean|string)",boundary:"(string|element)",display:"string",offset:"(array|string|function)",popperConfig:"(null|object|function)",reference:"(string|element|object)"};class Ki extends ve{constructor(t,e){super(t,e),this._popper=null,this._parent=this._element.parentNode,this._menu=we.next(this._element,Fi)[0]||we.prev(this._element,Fi)[0]||we.findOne(Fi,this._parent),this._inNavbar=this._detectNavbar()}static get Default(){return Vi}static get DefaultType(){return Yi}static get NAME(){return Ai}toggle(){return this._isShown()?this.hide():this.show()}show(){if(Wt(this._element)||this._isShown())return;const t={relatedTarget:this._element};if(!fe.trigger(this._element,Si,t).defaultPrevented){if(this._createPopper(),"ontouchstart"in document.documentElement&&!this._parent.closest(".navbar-nav"))for(const t of[].concat(...document.body.children))fe.on(t,"mouseover",Rt);this._element.focus(),this._element.setAttribute("aria-expanded",!0),this._menu.classList.add(Pi),this._element.classList.add(Pi),fe.trigger(this._element,Di,t)}}hide(){if(Wt(this._element)||!this._isShown())return;const t={relatedTarget:this._element};this._completeHide(t)}dispose(){this._popper&&this._popper.destroy(),super.dispose()}update(){this._inNavbar=this._detectNavbar(),this._popper&&this._popper.update()}_completeHide(t){if(!fe.trigger(this._element,ki,t).defaultPrevented){if("ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))fe.off(t,"mouseover",Rt);this._popper&&this._popper.destroy(),this._menu.classList.remove(Pi),this._element.classList.remove(Pi),this._element.setAttribute("aria-expanded","false"),_e.removeDataAttribute(this._menu,"popper"),fe.trigger(this._element,Li,t)}}_getConfig(t){if("object"==typeof(t=super._getConfig(t)).reference&&!Ft(t.reference)&&"function"!=typeof t.reference.getBoundingClientRect)throw new TypeError(`${Ai.toUpperCase()}: Option "reference" provided type "object" without a required "getBoundingClientRect" method.`);return t}_createPopper(){if(void 0===e)throw new TypeError("Bootstrap's dropdowns require Popper (https://popper.js.org)");let t=this._element;"parent"===this._config.reference?t=this._parent:Ft(this._config.reference)?t=Ht(this._config.reference):"object"==typeof this._config.reference&&(t=this._config.reference);const i=this._getPopperConfig();this._popper=Dt(t,this._menu,i)}_isShown(){return this._menu.classList.contains(Pi)}_getPlacement(){const t=this._parent;if(t.classList.contains("dropend"))return Ri;if(t.classList.contains("dropstart"))return qi;if(t.classList.contains("dropup-center"))return"top";if(t.classList.contains("dropdown-center"))return"bottom";const e="end"===getComputedStyle(this._menu).getPropertyValue("--bs-position").trim();return t.classList.contains("dropup")?e?Bi:Hi:e?zi:Wi}_detectNavbar(){return null!==this._element.closest(".navbar")}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_getPopperConfig(){const t={placement:this._getPlacement(),modifiers:[{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"offset",options:{offset:this._getOffset()}}]};return(this._inNavbar||"static"===this._config.display)&&(_e.setDataAttribute(this._menu,"popper","static"),t.modifiers=[{name:"applyStyles",enabled:!1}]),{...t,...Xt(this._config.popperConfig,[t])}}_selectMenuItem({key:t,target:e}){const i=we.find(".dropdown-menu .dropdown-item:not(.disabled):not(:disabled)",this._menu).filter((t=>Bt(t)));i.length&&Gt(i,e,t===xi,!i.includes(e)).focus()}static jQueryInterface(t){return this.each((function(){const e=Ki.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}static clearMenus(t){if(2===t.button||"keyup"===t.type&&"Tab"!==t.key)return;const e=we.find(ji);for(const i of e){const e=Ki.getInstance(i);if(!e||!1===e._config.autoClose)continue;const n=t.composedPath(),s=n.includes(e._menu);if(n.includes(e._element)||"inside"===e._config.autoClose&&!s||"outside"===e._config.autoClose&&s)continue;if(e._menu.contains(t.target)&&("keyup"===t.type&&"Tab"===t.key||/input|select|option|textarea|form/i.test(t.target.tagName)))continue;const o={relatedTarget:e._element};"click"===t.type&&(o.clickEvent=t),e._completeHide(o)}}static dataApiKeydownHandler(t){const e=/input|textarea/i.test(t.target.tagName),i="Escape"===t.key,n=[Oi,xi].includes(t.key);if(!n&&!i)return;if(e&&!i)return;t.preventDefault();const s=this.matches(Mi)?this:we.prev(this,Mi)[0]||we.next(this,Mi)[0]||we.findOne(Mi,t.delegateTarget.parentNode),o=Ki.getOrCreateInstance(s);if(n)return t.stopPropagation(),o.show(),void o._selectMenuItem(t);o._isShown()&&(t.stopPropagation(),o.hide(),s.focus())}}fe.on(document,Ii,Mi,Ki.dataApiKeydownHandler),fe.on(document,Ii,Fi,Ki.dataApiKeydownHandler),fe.on(document,$i,Ki.clearMenus),fe.on(document,Ni,Ki.clearMenus),fe.on(document,$i,Mi,(function(t){t.preventDefault(),Ki.getOrCreateInstance(this).toggle()})),Qt(Ki);const Qi="backdrop",Xi="show",Ui=`mousedown.bs.${Qi}`,Gi={className:"modal-backdrop",clickCallback:null,isAnimated:!1,isVisible:!0,rootElement:"body"},Ji={className:"string",clickCallback:"(function|null)",isAnimated:"boolean",isVisible:"boolean",rootElement:"(element|string)"};class Zi extends be{constructor(t){super(),this._config=this._getConfig(t),this._isAppended=!1,this._element=null}static get Default(){return Gi}static get DefaultType(){return Ji}static get NAME(){return Qi}show(t){if(!this._config.isVisible)return void Xt(t);this._append();const e=this._getElement();this._config.isAnimated&&qt(e),e.classList.add(Xi),this._emulateAnimation((()=>{Xt(t)}))}hide(t){this._config.isVisible?(this._getElement().classList.remove(Xi),this._emulateAnimation((()=>{this.dispose(),Xt(t)}))):Xt(t)}dispose(){this._isAppended&&(fe.off(this._element,Ui),this._element.remove(),this._isAppended=!1)}_getElement(){if(!this._element){const t=document.createElement("div");t.className=this._config.className,this._config.isAnimated&&t.classList.add("fade"),this._element=t}return this._element}_configAfterMerge(t){return t.rootElement=Ht(t.rootElement),t}_append(){if(this._isAppended)return;const t=this._getElement();this._config.rootElement.append(t),fe.on(t,Ui,(()=>{Xt(this._config.clickCallback)})),this._isAppended=!0}_emulateAnimation(t){Ut(t,this._getElement(),this._config.isAnimated)}}const tn=".bs.focustrap",en=`focusin${tn}`,nn=`keydown.tab${tn}`,sn="backward",on={autofocus:!0,trapElement:null},rn={autofocus:"boolean",trapElement:"element"};class an extends be{constructor(t){super(),this._config=this._getConfig(t),this._isActive=!1,this._lastTabNavDirection=null}static get Default(){return on}static get DefaultType(){return rn}static get NAME(){return"focustrap"}activate(){this._isActive||(this._config.autofocus&&this._config.trapElement.focus(),fe.off(document,tn),fe.on(document,en,(t=>this._handleFocusin(t))),fe.on(document,nn,(t=>this._handleKeydown(t))),this._isActive=!0)}deactivate(){this._isActive&&(this._isActive=!1,fe.off(document,tn))}_handleFocusin(t){const{trapElement:e}=this._config;if(t.target===document||t.target===e||e.contains(t.target))return;const i=we.focusableChildren(e);0===i.length?e.focus():this._lastTabNavDirection===sn?i[i.length-1].focus():i[0].focus()}_handleKeydown(t){"Tab"===t.key&&(this._lastTabNavDirection=t.shiftKey?sn:"forward")}}const ln=".fixed-top, .fixed-bottom, .is-fixed, .sticky-top",cn=".sticky-top",hn="padding-right",dn="margin-right";class un{constructor(){this._element=document.body}getWidth(){const t=document.documentElement.clientWidth;return Math.abs(window.innerWidth-t)}hide(){const t=this.getWidth();this._disableOverFlow(),this._setElementAttributes(this._element,hn,(e=>e+t)),this._setElementAttributes(ln,hn,(e=>e+t)),this._setElementAttributes(cn,dn,(e=>e-t))}reset(){this._resetElementAttributes(this._element,"overflow"),this._resetElementAttributes(this._element,hn),this._resetElementAttributes(ln,hn),this._resetElementAttributes(cn,dn)}isOverflowing(){return this.getWidth()>0}_disableOverFlow(){this._saveInitialAttribute(this._element,"overflow"),this._element.style.overflow="hidden"}_setElementAttributes(t,e,i){const n=this.getWidth();this._applyManipulationCallback(t,(t=>{if(t!==this._element&&window.innerWidth>t.clientWidth+n)return;this._saveInitialAttribute(t,e);const s=window.getComputedStyle(t).getPropertyValue(e);t.style.setProperty(e,`${i(Number.parseFloat(s))}px`)}))}_saveInitialAttribute(t,e){const i=t.style.getPropertyValue(e);i&&_e.setDataAttribute(t,e,i)}_resetElementAttributes(t,e){this._applyManipulationCallback(t,(t=>{const i=_e.getDataAttribute(t,e);null!==i?(_e.removeDataAttribute(t,e),t.style.setProperty(e,i)):t.style.removeProperty(e)}))}_applyManipulationCallback(t,e){if(Ft(t))e(t);else for(const i of we.find(t,this._element))e(i)}}const fn=".bs.modal",pn=`hide${fn}`,mn=`hidePrevented${fn}`,gn=`hidden${fn}`,_n=`show${fn}`,bn=`shown${fn}`,vn=`resize${fn}`,yn=`click.dismiss${fn}`,wn=`mousedown.dismiss${fn}`,En=`keydown.dismiss${fn}`,An=`click${fn}.data-api`,Tn="modal-open",Cn="show",On="modal-static",xn={backdrop:!0,focus:!0,keyboard:!0},kn={backdrop:"(boolean|string)",focus:"boolean",keyboard:"boolean"};class Ln extends ve{constructor(t,e){super(t,e),this._dialog=we.findOne(".modal-dialog",this._element),this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._isShown=!1,this._isTransitioning=!1,this._scrollBar=new un,this._addEventListeners()}static get Default(){return xn}static get DefaultType(){return kn}static get NAME(){return"modal"}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||this._isTransitioning||fe.trigger(this._element,_n,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._isTransitioning=!0,this._scrollBar.hide(),document.body.classList.add(Tn),this._adjustDialog(),this._backdrop.show((()=>this._showElement(t))))}hide(){this._isShown&&!this._isTransitioning&&(fe.trigger(this._element,pn).defaultPrevented||(this._isShown=!1,this._isTransitioning=!0,this._focustrap.deactivate(),this._element.classList.remove(Cn),this._queueCallback((()=>this._hideModal()),this._element,this._isAnimated())))}dispose(){fe.off(window,fn),fe.off(this._dialog,fn),this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}handleUpdate(){this._adjustDialog()}_initializeBackDrop(){return new Zi({isVisible:Boolean(this._config.backdrop),isAnimated:this._isAnimated()})}_initializeFocusTrap(){return new an({trapElement:this._element})}_showElement(t){document.body.contains(this._element)||document.body.append(this._element),this._element.style.display="block",this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.scrollTop=0;const e=we.findOne(".modal-body",this._dialog);e&&(e.scrollTop=0),qt(this._element),this._element.classList.add(Cn),this._queueCallback((()=>{this._config.focus&&this._focustrap.activate(),this._isTransitioning=!1,fe.trigger(this._element,bn,{relatedTarget:t})}),this._dialog,this._isAnimated())}_addEventListeners(){fe.on(this._element,En,(t=>{"Escape"===t.key&&(this._config.keyboard?this.hide():this._triggerBackdropTransition())})),fe.on(window,vn,(()=>{this._isShown&&!this._isTransitioning&&this._adjustDialog()})),fe.on(this._element,wn,(t=>{fe.one(this._element,yn,(e=>{this._element===t.target&&this._element===e.target&&("static"!==this._config.backdrop?this._config.backdrop&&this.hide():this._triggerBackdropTransition())}))}))}_hideModal(){this._element.style.display="none",this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._isTransitioning=!1,this._backdrop.hide((()=>{document.body.classList.remove(Tn),this._resetAdjustments(),this._scrollBar.reset(),fe.trigger(this._element,gn)}))}_isAnimated(){return this._element.classList.contains("fade")}_triggerBackdropTransition(){if(fe.trigger(this._element,mn).defaultPrevented)return;const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._element.style.overflowY;"hidden"===e||this._element.classList.contains(On)||(t||(this._element.style.overflowY="hidden"),this._element.classList.add(On),this._queueCallback((()=>{this._element.classList.remove(On),this._queueCallback((()=>{this._element.style.overflowY=e}),this._dialog)}),this._dialog),this._element.focus())}_adjustDialog(){const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._scrollBar.getWidth(),i=e>0;if(i&&!t){const t=Kt()?"paddingLeft":"paddingRight";this._element.style[t]=`${e}px`}if(!i&&t){const t=Kt()?"paddingRight":"paddingLeft";this._element.style[t]=`${e}px`}}_resetAdjustments(){this._element.style.paddingLeft="",this._element.style.paddingRight=""}static jQueryInterface(t,e){return this.each((function(){const i=Ln.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t](e)}}))}}fe.on(document,An,'[data-bs-toggle="modal"]',(function(t){const e=we.getElementFromSelector(this);["A","AREA"].includes(this.tagName)&&t.preventDefault(),fe.one(e,_n,(t=>{t.defaultPrevented||fe.one(e,gn,(()=>{Bt(this)&&this.focus()}))}));const i=we.findOne(".modal.show");i&&Ln.getInstance(i).hide(),Ln.getOrCreateInstance(e).toggle(this)})),Ee(Ln),Qt(Ln);const Sn=".bs.offcanvas",Dn=".data-api",$n=`load${Sn}${Dn}`,In="show",Nn="showing",Pn="hiding",Mn=".offcanvas.show",jn=`show${Sn}`,Fn=`shown${Sn}`,Hn=`hide${Sn}`,Bn=`hidePrevented${Sn}`,Wn=`hidden${Sn}`,zn=`resize${Sn}`,Rn=`click${Sn}${Dn}`,qn=`keydown.dismiss${Sn}`,Vn={backdrop:!0,keyboard:!0,scroll:!1},Yn={backdrop:"(boolean|string)",keyboard:"boolean",scroll:"boolean"};class Kn extends ve{constructor(t,e){super(t,e),this._isShown=!1,this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._addEventListeners()}static get Default(){return Vn}static get DefaultType(){return Yn}static get NAME(){return"offcanvas"}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||fe.trigger(this._element,jn,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._backdrop.show(),this._config.scroll||(new un).hide(),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.classList.add(Nn),this._queueCallback((()=>{this._config.scroll&&!this._config.backdrop||this._focustrap.activate(),this._element.classList.add(In),this._element.classList.remove(Nn),fe.trigger(this._element,Fn,{relatedTarget:t})}),this._element,!0))}hide(){this._isShown&&(fe.trigger(this._element,Hn).defaultPrevented||(this._focustrap.deactivate(),this._element.blur(),this._isShown=!1,this._element.classList.add(Pn),this._backdrop.hide(),this._queueCallback((()=>{this._element.classList.remove(In,Pn),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._config.scroll||(new un).reset(),fe.trigger(this._element,Wn)}),this._element,!0)))}dispose(){this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}_initializeBackDrop(){const t=Boolean(this._config.backdrop);return new Zi({className:"offcanvas-backdrop",isVisible:t,isAnimated:!0,rootElement:this._element.parentNode,clickCallback:t?()=>{"static"!==this._config.backdrop?this.hide():fe.trigger(this._element,Bn)}:null})}_initializeFocusTrap(){return new an({trapElement:this._element})}_addEventListeners(){fe.on(this._element,qn,(t=>{"Escape"===t.key&&(this._config.keyboard?this.hide():fe.trigger(this._element,Bn))}))}static jQueryInterface(t){return this.each((function(){const e=Kn.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}fe.on(document,Rn,'[data-bs-toggle="offcanvas"]',(function(t){const e=we.getElementFromSelector(this);if(["A","AREA"].includes(this.tagName)&&t.preventDefault(),Wt(this))return;fe.one(e,Wn,(()=>{Bt(this)&&this.focus()}));const i=we.findOne(Mn);i&&i!==e&&Kn.getInstance(i).hide(),Kn.getOrCreateInstance(e).toggle(this)})),fe.on(window,$n,(()=>{for(const t of we.find(Mn))Kn.getOrCreateInstance(t).show()})),fe.on(window,zn,(()=>{for(const t of we.find("[aria-modal][class*=show][class*=offcanvas-]"))"fixed"!==getComputedStyle(t).position&&Kn.getOrCreateInstance(t).hide()})),Ee(Kn),Qt(Kn);const Qn={"*":["class","dir","id","lang","role",/^aria-[\w-]*$/i],a:["target","href","title","rel"],area:[],b:[],br:[],col:[],code:[],dd:[],div:[],dl:[],dt:[],em:[],hr:[],h1:[],h2:[],h3:[],h4:[],h5:[],h6:[],i:[],img:["src","srcset","alt","title","width","height"],li:[],ol:[],p:[],pre:[],s:[],small:[],span:[],sub:[],sup:[],strong:[],u:[],ul:[]},Xn=new Set(["background","cite","href","itemtype","longdesc","poster","src","xlink:href"]),Un=/^(?!javascript:)(?:[a-z0-9+.-]+:|[^&:/?#]*(?:[/?#]|$))/i,Gn=(t,e)=>{const i=t.nodeName.toLowerCase();return e.includes(i)?!Xn.has(i)||Boolean(Un.test(t.nodeValue)):e.filter((t=>t instanceof RegExp)).some((t=>t.test(i)))},Jn={allowList:Qn,content:{},extraClass:"",html:!1,sanitize:!0,sanitizeFn:null,template:"<div></div>"},Zn={allowList:"object",content:"object",extraClass:"(string|function)",html:"boolean",sanitize:"boolean",sanitizeFn:"(null|function)",template:"string"},ts={entry:"(string|element|function|null)",selector:"(string|element)"};class es extends be{constructor(t){super(),this._config=this._getConfig(t)}static get Default(){return Jn}static get DefaultType(){return Zn}static get NAME(){return"TemplateFactory"}getContent(){return Object.values(this._config.content).map((t=>this._resolvePossibleFunction(t))).filter(Boolean)}hasContent(){return this.getContent().length>0}changeContent(t){return this._checkContent(t),this._config.content={...this._config.content,...t},this}toHtml(){const t=document.createElement("div");t.innerHTML=this._maybeSanitize(this._config.template);for(const[e,i]of Object.entries(this._config.content))this._setContent(t,i,e);const e=t.children[0],i=this._resolvePossibleFunction(this._config.extraClass);return i&&e.classList.add(...i.split(" ")),e}_typeCheckConfig(t){super._typeCheckConfig(t),this._checkContent(t.content)}_checkContent(t){for(const[e,i]of Object.entries(t))super._typeCheckConfig({selector:e,entry:i},ts)}_setContent(t,e,i){const n=we.findOne(i,t);n&&((e=this._resolvePossibleFunction(e))?Ft(e)?this._putElementInTemplate(Ht(e),n):this._config.html?n.innerHTML=this._maybeSanitize(e):n.textContent=e:n.remove())}_maybeSanitize(t){return this._config.sanitize?function(t,e,i){if(!t.length)return t;if(i&&"function"==typeof i)return i(t);const n=(new window.DOMParser).parseFromString(t,"text/html"),s=[].concat(...n.body.querySelectorAll("*"));for(const t of s){const i=t.nodeName.toLowerCase();if(!Object.keys(e).includes(i)){t.remove();continue}const n=[].concat(...t.attributes),s=[].concat(e["*"]||[],e[i]||[]);for(const e of n)Gn(e,s)||t.removeAttribute(e.nodeName)}return n.body.innerHTML}(t,this._config.allowList,this._config.sanitizeFn):t}_resolvePossibleFunction(t){return Xt(t,[this])}_putElementInTemplate(t,e){if(this._config.html)return e.innerHTML="",void e.append(t);e.textContent=t.textContent}}const is=new Set(["sanitize","allowList","sanitizeFn"]),ns="fade",ss="show",os=".tooltip-inner",rs=".modal",as="hide.bs.modal",ls="hover",cs="focus",hs={AUTO:"auto",TOP:"top",RIGHT:Kt()?"left":"right",BOTTOM:"bottom",LEFT:Kt()?"right":"left"},ds={allowList:Qn,animation:!0,boundary:"clippingParents",container:!1,customClass:"",delay:0,fallbackPlacements:["top","right","bottom","left"],html:!1,offset:[0,6],placement:"top",popperConfig:null,sanitize:!0,sanitizeFn:null,selector:!1,template:'<div class="tooltip" role="tooltip"><div class="tooltip-arrow"></div><div class="tooltip-inner"></div></div>',title:"",trigger:"hover focus"},us={allowList:"object",animation:"boolean",boundary:"(string|element)",container:"(string|element|boolean)",customClass:"(string|function)",delay:"(number|object)",fallbackPlacements:"array",html:"boolean",offset:"(array|string|function)",placement:"(string|function)",popperConfig:"(null|object|function)",sanitize:"boolean",sanitizeFn:"(null|function)",selector:"(string|boolean)",template:"string",title:"(string|element|function)",trigger:"string"};class fs extends ve{constructor(t,i){if(void 0===e)throw new TypeError("Bootstrap's tooltips require Popper (https://popper.js.org)");super(t,i),this._isEnabled=!0,this._timeout=0,this._isHovered=null,this._activeTrigger={},this._popper=null,this._templateFactory=null,this._newContent=null,this.tip=null,this._setListeners(),this._config.selector||this._fixTitle()}static get Default(){return ds}static get DefaultType(){return us}static get NAME(){return"tooltip"}enable(){this._isEnabled=!0}disable(){this._isEnabled=!1}toggleEnabled(){this._isEnabled=!this._isEnabled}toggle(){this._isEnabled&&(this._activeTrigger.click=!this._activeTrigger.click,this._isShown()?this._leave():this._enter())}dispose(){clearTimeout(this._timeout),fe.off(this._element.closest(rs),as,this._hideModalHandler),this._element.getAttribute("data-bs-original-title")&&this._element.setAttribute("title",this._element.getAttribute("data-bs-original-title")),this._disposePopper(),super.dispose()}show(){if("none"===this._element.style.display)throw new Error("Please use show on visible elements");if(!this._isWithContent()||!this._isEnabled)return;const t=fe.trigger(this._element,this.constructor.eventName("show")),e=(zt(this._element)||this._element.ownerDocument.documentElement).contains(this._element);if(t.defaultPrevented||!e)return;this._disposePopper();const i=this._getTipElement();this._element.setAttribute("aria-describedby",i.getAttribute("id"));const{container:n}=this._config;if(this._element.ownerDocument.documentElement.contains(this.tip)||(n.append(i),fe.trigger(this._element,this.constructor.eventName("inserted"))),this._popper=this._createPopper(i),i.classList.add(ss),"ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))fe.on(t,"mouseover",Rt);this._queueCallback((()=>{fe.trigger(this._element,this.constructor.eventName("shown")),!1===this._isHovered&&this._leave(),this._isHovered=!1}),this.tip,this._isAnimated())}hide(){if(this._isShown()&&!fe.trigger(this._element,this.constructor.eventName("hide")).defaultPrevented){if(this._getTipElement().classList.remove(ss),"ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))fe.off(t,"mouseover",Rt);this._activeTrigger.click=!1,this._activeTrigger[cs]=!1,this._activeTrigger[ls]=!1,this._isHovered=null,this._queueCallback((()=>{this._isWithActiveTrigger()||(this._isHovered||this._disposePopper(),this._element.removeAttribute("aria-describedby"),fe.trigger(this._element,this.constructor.eventName("hidden")))}),this.tip,this._isAnimated())}}update(){this._popper&&this._popper.update()}_isWithContent(){return Boolean(this._getTitle())}_getTipElement(){return this.tip||(this.tip=this._createTipElement(this._newContent||this._getContentForTemplate())),this.tip}_createTipElement(t){const e=this._getTemplateFactory(t).toHtml();if(!e)return null;e.classList.remove(ns,ss),e.classList.add(`bs-${this.constructor.NAME}-auto`);const i=(t=>{do{t+=Math.floor(1e6*Math.random())}while(document.getElementById(t));return t})(this.constructor.NAME).toString();return e.setAttribute("id",i),this._isAnimated()&&e.classList.add(ns),e}setContent(t){this._newContent=t,this._isShown()&&(this._disposePopper(),this.show())}_getTemplateFactory(t){return this._templateFactory?this._templateFactory.changeContent(t):this._templateFactory=new es({...this._config,content:t,extraClass:this._resolvePossibleFunction(this._config.customClass)}),this._templateFactory}_getContentForTemplate(){return{[os]:this._getTitle()}}_getTitle(){return this._resolvePossibleFunction(this._config.title)||this._element.getAttribute("data-bs-original-title")}_initializeOnDelegatedTarget(t){return this.constructor.getOrCreateInstance(t.delegateTarget,this._getDelegateConfig())}_isAnimated(){return this._config.animation||this.tip&&this.tip.classList.contains(ns)}_isShown(){return this.tip&&this.tip.classList.contains(ss)}_createPopper(t){const e=Xt(this._config.placement,[this,t,this._element]),i=hs[e.toUpperCase()];return Dt(this._element,t,this._getPopperConfig(i))}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_resolvePossibleFunction(t){return Xt(t,[this._element])}_getPopperConfig(t){const e={placement:t,modifiers:[{name:"flip",options:{fallbackPlacements:this._config.fallbackPlacements}},{name:"offset",options:{offset:this._getOffset()}},{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"arrow",options:{element:`.${this.constructor.NAME}-arrow`}},{name:"preSetPlacement",enabled:!0,phase:"beforeMain",fn:t=>{this._getTipElement().setAttribute("data-popper-placement",t.state.placement)}}]};return{...e,...Xt(this._config.popperConfig,[e])}}_setListeners(){const t=this._config.trigger.split(" ");for(const e of t)if("click"===e)fe.on(this._element,this.constructor.eventName("click"),this._config.selector,(t=>{this._initializeOnDelegatedTarget(t).toggle()}));else if("manual"!==e){const t=e===ls?this.constructor.eventName("mouseenter"):this.constructor.eventName("focusin"),i=e===ls?this.constructor.eventName("mouseleave"):this.constructor.eventName("focusout");fe.on(this._element,t,this._config.selector,(t=>{const e=this._initializeOnDelegatedTarget(t);e._activeTrigger["focusin"===t.type?cs:ls]=!0,e._enter()})),fe.on(this._element,i,this._config.selector,(t=>{const e=this._initializeOnDelegatedTarget(t);e._activeTrigger["focusout"===t.type?cs:ls]=e._element.contains(t.relatedTarget),e._leave()}))}this._hideModalHandler=()=>{this._element&&this.hide()},fe.on(this._element.closest(rs),as,this._hideModalHandler)}_fixTitle(){const t=this._element.getAttribute("title");t&&(this._element.getAttribute("aria-label")||this._element.textContent.trim()||this._element.setAttribute("aria-label",t),this._element.setAttribute("data-bs-original-title",t),this._element.removeAttribute("title"))}_enter(){this._isShown()||this._isHovered?this._isHovered=!0:(this._isHovered=!0,this._setTimeout((()=>{this._isHovered&&this.show()}),this._config.delay.show))}_leave(){this._isWithActiveTrigger()||(this._isHovered=!1,this._setTimeout((()=>{this._isHovered||this.hide()}),this._config.delay.hide))}_setTimeout(t,e){clearTimeout(this._timeout),this._timeout=setTimeout(t,e)}_isWithActiveTrigger(){return Object.values(this._activeTrigger).includes(!0)}_getConfig(t){const e=_e.getDataAttributes(this._element);for(const t of Object.keys(e))is.has(t)&&delete e[t];return t={...e,..."object"==typeof t&&t?t:{}},t=this._mergeConfigObj(t),t=this._configAfterMerge(t),this._typeCheckConfig(t),t}_configAfterMerge(t){return t.container=!1===t.container?document.body:Ht(t.container),"number"==typeof t.delay&&(t.delay={show:t.delay,hide:t.delay}),"number"==typeof t.title&&(t.title=t.title.toString()),"number"==typeof t.content&&(t.content=t.content.toString()),t}_getDelegateConfig(){const t={};for(const[e,i]of Object.entries(this._config))this.constructor.Default[e]!==i&&(t[e]=i);return t.selector=!1,t.trigger="manual",t}_disposePopper(){this._popper&&(this._popper.destroy(),this._popper=null),this.tip&&(this.tip.remove(),this.tip=null)}static jQueryInterface(t){return this.each((function(){const e=fs.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}Qt(fs);const ps=".popover-header",ms=".popover-body",gs={...fs.Default,content:"",offset:[0,8],placement:"right",template:'<div class="popover" role="tooltip"><div class="popover-arrow"></div><h3 class="popover-header"></h3><div class="popover-body"></div></div>',trigger:"click"},_s={...fs.DefaultType,content:"(null|string|element|function)"};class bs extends fs{static get Default(){return gs}static get DefaultType(){return _s}static get NAME(){return"popover"}_isWithContent(){return this._getTitle()||this._getContent()}_getContentForTemplate(){return{[ps]:this._getTitle(),[ms]:this._getContent()}}_getContent(){return this._resolvePossibleFunction(this._config.content)}static jQueryInterface(t){return this.each((function(){const e=bs.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}Qt(bs);const vs=".bs.scrollspy",ys=`activate${vs}`,ws=`click${vs}`,Es=`load${vs}.data-api`,As="active",Ts="[href]",Cs=".nav-link",Os=`${Cs}, .nav-item > ${Cs}, .list-group-item`,xs={offset:null,rootMargin:"0px 0px -25%",smoothScroll:!1,target:null,threshold:[.1,.5,1]},ks={offset:"(number|null)",rootMargin:"string",smoothScroll:"boolean",target:"element",threshold:"array"};class Ls extends ve{constructor(t,e){super(t,e),this._targetLinks=new Map,this._observableSections=new Map,this._rootElement="visible"===getComputedStyle(this._element).overflowY?null:this._element,this._activeTarget=null,this._observer=null,this._previousScrollData={visibleEntryTop:0,parentScrollTop:0},this.refresh()}static get Default(){return xs}static get DefaultType(){return ks}static get NAME(){return"scrollspy"}refresh(){this._initializeTargetsAndObservables(),this._maybeEnableSmoothScroll(),this._observer?this._observer.disconnect():this._observer=this._getNewObserver();for(const t of this._observableSections.values())this._observer.observe(t)}dispose(){this._observer.disconnect(),super.dispose()}_configAfterMerge(t){return t.target=Ht(t.target)||document.body,t.rootMargin=t.offset?`${t.offset}px 0px -30%`:t.rootMargin,"string"==typeof t.threshold&&(t.threshold=t.threshold.split(",").map((t=>Number.parseFloat(t)))),t}_maybeEnableSmoothScroll(){this._config.smoothScroll&&(fe.off(this._config.target,ws),fe.on(this._config.target,ws,Ts,(t=>{const e=this._observableSections.get(t.target.hash);if(e){t.preventDefault();const i=this._rootElement||window,n=e.offsetTop-this._element.offsetTop;if(i.scrollTo)return void i.scrollTo({top:n,behavior:"smooth"});i.scrollTop=n}})))}_getNewObserver(){const t={root:this._rootElement,threshold:this._config.threshold,rootMargin:this._config.rootMargin};return new IntersectionObserver((t=>this._observerCallback(t)),t)}_observerCallback(t){const e=t=>this._targetLinks.get(`#${t.target.id}`),i=t=>{this._previousScrollData.visibleEntryTop=t.target.offsetTop,this._process(e(t))},n=(this._rootElement||document.documentElement).scrollTop,s=n>=this._previousScrollData.parentScrollTop;this._previousScrollData.parentScrollTop=n;for(const o of t){if(!o.isIntersecting){this._activeTarget=null,this._clearActiveClass(e(o));continue}const t=o.target.offsetTop>=this._previousScrollData.visibleEntryTop;if(s&&t){if(i(o),!n)return}else s||t||i(o)}}_initializeTargetsAndObservables(){this._targetLinks=new Map,this._observableSections=new Map;const t=we.find(Ts,this._config.target);for(const e of t){if(!e.hash||Wt(e))continue;const t=we.findOne(decodeURI(e.hash),this._element);Bt(t)&&(this._targetLinks.set(decodeURI(e.hash),e),this._observableSections.set(e.hash,t))}}_process(t){this._activeTarget!==t&&(this._clearActiveClass(this._config.target),this._activeTarget=t,t.classList.add(As),this._activateParents(t),fe.trigger(this._element,ys,{relatedTarget:t}))}_activateParents(t){if(t.classList.contains("dropdown-item"))we.findOne(".dropdown-toggle",t.closest(".dropdown")).classList.add(As);else for(const e of we.parents(t,".nav, .list-group"))for(const t of we.prev(e,Os))t.classList.add(As)}_clearActiveClass(t){t.classList.remove(As);const e=we.find(`${Ts}.${As}`,t);for(const t of e)t.classList.remove(As)}static jQueryInterface(t){return this.each((function(){const e=Ls.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t]()}}))}}fe.on(window,Es,(()=>{for(const t of we.find('[data-bs-spy="scroll"]'))Ls.getOrCreateInstance(t)})),Qt(Ls);const Ss=".bs.tab",Ds=`hide${Ss}`,$s=`hidden${Ss}`,Is=`show${Ss}`,Ns=`shown${Ss}`,Ps=`click${Ss}`,Ms=`keydown${Ss}`,js=`load${Ss}`,Fs="ArrowLeft",Hs="ArrowRight",Bs="ArrowUp",Ws="ArrowDown",zs="Home",Rs="End",qs="active",Vs="fade",Ys="show",Ks=".dropdown-toggle",Qs=`:not(${Ks})`,Xs='[data-bs-toggle="tab"], [data-bs-toggle="pill"], [data-bs-toggle="list"]',Us=`.nav-link${Qs}, .list-group-item${Qs}, [role="tab"]${Qs}, ${Xs}`,Gs=`.${qs}[data-bs-toggle="tab"], .${qs}[data-bs-toggle="pill"], .${qs}[data-bs-toggle="list"]`;class Js extends ve{constructor(t){super(t),this._parent=this._element.closest('.list-group, .nav, [role="tablist"]'),this._parent&&(this._setInitialAttributes(this._parent,this._getChildren()),fe.on(this._element,Ms,(t=>this._keydown(t))))}static get NAME(){return"tab"}show(){const t=this._element;if(this._elemIsActive(t))return;const e=this._getActiveElem(),i=e?fe.trigger(e,Ds,{relatedTarget:t}):null;fe.trigger(t,Is,{relatedTarget:e}).defaultPrevented||i&&i.defaultPrevented||(this._deactivate(e,t),this._activate(t,e))}_activate(t,e){t&&(t.classList.add(qs),this._activate(we.getElementFromSelector(t)),this._queueCallback((()=>{"tab"===t.getAttribute("role")?(t.removeAttribute("tabindex"),t.setAttribute("aria-selected",!0),this._toggleDropDown(t,!0),fe.trigger(t,Ns,{relatedTarget:e})):t.classList.add(Ys)}),t,t.classList.contains(Vs)))}_deactivate(t,e){t&&(t.classList.remove(qs),t.blur(),this._deactivate(we.getElementFromSelector(t)),this._queueCallback((()=>{"tab"===t.getAttribute("role")?(t.setAttribute("aria-selected",!1),t.setAttribute("tabindex","-1"),this._toggleDropDown(t,!1),fe.trigger(t,$s,{relatedTarget:e})):t.classList.remove(Ys)}),t,t.classList.contains(Vs)))}_keydown(t){if(![Fs,Hs,Bs,Ws,zs,Rs].includes(t.key))return;t.stopPropagation(),t.preventDefault();const e=this._getChildren().filter((t=>!Wt(t)));let i;if([zs,Rs].includes(t.key))i=e[t.key===zs?0:e.length-1];else{const n=[Hs,Ws].includes(t.key);i=Gt(e,t.target,n,!0)}i&&(i.focus({preventScroll:!0}),Js.getOrCreateInstance(i).show())}_getChildren(){return we.find(Us,this._parent)}_getActiveElem(){return this._getChildren().find((t=>this._elemIsActive(t)))||null}_setInitialAttributes(t,e){this._setAttributeIfNotExists(t,"role","tablist");for(const t of e)this._setInitialAttributesOnChild(t)}_setInitialAttributesOnChild(t){t=this._getInnerElement(t);const e=this._elemIsActive(t),i=this._getOuterElement(t);t.setAttribute("aria-selected",e),i!==t&&this._setAttributeIfNotExists(i,"role","presentation"),e||t.setAttribute("tabindex","-1"),this._setAttributeIfNotExists(t,"role","tab"),this._setInitialAttributesOnTargetPanel(t)}_setInitialAttributesOnTargetPanel(t){const e=we.getElementFromSelector(t);e&&(this._setAttributeIfNotExists(e,"role","tabpanel"),t.id&&this._setAttributeIfNotExists(e,"aria-labelledby",`${t.id}`))}_toggleDropDown(t,e){const i=this._getOuterElement(t);if(!i.classList.contains("dropdown"))return;const n=(t,n)=>{const s=we.findOne(t,i);s&&s.classList.toggle(n,e)};n(Ks,qs),n(".dropdown-menu",Ys),i.setAttribute("aria-expanded",e)}_setAttributeIfNotExists(t,e,i){t.hasAttribute(e)||t.setAttribute(e,i)}_elemIsActive(t){return t.classList.contains(qs)}_getInnerElement(t){return t.matches(Us)?t:we.findOne(Us,t)}_getOuterElement(t){return t.closest(".nav-item, .list-group-item")||t}static jQueryInterface(t){return this.each((function(){const e=Js.getOrCreateInstance(this);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t]()}}))}}fe.on(document,Ps,Xs,(function(t){["A","AREA"].includes(this.tagName)&&t.preventDefault(),Wt(this)||Js.getOrCreateInstance(this).show()})),fe.on(window,js,(()=>{for(const t of we.find(Gs))Js.getOrCreateInstance(t)})),Qt(Js);const Zs=".bs.toast",to=`mouseover${Zs}`,eo=`mouseout${Zs}`,io=`focusin${Zs}`,no=`focusout${Zs}`,so=`hide${Zs}`,oo=`hidden${Zs}`,ro=`show${Zs}`,ao=`shown${Zs}`,lo="hide",co="show",ho="showing",uo={animation:"boolean",autohide:"boolean",delay:"number"},fo={animation:!0,autohide:!0,delay:5e3};class po extends ve{constructor(t,e){super(t,e),this._timeout=null,this._hasMouseInteraction=!1,this._hasKeyboardInteraction=!1,this._setListeners()}static get Default(){return fo}static get DefaultType(){return uo}static get NAME(){return"toast"}show(){fe.trigger(this._element,ro).defaultPrevented||(this._clearTimeout(),this._config.animation&&this._element.classList.add("fade"),this._element.classList.remove(lo),qt(this._element),this._element.classList.add(co,ho),this._queueCallback((()=>{this._element.classList.remove(ho),fe.trigger(this._element,ao),this._maybeScheduleHide()}),this._element,this._config.animation))}hide(){this.isShown()&&(fe.trigger(this._element,so).defaultPrevented||(this._element.classList.add(ho),this._queueCallback((()=>{this._element.classList.add(lo),this._element.classList.remove(ho,co),fe.trigger(this._element,oo)}),this._element,this._config.animation)))}dispose(){this._clearTimeout(),this.isShown()&&this._element.classList.remove(co),super.dispose()}isShown(){return this._element.classList.contains(co)}_maybeScheduleHide(){this._config.autohide&&(this._hasMouseInteraction||this._hasKeyboardInteraction||(this._timeout=setTimeout((()=>{this.hide()}),this._config.delay)))}_onInteraction(t,e){switch(t.type){case"mouseover":case"mouseout":this._hasMouseInteraction=e;break;case"focusin":case"focusout":this._hasKeyboardInteraction=e}if(e)return void this._clearTimeout();const i=t.relatedTarget;this._element===i||this._element.contains(i)||this._maybeScheduleHide()}_setListeners(){fe.on(this._element,to,(t=>this._onInteraction(t,!0))),fe.on(this._element,eo,(t=>this._onInteraction(t,!1))),fe.on(this._element,io,(t=>this._onInteraction(t,!0))),fe.on(this._element,no,(t=>this._onInteraction(t,!1)))}_clearTimeout(){clearTimeout(this._timeout),this._timeout=null}static jQueryInterface(t){return this.each((function(){const e=po.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}function mo(t){"loading"!=document.readyState?t():document.addEventListener("DOMContentLoaded",t)}Ee(po),Qt(po),mo((function(){[].slice.call(document.querySelectorAll('[data-bs-toggle="tooltip"]')).map((function(t){return new fs(t,{delay:{show:500,hide:100}})}))})),mo((function(){document.getElementById("pst-back-to-top").addEventListener("click",(function(){document.body.scrollTop=0,document.documentElement.scrollTop=0}))})),mo((function(){var t=document.getElementById("pst-back-to-top"),e=document.getElementsByClassName("bd-header")[0].getBoundingClientRect();window.addEventListener("scroll",(function(){this.oldScroll>this.scrollY&&this.scrollY>e.bottom?t.style.display="block":t.style.display="none",this.oldScroll=this.scrollY}))})),window.bootstrap=i})();
+//# sourceMappingURL=bootstrap.js.map
\ No newline at end of file
diff --git a/docs/html/_static/scripts/bootstrap.js.LICENSE.txt b/docs/html/_static/scripts/bootstrap.js.LICENSE.txt
new file mode 100644
index 0000000..28755c2
--- /dev/null
+++ b/docs/html/_static/scripts/bootstrap.js.LICENSE.txt
@@ -0,0 +1,5 @@
+/*!
+  * Bootstrap v5.3.3 (https://getbootstrap.com/)
+  * Copyright 2011-2024 The Bootstrap Authors (https://github.com/twbs/bootstrap/graphs/contributors)
+  * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)
+  */
diff --git a/docs/html/_static/scripts/bootstrap.js.map b/docs/html/_static/scripts/bootstrap.js.map
new file mode 100644
index 0000000..e9e8158
--- /dev/null
+++ b/docs/html/_static/scripts/bootstrap.js.map
@@ -0,0 +1 @@
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(element.nodeName || '').toLowerCase() : null;\n}","export default function getWindow(node) {\n  if (node == null) {\n    return window;\n  }\n\n  if (node.toString() !== '[object Window]') {\n    var ownerDocument = node.ownerDocument;\n    return ownerDocument ? ownerDocument.defaultView || window : window;\n  }\n\n  return node;\n}","import getWindow from \"./getWindow.js\";\n\nfunction isElement(node) {\n  var OwnElement = getWindow(node).Element;\n  return node instanceof OwnElement || node instanceof Element;\n}\n\nfunction isHTMLElement(node) {\n  var OwnElement = getWindow(node).HTMLElement;\n  return node instanceof OwnElement || node instanceof HTMLElement;\n}\n\nfunction isShadowRoot(node) {\n  // IE 11 has no ShadowRoot\n  if (typeof ShadowRoot === 'undefined') {\n    return false;\n  }\n\n  var OwnElement = getWindow(node).ShadowRoot;\n  return node instanceof OwnElement || node instanceof ShadowRoot;\n}\n\nexport { isElement, isHTMLElement, isShadowRoot };","import getNodeName from \"../dom-utils/getNodeName.js\";\nimport { isHTMLElement } from \"../dom-utils/instanceOf.js\"; // This modifier takes the styles prepared by the `computeStyles` modifier\n// and applies them to the HTMLElements such as popper and arrow\n\nfunction applyStyles(_ref) {\n  var state = _ref.state;\n  Object.keys(state.elements).forEach(function (name) {\n    var style = state.styles[name] || {};\n    var attributes = state.attributes[name] || {};\n    var element = state.elements[name]; // arrow is optional + virtual elements\n\n    if (!isHTMLElement(element) || !getNodeName(element)) {\n      return;\n    } // Flow doesn't support to extend this property, but it's the most\n    // effective way to apply styles to an HTMLElement\n    // $FlowFixMe[cannot-write]\n\n\n    Object.assign(element.style, style);\n    Object.keys(attributes).forEach(function (name) {\n      var value = attributes[name];\n\n      if (value === false) {\n        element.removeAttribute(name);\n      } else {\n        element.setAttribute(name, value === true ? '' : value);\n      }\n    });\n  });\n}\n\nfunction effect(_ref2) {\n  var state = _ref2.state;\n  var initialStyles = {\n    popper: {\n      position: state.options.strategy,\n      left: '0',\n      top: '0',\n      margin: '0'\n    },\n    arrow: {\n      position: 'absolute'\n    },\n    reference: {}\n  };\n  Object.assign(state.elements.popper.style, initialStyles.popper);\n  state.styles = initialStyles;\n\n  if (state.elements.arrow) {\n    Object.assign(state.elements.arrow.style, initialStyles.arrow);\n  }\n\n  return function () {\n    Object.keys(state.elements).forEach(function (name) {\n      var element = state.elements[name];\n      var attributes = state.attributes[name] || {};\n      var styleProperties = Object.keys(state.styles.hasOwnProperty(name) ? state.styles[name] : initialStyles[name]); // Set all values to an empty string to unset them\n\n      var style = styleProperties.reduce(function (style, property) {\n        style[property] = '';\n        return style;\n      }, {}); // arrow is optional + virtual elements\n\n      if (!isHTMLElement(element) || !getNodeName(element)) {\n        return;\n      }\n\n      Object.assign(element.style, style);\n      Object.keys(attributes).forEach(function (attribute) {\n        element.removeAttribute(attribute);\n      });\n    });\n  };\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'applyStyles',\n  enabled: true,\n  phase: 'write',\n  fn: applyStyles,\n  effect: effect,\n  requires: ['computeStyles']\n};","import { auto } from \"../enums.js\";\nexport default function getBasePlacement(placement) {\n  return placement.split('-')[0];\n}","export var max = Math.max;\nexport var min = Math.min;\nexport var round = Math.round;","export default function getUAString() {\n  var uaData = navigator.userAgentData;\n\n  if (uaData != null && uaData.brands && Array.isArray(uaData.brands)) {\n    return uaData.brands.map(function (item) {\n      return item.brand + \"/\" + item.version;\n    }).join(' ');\n  }\n\n  return navigator.userAgent;\n}","import getUAString from \"../utils/userAgent.js\";\nexport default function isLayoutViewport() {\n  return !/^((?!chrome|android).)*safari/i.test(getUAString());\n}","import { isElement, isHTMLElement } from \"./instanceOf.js\";\nimport { round } from \"../utils/math.js\";\nimport getWindow from \"./getWindow.js\";\nimport isLayoutViewport from \"./isLayoutViewport.js\";\nexport default function getBoundingClientRect(element, includeScale, isFixedStrategy) {\n  if (includeScale === void 0) {\n    includeScale = false;\n  }\n\n  if (isFixedStrategy === void 0) {\n    isFixedStrategy = false;\n  }\n\n  var clientRect = element.getBoundingClientRect();\n  var scaleX = 1;\n  var scaleY = 1;\n\n  if (includeScale && isHTMLElement(element)) {\n    scaleX = element.offsetWidth > 0 ? round(clientRect.width) / element.offsetWidth || 1 : 1;\n    scaleY = element.offsetHeight > 0 ? round(clientRect.height) / element.offsetHeight || 1 : 1;\n  }\n\n  var _ref = isElement(element) ? getWindow(element) : window,\n      visualViewport = _ref.visualViewport;\n\n  var addVisualOffsets = !isLayoutViewport() && isFixedStrategy;\n  var x = (clientRect.left + (addVisualOffsets && visualViewport ? visualViewport.offsetLeft : 0)) / scaleX;\n  var y = (clientRect.top + (addVisualOffsets && visualViewport ? visualViewport.offsetTop : 0)) / scaleY;\n  var width = clientRect.width / scaleX;\n  var height = clientRect.height / scaleY;\n  return {\n    width: width,\n    height: height,\n    top: y,\n    right: x + width,\n    bottom: y + height,\n    left: x,\n    x: x,\n    y: y\n  };\n}","import getBoundingClientRect from \"./getBoundingClientRect.js\"; // Returns the layout rect of an element relative to its offsetParent. Layout\n// means it doesn't take into account transforms.\n\nexport default function getLayoutRect(element) {\n  var clientRect = getBoundingClientRect(element); // Use the clientRect sizes if it's not been transformed.\n  // Fixes https://github.com/popperjs/popper-core/issues/1223\n\n  var width = element.offsetWidth;\n  var height = element.offsetHeight;\n\n  if (Math.abs(clientRect.width - width) <= 1) {\n    width = clientRect.width;\n  }\n\n  if (Math.abs(clientRect.height - height) <= 1) {\n    height = clientRect.height;\n  }\n\n  return {\n    x: element.offsetLeft,\n    y: element.offsetTop,\n    width: width,\n    height: height\n  };\n}","import { isShadowRoot } from \"./instanceOf.js\";\nexport default function contains(parent, child) {\n  var rootNode = child.getRootNode && child.getRootNode(); // First, attempt with faster native method\n\n  if (parent.contains(child)) {\n    return true;\n  } // then fallback to custom implementation with Shadow DOM support\n  else if (rootNode && isShadowRoot(rootNode)) {\n      var next = child;\n\n      do {\n        if (next && parent.isSameNode(next)) {\n          return true;\n        } // $FlowFixMe[prop-missing]: need a better way to handle this...\n\n\n        next = next.parentNode || next.host;\n      } while (next);\n    } // Give up, the result is false\n\n\n  return false;\n}","import getWindow from \"./getWindow.js\";\nexport default function getComputedStyle(element) {\n  return getWindow(element).getComputedStyle(element);\n}","import getNodeName from \"./getNodeName.js\";\nexport default function isTableElement(element) {\n  return ['table', 'td', 'th'].indexOf(getNodeName(element)) >= 0;\n}","import { isElement } from \"./instanceOf.js\";\nexport default function getDocumentElement(element) {\n  // $FlowFixMe[incompatible-return]: assume body is always available\n  return ((isElement(element) ? element.ownerDocument : // $FlowFixMe[prop-missing]\n  element.document) || window.document).documentElement;\n}","import getNodeName from \"./getNodeName.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport { isShadowRoot } from \"./instanceOf.js\";\nexport default function getParentNode(element) {\n  if (getNodeName(element) === 'html') {\n    return element;\n  }\n\n  return (// this is a quicker (but less type safe) way to save quite some bytes from the bundle\n    // $FlowFixMe[incompatible-return]\n    // $FlowFixMe[prop-missing]\n    element.assignedSlot || // step into the shadow DOM of the parent of a slotted node\n    element.parentNode || ( // DOM Element detected\n    isShadowRoot(element) ? element.host : null) || // ShadowRoot detected\n    // $FlowFixMe[incompatible-call]: HTMLElement is a Node\n    getDocumentElement(element) // fallback\n\n  );\n}","import getWindow from \"./getWindow.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport getComputedStyle from \"./getComputedStyle.js\";\nimport { isHTMLElement, isShadowRoot } from \"./instanceOf.js\";\nimport isTableElement from \"./isTableElement.js\";\nimport getParentNode from \"./getParentNode.js\";\nimport getUAString from \"../utils/userAgent.js\";\n\nfunction getTrueOffsetParent(element) {\n  if (!isHTMLElement(element) || // https://github.com/popperjs/popper-core/issues/837\n  getComputedStyle(element).position === 'fixed') {\n    return null;\n  }\n\n  return element.offsetParent;\n} // `.offsetParent` reports `null` for fixed elements, while absolute elements\n// return the containing block\n\n\nfunction getContainingBlock(element) {\n  var isFirefox = /firefox/i.test(getUAString());\n  var isIE = /Trident/i.test(getUAString());\n\n  if (isIE && isHTMLElement(element)) {\n    // In IE 9, 10 and 11 fixed elements containing block is always established by the viewport\n    var elementCss = getComputedStyle(element);\n\n    if (elementCss.position === 'fixed') {\n      return null;\n    }\n  }\n\n  var currentNode = getParentNode(element);\n\n  if (isShadowRoot(currentNode)) {\n    currentNode = currentNode.host;\n  }\n\n  while (isHTMLElement(currentNode) && ['html', 'body'].indexOf(getNodeName(currentNode)) < 0) {\n    var css = getComputedStyle(currentNode); // This is non-exhaustive but covers the most common CSS properties that\n    // create a containing block.\n    // https://developer.mozilla.org/en-US/docs/Web/CSS/Containing_block#identifying_the_containing_block\n\n    if (css.transform !== 'none' || css.perspective !== 'none' || css.contain === 'paint' || ['transform', 'perspective'].indexOf(css.willChange) !== -1 || isFirefox && css.willChange === 'filter' || isFirefox && css.filter && css.filter !== 'none') {\n      return currentNode;\n    } else {\n      currentNode = currentNode.parentNode;\n    }\n  }\n\n  return null;\n} // Gets the closest ancestor positioned element. Handles some edge cases,\n// such as table ancestors and cross browser bugs.\n\n\nexport default function getOffsetParent(element) {\n  var window = getWindow(element);\n  var offsetParent = getTrueOffsetParent(element);\n\n  while (offsetParent && isTableElement(offsetParent) && getComputedStyle(offsetParent).position === 'static') {\n    offsetParent = getTrueOffsetParent(offsetParent);\n  }\n\n  if (offsetParent && (getNodeName(offsetParent) === 'html' || getNodeName(offsetParent) === 'body' && getComputedStyle(offsetParent).position === 'static')) {\n    return window;\n  }\n\n  return offsetParent || getContainingBlock(element) || window;\n}","export default function getMainAxisFromPlacement(placement) {\n  return ['top', 'bottom'].indexOf(placement) >= 0 ? 'x' : 'y';\n}","import { max as mathMax, min as mathMin } from \"./math.js\";\nexport function within(min, value, max) {\n  return mathMax(min, mathMin(value, max));\n}\nexport function withinMaxClamp(min, value, max) {\n  var v = within(min, value, max);\n  return v > max ? max : v;\n}","import getFreshSideObject from \"./getFreshSideObject.js\";\nexport default function mergePaddingObject(paddingObject) {\n  return Object.assign({}, getFreshSideObject(), paddingObject);\n}","export default function getFreshSideObject() {\n  return {\n    top: 0,\n    right: 0,\n    bottom: 0,\n    left: 0\n  };\n}","export default function expandToHashMap(value, keys) {\n  return keys.reduce(function (hashMap, key) {\n    hashMap[key] = value;\n    return hashMap;\n  }, {});\n}","import getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getLayoutRect from \"../dom-utils/getLayoutRect.js\";\nimport contains from \"../dom-utils/contains.js\";\nimport getOffsetParent from \"../dom-utils/getOffsetParent.js\";\nimport getMainAxisFromPlacement from \"../utils/getMainAxisFromPlacement.js\";\nimport { within } from \"../utils/within.js\";\nimport mergePaddingObject from \"../utils/mergePaddingObject.js\";\nimport expandToHashMap from \"../utils/expandToHashMap.js\";\nimport { left, right, basePlacements, top, bottom } from \"../enums.js\"; // eslint-disable-next-line import/no-unused-modules\n\nvar toPaddingObject = function toPaddingObject(padding, state) {\n  padding = typeof padding === 'function' ? padding(Object.assign({}, state.rects, {\n    placement: state.placement\n  })) : padding;\n  return mergePaddingObject(typeof padding !== 'number' ? padding : expandToHashMap(padding, basePlacements));\n};\n\nfunction arrow(_ref) {\n  var _state$modifiersData$;\n\n  var state = _ref.state,\n      name = _ref.name,\n      options = _ref.options;\n  var arrowElement = state.elements.arrow;\n  var popperOffsets = state.modifiersData.popperOffsets;\n  var basePlacement = getBasePlacement(state.placement);\n  var axis = getMainAxisFromPlacement(basePlacement);\n  var isVertical = [left, right].indexOf(basePlacement) >= 0;\n  var len = isVertical ? 'height' : 'width';\n\n  if (!arrowElement || !popperOffsets) {\n    return;\n  }\n\n  var paddingObject = toPaddingObject(options.padding, state);\n  var arrowRect = getLayoutRect(arrowElement);\n  var minProp = axis === 'y' ? top : left;\n  var maxProp = axis === 'y' ? bottom : right;\n  var endDiff = state.rects.reference[len] + state.rects.reference[axis] - popperOffsets[axis] - state.rects.popper[len];\n  var startDiff = popperOffsets[axis] - state.rects.reference[axis];\n  var arrowOffsetParent = getOffsetParent(arrowElement);\n  var clientSize = arrowOffsetParent ? axis === 'y' ? arrowOffsetParent.clientHeight || 0 : arrowOffsetParent.clientWidth || 0 : 0;\n  var centerToReference = endDiff / 2 - startDiff / 2; // Make sure the arrow doesn't overflow the popper if the center point is\n  // outside of the popper bounds\n\n  var min = paddingObject[minProp];\n  var max = clientSize - arrowRect[len] - paddingObject[maxProp];\n  var center = clientSize / 2 - arrowRect[len] / 2 + centerToReference;\n  var offset = within(min, center, max); // Prevents breaking syntax highlighting...\n\n  var axisProp = axis;\n  state.modifiersData[name] = (_state$modifiersData$ = {}, _state$modifiersData$[axisProp] = offset, _state$modifiersData$.centerOffset = offset - center, _state$modifiersData$);\n}\n\nfunction effect(_ref2) {\n  var state = _ref2.state,\n      options = _ref2.options;\n  var _options$element = options.element,\n      arrowElement = _options$element === void 0 ? '[data-popper-arrow]' : _options$element;\n\n  if (arrowElement == null) {\n    return;\n  } // CSS selector\n\n\n  if (typeof arrowElement === 'string') {\n    arrowElement = state.elements.popper.querySelector(arrowElement);\n\n    if (!arrowElement) {\n      return;\n    }\n  }\n\n  if (!contains(state.elements.popper, arrowElement)) {\n    return;\n  }\n\n  state.elements.arrow = arrowElement;\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'arrow',\n  enabled: true,\n  phase: 'main',\n  fn: arrow,\n  effect: effect,\n  requires: ['popperOffsets'],\n  requiresIfExists: ['preventOverflow']\n};","export default function getVariation(placement) {\n  return placement.split('-')[1];\n}","import { top, left, right, bottom, end } from \"../enums.js\";\nimport getOffsetParent from \"../dom-utils/getOffsetParent.js\";\nimport getWindow from \"../dom-utils/getWindow.js\";\nimport getDocumentElement from \"../dom-utils/getDocumentElement.js\";\nimport getComputedStyle from \"../dom-utils/getComputedStyle.js\";\nimport getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getVariation from \"../utils/getVariation.js\";\nimport { round } from \"../utils/math.js\"; // eslint-disable-next-line import/no-unused-modules\n\nvar unsetSides = {\n  top: 'auto',\n  right: 'auto',\n  bottom: 'auto',\n  left: 'auto'\n}; // Round the offsets to the nearest suitable subpixel based on the DPR.\n// Zooming can change the DPR, but it seems to report a value that will\n// cleanly divide the values into the appropriate subpixels.\n\nfunction roundOffsetsByDPR(_ref, win) {\n  var x = _ref.x,\n      y = _ref.y;\n  var dpr = win.devicePixelRatio || 1;\n  return {\n    x: round(x * dpr) / dpr || 0,\n    y: round(y * dpr) / dpr || 0\n  };\n}\n\nexport function mapToStyles(_ref2) {\n  var _Object$assign2;\n\n  var popper = _ref2.popper,\n      popperRect = _ref2.popperRect,\n      placement = _ref2.placement,\n      variation = _ref2.variation,\n      offsets = _ref2.offsets,\n      position = _ref2.position,\n      gpuAcceleration = _ref2.gpuAcceleration,\n      adaptive = _ref2.adaptive,\n      roundOffsets = _ref2.roundOffsets,\n      isFixed = _ref2.isFixed;\n  var _offsets$x = offsets.x,\n      x = _offsets$x === void 0 ? 0 : _offsets$x,\n      _offsets$y = offsets.y,\n      y = _offsets$y === void 0 ? 0 : _offsets$y;\n\n  var _ref3 = typeof roundOffsets === 'function' ? roundOffsets({\n    x: x,\n    y: y\n  }) : {\n    x: x,\n    y: y\n  };\n\n  x = _ref3.x;\n  y = _ref3.y;\n  var hasX = offsets.hasOwnProperty('x');\n  var hasY = offsets.hasOwnProperty('y');\n  var sideX = left;\n  var sideY = top;\n  var win = window;\n\n  if (adaptive) {\n    var offsetParent = getOffsetParent(popper);\n    var heightProp = 'clientHeight';\n    var widthProp = 'clientWidth';\n\n    if (offsetParent === getWindow(popper)) {\n      offsetParent = getDocumentElement(popper);\n\n      if (getComputedStyle(offsetParent).position !== 'static' && position === 'absolute') {\n        heightProp = 'scrollHeight';\n        widthProp = 'scrollWidth';\n      }\n    } // $FlowFixMe[incompatible-cast]: force type refinement, we compare offsetParent with window above, but Flow doesn't detect it\n\n\n    offsetParent = offsetParent;\n\n    if (placement === top || (placement === left || placement === right) && variation === end) {\n      sideY = bottom;\n      var offsetY = isFixed && offsetParent === win && win.visualViewport ? win.visualViewport.height : // $FlowFixMe[prop-missing]\n      offsetParent[heightProp];\n      y -= offsetY - popperRect.height;\n      y *= gpuAcceleration ? 1 : -1;\n    }\n\n    if (placement === left || (placement === top || placement === bottom) && variation === end) {\n      sideX = right;\n      var offsetX = isFixed && offsetParent === win && win.visualViewport ? win.visualViewport.width : // $FlowFixMe[prop-missing]\n      offsetParent[widthProp];\n      x -= offsetX - popperRect.width;\n      x *= gpuAcceleration ? 1 : -1;\n    }\n  }\n\n  var commonStyles = Object.assign({\n    position: position\n  }, adaptive && unsetSides);\n\n  var _ref4 = roundOffsets === true ? roundOffsetsByDPR({\n    x: x,\n    y: y\n  }, getWindow(popper)) : {\n    x: x,\n    y: y\n  };\n\n  x = _ref4.x;\n  y = _ref4.y;\n\n  if (gpuAcceleration) {\n    var _Object$assign;\n\n    return Object.assign({}, commonStyles, (_Object$assign = {}, _Object$assign[sideY] = hasY ? '0' : '', _Object$assign[sideX] = hasX ? '0' : '', _Object$assign.transform = (win.devicePixelRatio || 1) <= 1 ? \"translate(\" + x + \"px, \" + y + \"px)\" : \"translate3d(\" + x + \"px, \" + y + \"px, 0)\", _Object$assign));\n  }\n\n  return Object.assign({}, commonStyles, (_Object$assign2 = {}, _Object$assign2[sideY] = hasY ? y + \"px\" : '', _Object$assign2[sideX] = hasX ? x + \"px\" : '', _Object$assign2.transform = '', _Object$assign2));\n}\n\nfunction computeStyles(_ref5) {\n  var state = _ref5.state,\n      options = _ref5.options;\n  var _options$gpuAccelerat = options.gpuAcceleration,\n      gpuAcceleration = _options$gpuAccelerat === void 0 ? true : _options$gpuAccelerat,\n      _options$adaptive = options.adaptive,\n      adaptive = _options$adaptive === void 0 ? true : _options$adaptive,\n      _options$roundOffsets = options.roundOffsets,\n      roundOffsets = _options$roundOffsets === void 0 ? true : _options$roundOffsets;\n  var commonStyles = {\n    placement: getBasePlacement(state.placement),\n    variation: getVariation(state.placement),\n    popper: state.elements.popper,\n    popperRect: state.rects.popper,\n    gpuAcceleration: gpuAcceleration,\n    isFixed: state.options.strategy === 'fixed'\n  };\n\n  if (state.modifiersData.popperOffsets != null) {\n    state.styles.popper = Object.assign({}, state.styles.popper, mapToStyles(Object.assign({}, commonStyles, {\n      offsets: state.modifiersData.popperOffsets,\n      position: state.options.strategy,\n      adaptive: adaptive,\n      roundOffsets: roundOffsets\n    })));\n  }\n\n  if (state.modifiersData.arrow != null) {\n    state.styles.arrow = Object.assign({}, state.styles.arrow, mapToStyles(Object.assign({}, commonStyles, {\n      offsets: state.modifiersData.arrow,\n      position: 'absolute',\n      adaptive: false,\n      roundOffsets: roundOffsets\n    })));\n  }\n\n  state.attributes.popper = Object.assign({}, state.attributes.popper, {\n    'data-popper-placement': state.placement\n  });\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'computeStyles',\n  enabled: true,\n  phase: 'beforeWrite',\n  fn: computeStyles,\n  data: {}\n};","import getWindow from \"../dom-utils/getWindow.js\"; // eslint-disable-next-line import/no-unused-modules\n\nvar passive = {\n  passive: true\n};\n\nfunction effect(_ref) {\n  var state = _ref.state,\n      instance = _ref.instance,\n      options = _ref.options;\n  var _options$scroll = options.scroll,\n      scroll = _options$scroll === void 0 ? true : _options$scroll,\n      _options$resize = options.resize,\n      resize = _options$resize === void 0 ? true : _options$resize;\n  var window = getWindow(state.elements.popper);\n  var scrollParents = [].concat(state.scrollParents.reference, state.scrollParents.popper);\n\n  if (scroll) {\n    scrollParents.forEach(function (scrollParent) {\n      scrollParent.addEventListener('scroll', instance.update, passive);\n    });\n  }\n\n  if (resize) {\n    window.addEventListener('resize', instance.update, passive);\n  }\n\n  return function () {\n    if (scroll) {\n      scrollParents.forEach(function (scrollParent) {\n        scrollParent.removeEventListener('scroll', instance.update, passive);\n      });\n    }\n\n    if (resize) {\n      window.removeEventListener('resize', instance.update, passive);\n    }\n  };\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'eventListeners',\n  enabled: true,\n  phase: 'write',\n  fn: function fn() {},\n  effect: effect,\n  data: {}\n};","var hash = {\n  left: 'right',\n  right: 'left',\n  bottom: 'top',\n  top: 'bottom'\n};\nexport default function getOppositePlacement(placement) {\n  return placement.replace(/left|right|bottom|top/g, function (matched) {\n    return hash[matched];\n  });\n}","var hash = {\n  start: 'end',\n  end: 'start'\n};\nexport default function getOppositeVariationPlacement(placement) {\n  return placement.replace(/start|end/g, function (matched) {\n    return hash[matched];\n  });\n}","import getWindow from \"./getWindow.js\";\nexport default function getWindowScroll(node) {\n  var win = getWindow(node);\n  var scrollLeft = win.pageXOffset;\n  var scrollTop = win.pageYOffset;\n  return {\n    scrollLeft: scrollLeft,\n    scrollTop: scrollTop\n  };\n}","import getBoundingClientRect from \"./getBoundingClientRect.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport getWindowScroll from \"./getWindowScroll.js\";\nexport default function getWindowScrollBarX(element) {\n  // If <html> has a CSS width greater than the viewport, then this will be\n  // incorrect for RTL.\n  // Popper 1 is broken in this case and never had a bug report so let's assume\n  // it's not an issue. I don't think anyone ever specifies width on <html>\n  // anyway.\n  // Browsers where the left scrollbar doesn't cause an issue report `0` for\n  // this (e.g. Edge 2019, IE11, Safari)\n  return getBoundingClientRect(getDocumentElement(element)).left + getWindowScroll(element).scrollLeft;\n}","import getComputedStyle from \"./getComputedStyle.js\";\nexport default function isScrollParent(element) {\n  // Firefox wants us to check `-x` and `-y` variations as well\n  var _getComputedStyle = getComputedStyle(element),\n      overflow = _getComputedStyle.overflow,\n      overflowX = _getComputedStyle.overflowX,\n      overflowY = _getComputedStyle.overflowY;\n\n  return /auto|scroll|overlay|hidden/.test(overflow + overflowY + overflowX);\n}","import getParentNode from \"./getParentNode.js\";\nimport isScrollParent from \"./isScrollParent.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport { isHTMLElement } from \"./instanceOf.js\";\nexport default function getScrollParent(node) {\n  if (['html', 'body', '#document'].indexOf(getNodeName(node)) >= 0) {\n    // $FlowFixMe[incompatible-return]: assume body is always available\n    return node.ownerDocument.body;\n  }\n\n  if (isHTMLElement(node) && isScrollParent(node)) {\n    return node;\n  }\n\n  return getScrollParent(getParentNode(node));\n}","import getScrollParent from \"./getScrollParent.js\";\nimport getParentNode from \"./getParentNode.js\";\nimport getWindow from \"./getWindow.js\";\nimport isScrollParent from \"./isScrollParent.js\";\n/*\ngiven a DOM element, return the list of all scroll parents, up the list of ancesors\nuntil we get to the top window object. This list is what we attach scroll listeners\nto, because if any of these parent elements scroll, we'll need to re-calculate the\nreference element's position.\n*/\n\nexport default function listScrollParents(element, list) {\n  var _element$ownerDocumen;\n\n  if (list === void 0) {\n    list = [];\n  }\n\n  var scrollParent = getScrollParent(element);\n  var isBody = scrollParent === ((_element$ownerDocumen = element.ownerDocument) == null ? void 0 : _element$ownerDocumen.body);\n  var win = getWindow(scrollParent);\n  var target = isBody ? [win].concat(win.visualViewport || [], isScrollParent(scrollParent) ? scrollParent : []) : scrollParent;\n  var updatedList = list.concat(target);\n  return isBody ? updatedList : // $FlowFixMe[incompatible-call]: isBody tells us target will be an HTMLElement here\n  updatedList.concat(listScrollParents(getParentNode(target)));\n}","export default function rectToClientRect(rect) {\n  return Object.assign({}, rect, {\n    left: rect.x,\n    top: rect.y,\n    right: rect.x + rect.width,\n    bottom: rect.y + rect.height\n  });\n}","import { viewport } from \"../enums.js\";\nimport getViewportRect from \"./getViewportRect.js\";\nimport getDocumentRect from \"./getDocumentRect.js\";\nimport listScrollParents from \"./listScrollParents.js\";\nimport getOffsetParent from \"./getOffsetParent.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport getComputedStyle from \"./getComputedStyle.js\";\nimport { isElement, isHTMLElement } from \"./instanceOf.js\";\nimport getBoundingClientRect from \"./getBoundingClientRect.js\";\nimport getParentNode from \"./getParentNode.js\";\nimport contains from \"./contains.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport rectToClientRect from \"../utils/rectToClientRect.js\";\nimport { max, min } from \"../utils/math.js\";\n\nfunction getInnerBoundingClientRect(element, strategy) {\n  var rect = getBoundingClientRect(element, false, strategy === 'fixed');\n  rect.top = rect.top + element.clientTop;\n  rect.left = rect.left + element.clientLeft;\n  rect.bottom = rect.top + element.clientHeight;\n  rect.right = rect.left + element.clientWidth;\n  rect.width = element.clientWidth;\n  rect.height = element.clientHeight;\n  rect.x = rect.left;\n  rect.y = rect.top;\n  return rect;\n}\n\nfunction getClientRectFromMixedType(element, clippingParent, strategy) {\n  return clippingParent === viewport ? rectToClientRect(getViewportRect(element, strategy)) : isElement(clippingParent) ? getInnerBoundingClientRect(clippingParent, strategy) : rectToClientRect(getDocumentRect(getDocumentElement(element)));\n} // A \"clipping parent\" is an overflowable container with the characteristic of\n// clipping (or hiding) overflowing elements with a position different from\n// `initial`\n\n\nfunction getClippingParents(element) {\n  var clippingParents = listScrollParents(getParentNode(element));\n  var canEscapeClipping = ['absolute', 'fixed'].indexOf(getComputedStyle(element).position) >= 0;\n  var clipperElement = canEscapeClipping && isHTMLElement(element) ? getOffsetParent(element) : element;\n\n  if (!isElement(clipperElement)) {\n    return [];\n  } // $FlowFixMe[incompatible-return]: https://github.com/facebook/flow/issues/1414\n\n\n  return clippingParents.filter(function (clippingParent) {\n    return isElement(clippingParent) && contains(clippingParent, clipperElement) && getNodeName(clippingParent) !== 'body';\n  });\n} // Gets the maximum area that the element is visible in due to any number of\n// clipping parents\n\n\nexport default function getClippingRect(element, boundary, rootBoundary, strategy) {\n  var mainClippingParents = boundary === 'clippingParents' ? getClippingParents(element) : [].concat(boundary);\n  var clippingParents = [].concat(mainClippingParents, [rootBoundary]);\n  var firstClippingParent = clippingParents[0];\n  var clippingRect = clippingParents.reduce(function (accRect, clippingParent) {\n    var rect = getClientRectFromMixedType(element, clippingParent, strategy);\n    accRect.top = max(rect.top, accRect.top);\n    accRect.right = min(rect.right, accRect.right);\n    accRect.bottom = min(rect.bottom, accRect.bottom);\n    accRect.left = max(rect.left, accRect.left);\n    return accRect;\n  }, getClientRectFromMixedType(element, firstClippingParent, strategy));\n  clippingRect.width = clippingRect.right - clippingRect.left;\n  clippingRect.height = clippingRect.bottom - clippingRect.top;\n  clippingRect.x = clippingRect.left;\n  clippingRect.y = clippingRect.top;\n  return clippingRect;\n}","import getWindow from \"./getWindow.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport getWindowScrollBarX from \"./getWindowScrollBarX.js\";\nimport isLayoutViewport from \"./isLayoutViewport.js\";\nexport default function getViewportRect(element, strategy) {\n  var win = getWindow(element);\n  var html = getDocumentElement(element);\n  var visualViewport = win.visualViewport;\n  var width = html.clientWidth;\n  var height = html.clientHeight;\n  var x = 0;\n  var y = 0;\n\n  if (visualViewport) {\n    width = visualViewport.width;\n    height = visualViewport.height;\n    var layoutViewport = isLayoutViewport();\n\n    if (layoutViewport || !layoutViewport && strategy === 'fixed') {\n      x = visualViewport.offsetLeft;\n      y = visualViewport.offsetTop;\n    }\n  }\n\n  return {\n    width: width,\n    height: height,\n    x: x + getWindowScrollBarX(element),\n    y: y\n  };\n}","import getDocumentElement from \"./getDocumentElement.js\";\nimport getComputedStyle from \"./getComputedStyle.js\";\nimport getWindowScrollBarX from \"./getWindowScrollBarX.js\";\nimport getWindowScroll from \"./getWindowScroll.js\";\nimport { max } from \"../utils/math.js\"; // Gets the entire size of the scrollable document area, even extending outside\n// of the `<html>` and `<body>` rect bounds if horizontally scrollable\n\nexport default function getDocumentRect(element) {\n  var _element$ownerDocumen;\n\n  var html = getDocumentElement(element);\n  var winScroll = getWindowScroll(element);\n  var body = (_element$ownerDocumen = element.ownerDocument) == null ? void 0 : _element$ownerDocumen.body;\n  var width = max(html.scrollWidth, html.clientWidth, body ? body.scrollWidth : 0, body ? body.clientWidth : 0);\n  var height = max(html.scrollHeight, html.clientHeight, body ? body.scrollHeight : 0, body ? body.clientHeight : 0);\n  var x = -winScroll.scrollLeft + getWindowScrollBarX(element);\n  var y = -winScroll.scrollTop;\n\n  if (getComputedStyle(body || html).direction === 'rtl') {\n    x += max(html.clientWidth, body ? body.clientWidth : 0) - width;\n  }\n\n  return {\n    width: width,\n    height: height,\n    x: x,\n    y: y\n  };\n}","import getBasePlacement from \"./getBasePlacement.js\";\nimport getVariation from \"./getVariation.js\";\nimport getMainAxisFromPlacement from \"./getMainAxisFromPlacement.js\";\nimport { top, right, bottom, left, start, end } from \"../enums.js\";\nexport default function computeOffsets(_ref) {\n  var reference = _ref.reference,\n      element = _ref.element,\n      placement = _ref.placement;\n  var basePlacement = placement ? getBasePlacement(placement) : null;\n  var variation = placement ? getVariation(placement) : null;\n  var commonX = reference.x + reference.width / 2 - element.width / 2;\n  var commonY = reference.y + reference.height / 2 - element.height / 2;\n  var offsets;\n\n  switch (basePlacement) {\n    case top:\n      offsets = {\n        x: commonX,\n        y: reference.y - element.height\n      };\n      break;\n\n    case bottom:\n      offsets = {\n        x: commonX,\n        y: reference.y + reference.height\n      };\n      break;\n\n    case right:\n      offsets = {\n        x: reference.x + reference.width,\n        y: commonY\n      };\n      break;\n\n    case left:\n      offsets = {\n        x: reference.x - element.width,\n        y: commonY\n      };\n      break;\n\n    default:\n      offsets = {\n        x: reference.x,\n        y: reference.y\n      };\n  }\n\n  var mainAxis = basePlacement ? getMainAxisFromPlacement(basePlacement) : null;\n\n  if (mainAxis != null) {\n    var len = mainAxis === 'y' ? 'height' : 'width';\n\n    switch (variation) {\n      case start:\n        offsets[mainAxis] = offsets[mainAxis] - (reference[len] / 2 - element[len] / 2);\n        break;\n\n      case end:\n        offsets[mainAxis] = offsets[mainAxis] + (reference[len] / 2 - element[len] / 2);\n        break;\n\n      default:\n    }\n  }\n\n  return offsets;\n}","import getClippingRect from \"../dom-utils/getClippingRect.js\";\nimport getDocumentElement from \"../dom-utils/getDocumentElement.js\";\nimport getBoundingClientRect from \"../dom-utils/getBoundingClientRect.js\";\nimport computeOffsets from \"./computeOffsets.js\";\nimport rectToClientRect from \"./rectToClientRect.js\";\nimport { clippingParents, reference, popper, bottom, top, right, basePlacements, viewport } from \"../enums.js\";\nimport { isElement } from \"../dom-utils/instanceOf.js\";\nimport mergePaddingObject from \"./mergePaddingObject.js\";\nimport expandToHashMap from \"./expandToHashMap.js\"; // eslint-disable-next-line import/no-unused-modules\n\nexport default function detectOverflow(state, options) {\n  if (options === void 0) {\n    options = {};\n  }\n\n  var _options = options,\n      _options$placement = _options.placement,\n      placement = _options$placement === void 0 ? state.placement : _options$placement,\n      _options$strategy = _options.strategy,\n      strategy = _options$strategy === void 0 ? state.strategy : _options$strategy,\n      _options$boundary = _options.boundary,\n      boundary = _options$boundary === void 0 ? clippingParents : _options$boundary,\n      _options$rootBoundary = _options.rootBoundary,\n      rootBoundary = _options$rootBoundary === void 0 ? viewport : _options$rootBoundary,\n      _options$elementConte = _options.elementContext,\n      elementContext = _options$elementConte === void 0 ? popper : _options$elementConte,\n      _options$altBoundary = _options.altBoundary,\n      altBoundary = _options$altBoundary === void 0 ? false : _options$altBoundary,\n      _options$padding = _options.padding,\n      padding = _options$padding === void 0 ? 0 : _options$padding;\n  var paddingObject = mergePaddingObject(typeof padding !== 'number' ? padding : expandToHashMap(padding, basePlacements));\n  var altContext = elementContext === popper ? reference : popper;\n  var popperRect = state.rects.popper;\n  var element = state.elements[altBoundary ? altContext : elementContext];\n  var clippingClientRect = getClippingRect(isElement(element) ? element : element.contextElement || getDocumentElement(state.elements.popper), boundary, rootBoundary, strategy);\n  var referenceClientRect = getBoundingClientRect(state.elements.reference);\n  var popperOffsets = computeOffsets({\n    reference: referenceClientRect,\n    element: popperRect,\n    strategy: 'absolute',\n    placement: placement\n  });\n  var popperClientRect = rectToClientRect(Object.assign({}, popperRect, popperOffsets));\n  var elementClientRect = elementContext === popper ? popperClientRect : referenceClientRect; // positive = overflowing the clipping rect\n  // 0 or negative = within the clipping rect\n\n  var overflowOffsets = {\n    top: clippingClientRect.top - elementClientRect.top + paddingObject.top,\n    bottom: elementClientRect.bottom - clippingClientRect.bottom + paddingObject.bottom,\n    left: clippingClientRect.left - elementClientRect.left + paddingObject.left,\n    right: elementClientRect.right - clippingClientRect.right + paddingObject.right\n  };\n  var offsetData = state.modifiersData.offset; // Offsets can be applied only to the popper element\n\n  if (elementContext === popper && offsetData) {\n    var offset = offsetData[placement];\n    Object.keys(overflowOffsets).forEach(function (key) {\n      var multiply = [right, bottom].indexOf(key) >= 0 ? 1 : -1;\n      var axis = [top, bottom].indexOf(key) >= 0 ? 'y' : 'x';\n      overflowOffsets[key] += offset[axis] * multiply;\n    });\n  }\n\n  return overflowOffsets;\n}","import getOppositePlacement from \"../utils/getOppositePlacement.js\";\nimport getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getOppositeVariationPlacement from \"../utils/getOppositeVariationPlacement.js\";\nimport detectOverflow from \"../utils/detectOverflow.js\";\nimport computeAutoPlacement from \"../utils/computeAutoPlacement.js\";\nimport { bottom, top, start, right, left, auto } from \"../enums.js\";\nimport getVariation from \"../utils/getVariation.js\"; // eslint-disable-next-line import/no-unused-modules\n\nfunction getExpandedFallbackPlacements(placement) {\n  if (getBasePlacement(placement) === auto) {\n    return [];\n  }\n\n  var oppositePlacement = getOppositePlacement(placement);\n  return [getOppositeVariationPlacement(placement), oppositePlacement, getOppositeVariationPlacement(oppositePlacement)];\n}\n\nfunction flip(_ref) {\n  var state = _ref.state,\n      options = _ref.options,\n      name = _ref.name;\n\n  if (state.modifiersData[name]._skip) {\n    return;\n  }\n\n  var _options$mainAxis = options.mainAxis,\n      checkMainAxis = _options$mainAxis === void 0 ? true : _options$mainAxis,\n      _options$altAxis = options.altAxis,\n      checkAltAxis = _options$altAxis === void 0 ? true : _options$altAxis,\n      specifiedFallbackPlacements = options.fallbackPlacements,\n      padding = options.padding,\n      boundary = options.boundary,\n      rootBoundary = options.rootBoundary,\n      altBoundary = options.altBoundary,\n      _options$flipVariatio = options.flipVariations,\n      flipVariations = _options$flipVariatio === void 0 ? true : _options$flipVariatio,\n      allowedAutoPlacements = options.allowedAutoPlacements;\n  var preferredPlacement = state.options.placement;\n  var basePlacement = getBasePlacement(preferredPlacement);\n  var isBasePlacement = basePlacement === preferredPlacement;\n  var fallbackPlacements = specifiedFallbackPlacements || (isBasePlacement || !flipVariations ? [getOppositePlacement(preferredPlacement)] : getExpandedFallbackPlacements(preferredPlacement));\n  var placements = [preferredPlacement].concat(fallbackPlacements).reduce(function (acc, placement) {\n    return acc.concat(getBasePlacement(placement) === auto ? computeAutoPlacement(state, {\n      placement: placement,\n      boundary: boundary,\n      rootBoundary: rootBoundary,\n      padding: padding,\n      flipVariations: flipVariations,\n      allowedAutoPlacements: allowedAutoPlacements\n    }) : placement);\n  }, []);\n  var referenceRect = state.rects.reference;\n  var popperRect = state.rects.popper;\n  var checksMap = new Map();\n  var makeFallbackChecks = true;\n  var firstFittingPlacement = placements[0];\n\n  for (var i = 0; i < placements.length; i++) {\n    var placement = placements[i];\n\n    var _basePlacement = getBasePlacement(placement);\n\n    var isStartVariation = getVariation(placement) === start;\n    var isVertical = [top, bottom].indexOf(_basePlacement) >= 0;\n    var len = isVertical ? 'width' : 'height';\n    var overflow = detectOverflow(state, {\n      placement: placement,\n      boundary: boundary,\n      rootBoundary: rootBoundary,\n      altBoundary: altBoundary,\n      padding: padding\n    });\n    var mainVariationSide = isVertical ? isStartVariation ? right : left : isStartVariation ? bottom : top;\n\n    if (referenceRect[len] > popperRect[len]) {\n      mainVariationSide = getOppositePlacement(mainVariationSide);\n    }\n\n    var altVariationSide = getOppositePlacement(mainVariationSide);\n    var checks = [];\n\n    if (checkMainAxis) {\n      checks.push(overflow[_basePlacement] <= 0);\n    }\n\n    if (checkAltAxis) {\n      checks.push(overflow[mainVariationSide] <= 0, overflow[altVariationSide] <= 0);\n    }\n\n    if (checks.every(function (check) {\n      return check;\n    })) {\n      firstFittingPlacement = placement;\n      makeFallbackChecks = false;\n      break;\n    }\n\n    checksMap.set(placement, checks);\n  }\n\n  if (makeFallbackChecks) {\n    // `2` may be desired in some cases – research later\n    var numberOfChecks = flipVariations ? 3 : 1;\n\n    var _loop = function _loop(_i) {\n      var fittingPlacement = placements.find(function (placement) {\n        var checks = checksMap.get(placement);\n\n        if (checks) {\n          return checks.slice(0, _i).every(function (check) {\n            return check;\n          });\n        }\n      });\n\n      if (fittingPlacement) {\n        firstFittingPlacement = fittingPlacement;\n        return \"break\";\n      }\n    };\n\n    for (var _i = numberOfChecks; _i > 0; _i--) {\n      var _ret = _loop(_i);\n\n      if (_ret === \"break\") break;\n    }\n  }\n\n  if (state.placement !== firstFittingPlacement) {\n    state.modifiersData[name]._skip = true;\n    state.placement = firstFittingPlacement;\n    state.reset = true;\n  }\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'flip',\n  enabled: true,\n  phase: 'main',\n  fn: flip,\n  requiresIfExists: ['offset'],\n  data: {\n    _skip: false\n  }\n};","import getVariation from \"./getVariation.js\";\nimport { variationPlacements, basePlacements, placements as allPlacements } from \"../enums.js\";\nimport detectOverflow from \"./detectOverflow.js\";\nimport getBasePlacement from \"./getBasePlacement.js\";\nexport default function computeAutoPlacement(state, options) {\n  if (options === void 0) {\n    options = {};\n  }\n\n  var _options = options,\n      placement = _options.placement,\n      boundary = _options.boundary,\n      rootBoundary = _options.rootBoundary,\n      padding = _options.padding,\n      flipVariations = _options.flipVariations,\n      _options$allowedAutoP = _options.allowedAutoPlacements,\n      allowedAutoPlacements = _options$allowedAutoP === void 0 ? allPlacements : _options$allowedAutoP;\n  var variation = getVariation(placement);\n  var placements = variation ? flipVariations ? variationPlacements : variationPlacements.filter(function (placement) {\n    return getVariation(placement) === variation;\n  }) : basePlacements;\n  var allowedPlacements = placements.filter(function (placement) {\n    return allowedAutoPlacements.indexOf(placement) >= 0;\n  });\n\n  if (allowedPlacements.length === 0) {\n    allowedPlacements = placements;\n  } // $FlowFixMe[incompatible-type]: Flow seems to have problems with two array unions...\n\n\n  var overflows = allowedPlacements.reduce(function (acc, placement) {\n    acc[placement] = detectOverflow(state, {\n      placement: placement,\n      boundary: boundary,\n      rootBoundary: rootBoundary,\n      padding: padding\n    })[getBasePlacement(placement)];\n    return acc;\n  }, {});\n  return Object.keys(overflows).sort(function (a, b) {\n    return overflows[a] - overflows[b];\n  });\n}","import { top, bottom, left, right } from \"../enums.js\";\nimport detectOverflow from \"../utils/detectOverflow.js\";\n\nfunction getSideOffsets(overflow, rect, preventedOffsets) {\n  if (preventedOffsets === void 0) {\n    preventedOffsets = {\n      x: 0,\n      y: 0\n    };\n  }\n\n  return {\n    top: overflow.top - rect.height - preventedOffsets.y,\n    right: overflow.right - rect.width + preventedOffsets.x,\n    bottom: overflow.bottom - rect.height + preventedOffsets.y,\n    left: overflow.left - rect.width - preventedOffsets.x\n  };\n}\n\nfunction isAnySideFullyClipped(overflow) {\n  return [top, right, bottom, left].some(function (side) {\n    return overflow[side] >= 0;\n  });\n}\n\nfunction hide(_ref) {\n  var state = _ref.state,\n      name = _ref.name;\n  var referenceRect = state.rects.reference;\n  var popperRect = state.rects.popper;\n  var preventedOffsets = state.modifiersData.preventOverflow;\n  var referenceOverflow = detectOverflow(state, {\n    elementContext: 'reference'\n  });\n  var popperAltOverflow = detectOverflow(state, {\n    altBoundary: true\n  });\n  var referenceClippingOffsets = getSideOffsets(referenceOverflow, referenceRect);\n  var popperEscapeOffsets = getSideOffsets(popperAltOverflow, popperRect, preventedOffsets);\n  var isReferenceHidden = isAnySideFullyClipped(referenceClippingOffsets);\n  var hasPopperEscaped = isAnySideFullyClipped(popperEscapeOffsets);\n  state.modifiersData[name] = {\n    referenceClippingOffsets: referenceClippingOffsets,\n    popperEscapeOffsets: popperEscapeOffsets,\n    isReferenceHidden: isReferenceHidden,\n    hasPopperEscaped: hasPopperEscaped\n  };\n  state.attributes.popper = Object.assign({}, state.attributes.popper, {\n    'data-popper-reference-hidden': isReferenceHidden,\n    'data-popper-escaped': hasPopperEscaped\n  });\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'hide',\n  enabled: true,\n  phase: 'main',\n  requiresIfExists: ['preventOverflow'],\n  fn: hide\n};","import getBasePlacement from \"../utils/getBasePlacement.js\";\nimport { top, left, right, placements } from \"../enums.js\"; // eslint-disable-next-line import/no-unused-modules\n\nexport function distanceAndSkiddingToXY(placement, rects, offset) {\n  var basePlacement = getBasePlacement(placement);\n  var invertDistance = [left, top].indexOf(basePlacement) >= 0 ? -1 : 1;\n\n  var _ref = typeof offset === 'function' ? offset(Object.assign({}, rects, {\n    placement: placement\n  })) : offset,\n      skidding = _ref[0],\n      distance = _ref[1];\n\n  skidding = skidding || 0;\n  distance = (distance || 0) * invertDistance;\n  return [left, right].indexOf(basePlacement) >= 0 ? {\n    x: distance,\n    y: skidding\n  } : {\n    x: skidding,\n    y: distance\n  };\n}\n\nfunction offset(_ref2) {\n  var state = _ref2.state,\n      options = _ref2.options,\n      name = _ref2.name;\n  var _options$offset = options.offset,\n      offset = _options$offset === void 0 ? [0, 0] : _options$offset;\n  var data = placements.reduce(function (acc, placement) {\n    acc[placement] = distanceAndSkiddingToXY(placement, state.rects, offset);\n    return acc;\n  }, {});\n  var _data$state$placement = data[state.placement],\n      x = _data$state$placement.x,\n      y = _data$state$placement.y;\n\n  if (state.modifiersData.popperOffsets != null) {\n    state.modifiersData.popperOffsets.x += x;\n    state.modifiersData.popperOffsets.y += y;\n  }\n\n  state.modifiersData[name] = data;\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'offset',\n  enabled: true,\n  phase: 'main',\n  requires: ['popperOffsets'],\n  fn: offset\n};","import computeOffsets from \"../utils/computeOffsets.js\";\n\nfunction popperOffsets(_ref) {\n  var state = _ref.state,\n      name = _ref.name;\n  // Offsets are the actual position the popper needs to have to be\n  // properly positioned near its reference element\n  // This is the most basic placement, and will be adjusted by\n  // the modifiers in the next step\n  state.modifiersData[name] = computeOffsets({\n    reference: state.rects.reference,\n    element: state.rects.popper,\n    strategy: 'absolute',\n    placement: state.placement\n  });\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'popperOffsets',\n  enabled: true,\n  phase: 'read',\n  fn: popperOffsets,\n  data: {}\n};","import { top, left, right, bottom, start } from \"../enums.js\";\nimport getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getMainAxisFromPlacement from \"../utils/getMainAxisFromPlacement.js\";\nimport getAltAxis from \"../utils/getAltAxis.js\";\nimport { within, withinMaxClamp } from \"../utils/within.js\";\nimport getLayoutRect from \"../dom-utils/getLayoutRect.js\";\nimport getOffsetParent from \"../dom-utils/getOffsetParent.js\";\nimport detectOverflow from \"../utils/detectOverflow.js\";\nimport getVariation from \"../utils/getVariation.js\";\nimport getFreshSideObject from \"../utils/getFreshSideObject.js\";\nimport { min as mathMin, max as mathMax } from \"../utils/math.js\";\n\nfunction preventOverflow(_ref) {\n  var state = _ref.state,\n      options = _ref.options,\n      name = _ref.name;\n  var _options$mainAxis = options.mainAxis,\n      checkMainAxis = _options$mainAxis === void 0 ? true : _options$mainAxis,\n      _options$altAxis = options.altAxis,\n      checkAltAxis = _options$altAxis === void 0 ? false : _options$altAxis,\n      boundary = options.boundary,\n      rootBoundary = options.rootBoundary,\n      altBoundary = options.altBoundary,\n      padding = options.padding,\n      _options$tether = options.tether,\n      tether = _options$tether === void 0 ? true : _options$tether,\n      _options$tetherOffset = options.tetherOffset,\n      tetherOffset = _options$tetherOffset === void 0 ? 0 : _options$tetherOffset;\n  var overflow = detectOverflow(state, {\n    boundary: boundary,\n    rootBoundary: rootBoundary,\n    padding: padding,\n    altBoundary: altBoundary\n  });\n  var basePlacement = getBasePlacement(state.placement);\n  var variation = getVariation(state.placement);\n  var isBasePlacement = !variation;\n  var mainAxis = getMainAxisFromPlacement(basePlacement);\n  var altAxis = getAltAxis(mainAxis);\n  var popperOffsets = state.modifiersData.popperOffsets;\n  var referenceRect = state.rects.reference;\n  var popperRect = state.rects.popper;\n  var tetherOffsetValue = typeof tetherOffset === 'function' ? tetherOffset(Object.assign({}, state.rects, {\n    placement: state.placement\n  })) : tetherOffset;\n  var normalizedTetherOffsetValue = typeof tetherOffsetValue === 'number' ? {\n    mainAxis: tetherOffsetValue,\n    altAxis: tetherOffsetValue\n  } : Object.assign({\n    mainAxis: 0,\n    altAxis: 0\n  }, tetherOffsetValue);\n  var offsetModifierState = state.modifiersData.offset ? state.modifiersData.offset[state.placement] : null;\n  var data = {\n    x: 0,\n    y: 0\n  };\n\n  if (!popperOffsets) {\n    return;\n  }\n\n  if (checkMainAxis) {\n    var _offsetModifierState$;\n\n    var mainSide = mainAxis === 'y' ? top : left;\n    var altSide = mainAxis === 'y' ? bottom : right;\n    var len = mainAxis === 'y' ? 'height' : 'width';\n    var offset = popperOffsets[mainAxis];\n    var min = offset + overflow[mainSide];\n    var max = offset - overflow[altSide];\n    var additive = tether ? -popperRect[len] / 2 : 0;\n    var minLen = variation === start ? referenceRect[len] : popperRect[len];\n    var maxLen = variation === start ? -popperRect[len] : -referenceRect[len]; // We need to include the arrow in the calculation so the arrow doesn't go\n    // outside the reference bounds\n\n    var arrowElement = state.elements.arrow;\n    var arrowRect = tether && arrowElement ? getLayoutRect(arrowElement) : {\n      width: 0,\n      height: 0\n    };\n    var arrowPaddingObject = state.modifiersData['arrow#persistent'] ? state.modifiersData['arrow#persistent'].padding : getFreshSideObject();\n    var arrowPaddingMin = arrowPaddingObject[mainSide];\n    var arrowPaddingMax = arrowPaddingObject[altSide]; // If the reference length is smaller than the arrow length, we don't want\n    // to include its full size in the calculation. If the reference is small\n    // and near the edge of a boundary, the popper can overflow even if the\n    // reference is not overflowing as well (e.g. virtual elements with no\n    // width or height)\n\n    var arrowLen = within(0, referenceRect[len], arrowRect[len]);\n    var minOffset = isBasePlacement ? referenceRect[len] / 2 - additive - arrowLen - arrowPaddingMin - normalizedTetherOffsetValue.mainAxis : minLen - arrowLen - arrowPaddingMin - normalizedTetherOffsetValue.mainAxis;\n    var maxOffset = isBasePlacement ? -referenceRect[len] / 2 + additive + arrowLen + arrowPaddingMax + normalizedTetherOffsetValue.mainAxis : maxLen + arrowLen + arrowPaddingMax + normalizedTetherOffsetValue.mainAxis;\n    var arrowOffsetParent = state.elements.arrow && getOffsetParent(state.elements.arrow);\n    var clientOffset = arrowOffsetParent ? mainAxis === 'y' ? arrowOffsetParent.clientTop || 0 : arrowOffsetParent.clientLeft || 0 : 0;\n    var offsetModifierValue = (_offsetModifierState$ = offsetModifierState == null ? void 0 : offsetModifierState[mainAxis]) != null ? _offsetModifierState$ : 0;\n    var tetherMin = offset + minOffset - offsetModifierValue - clientOffset;\n    var tetherMax = offset + maxOffset - offsetModifierValue;\n    var preventedOffset = within(tether ? mathMin(min, tetherMin) : min, offset, tether ? mathMax(max, tetherMax) : max);\n    popperOffsets[mainAxis] = preventedOffset;\n    data[mainAxis] = preventedOffset - offset;\n  }\n\n  if (checkAltAxis) {\n    var _offsetModifierState$2;\n\n    var _mainSide = mainAxis === 'x' ? top : left;\n\n    var _altSide = mainAxis === 'x' ? bottom : right;\n\n    var _offset = popperOffsets[altAxis];\n\n    var _len = altAxis === 'y' ? 'height' : 'width';\n\n    var _min = _offset + overflow[_mainSide];\n\n    var _max = _offset - overflow[_altSide];\n\n    var isOriginSide = [top, left].indexOf(basePlacement) !== -1;\n\n    var _offsetModifierValue = (_offsetModifierState$2 = offsetModifierState == null ? void 0 : offsetModifierState[altAxis]) != null ? _offsetModifierState$2 : 0;\n\n    var _tetherMin = isOriginSide ? _min : _offset - referenceRect[_len] - popperRect[_len] - _offsetModifierValue + normalizedTetherOffsetValue.altAxis;\n\n    var _tetherMax = isOriginSide ? _offset + referenceRect[_len] + popperRect[_len] - _offsetModifierValue - normalizedTetherOffsetValue.altAxis : _max;\n\n    var _preventedOffset = tether && isOriginSide ? withinMaxClamp(_tetherMin, _offset, _tetherMax) : within(tether ? _tetherMin : _min, _offset, tether ? _tetherMax : _max);\n\n    popperOffsets[altAxis] = _preventedOffset;\n    data[altAxis] = _preventedOffset - _offset;\n  }\n\n  state.modifiersData[name] = data;\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'preventOverflow',\n  enabled: true,\n  phase: 'main',\n  fn: preventOverflow,\n  requiresIfExists: ['offset']\n};","export default function getAltAxis(axis) {\n  return axis === 'x' ? 'y' : 'x';\n}","import getBoundingClientRect from \"./getBoundingClientRect.js\";\nimport getNodeScroll from \"./getNodeScroll.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport { isHTMLElement } from \"./instanceOf.js\";\nimport getWindowScrollBarX from \"./getWindowScrollBarX.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport isScrollParent from \"./isScrollParent.js\";\nimport { round } from \"../utils/math.js\";\n\nfunction isElementScaled(element) {\n  var rect = element.getBoundingClientRect();\n  var scaleX = round(rect.width) / element.offsetWidth || 1;\n  var scaleY = round(rect.height) / element.offsetHeight || 1;\n  return scaleX !== 1 || scaleY !== 1;\n} // Returns the composite rect of an element relative to its offsetParent.\n// Composite means it takes into account transforms as well as layout.\n\n\nexport default function getCompositeRect(elementOrVirtualElement, offsetParent, isFixed) {\n  if (isFixed === void 0) {\n    isFixed = false;\n  }\n\n  var isOffsetParentAnElement = isHTMLElement(offsetParent);\n  var offsetParentIsScaled = isHTMLElement(offsetParent) && isElementScaled(offsetParent);\n  var documentElement = getDocumentElement(offsetParent);\n  var rect = getBoundingClientRect(elementOrVirtualElement, offsetParentIsScaled, isFixed);\n  var scroll = {\n    scrollLeft: 0,\n    scrollTop: 0\n  };\n  var offsets = {\n    x: 0,\n    y: 0\n  };\n\n  if (isOffsetParentAnElement || !isOffsetParentAnElement && !isFixed) {\n    if (getNodeName(offsetParent) !== 'body' || // https://github.com/popperjs/popper-core/issues/1078\n    isScrollParent(documentElement)) {\n      scroll = getNodeScroll(offsetParent);\n    }\n\n    if (isHTMLElement(offsetParent)) {\n      offsets = getBoundingClientRect(offsetParent, true);\n      offsets.x += offsetParent.clientLeft;\n      offsets.y += offsetParent.clientTop;\n    } else if (documentElement) {\n      offsets.x = getWindowScrollBarX(documentElement);\n    }\n  }\n\n  return {\n    x: rect.left + scroll.scrollLeft - offsets.x,\n    y: rect.top + scroll.scrollTop - offsets.y,\n    width: rect.width,\n    height: rect.height\n  };\n}","import getWindowScroll from \"./getWindowScroll.js\";\nimport getWindow from \"./getWindow.js\";\nimport { isHTMLElement } from \"./instanceOf.js\";\nimport getHTMLElementScroll from \"./getHTMLElementScroll.js\";\nexport default function getNodeScroll(node) {\n  if (node === getWindow(node) || !isHTMLElement(node)) {\n    return getWindowScroll(node);\n  } else {\n    return getHTMLElementScroll(node);\n  }\n}","export default function getHTMLElementScroll(element) {\n  return {\n    scrollLeft: element.scrollLeft,\n    scrollTop: element.scrollTop\n  };\n}","import { modifierPhases } from \"../enums.js\"; // source: https://stackoverflow.com/questions/49875255\n\nfunction order(modifiers) {\n  var map = new Map();\n  var visited = new Set();\n  var result = [];\n  modifiers.forEach(function (modifier) {\n    map.set(modifier.name, modifier);\n  }); // On visiting object, check for its dependencies and visit them recursively\n\n  function sort(modifier) {\n    visited.add(modifier.name);\n    var requires = [].concat(modifier.requires || [], modifier.requiresIfExists || []);\n    requires.forEach(function (dep) {\n      if (!visited.has(dep)) {\n        var depModifier = map.get(dep);\n\n        if (depModifier) {\n          sort(depModifier);\n        }\n      }\n    });\n    result.push(modifier);\n  }\n\n  modifiers.forEach(function (modifier) {\n    if (!visited.has(modifier.name)) {\n      // check for visited object\n      sort(modifier);\n    }\n  });\n  return result;\n}\n\nexport default function orderModifiers(modifiers) {\n  // order based on dependencies\n  var orderedModifiers = order(modifiers); // order based on phase\n\n  return modifierPhases.reduce(function (acc, phase) {\n    return acc.concat(orderedModifiers.filter(function (modifier) {\n      return modifier.phase === phase;\n    }));\n  }, []);\n}","import getCompositeRect from \"./dom-utils/getCompositeRect.js\";\nimport getLayoutRect from \"./dom-utils/getLayoutRect.js\";\nimport listScrollParents from \"./dom-utils/listScrollParents.js\";\nimport getOffsetParent from \"./dom-utils/getOffsetParent.js\";\nimport orderModifiers from \"./utils/orderModifiers.js\";\nimport debounce from \"./utils/debounce.js\";\nimport mergeByName from \"./utils/mergeByName.js\";\nimport detectOverflow from \"./utils/detectOverflow.js\";\nimport { isElement } from \"./dom-utils/instanceOf.js\";\nvar DEFAULT_OPTIONS = {\n  placement: 'bottom',\n  modifiers: [],\n  strategy: 'absolute'\n};\n\nfunction areValidElements() {\n  for (var _len = arguments.length, args = new Array(_len), _key = 0; _key < _len; _key++) {\n    args[_key] = arguments[_key];\n  }\n\n  return !args.some(function (element) {\n    return !(element && typeof element.getBoundingClientRect === 'function');\n  });\n}\n\nexport function popperGenerator(generatorOptions) {\n  if (generatorOptions === void 0) {\n    generatorOptions = {};\n  }\n\n  var _generatorOptions = generatorOptions,\n      _generatorOptions$def = _generatorOptions.defaultModifiers,\n      defaultModifiers = _generatorOptions$def === void 0 ? [] : _generatorOptions$def,\n      _generatorOptions$def2 = _generatorOptions.defaultOptions,\n      defaultOptions = _generatorOptions$def2 === void 0 ? DEFAULT_OPTIONS : _generatorOptions$def2;\n  return function createPopper(reference, popper, options) {\n    if (options === void 0) {\n      options = defaultOptions;\n    }\n\n    var state = {\n      placement: 'bottom',\n      orderedModifiers: [],\n      options: Object.assign({}, DEFAULT_OPTIONS, defaultOptions),\n      modifiersData: {},\n      elements: {\n        reference: reference,\n        popper: popper\n      },\n      attributes: {},\n      styles: {}\n    };\n    var effectCleanupFns = [];\n    var isDestroyed = false;\n    var instance = {\n      state: state,\n      setOptions: function setOptions(setOptionsAction) {\n        var options = typeof setOptionsAction === 'function' ? setOptionsAction(state.options) : setOptionsAction;\n        cleanupModifierEffects();\n        state.options = Object.assign({}, defaultOptions, state.options, options);\n        state.scrollParents = {\n          reference: isElement(reference) ? listScrollParents(reference) : reference.contextElement ? listScrollParents(reference.contextElement) : [],\n          popper: listScrollParents(popper)\n        }; // Orders the modifiers based on their dependencies and `phase`\n        // properties\n\n        var orderedModifiers = orderModifiers(mergeByName([].concat(defaultModifiers, state.options.modifiers))); // Strip out disabled modifiers\n\n        state.orderedModifiers = orderedModifiers.filter(function (m) {\n          return m.enabled;\n        });\n        runModifierEffects();\n        return instance.update();\n      },\n      // Sync update – it will always be executed, even if not necessary. This\n      // is useful for low frequency updates where sync behavior simplifies the\n      // logic.\n      // For high frequency updates (e.g. `resize` and `scroll` events), always\n      // prefer the async Popper#update method\n      forceUpdate: function forceUpdate() {\n        if (isDestroyed) {\n          return;\n        }\n\n        var _state$elements = state.elements,\n            reference = _state$elements.reference,\n            popper = _state$elements.popper; // Don't proceed if `reference` or `popper` are not valid elements\n        // anymore\n\n        if (!areValidElements(reference, popper)) {\n          return;\n        } // Store the reference and popper rects to be read by modifiers\n\n\n        state.rects = {\n          reference: getCompositeRect(reference, getOffsetParent(popper), state.options.strategy === 'fixed'),\n          popper: getLayoutRect(popper)\n        }; // Modifiers have the ability to reset the current update cycle. The\n        // most common use case for this is the `flip` modifier changing the\n        // placement, which then needs to re-run all the modifiers, because the\n        // logic was previously ran for the previous placement and is therefore\n        // stale/incorrect\n\n        state.reset = false;\n        state.placement = state.options.placement; // On each update cycle, the `modifiersData` property for each modifier\n        // is filled with the initial data specified by the modifier. This means\n        // it doesn't persist and is fresh on each update.\n        // To ensure persistent data, use `${name}#persistent`\n\n        state.orderedModifiers.forEach(function (modifier) {\n          return state.modifiersData[modifier.name] = Object.assign({}, modifier.data);\n        });\n\n        for (var index = 0; index < state.orderedModifiers.length; index++) {\n          if (state.reset === true) {\n            state.reset = false;\n            index = -1;\n            continue;\n          }\n\n          var _state$orderedModifie = state.orderedModifiers[index],\n              fn = _state$orderedModifie.fn,\n              _state$orderedModifie2 = _state$orderedModifie.options,\n              _options = _state$orderedModifie2 === void 0 ? {} : _state$orderedModifie2,\n              name = _state$orderedModifie.name;\n\n          if (typeof fn === 'function') {\n            state = fn({\n              state: state,\n              options: _options,\n              name: name,\n              instance: instance\n            }) || state;\n          }\n        }\n      },\n      // Async and optimistically optimized update – it will not be executed if\n      // not necessary (debounced to run at most once-per-tick)\n      update: debounce(function () {\n        return new Promise(function (resolve) {\n          instance.forceUpdate();\n          resolve(state);\n        });\n      }),\n      destroy: function destroy() {\n        cleanupModifierEffects();\n        isDestroyed = true;\n      }\n    };\n\n    if (!areValidElements(reference, popper)) {\n      return instance;\n    }\n\n    instance.setOptions(options).then(function (state) {\n      if (!isDestroyed && options.onFirstUpdate) {\n        options.onFirstUpdate(state);\n      }\n    }); // Modifiers have the ability to execute arbitrary code before the first\n    // update cycle runs. They will be executed in the same order as the update\n    // cycle. This is useful when a modifier adds some persistent data that\n    // other modifiers need to use, but the modifier is run after the dependent\n    // one.\n\n    function runModifierEffects() {\n      state.orderedModifiers.forEach(function (_ref) {\n        var name = _ref.name,\n            _ref$options = _ref.options,\n            options = _ref$options === void 0 ? {} : _ref$options,\n            effect = _ref.effect;\n\n        if (typeof effect === 'function') {\n          var cleanupFn = effect({\n            state: state,\n            name: name,\n            instance: instance,\n            options: options\n          });\n\n          var noopFn = function noopFn() {};\n\n          effectCleanupFns.push(cleanupFn || noopFn);\n        }\n      });\n    }\n\n    function cleanupModifierEffects() {\n      effectCleanupFns.forEach(function (fn) {\n        return fn();\n      });\n      effectCleanupFns = [];\n    }\n\n    return instance;\n  };\n}\nexport var createPopper = /*#__PURE__*/popperGenerator(); // eslint-disable-next-line import/no-unused-modules\n\nexport { detectOverflow };","export default function debounce(fn) {\n  var pending;\n  return function () {\n    if (!pending) {\n      pending = new Promise(function (resolve) {\n        Promise.resolve().then(function () {\n          pending = undefined;\n          resolve(fn());\n        });\n      });\n    }\n\n    return pending;\n  };\n}","export default function mergeByName(modifiers) {\n  var merged = modifiers.reduce(function (merged, current) {\n    var existing = merged[current.name];\n    merged[current.name] = existing ? Object.assign({}, existing, current, {\n      options: Object.assign({}, existing.options, current.options),\n      data: Object.assign({}, existing.data, current.data)\n    }) : current;\n    return merged;\n  }, {}); // IE11 does not support Object.values\n\n  return Object.keys(merged).map(function (key) {\n    return merged[key];\n  });\n}","import { popperGenerator, detectOverflow } from \"./createPopper.js\";\nimport eventListeners from \"./modifiers/eventListeners.js\";\nimport popperOffsets from \"./modifiers/popperOffsets.js\";\nimport computeStyles from \"./modifiers/computeStyles.js\";\nimport applyStyles from \"./modifiers/applyStyles.js\";\nimport offset from \"./modifiers/offset.js\";\nimport flip from \"./modifiers/flip.js\";\nimport preventOverflow from \"./modifiers/preventOverflow.js\";\nimport arrow from \"./modifiers/arrow.js\";\nimport hide from \"./modifiers/hide.js\";\nvar defaultModifiers = [eventListeners, popperOffsets, computeStyles, applyStyles, offset, flip, preventOverflow, arrow, hide];\nvar createPopper = /*#__PURE__*/popperGenerator({\n  defaultModifiers: defaultModifiers\n}); // eslint-disable-next-line import/no-unused-modules\n\nexport { createPopper, popperGenerator, defaultModifiers, detectOverflow }; // eslint-disable-next-line import/no-unused-modules\n\nexport { createPopper as createPopperLite } from \"./popper-lite.js\"; // eslint-disable-next-line import/no-unused-modules\n\nexport * from \"./modifiers/index.js\";","import { popperGenerator, detectOverflow } from \"./createPopper.js\";\nimport eventListeners from \"./modifiers/eventListeners.js\";\nimport popperOffsets from \"./modifiers/popperOffsets.js\";\nimport computeStyles from \"./modifiers/computeStyles.js\";\nimport applyStyles from \"./modifiers/applyStyles.js\";\nvar defaultModifiers = [eventListeners, popperOffsets, computeStyles, applyStyles];\nvar createPopper = /*#__PURE__*/popperGenerator({\n  defaultModifiers: defaultModifiers\n}); // eslint-disable-next-line import/no-unused-modules\n\nexport { createPopper, popperGenerator, defaultModifiers, detectOverflow };","/*!\n  * Bootstrap v5.3.3 (https://getbootstrap.com/)\n  * Copyright 2011-2024 The Bootstrap Authors (https://github.com/twbs/bootstrap/graphs/contributors)\n  * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n  */\nimport * as Popper from '@popperjs/core';\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dom/data.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n/**\n * Constants\n */\n\nconst elementMap = new Map();\nconst Data = {\n  set(element, key, instance) {\n    if (!elementMap.has(element)) {\n      elementMap.set(element, new Map());\n    }\n    const instanceMap = elementMap.get(element);\n\n    // make it clear we only want one instance per element\n    // can be removed later when multiple key/instances are fine to be used\n    if (!instanceMap.has(key) && instanceMap.size !== 0) {\n      // eslint-disable-next-line no-console\n      console.error(`Bootstrap doesn't allow more than one instance per element. Bound instance: ${Array.from(instanceMap.keys())[0]}.`);\n      return;\n    }\n    instanceMap.set(key, instance);\n  },\n  get(element, key) {\n    if (elementMap.has(element)) {\n      return elementMap.get(element).get(key) || null;\n    }\n    return null;\n  },\n  remove(element, key) {\n    if (!elementMap.has(element)) {\n      return;\n    }\n    const instanceMap = elementMap.get(element);\n    instanceMap.delete(key);\n\n    // free up element references if there are no instances left for an element\n    if (instanceMap.size === 0) {\n      elementMap.delete(element);\n    }\n  }\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/index.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nconst MAX_UID = 1000000;\nconst MILLISECONDS_MULTIPLIER = 1000;\nconst TRANSITION_END = 'transitionend';\n\n/**\n * Properly escape IDs selectors to handle weird IDs\n * @param {string} selector\n * @returns {string}\n */\nconst parseSelector = selector => {\n  if (selector && window.CSS && window.CSS.escape) {\n    // document.querySelector needs escaping to handle IDs (html5+) containing for instance /\n    selector = selector.replace(/#([^\\s\"#']+)/g, (match, id) => `#${CSS.escape(id)}`);\n  }\n  return selector;\n};\n\n// Shout-out Angus Croll (https://goo.gl/pxwQGp)\nconst toType = object => {\n  if (object === null || object === undefined) {\n    return `${object}`;\n  }\n  return Object.prototype.toString.call(object).match(/\\s([a-z]+)/i)[1].toLowerCase();\n};\n\n/**\n * Public Util API\n */\n\nconst getUID = prefix => {\n  do {\n    prefix += Math.floor(Math.random() * MAX_UID);\n  } while (document.getElementById(prefix));\n  return prefix;\n};\nconst getTransitionDurationFromElement = element => {\n  if (!element) {\n    return 0;\n  }\n\n  // Get transition-duration of the element\n  let {\n    transitionDuration,\n    transitionDelay\n  } = window.getComputedStyle(element);\n  const floatTransitionDuration = Number.parseFloat(transitionDuration);\n  const floatTransitionDelay = Number.parseFloat(transitionDelay);\n\n  // Return 0 if element or transition duration is not found\n  if (!floatTransitionDuration && !floatTransitionDelay) {\n    return 0;\n  }\n\n  // If multiple durations are defined, take the first\n  transitionDuration = transitionDuration.split(',')[0];\n  transitionDelay = transitionDelay.split(',')[0];\n  return (Number.parseFloat(transitionDuration) + Number.parseFloat(transitionDelay)) * MILLISECONDS_MULTIPLIER;\n};\nconst triggerTransitionEnd = element => {\n  element.dispatchEvent(new Event(TRANSITION_END));\n};\nconst isElement = object => {\n  if (!object || typeof object !== 'object') {\n    return false;\n  }\n  if (typeof object.jquery !== 'undefined') {\n    object = object[0];\n  }\n  return typeof object.nodeType !== 'undefined';\n};\nconst getElement = object => {\n  // it's a jQuery object or a node element\n  if (isElement(object)) {\n    return object.jquery ? object[0] : object;\n  }\n  if (typeof object === 'string' && object.length > 0) {\n    return document.querySelector(parseSelector(object));\n  }\n  return null;\n};\nconst isVisible = element => {\n  if (!isElement(element) || element.getClientRects().length === 0) {\n    return false;\n  }\n  const elementIsVisible = getComputedStyle(element).getPropertyValue('visibility') === 'visible';\n  // Handle `details` element as its content may falsie appear visible when it is closed\n  const closedDetails = element.closest('details:not([open])');\n  if (!closedDetails) {\n    return elementIsVisible;\n  }\n  if (closedDetails !== element) {\n    const summary = element.closest('summary');\n    if (summary && summary.parentNode !== closedDetails) {\n      return false;\n    }\n    if (summary === null) {\n      return false;\n    }\n  }\n  return elementIsVisible;\n};\nconst isDisabled = element => {\n  if (!element || element.nodeType !== Node.ELEMENT_NODE) {\n    return true;\n  }\n  if (element.classList.contains('disabled')) {\n    return true;\n  }\n  if (typeof element.disabled !== 'undefined') {\n    return element.disabled;\n  }\n  return element.hasAttribute('disabled') && element.getAttribute('disabled') !== 'false';\n};\nconst findShadowRoot = element => {\n  if (!document.documentElement.attachShadow) {\n    return null;\n  }\n\n  // Can find the shadow root otherwise it'll return the document\n  if (typeof element.getRootNode === 'function') {\n    const root = element.getRootNode();\n    return root instanceof ShadowRoot ? root : null;\n  }\n  if (element instanceof ShadowRoot) {\n    return element;\n  }\n\n  // when we don't find a shadow root\n  if (!element.parentNode) {\n    return null;\n  }\n  return findShadowRoot(element.parentNode);\n};\nconst noop = () => {};\n\n/**\n * Trick to restart an element's animation\n *\n * @param {HTMLElement} element\n * @return void\n *\n * @see https://www.charistheo.io/blog/2021/02/restart-a-css-animation-with-javascript/#restarting-a-css-animation\n */\nconst reflow = element => {\n  element.offsetHeight; // eslint-disable-line no-unused-expressions\n};\nconst getjQuery = () => {\n  if (window.jQuery && !document.body.hasAttribute('data-bs-no-jquery')) {\n    return window.jQuery;\n  }\n  return null;\n};\nconst DOMContentLoadedCallbacks = [];\nconst onDOMContentLoaded = callback => {\n  if (document.readyState === 'loading') {\n    // add listener on the first call when the document is in loading state\n    if (!DOMContentLoadedCallbacks.length) {\n      document.addEventListener('DOMContentLoaded', () => {\n        for (const callback of DOMContentLoadedCallbacks) {\n          callback();\n        }\n      });\n    }\n    DOMContentLoadedCallbacks.push(callback);\n  } else {\n    callback();\n  }\n};\nconst isRTL = () => document.documentElement.dir === 'rtl';\nconst defineJQueryPlugin = plugin => {\n  onDOMContentLoaded(() => {\n    const $ = getjQuery();\n    /* istanbul ignore if */\n    if ($) {\n      const name = plugin.NAME;\n      const JQUERY_NO_CONFLICT = $.fn[name];\n      $.fn[name] = plugin.jQueryInterface;\n      $.fn[name].Constructor = plugin;\n      $.fn[name].noConflict = () => {\n        $.fn[name] = JQUERY_NO_CONFLICT;\n        return plugin.jQueryInterface;\n      };\n    }\n  });\n};\nconst execute = (possibleCallback, args = [], defaultValue = possibleCallback) => {\n  return typeof possibleCallback === 'function' ? possibleCallback(...args) : defaultValue;\n};\nconst executeAfterTransition = (callback, transitionElement, waitForTransition = true) => {\n  if (!waitForTransition) {\n    execute(callback);\n    return;\n  }\n  const durationPadding = 5;\n  const emulatedDuration = getTransitionDurationFromElement(transitionElement) + durationPadding;\n  let called = false;\n  const handler = ({\n    target\n  }) => {\n    if (target !== transitionElement) {\n      return;\n    }\n    called = true;\n    transitionElement.removeEventListener(TRANSITION_END, handler);\n    execute(callback);\n  };\n  transitionElement.addEventListener(TRANSITION_END, handler);\n  setTimeout(() => {\n    if (!called) {\n      triggerTransitionEnd(transitionElement);\n    }\n  }, emulatedDuration);\n};\n\n/**\n * Return the previous/next element of a list.\n *\n * @param {array} list    The list of elements\n * @param activeElement   The active element\n * @param shouldGetNext   Choose to get next or previous element\n * @param isCycleAllowed\n * @return {Element|elem} The proper element\n */\nconst getNextActiveElement = (list, activeElement, shouldGetNext, isCycleAllowed) => {\n  const listLength = list.length;\n  let index = list.indexOf(activeElement);\n\n  // if the element does not exist in the list return an element\n  // depending on the direction and if cycle is allowed\n  if (index === -1) {\n    return !shouldGetNext && isCycleAllowed ? list[listLength - 1] : list[0];\n  }\n  index += shouldGetNext ? 1 : -1;\n  if (isCycleAllowed) {\n    index = (index + listLength) % listLength;\n  }\n  return list[Math.max(0, Math.min(index, listLength - 1))];\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dom/event-handler.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst namespaceRegex = /[^.]*(?=\\..*)\\.|.*/;\nconst stripNameRegex = /\\..*/;\nconst stripUidRegex = /::\\d+$/;\nconst eventRegistry = {}; // Events storage\nlet uidEvent = 1;\nconst customEvents = {\n  mouseenter: 'mouseover',\n  mouseleave: 'mouseout'\n};\nconst nativeEvents = new Set(['click', 'dblclick', 'mouseup', 'mousedown', 'contextmenu', 'mousewheel', 'DOMMouseScroll', 'mouseover', 'mouseout', 'mousemove', 'selectstart', 'selectend', 'keydown', 'keypress', 'keyup', 'orientationchange', 'touchstart', 'touchmove', 'touchend', 'touchcancel', 'pointerdown', 'pointermove', 'pointerup', 'pointerleave', 'pointercancel', 'gesturestart', 'gesturechange', 'gestureend', 'focus', 'blur', 'change', 'reset', 'select', 'submit', 'focusin', 'focusout', 'load', 'unload', 'beforeunload', 'resize', 'move', 'DOMContentLoaded', 'readystatechange', 'error', 'abort', 'scroll']);\n\n/**\n * Private methods\n */\n\nfunction makeEventUid(element, uid) {\n  return uid && `${uid}::${uidEvent++}` || element.uidEvent || uidEvent++;\n}\nfunction getElementEvents(element) {\n  const uid = makeEventUid(element);\n  element.uidEvent = uid;\n  eventRegistry[uid] = eventRegistry[uid] || {};\n  return eventRegistry[uid];\n}\nfunction bootstrapHandler(element, fn) {\n  return function handler(event) {\n    hydrateObj(event, {\n      delegateTarget: element\n    });\n    if (handler.oneOff) {\n      EventHandler.off(element, event.type, fn);\n    }\n    return fn.apply(element, [event]);\n  };\n}\nfunction bootstrapDelegationHandler(element, selector, fn) {\n  return function handler(event) {\n    const domElements = element.querySelectorAll(selector);\n    for (let {\n      target\n    } = event; target && target !== this; target = target.parentNode) {\n      for (const domElement of domElements) {\n        if (domElement !== target) {\n          continue;\n        }\n        hydrateObj(event, {\n          delegateTarget: target\n        });\n        if (handler.oneOff) {\n          EventHandler.off(element, event.type, selector, fn);\n        }\n        return fn.apply(target, [event]);\n      }\n    }\n  };\n}\nfunction findHandler(events, callable, delegationSelector = null) {\n  return Object.values(events).find(event => event.callable === callable && event.delegationSelector === delegationSelector);\n}\nfunction normalizeParameters(originalTypeEvent, handler, delegationFunction) {\n  const isDelegated = typeof handler === 'string';\n  // TODO: tooltip passes `false` instead of selector, so we need to check\n  const callable = isDelegated ? delegationFunction : handler || delegationFunction;\n  let typeEvent = getTypeEvent(originalTypeEvent);\n  if (!nativeEvents.has(typeEvent)) {\n    typeEvent = originalTypeEvent;\n  }\n  return [isDelegated, callable, typeEvent];\n}\nfunction addHandler(element, originalTypeEvent, handler, delegationFunction, oneOff) {\n  if (typeof originalTypeEvent !== 'string' || !element) {\n    return;\n  }\n  let [isDelegated, callable, typeEvent] = normalizeParameters(originalTypeEvent, handler, delegationFunction);\n\n  // in case of mouseenter or mouseleave wrap the handler within a function that checks for its DOM position\n  // this prevents the handler from being dispatched the same way as mouseover or mouseout does\n  if (originalTypeEvent in customEvents) {\n    const wrapFunction = fn => {\n      return function (event) {\n        if (!event.relatedTarget || event.relatedTarget !== event.delegateTarget && !event.delegateTarget.contains(event.relatedTarget)) {\n          return fn.call(this, event);\n        }\n      };\n    };\n    callable = wrapFunction(callable);\n  }\n  const events = getElementEvents(element);\n  const handlers = events[typeEvent] || (events[typeEvent] = {});\n  const previousFunction = findHandler(handlers, callable, isDelegated ? handler : null);\n  if (previousFunction) {\n    previousFunction.oneOff = previousFunction.oneOff && oneOff;\n    return;\n  }\n  const uid = makeEventUid(callable, originalTypeEvent.replace(namespaceRegex, ''));\n  const fn = isDelegated ? bootstrapDelegationHandler(element, handler, callable) : bootstrapHandler(element, callable);\n  fn.delegationSelector = isDelegated ? handler : null;\n  fn.callable = callable;\n  fn.oneOff = oneOff;\n  fn.uidEvent = uid;\n  handlers[uid] = fn;\n  element.addEventListener(typeEvent, fn, isDelegated);\n}\nfunction removeHandler(element, events, typeEvent, handler, delegationSelector) {\n  const fn = findHandler(events[typeEvent], handler, delegationSelector);\n  if (!fn) {\n    return;\n  }\n  element.removeEventListener(typeEvent, fn, Boolean(delegationSelector));\n  delete events[typeEvent][fn.uidEvent];\n}\nfunction removeNamespacedHandlers(element, events, typeEvent, namespace) {\n  const storeElementEvent = events[typeEvent] || {};\n  for (const [handlerKey, event] of Object.entries(storeElementEvent)) {\n    if (handlerKey.includes(namespace)) {\n      removeHandler(element, events, typeEvent, event.callable, event.delegationSelector);\n    }\n  }\n}\nfunction getTypeEvent(event) {\n  // allow to get the native events from namespaced events ('click.bs.button' --> 'click')\n  event = event.replace(stripNameRegex, '');\n  return customEvents[event] || event;\n}\nconst EventHandler = {\n  on(element, event, handler, delegationFunction) {\n    addHandler(element, event, handler, delegationFunction, false);\n  },\n  one(element, event, handler, delegationFunction) {\n    addHandler(element, event, handler, delegationFunction, true);\n  },\n  off(element, originalTypeEvent, handler, delegationFunction) {\n    if (typeof originalTypeEvent !== 'string' || !element) {\n      return;\n    }\n    const [isDelegated, callable, typeEvent] = normalizeParameters(originalTypeEvent, handler, delegationFunction);\n    const inNamespace = typeEvent !== originalTypeEvent;\n    const events = getElementEvents(element);\n    const storeElementEvent = events[typeEvent] || {};\n    const isNamespace = originalTypeEvent.startsWith('.');\n    if (typeof callable !== 'undefined') {\n      // Simplest case: handler is passed, remove that listener ONLY.\n      if (!Object.keys(storeElementEvent).length) {\n        return;\n      }\n      removeHandler(element, events, typeEvent, callable, isDelegated ? handler : null);\n      return;\n    }\n    if (isNamespace) {\n      for (const elementEvent of Object.keys(events)) {\n        removeNamespacedHandlers(element, events, elementEvent, originalTypeEvent.slice(1));\n      }\n    }\n    for (const [keyHandlers, event] of Object.entries(storeElementEvent)) {\n      const handlerKey = keyHandlers.replace(stripUidRegex, '');\n      if (!inNamespace || originalTypeEvent.includes(handlerKey)) {\n        removeHandler(element, events, typeEvent, event.callable, event.delegationSelector);\n      }\n    }\n  },\n  trigger(element, event, args) {\n    if (typeof event !== 'string' || !element) {\n      return null;\n    }\n    const $ = getjQuery();\n    const typeEvent = getTypeEvent(event);\n    const inNamespace = event !== typeEvent;\n    let jQueryEvent = null;\n    let bubbles = true;\n    let nativeDispatch = true;\n    let defaultPrevented = false;\n    if (inNamespace && $) {\n      jQueryEvent = $.Event(event, args);\n      $(element).trigger(jQueryEvent);\n      bubbles = !jQueryEvent.isPropagationStopped();\n      nativeDispatch = !jQueryEvent.isImmediatePropagationStopped();\n      defaultPrevented = jQueryEvent.isDefaultPrevented();\n    }\n    const evt = hydrateObj(new Event(event, {\n      bubbles,\n      cancelable: true\n    }), args);\n    if (defaultPrevented) {\n      evt.preventDefault();\n    }\n    if (nativeDispatch) {\n      element.dispatchEvent(evt);\n    }\n    if (evt.defaultPrevented && jQueryEvent) {\n      jQueryEvent.preventDefault();\n    }\n    return evt;\n  }\n};\nfunction hydrateObj(obj, meta = {}) {\n  for (const [key, value] of Object.entries(meta)) {\n    try {\n      obj[key] = value;\n    } catch (_unused) {\n      Object.defineProperty(obj, key, {\n        configurable: true,\n        get() {\n          return value;\n        }\n      });\n    }\n  }\n  return obj;\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dom/manipulator.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nfunction normalizeData(value) {\n  if (value === 'true') {\n    return true;\n  }\n  if (value === 'false') {\n    return false;\n  }\n  if (value === Number(value).toString()) {\n    return Number(value);\n  }\n  if (value === '' || value === 'null') {\n    return null;\n  }\n  if (typeof value !== 'string') {\n    return value;\n  }\n  try {\n    return JSON.parse(decodeURIComponent(value));\n  } catch (_unused) {\n    return value;\n  }\n}\nfunction normalizeDataKey(key) {\n  return key.replace(/[A-Z]/g, chr => `-${chr.toLowerCase()}`);\n}\nconst Manipulator = {\n  setDataAttribute(element, key, value) {\n    element.setAttribute(`data-bs-${normalizeDataKey(key)}`, value);\n  },\n  removeDataAttribute(element, key) {\n    element.removeAttribute(`data-bs-${normalizeDataKey(key)}`);\n  },\n  getDataAttributes(element) {\n    if (!element) {\n      return {};\n    }\n    const attributes = {};\n    const bsKeys = Object.keys(element.dataset).filter(key => key.startsWith('bs') && !key.startsWith('bsConfig'));\n    for (const key of bsKeys) {\n      let pureKey = key.replace(/^bs/, '');\n      pureKey = pureKey.charAt(0).toLowerCase() + pureKey.slice(1, pureKey.length);\n      attributes[pureKey] = normalizeData(element.dataset[key]);\n    }\n    return attributes;\n  },\n  getDataAttribute(element, key) {\n    return normalizeData(element.getAttribute(`data-bs-${normalizeDataKey(key)}`));\n  }\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/config.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Class definition\n */\n\nclass Config {\n  // Getters\n  static get Default() {\n    return {};\n  }\n  static get DefaultType() {\n    return {};\n  }\n  static get NAME() {\n    throw new Error('You have to implement the static method \"NAME\", for each component!');\n  }\n  _getConfig(config) {\n    config = this._mergeConfigObj(config);\n    config = this._configAfterMerge(config);\n    this._typeCheckConfig(config);\n    return config;\n  }\n  _configAfterMerge(config) {\n    return config;\n  }\n  _mergeConfigObj(config, element) {\n    const jsonConfig = isElement(element) ? Manipulator.getDataAttribute(element, 'config') : {}; // try to parse\n\n    return {\n      ...this.constructor.Default,\n      ...(typeof jsonConfig === 'object' ? jsonConfig : {}),\n      ...(isElement(element) ? Manipulator.getDataAttributes(element) : {}),\n      ...(typeof config === 'object' ? config : {})\n    };\n  }\n  _typeCheckConfig(config, configTypes = this.constructor.DefaultType) {\n    for (const [property, expectedTypes] of Object.entries(configTypes)) {\n      const value = config[property];\n      const valueType = isElement(value) ? 'element' : toType(value);\n      if (!new RegExp(expectedTypes).test(valueType)) {\n        throw new TypeError(`${this.constructor.NAME.toUpperCase()}: Option \"${property}\" provided type \"${valueType}\" but expected type \"${expectedTypes}\".`);\n      }\n    }\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap base-component.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst VERSION = '5.3.3';\n\n/**\n * Class definition\n */\n\nclass BaseComponent extends Config {\n  constructor(element, config) {\n    super();\n    element = getElement(element);\n    if (!element) {\n      return;\n    }\n    this._element = element;\n    this._config = this._getConfig(config);\n    Data.set(this._element, this.constructor.DATA_KEY, this);\n  }\n\n  // Public\n  dispose() {\n    Data.remove(this._element, this.constructor.DATA_KEY);\n    EventHandler.off(this._element, this.constructor.EVENT_KEY);\n    for (const propertyName of Object.getOwnPropertyNames(this)) {\n      this[propertyName] = null;\n    }\n  }\n  _queueCallback(callback, element, isAnimated = true) {\n    executeAfterTransition(callback, element, isAnimated);\n  }\n  _getConfig(config) {\n    config = this._mergeConfigObj(config, this._element);\n    config = this._configAfterMerge(config);\n    this._typeCheckConfig(config);\n    return config;\n  }\n\n  // Static\n  static getInstance(element) {\n    return Data.get(getElement(element), this.DATA_KEY);\n  }\n  static getOrCreateInstance(element, config = {}) {\n    return this.getInstance(element) || new this(element, typeof config === 'object' ? config : null);\n  }\n  static get VERSION() {\n    return VERSION;\n  }\n  static get DATA_KEY() {\n    return `bs.${this.NAME}`;\n  }\n  static get EVENT_KEY() {\n    return `.${this.DATA_KEY}`;\n  }\n  static eventName(name) {\n    return `${name}${this.EVENT_KEY}`;\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dom/selector-engine.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nconst getSelector = element => {\n  let selector = element.getAttribute('data-bs-target');\n  if (!selector || selector === '#') {\n    let hrefAttribute = element.getAttribute('href');\n\n    // The only valid content that could double as a selector are IDs or classes,\n    // so everything starting with `#` or `.`. If a \"real\" URL is used as the selector,\n    // `document.querySelector` will rightfully complain it is invalid.\n    // See https://github.com/twbs/bootstrap/issues/32273\n    if (!hrefAttribute || !hrefAttribute.includes('#') && !hrefAttribute.startsWith('.')) {\n      return null;\n    }\n\n    // Just in case some CMS puts out a full URL with the anchor appended\n    if (hrefAttribute.includes('#') && !hrefAttribute.startsWith('#')) {\n      hrefAttribute = `#${hrefAttribute.split('#')[1]}`;\n    }\n    selector = hrefAttribute && hrefAttribute !== '#' ? hrefAttribute.trim() : null;\n  }\n  return selector ? selector.split(',').map(sel => parseSelector(sel)).join(',') : null;\n};\nconst SelectorEngine = {\n  find(selector, element = document.documentElement) {\n    return [].concat(...Element.prototype.querySelectorAll.call(element, selector));\n  },\n  findOne(selector, element = document.documentElement) {\n    return Element.prototype.querySelector.call(element, selector);\n  },\n  children(element, selector) {\n    return [].concat(...element.children).filter(child => child.matches(selector));\n  },\n  parents(element, selector) {\n    const parents = [];\n    let ancestor = element.parentNode.closest(selector);\n    while (ancestor) {\n      parents.push(ancestor);\n      ancestor = ancestor.parentNode.closest(selector);\n    }\n    return parents;\n  },\n  prev(element, selector) {\n    let previous = element.previousElementSibling;\n    while (previous) {\n      if (previous.matches(selector)) {\n        return [previous];\n      }\n      previous = previous.previousElementSibling;\n    }\n    return [];\n  },\n  // TODO: this is now unused; remove later along with prev()\n  next(element, selector) {\n    let next = element.nextElementSibling;\n    while (next) {\n      if (next.matches(selector)) {\n        return [next];\n      }\n      next = next.nextElementSibling;\n    }\n    return [];\n  },\n  focusableChildren(element) {\n    const focusables = ['a', 'button', 'input', 'textarea', 'select', 'details', '[tabindex]', '[contenteditable=\"true\"]'].map(selector => `${selector}:not([tabindex^=\"-\"])`).join(',');\n    return this.find(focusables, element).filter(el => !isDisabled(el) && isVisible(el));\n  },\n  getSelectorFromElement(element) {\n    const selector = getSelector(element);\n    if (selector) {\n      return SelectorEngine.findOne(selector) ? selector : null;\n    }\n    return null;\n  },\n  getElementFromSelector(element) {\n    const selector = getSelector(element);\n    return selector ? SelectorEngine.findOne(selector) : null;\n  },\n  getMultipleElementsFromSelector(element) {\n    const selector = getSelector(element);\n    return selector ? SelectorEngine.find(selector) : [];\n  }\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/component-functions.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nconst enableDismissTrigger = (component, method = 'hide') => {\n  const clickEvent = `click.dismiss${component.EVENT_KEY}`;\n  const name = component.NAME;\n  EventHandler.on(document, clickEvent, `[data-bs-dismiss=\"${name}\"]`, function (event) {\n    if (['A', 'AREA'].includes(this.tagName)) {\n      event.preventDefault();\n    }\n    if (isDisabled(this)) {\n      return;\n    }\n    const target = SelectorEngine.getElementFromSelector(this) || this.closest(`.${name}`);\n    const instance = component.getOrCreateInstance(target);\n\n    // Method argument is left, for Alert and only, as it doesn't implement the 'hide' method\n    instance[method]();\n  });\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap alert.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$f = 'alert';\nconst DATA_KEY$a = 'bs.alert';\nconst EVENT_KEY$b = `.${DATA_KEY$a}`;\nconst EVENT_CLOSE = `close${EVENT_KEY$b}`;\nconst EVENT_CLOSED = `closed${EVENT_KEY$b}`;\nconst CLASS_NAME_FADE$5 = 'fade';\nconst CLASS_NAME_SHOW$8 = 'show';\n\n/**\n * Class definition\n */\n\nclass Alert extends BaseComponent {\n  // Getters\n  static get NAME() {\n    return NAME$f;\n  }\n\n  // Public\n  close() {\n    const closeEvent = EventHandler.trigger(this._element, EVENT_CLOSE);\n    if (closeEvent.defaultPrevented) {\n      return;\n    }\n    this._element.classList.remove(CLASS_NAME_SHOW$8);\n    const isAnimated = this._element.classList.contains(CLASS_NAME_FADE$5);\n    this._queueCallback(() => this._destroyElement(), this._element, isAnimated);\n  }\n\n  // Private\n  _destroyElement() {\n    this._element.remove();\n    EventHandler.trigger(this._element, EVENT_CLOSED);\n    this.dispose();\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Alert.getOrCreateInstance(this);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config](this);\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nenableDismissTrigger(Alert, 'close');\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Alert);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap button.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$e = 'button';\nconst DATA_KEY$9 = 'bs.button';\nconst EVENT_KEY$a = `.${DATA_KEY$9}`;\nconst DATA_API_KEY$6 = '.data-api';\nconst CLASS_NAME_ACTIVE$3 = 'active';\nconst SELECTOR_DATA_TOGGLE$5 = '[data-bs-toggle=\"button\"]';\nconst EVENT_CLICK_DATA_API$6 = `click${EVENT_KEY$a}${DATA_API_KEY$6}`;\n\n/**\n * Class definition\n */\n\nclass Button extends BaseComponent {\n  // Getters\n  static get NAME() {\n    return NAME$e;\n  }\n\n  // Public\n  toggle() {\n    // Toggle class and sync the `aria-pressed` attribute with the return value of the `.toggle()` method\n    this._element.setAttribute('aria-pressed', this._element.classList.toggle(CLASS_NAME_ACTIVE$3));\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Button.getOrCreateInstance(this);\n      if (config === 'toggle') {\n        data[config]();\n      }\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$6, SELECTOR_DATA_TOGGLE$5, event => {\n  event.preventDefault();\n  const button = event.target.closest(SELECTOR_DATA_TOGGLE$5);\n  const data = Button.getOrCreateInstance(button);\n  data.toggle();\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Button);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/swipe.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$d = 'swipe';\nconst EVENT_KEY$9 = '.bs.swipe';\nconst EVENT_TOUCHSTART = `touchstart${EVENT_KEY$9}`;\nconst EVENT_TOUCHMOVE = `touchmove${EVENT_KEY$9}`;\nconst EVENT_TOUCHEND = `touchend${EVENT_KEY$9}`;\nconst EVENT_POINTERDOWN = `pointerdown${EVENT_KEY$9}`;\nconst EVENT_POINTERUP = `pointerup${EVENT_KEY$9}`;\nconst POINTER_TYPE_TOUCH = 'touch';\nconst POINTER_TYPE_PEN = 'pen';\nconst CLASS_NAME_POINTER_EVENT = 'pointer-event';\nconst SWIPE_THRESHOLD = 40;\nconst Default$c = {\n  endCallback: null,\n  leftCallback: null,\n  rightCallback: null\n};\nconst DefaultType$c = {\n  endCallback: '(function|null)',\n  leftCallback: '(function|null)',\n  rightCallback: '(function|null)'\n};\n\n/**\n * Class definition\n */\n\nclass Swipe extends Config {\n  constructor(element, config) {\n    super();\n    this._element = element;\n    if (!element || !Swipe.isSupported()) {\n      return;\n    }\n    this._config = this._getConfig(config);\n    this._deltaX = 0;\n    this._supportPointerEvents = Boolean(window.PointerEvent);\n    this._initEvents();\n  }\n\n  // Getters\n  static get Default() {\n    return Default$c;\n  }\n  static get DefaultType() {\n    return DefaultType$c;\n  }\n  static get NAME() {\n    return NAME$d;\n  }\n\n  // Public\n  dispose() {\n    EventHandler.off(this._element, EVENT_KEY$9);\n  }\n\n  // Private\n  _start(event) {\n    if (!this._supportPointerEvents) {\n      this._deltaX = event.touches[0].clientX;\n      return;\n    }\n    if (this._eventIsPointerPenTouch(event)) {\n      this._deltaX = event.clientX;\n    }\n  }\n  _end(event) {\n    if (this._eventIsPointerPenTouch(event)) {\n      this._deltaX = event.clientX - this._deltaX;\n    }\n    this._handleSwipe();\n    execute(this._config.endCallback);\n  }\n  _move(event) {\n    this._deltaX = event.touches && event.touches.length > 1 ? 0 : event.touches[0].clientX - this._deltaX;\n  }\n  _handleSwipe() {\n    const absDeltaX = Math.abs(this._deltaX);\n    if (absDeltaX <= SWIPE_THRESHOLD) {\n      return;\n    }\n    const direction = absDeltaX / this._deltaX;\n    this._deltaX = 0;\n    if (!direction) {\n      return;\n    }\n    execute(direction > 0 ? this._config.rightCallback : this._config.leftCallback);\n  }\n  _initEvents() {\n    if (this._supportPointerEvents) {\n      EventHandler.on(this._element, EVENT_POINTERDOWN, event => this._start(event));\n      EventHandler.on(this._element, EVENT_POINTERUP, event => this._end(event));\n      this._element.classList.add(CLASS_NAME_POINTER_EVENT);\n    } else {\n      EventHandler.on(this._element, EVENT_TOUCHSTART, event => this._start(event));\n      EventHandler.on(this._element, EVENT_TOUCHMOVE, event => this._move(event));\n      EventHandler.on(this._element, EVENT_TOUCHEND, event => this._end(event));\n    }\n  }\n  _eventIsPointerPenTouch(event) {\n    return this._supportPointerEvents && (event.pointerType === POINTER_TYPE_PEN || event.pointerType === POINTER_TYPE_TOUCH);\n  }\n\n  // Static\n  static isSupported() {\n    return 'ontouchstart' in document.documentElement || navigator.maxTouchPoints > 0;\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap carousel.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$c = 'carousel';\nconst DATA_KEY$8 = 'bs.carousel';\nconst EVENT_KEY$8 = `.${DATA_KEY$8}`;\nconst DATA_API_KEY$5 = '.data-api';\nconst ARROW_LEFT_KEY$1 = 'ArrowLeft';\nconst ARROW_RIGHT_KEY$1 = 'ArrowRight';\nconst TOUCHEVENT_COMPAT_WAIT = 500; // Time for mouse compat events to fire after touch\n\nconst ORDER_NEXT = 'next';\nconst ORDER_PREV = 'prev';\nconst DIRECTION_LEFT = 'left';\nconst DIRECTION_RIGHT = 'right';\nconst EVENT_SLIDE = `slide${EVENT_KEY$8}`;\nconst EVENT_SLID = `slid${EVENT_KEY$8}`;\nconst EVENT_KEYDOWN$1 = `keydown${EVENT_KEY$8}`;\nconst EVENT_MOUSEENTER$1 = `mouseenter${EVENT_KEY$8}`;\nconst EVENT_MOUSELEAVE$1 = `mouseleave${EVENT_KEY$8}`;\nconst EVENT_DRAG_START = `dragstart${EVENT_KEY$8}`;\nconst EVENT_LOAD_DATA_API$3 = `load${EVENT_KEY$8}${DATA_API_KEY$5}`;\nconst EVENT_CLICK_DATA_API$5 = `click${EVENT_KEY$8}${DATA_API_KEY$5}`;\nconst CLASS_NAME_CAROUSEL = 'carousel';\nconst CLASS_NAME_ACTIVE$2 = 'active';\nconst CLASS_NAME_SLIDE = 'slide';\nconst CLASS_NAME_END = 'carousel-item-end';\nconst CLASS_NAME_START = 'carousel-item-start';\nconst CLASS_NAME_NEXT = 'carousel-item-next';\nconst CLASS_NAME_PREV = 'carousel-item-prev';\nconst SELECTOR_ACTIVE = '.active';\nconst SELECTOR_ITEM = '.carousel-item';\nconst SELECTOR_ACTIVE_ITEM = SELECTOR_ACTIVE + SELECTOR_ITEM;\nconst SELECTOR_ITEM_IMG = '.carousel-item img';\nconst SELECTOR_INDICATORS = '.carousel-indicators';\nconst SELECTOR_DATA_SLIDE = '[data-bs-slide], [data-bs-slide-to]';\nconst SELECTOR_DATA_RIDE = '[data-bs-ride=\"carousel\"]';\nconst KEY_TO_DIRECTION = {\n  [ARROW_LEFT_KEY$1]: DIRECTION_RIGHT,\n  [ARROW_RIGHT_KEY$1]: DIRECTION_LEFT\n};\nconst Default$b = {\n  interval: 5000,\n  keyboard: true,\n  pause: 'hover',\n  ride: false,\n  touch: true,\n  wrap: true\n};\nconst DefaultType$b = {\n  interval: '(number|boolean)',\n  // TODO:v6 remove boolean support\n  keyboard: 'boolean',\n  pause: '(string|boolean)',\n  ride: '(boolean|string)',\n  touch: 'boolean',\n  wrap: 'boolean'\n};\n\n/**\n * Class definition\n */\n\nclass Carousel extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._interval = null;\n    this._activeElement = null;\n    this._isSliding = false;\n    this.touchTimeout = null;\n    this._swipeHelper = null;\n    this._indicatorsElement = SelectorEngine.findOne(SELECTOR_INDICATORS, this._element);\n    this._addEventListeners();\n    if (this._config.ride === CLASS_NAME_CAROUSEL) {\n      this.cycle();\n    }\n  }\n\n  // Getters\n  static get Default() {\n    return Default$b;\n  }\n  static get DefaultType() {\n    return DefaultType$b;\n  }\n  static get NAME() {\n    return NAME$c;\n  }\n\n  // Public\n  next() {\n    this._slide(ORDER_NEXT);\n  }\n  nextWhenVisible() {\n    // FIXME TODO use `document.visibilityState`\n    // Don't call next when the page isn't visible\n    // or the carousel or its parent isn't visible\n    if (!document.hidden && isVisible(this._element)) {\n      this.next();\n    }\n  }\n  prev() {\n    this._slide(ORDER_PREV);\n  }\n  pause() {\n    if (this._isSliding) {\n      triggerTransitionEnd(this._element);\n    }\n    this._clearInterval();\n  }\n  cycle() {\n    this._clearInterval();\n    this._updateInterval();\n    this._interval = setInterval(() => this.nextWhenVisible(), this._config.interval);\n  }\n  _maybeEnableCycle() {\n    if (!this._config.ride) {\n      return;\n    }\n    if (this._isSliding) {\n      EventHandler.one(this._element, EVENT_SLID, () => this.cycle());\n      return;\n    }\n    this.cycle();\n  }\n  to(index) {\n    const items = this._getItems();\n    if (index > items.length - 1 || index < 0) {\n      return;\n    }\n    if (this._isSliding) {\n      EventHandler.one(this._element, EVENT_SLID, () => this.to(index));\n      return;\n    }\n    const activeIndex = this._getItemIndex(this._getActive());\n    if (activeIndex === index) {\n      return;\n    }\n    const order = index > activeIndex ? ORDER_NEXT : ORDER_PREV;\n    this._slide(order, items[index]);\n  }\n  dispose() {\n    if (this._swipeHelper) {\n      this._swipeHelper.dispose();\n    }\n    super.dispose();\n  }\n\n  // Private\n  _configAfterMerge(config) {\n    config.defaultInterval = config.interval;\n    return config;\n  }\n  _addEventListeners() {\n    if (this._config.keyboard) {\n      EventHandler.on(this._element, EVENT_KEYDOWN$1, event => this._keydown(event));\n    }\n    if (this._config.pause === 'hover') {\n      EventHandler.on(this._element, EVENT_MOUSEENTER$1, () => this.pause());\n      EventHandler.on(this._element, EVENT_MOUSELEAVE$1, () => this._maybeEnableCycle());\n    }\n    if (this._config.touch && Swipe.isSupported()) {\n      this._addTouchEventListeners();\n    }\n  }\n  _addTouchEventListeners() {\n    for (const img of SelectorEngine.find(SELECTOR_ITEM_IMG, this._element)) {\n      EventHandler.on(img, EVENT_DRAG_START, event => event.preventDefault());\n    }\n    const endCallBack = () => {\n      if (this._config.pause !== 'hover') {\n        return;\n      }\n\n      // If it's a touch-enabled device, mouseenter/leave are fired as\n      // part of the mouse compatibility events on first tap - the carousel\n      // would stop cycling until user tapped out of it;\n      // here, we listen for touchend, explicitly pause the carousel\n      // (as if it's the second time we tap on it, mouseenter compat event\n      // is NOT fired) and after a timeout (to allow for mouse compatibility\n      // events to fire) we explicitly restart cycling\n\n      this.pause();\n      if (this.touchTimeout) {\n        clearTimeout(this.touchTimeout);\n      }\n      this.touchTimeout = setTimeout(() => this._maybeEnableCycle(), TOUCHEVENT_COMPAT_WAIT + this._config.interval);\n    };\n    const swipeConfig = {\n      leftCallback: () => this._slide(this._directionToOrder(DIRECTION_LEFT)),\n      rightCallback: () => this._slide(this._directionToOrder(DIRECTION_RIGHT)),\n      endCallback: endCallBack\n    };\n    this._swipeHelper = new Swipe(this._element, swipeConfig);\n  }\n  _keydown(event) {\n    if (/input|textarea/i.test(event.target.tagName)) {\n      return;\n    }\n    const direction = KEY_TO_DIRECTION[event.key];\n    if (direction) {\n      event.preventDefault();\n      this._slide(this._directionToOrder(direction));\n    }\n  }\n  _getItemIndex(element) {\n    return this._getItems().indexOf(element);\n  }\n  _setActiveIndicatorElement(index) {\n    if (!this._indicatorsElement) {\n      return;\n    }\n    const activeIndicator = SelectorEngine.findOne(SELECTOR_ACTIVE, this._indicatorsElement);\n    activeIndicator.classList.remove(CLASS_NAME_ACTIVE$2);\n    activeIndicator.removeAttribute('aria-current');\n    const newActiveIndicator = SelectorEngine.findOne(`[data-bs-slide-to=\"${index}\"]`, this._indicatorsElement);\n    if (newActiveIndicator) {\n      newActiveIndicator.classList.add(CLASS_NAME_ACTIVE$2);\n      newActiveIndicator.setAttribute('aria-current', 'true');\n    }\n  }\n  _updateInterval() {\n    const element = this._activeElement || this._getActive();\n    if (!element) {\n      return;\n    }\n    const elementInterval = Number.parseInt(element.getAttribute('data-bs-interval'), 10);\n    this._config.interval = elementInterval || this._config.defaultInterval;\n  }\n  _slide(order, element = null) {\n    if (this._isSliding) {\n      return;\n    }\n    const activeElement = this._getActive();\n    const isNext = order === ORDER_NEXT;\n    const nextElement = element || getNextActiveElement(this._getItems(), activeElement, isNext, this._config.wrap);\n    if (nextElement === activeElement) {\n      return;\n    }\n    const nextElementIndex = this._getItemIndex(nextElement);\n    const triggerEvent = eventName => {\n      return EventHandler.trigger(this._element, eventName, {\n        relatedTarget: nextElement,\n        direction: this._orderToDirection(order),\n        from: this._getItemIndex(activeElement),\n        to: nextElementIndex\n      });\n    };\n    const slideEvent = triggerEvent(EVENT_SLIDE);\n    if (slideEvent.defaultPrevented) {\n      return;\n    }\n    if (!activeElement || !nextElement) {\n      // Some weirdness is happening, so we bail\n      // TODO: change tests that use empty divs to avoid this check\n      return;\n    }\n    const isCycling = Boolean(this._interval);\n    this.pause();\n    this._isSliding = true;\n    this._setActiveIndicatorElement(nextElementIndex);\n    this._activeElement = nextElement;\n    const directionalClassName = isNext ? CLASS_NAME_START : CLASS_NAME_END;\n    const orderClassName = isNext ? CLASS_NAME_NEXT : CLASS_NAME_PREV;\n    nextElement.classList.add(orderClassName);\n    reflow(nextElement);\n    activeElement.classList.add(directionalClassName);\n    nextElement.classList.add(directionalClassName);\n    const completeCallBack = () => {\n      nextElement.classList.remove(directionalClassName, orderClassName);\n      nextElement.classList.add(CLASS_NAME_ACTIVE$2);\n      activeElement.classList.remove(CLASS_NAME_ACTIVE$2, orderClassName, directionalClassName);\n      this._isSliding = false;\n      triggerEvent(EVENT_SLID);\n    };\n    this._queueCallback(completeCallBack, activeElement, this._isAnimated());\n    if (isCycling) {\n      this.cycle();\n    }\n  }\n  _isAnimated() {\n    return this._element.classList.contains(CLASS_NAME_SLIDE);\n  }\n  _getActive() {\n    return SelectorEngine.findOne(SELECTOR_ACTIVE_ITEM, this._element);\n  }\n  _getItems() {\n    return SelectorEngine.find(SELECTOR_ITEM, this._element);\n  }\n  _clearInterval() {\n    if (this._interval) {\n      clearInterval(this._interval);\n      this._interval = null;\n    }\n  }\n  _directionToOrder(direction) {\n    if (isRTL()) {\n      return direction === DIRECTION_LEFT ? ORDER_PREV : ORDER_NEXT;\n    }\n    return direction === DIRECTION_LEFT ? ORDER_NEXT : ORDER_PREV;\n  }\n  _orderToDirection(order) {\n    if (isRTL()) {\n      return order === ORDER_PREV ? DIRECTION_LEFT : DIRECTION_RIGHT;\n    }\n    return order === ORDER_PREV ? DIRECTION_RIGHT : DIRECTION_LEFT;\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Carousel.getOrCreateInstance(this, config);\n      if (typeof config === 'number') {\n        data.to(config);\n        return;\n      }\n      if (typeof config === 'string') {\n        if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n          throw new TypeError(`No method named \"${config}\"`);\n        }\n        data[config]();\n      }\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$5, SELECTOR_DATA_SLIDE, function (event) {\n  const target = SelectorEngine.getElementFromSelector(this);\n  if (!target || !target.classList.contains(CLASS_NAME_CAROUSEL)) {\n    return;\n  }\n  event.preventDefault();\n  const carousel = Carousel.getOrCreateInstance(target);\n  const slideIndex = this.getAttribute('data-bs-slide-to');\n  if (slideIndex) {\n    carousel.to(slideIndex);\n    carousel._maybeEnableCycle();\n    return;\n  }\n  if (Manipulator.getDataAttribute(this, 'slide') === 'next') {\n    carousel.next();\n    carousel._maybeEnableCycle();\n    return;\n  }\n  carousel.prev();\n  carousel._maybeEnableCycle();\n});\nEventHandler.on(window, EVENT_LOAD_DATA_API$3, () => {\n  const carousels = SelectorEngine.find(SELECTOR_DATA_RIDE);\n  for (const carousel of carousels) {\n    Carousel.getOrCreateInstance(carousel);\n  }\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Carousel);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap collapse.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$b = 'collapse';\nconst DATA_KEY$7 = 'bs.collapse';\nconst EVENT_KEY$7 = `.${DATA_KEY$7}`;\nconst DATA_API_KEY$4 = '.data-api';\nconst EVENT_SHOW$6 = `show${EVENT_KEY$7}`;\nconst EVENT_SHOWN$6 = `shown${EVENT_KEY$7}`;\nconst EVENT_HIDE$6 = `hide${EVENT_KEY$7}`;\nconst EVENT_HIDDEN$6 = `hidden${EVENT_KEY$7}`;\nconst EVENT_CLICK_DATA_API$4 = `click${EVENT_KEY$7}${DATA_API_KEY$4}`;\nconst CLASS_NAME_SHOW$7 = 'show';\nconst CLASS_NAME_COLLAPSE = 'collapse';\nconst CLASS_NAME_COLLAPSING = 'collapsing';\nconst CLASS_NAME_COLLAPSED = 'collapsed';\nconst CLASS_NAME_DEEPER_CHILDREN = `:scope .${CLASS_NAME_COLLAPSE} .${CLASS_NAME_COLLAPSE}`;\nconst CLASS_NAME_HORIZONTAL = 'collapse-horizontal';\nconst WIDTH = 'width';\nconst HEIGHT = 'height';\nconst SELECTOR_ACTIVES = '.collapse.show, .collapse.collapsing';\nconst SELECTOR_DATA_TOGGLE$4 = '[data-bs-toggle=\"collapse\"]';\nconst Default$a = {\n  parent: null,\n  toggle: true\n};\nconst DefaultType$a = {\n  parent: '(null|element)',\n  toggle: 'boolean'\n};\n\n/**\n * Class definition\n */\n\nclass Collapse extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._isTransitioning = false;\n    this._triggerArray = [];\n    const toggleList = SelectorEngine.find(SELECTOR_DATA_TOGGLE$4);\n    for (const elem of toggleList) {\n      const selector = SelectorEngine.getSelectorFromElement(elem);\n      const filterElement = SelectorEngine.find(selector).filter(foundElement => foundElement === this._element);\n      if (selector !== null && filterElement.length) {\n        this._triggerArray.push(elem);\n      }\n    }\n    this._initializeChildren();\n    if (!this._config.parent) {\n      this._addAriaAndCollapsedClass(this._triggerArray, this._isShown());\n    }\n    if (this._config.toggle) {\n      this.toggle();\n    }\n  }\n\n  // Getters\n  static get Default() {\n    return Default$a;\n  }\n  static get DefaultType() {\n    return DefaultType$a;\n  }\n  static get NAME() {\n    return NAME$b;\n  }\n\n  // Public\n  toggle() {\n    if (this._isShown()) {\n      this.hide();\n    } else {\n      this.show();\n    }\n  }\n  show() {\n    if (this._isTransitioning || this._isShown()) {\n      return;\n    }\n    let activeChildren = [];\n\n    // find active children\n    if (this._config.parent) {\n      activeChildren = this._getFirstLevelChildren(SELECTOR_ACTIVES).filter(element => element !== this._element).map(element => Collapse.getOrCreateInstance(element, {\n        toggle: false\n      }));\n    }\n    if (activeChildren.length && activeChildren[0]._isTransitioning) {\n      return;\n    }\n    const startEvent = EventHandler.trigger(this._element, EVENT_SHOW$6);\n    if (startEvent.defaultPrevented) {\n      return;\n    }\n    for (const activeInstance of activeChildren) {\n      activeInstance.hide();\n    }\n    const dimension = this._getDimension();\n    this._element.classList.remove(CLASS_NAME_COLLAPSE);\n    this._element.classList.add(CLASS_NAME_COLLAPSING);\n    this._element.style[dimension] = 0;\n    this._addAriaAndCollapsedClass(this._triggerArray, true);\n    this._isTransitioning = true;\n    const complete = () => {\n      this._isTransitioning = false;\n      this._element.classList.remove(CLASS_NAME_COLLAPSING);\n      this._element.classList.add(CLASS_NAME_COLLAPSE, CLASS_NAME_SHOW$7);\n      this._element.style[dimension] = '';\n      EventHandler.trigger(this._element, EVENT_SHOWN$6);\n    };\n    const capitalizedDimension = dimension[0].toUpperCase() + dimension.slice(1);\n    const scrollSize = `scroll${capitalizedDimension}`;\n    this._queueCallback(complete, this._element, true);\n    this._element.style[dimension] = `${this._element[scrollSize]}px`;\n  }\n  hide() {\n    if (this._isTransitioning || !this._isShown()) {\n      return;\n    }\n    const startEvent = EventHandler.trigger(this._element, EVENT_HIDE$6);\n    if (startEvent.defaultPrevented) {\n      return;\n    }\n    const dimension = this._getDimension();\n    this._element.style[dimension] = `${this._element.getBoundingClientRect()[dimension]}px`;\n    reflow(this._element);\n    this._element.classList.add(CLASS_NAME_COLLAPSING);\n    this._element.classList.remove(CLASS_NAME_COLLAPSE, CLASS_NAME_SHOW$7);\n    for (const trigger of this._triggerArray) {\n      const element = SelectorEngine.getElementFromSelector(trigger);\n      if (element && !this._isShown(element)) {\n        this._addAriaAndCollapsedClass([trigger], false);\n      }\n    }\n    this._isTransitioning = true;\n    const complete = () => {\n      this._isTransitioning = false;\n      this._element.classList.remove(CLASS_NAME_COLLAPSING);\n      this._element.classList.add(CLASS_NAME_COLLAPSE);\n      EventHandler.trigger(this._element, EVENT_HIDDEN$6);\n    };\n    this._element.style[dimension] = '';\n    this._queueCallback(complete, this._element, true);\n  }\n  _isShown(element = this._element) {\n    return element.classList.contains(CLASS_NAME_SHOW$7);\n  }\n\n  // Private\n  _configAfterMerge(config) {\n    config.toggle = Boolean(config.toggle); // Coerce string values\n    config.parent = getElement(config.parent);\n    return config;\n  }\n  _getDimension() {\n    return this._element.classList.contains(CLASS_NAME_HORIZONTAL) ? WIDTH : HEIGHT;\n  }\n  _initializeChildren() {\n    if (!this._config.parent) {\n      return;\n    }\n    const children = this._getFirstLevelChildren(SELECTOR_DATA_TOGGLE$4);\n    for (const element of children) {\n      const selected = SelectorEngine.getElementFromSelector(element);\n      if (selected) {\n        this._addAriaAndCollapsedClass([element], this._isShown(selected));\n      }\n    }\n  }\n  _getFirstLevelChildren(selector) {\n    const children = SelectorEngine.find(CLASS_NAME_DEEPER_CHILDREN, this._config.parent);\n    // remove children if greater depth\n    return SelectorEngine.find(selector, this._config.parent).filter(element => !children.includes(element));\n  }\n  _addAriaAndCollapsedClass(triggerArray, isOpen) {\n    if (!triggerArray.length) {\n      return;\n    }\n    for (const element of triggerArray) {\n      element.classList.toggle(CLASS_NAME_COLLAPSED, !isOpen);\n      element.setAttribute('aria-expanded', isOpen);\n    }\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    const _config = {};\n    if (typeof config === 'string' && /show|hide/.test(config)) {\n      _config.toggle = false;\n    }\n    return this.each(function () {\n      const data = Collapse.getOrCreateInstance(this, _config);\n      if (typeof config === 'string') {\n        if (typeof data[config] === 'undefined') {\n          throw new TypeError(`No method named \"${config}\"`);\n        }\n        data[config]();\n      }\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$4, SELECTOR_DATA_TOGGLE$4, function (event) {\n  // preventDefault only for <a> elements (which change the URL) not inside the collapsible element\n  if (event.target.tagName === 'A' || event.delegateTarget && event.delegateTarget.tagName === 'A') {\n    event.preventDefault();\n  }\n  for (const element of SelectorEngine.getMultipleElementsFromSelector(this)) {\n    Collapse.getOrCreateInstance(element, {\n      toggle: false\n    }).toggle();\n  }\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Collapse);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dropdown.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$a = 'dropdown';\nconst DATA_KEY$6 = 'bs.dropdown';\nconst EVENT_KEY$6 = `.${DATA_KEY$6}`;\nconst DATA_API_KEY$3 = '.data-api';\nconst ESCAPE_KEY$2 = 'Escape';\nconst TAB_KEY$1 = 'Tab';\nconst ARROW_UP_KEY$1 = 'ArrowUp';\nconst ARROW_DOWN_KEY$1 = 'ArrowDown';\nconst RIGHT_MOUSE_BUTTON = 2; // MouseEvent.button value for the secondary button, usually the right button\n\nconst EVENT_HIDE$5 = `hide${EVENT_KEY$6}`;\nconst EVENT_HIDDEN$5 = `hidden${EVENT_KEY$6}`;\nconst EVENT_SHOW$5 = `show${EVENT_KEY$6}`;\nconst EVENT_SHOWN$5 = `shown${EVENT_KEY$6}`;\nconst EVENT_CLICK_DATA_API$3 = `click${EVENT_KEY$6}${DATA_API_KEY$3}`;\nconst EVENT_KEYDOWN_DATA_API = `keydown${EVENT_KEY$6}${DATA_API_KEY$3}`;\nconst EVENT_KEYUP_DATA_API = `keyup${EVENT_KEY$6}${DATA_API_KEY$3}`;\nconst CLASS_NAME_SHOW$6 = 'show';\nconst CLASS_NAME_DROPUP = 'dropup';\nconst CLASS_NAME_DROPEND = 'dropend';\nconst CLASS_NAME_DROPSTART = 'dropstart';\nconst CLASS_NAME_DROPUP_CENTER = 'dropup-center';\nconst CLASS_NAME_DROPDOWN_CENTER = 'dropdown-center';\nconst SELECTOR_DATA_TOGGLE$3 = '[data-bs-toggle=\"dropdown\"]:not(.disabled):not(:disabled)';\nconst SELECTOR_DATA_TOGGLE_SHOWN = `${SELECTOR_DATA_TOGGLE$3}.${CLASS_NAME_SHOW$6}`;\nconst SELECTOR_MENU = '.dropdown-menu';\nconst SELECTOR_NAVBAR = '.navbar';\nconst SELECTOR_NAVBAR_NAV = '.navbar-nav';\nconst SELECTOR_VISIBLE_ITEMS = '.dropdown-menu .dropdown-item:not(.disabled):not(:disabled)';\nconst PLACEMENT_TOP = isRTL() ? 'top-end' : 'top-start';\nconst PLACEMENT_TOPEND = isRTL() ? 'top-start' : 'top-end';\nconst PLACEMENT_BOTTOM = isRTL() ? 'bottom-end' : 'bottom-start';\nconst PLACEMENT_BOTTOMEND = isRTL() ? 'bottom-start' : 'bottom-end';\nconst PLACEMENT_RIGHT = isRTL() ? 'left-start' : 'right-start';\nconst PLACEMENT_LEFT = isRTL() ? 'right-start' : 'left-start';\nconst PLACEMENT_TOPCENTER = 'top';\nconst PLACEMENT_BOTTOMCENTER = 'bottom';\nconst Default$9 = {\n  autoClose: true,\n  boundary: 'clippingParents',\n  display: 'dynamic',\n  offset: [0, 2],\n  popperConfig: null,\n  reference: 'toggle'\n};\nconst DefaultType$9 = {\n  autoClose: '(boolean|string)',\n  boundary: '(string|element)',\n  display: 'string',\n  offset: '(array|string|function)',\n  popperConfig: '(null|object|function)',\n  reference: '(string|element|object)'\n};\n\n/**\n * Class definition\n */\n\nclass Dropdown extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._popper = null;\n    this._parent = this._element.parentNode; // dropdown wrapper\n    // TODO: v6 revert #37011 & change markup https://getbootstrap.com/docs/5.3/forms/input-group/\n    this._menu = SelectorEngine.next(this._element, SELECTOR_MENU)[0] || SelectorEngine.prev(this._element, SELECTOR_MENU)[0] || SelectorEngine.findOne(SELECTOR_MENU, this._parent);\n    this._inNavbar = this._detectNavbar();\n  }\n\n  // Getters\n  static get Default() {\n    return Default$9;\n  }\n  static get DefaultType() {\n    return DefaultType$9;\n  }\n  static get NAME() {\n    return NAME$a;\n  }\n\n  // Public\n  toggle() {\n    return this._isShown() ? this.hide() : this.show();\n  }\n  show() {\n    if (isDisabled(this._element) || this._isShown()) {\n      return;\n    }\n    const relatedTarget = {\n      relatedTarget: this._element\n    };\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW$5, relatedTarget);\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n    this._createPopper();\n\n    // If this is a touch-enabled device we add extra\n    // empty mouseover listeners to the body's immediate children;\n    // only needed because of broken event delegation on iOS\n    // https://www.quirksmode.org/blog/archives/2014/02/mouse_event_bub.html\n    if ('ontouchstart' in document.documentElement && !this._parent.closest(SELECTOR_NAVBAR_NAV)) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.on(element, 'mouseover', noop);\n      }\n    }\n    this._element.focus();\n    this._element.setAttribute('aria-expanded', true);\n    this._menu.classList.add(CLASS_NAME_SHOW$6);\n    this._element.classList.add(CLASS_NAME_SHOW$6);\n    EventHandler.trigger(this._element, EVENT_SHOWN$5, relatedTarget);\n  }\n  hide() {\n    if (isDisabled(this._element) || !this._isShown()) {\n      return;\n    }\n    const relatedTarget = {\n      relatedTarget: this._element\n    };\n    this._completeHide(relatedTarget);\n  }\n  dispose() {\n    if (this._popper) {\n      this._popper.destroy();\n    }\n    super.dispose();\n  }\n  update() {\n    this._inNavbar = this._detectNavbar();\n    if (this._popper) {\n      this._popper.update();\n    }\n  }\n\n  // Private\n  _completeHide(relatedTarget) {\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE$5, relatedTarget);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n\n    // If this is a touch-enabled device we remove the extra\n    // empty mouseover listeners we added for iOS support\n    if ('ontouchstart' in document.documentElement) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.off(element, 'mouseover', noop);\n      }\n    }\n    if (this._popper) {\n      this._popper.destroy();\n    }\n    this._menu.classList.remove(CLASS_NAME_SHOW$6);\n    this._element.classList.remove(CLASS_NAME_SHOW$6);\n    this._element.setAttribute('aria-expanded', 'false');\n    Manipulator.removeDataAttribute(this._menu, 'popper');\n    EventHandler.trigger(this._element, EVENT_HIDDEN$5, relatedTarget);\n  }\n  _getConfig(config) {\n    config = super._getConfig(config);\n    if (typeof config.reference === 'object' && !isElement(config.reference) && typeof config.reference.getBoundingClientRect !== 'function') {\n      // Popper virtual elements require a getBoundingClientRect method\n      throw new TypeError(`${NAME$a.toUpperCase()}: Option \"reference\" provided type \"object\" without a required \"getBoundingClientRect\" method.`);\n    }\n    return config;\n  }\n  _createPopper() {\n    if (typeof Popper === 'undefined') {\n      throw new TypeError('Bootstrap\\'s dropdowns require Popper (https://popper.js.org)');\n    }\n    let referenceElement = this._element;\n    if (this._config.reference === 'parent') {\n      referenceElement = this._parent;\n    } else if (isElement(this._config.reference)) {\n      referenceElement = getElement(this._config.reference);\n    } else if (typeof this._config.reference === 'object') {\n      referenceElement = this._config.reference;\n    }\n    const popperConfig = this._getPopperConfig();\n    this._popper = Popper.createPopper(referenceElement, this._menu, popperConfig);\n  }\n  _isShown() {\n    return this._menu.classList.contains(CLASS_NAME_SHOW$6);\n  }\n  _getPlacement() {\n    const parentDropdown = this._parent;\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPEND)) {\n      return PLACEMENT_RIGHT;\n    }\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPSTART)) {\n      return PLACEMENT_LEFT;\n    }\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPUP_CENTER)) {\n      return PLACEMENT_TOPCENTER;\n    }\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPDOWN_CENTER)) {\n      return PLACEMENT_BOTTOMCENTER;\n    }\n\n    // We need to trim the value because custom properties can also include spaces\n    const isEnd = getComputedStyle(this._menu).getPropertyValue('--bs-position').trim() === 'end';\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPUP)) {\n      return isEnd ? PLACEMENT_TOPEND : PLACEMENT_TOP;\n    }\n    return isEnd ? PLACEMENT_BOTTOMEND : PLACEMENT_BOTTOM;\n  }\n  _detectNavbar() {\n    return this._element.closest(SELECTOR_NAVBAR) !== null;\n  }\n  _getOffset() {\n    const {\n      offset\n    } = this._config;\n    if (typeof offset === 'string') {\n      return offset.split(',').map(value => Number.parseInt(value, 10));\n    }\n    if (typeof offset === 'function') {\n      return popperData => offset(popperData, this._element);\n    }\n    return offset;\n  }\n  _getPopperConfig() {\n    const defaultBsPopperConfig = {\n      placement: this._getPlacement(),\n      modifiers: [{\n        name: 'preventOverflow',\n        options: {\n          boundary: this._config.boundary\n        }\n      }, {\n        name: 'offset',\n        options: {\n          offset: this._getOffset()\n        }\n      }]\n    };\n\n    // Disable Popper if we have a static display or Dropdown is in Navbar\n    if (this._inNavbar || this._config.display === 'static') {\n      Manipulator.setDataAttribute(this._menu, 'popper', 'static'); // TODO: v6 remove\n      defaultBsPopperConfig.modifiers = [{\n        name: 'applyStyles',\n        enabled: false\n      }];\n    }\n    return {\n      ...defaultBsPopperConfig,\n      ...execute(this._config.popperConfig, [defaultBsPopperConfig])\n    };\n  }\n  _selectMenuItem({\n    key,\n    target\n  }) {\n    const items = SelectorEngine.find(SELECTOR_VISIBLE_ITEMS, this._menu).filter(element => isVisible(element));\n    if (!items.length) {\n      return;\n    }\n\n    // if target isn't included in items (e.g. when expanding the dropdown)\n    // allow cycling to get the last item in case key equals ARROW_UP_KEY\n    getNextActiveElement(items, target, key === ARROW_DOWN_KEY$1, !items.includes(target)).focus();\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Dropdown.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n  static clearMenus(event) {\n    if (event.button === RIGHT_MOUSE_BUTTON || event.type === 'keyup' && event.key !== TAB_KEY$1) {\n      return;\n    }\n    const openToggles = SelectorEngine.find(SELECTOR_DATA_TOGGLE_SHOWN);\n    for (const toggle of openToggles) {\n      const context = Dropdown.getInstance(toggle);\n      if (!context || context._config.autoClose === false) {\n        continue;\n      }\n      const composedPath = event.composedPath();\n      const isMenuTarget = composedPath.includes(context._menu);\n      if (composedPath.includes(context._element) || context._config.autoClose === 'inside' && !isMenuTarget || context._config.autoClose === 'outside' && isMenuTarget) {\n        continue;\n      }\n\n      // Tab navigation through the dropdown menu or events from contained inputs shouldn't close the menu\n      if (context._menu.contains(event.target) && (event.type === 'keyup' && event.key === TAB_KEY$1 || /input|select|option|textarea|form/i.test(event.target.tagName))) {\n        continue;\n      }\n      const relatedTarget = {\n        relatedTarget: context._element\n      };\n      if (event.type === 'click') {\n        relatedTarget.clickEvent = event;\n      }\n      context._completeHide(relatedTarget);\n    }\n  }\n  static dataApiKeydownHandler(event) {\n    // If not an UP | DOWN | ESCAPE key => not a dropdown command\n    // If input/textarea && if key is other than ESCAPE => not a dropdown command\n\n    const isInput = /input|textarea/i.test(event.target.tagName);\n    const isEscapeEvent = event.key === ESCAPE_KEY$2;\n    const isUpOrDownEvent = [ARROW_UP_KEY$1, ARROW_DOWN_KEY$1].includes(event.key);\n    if (!isUpOrDownEvent && !isEscapeEvent) {\n      return;\n    }\n    if (isInput && !isEscapeEvent) {\n      return;\n    }\n    event.preventDefault();\n\n    // TODO: v6 revert #37011 & change markup https://getbootstrap.com/docs/5.3/forms/input-group/\n    const getToggleButton = this.matches(SELECTOR_DATA_TOGGLE$3) ? this : SelectorEngine.prev(this, SELECTOR_DATA_TOGGLE$3)[0] || SelectorEngine.next(this, SELECTOR_DATA_TOGGLE$3)[0] || SelectorEngine.findOne(SELECTOR_DATA_TOGGLE$3, event.delegateTarget.parentNode);\n    const instance = Dropdown.getOrCreateInstance(getToggleButton);\n    if (isUpOrDownEvent) {\n      event.stopPropagation();\n      instance.show();\n      instance._selectMenuItem(event);\n      return;\n    }\n    if (instance._isShown()) {\n      // else is escape and we check if it is shown\n      event.stopPropagation();\n      instance.hide();\n      getToggleButton.focus();\n    }\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_KEYDOWN_DATA_API, SELECTOR_DATA_TOGGLE$3, Dropdown.dataApiKeydownHandler);\nEventHandler.on(document, EVENT_KEYDOWN_DATA_API, SELECTOR_MENU, Dropdown.dataApiKeydownHandler);\nEventHandler.on(document, EVENT_CLICK_DATA_API$3, Dropdown.clearMenus);\nEventHandler.on(document, EVENT_KEYUP_DATA_API, Dropdown.clearMenus);\nEventHandler.on(document, EVENT_CLICK_DATA_API$3, SELECTOR_DATA_TOGGLE$3, function (event) {\n  event.preventDefault();\n  Dropdown.getOrCreateInstance(this).toggle();\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Dropdown);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/backdrop.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$9 = 'backdrop';\nconst CLASS_NAME_FADE$4 = 'fade';\nconst CLASS_NAME_SHOW$5 = 'show';\nconst EVENT_MOUSEDOWN = `mousedown.bs.${NAME$9}`;\nconst Default$8 = {\n  className: 'modal-backdrop',\n  clickCallback: null,\n  isAnimated: false,\n  isVisible: true,\n  // if false, we use the backdrop helper without adding any element to the dom\n  rootElement: 'body' // give the choice to place backdrop under different elements\n};\nconst DefaultType$8 = {\n  className: 'string',\n  clickCallback: '(function|null)',\n  isAnimated: 'boolean',\n  isVisible: 'boolean',\n  rootElement: '(element|string)'\n};\n\n/**\n * Class definition\n */\n\nclass Backdrop extends Config {\n  constructor(config) {\n    super();\n    this._config = this._getConfig(config);\n    this._isAppended = false;\n    this._element = null;\n  }\n\n  // Getters\n  static get Default() {\n    return Default$8;\n  }\n  static get DefaultType() {\n    return DefaultType$8;\n  }\n  static get NAME() {\n    return NAME$9;\n  }\n\n  // Public\n  show(callback) {\n    if (!this._config.isVisible) {\n      execute(callback);\n      return;\n    }\n    this._append();\n    const element = this._getElement();\n    if (this._config.isAnimated) {\n      reflow(element);\n    }\n    element.classList.add(CLASS_NAME_SHOW$5);\n    this._emulateAnimation(() => {\n      execute(callback);\n    });\n  }\n  hide(callback) {\n    if (!this._config.isVisible) {\n      execute(callback);\n      return;\n    }\n    this._getElement().classList.remove(CLASS_NAME_SHOW$5);\n    this._emulateAnimation(() => {\n      this.dispose();\n      execute(callback);\n    });\n  }\n  dispose() {\n    if (!this._isAppended) {\n      return;\n    }\n    EventHandler.off(this._element, EVENT_MOUSEDOWN);\n    this._element.remove();\n    this._isAppended = false;\n  }\n\n  // Private\n  _getElement() {\n    if (!this._element) {\n      const backdrop = document.createElement('div');\n      backdrop.className = this._config.className;\n      if (this._config.isAnimated) {\n        backdrop.classList.add(CLASS_NAME_FADE$4);\n      }\n      this._element = backdrop;\n    }\n    return this._element;\n  }\n  _configAfterMerge(config) {\n    // use getElement() with the default \"body\" to get a fresh Element on each instantiation\n    config.rootElement = getElement(config.rootElement);\n    return config;\n  }\n  _append() {\n    if (this._isAppended) {\n      return;\n    }\n    const element = this._getElement();\n    this._config.rootElement.append(element);\n    EventHandler.on(element, EVENT_MOUSEDOWN, () => {\n      execute(this._config.clickCallback);\n    });\n    this._isAppended = true;\n  }\n  _emulateAnimation(callback) {\n    executeAfterTransition(callback, this._getElement(), this._config.isAnimated);\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/focustrap.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$8 = 'focustrap';\nconst DATA_KEY$5 = 'bs.focustrap';\nconst EVENT_KEY$5 = `.${DATA_KEY$5}`;\nconst EVENT_FOCUSIN$2 = `focusin${EVENT_KEY$5}`;\nconst EVENT_KEYDOWN_TAB = `keydown.tab${EVENT_KEY$5}`;\nconst TAB_KEY = 'Tab';\nconst TAB_NAV_FORWARD = 'forward';\nconst TAB_NAV_BACKWARD = 'backward';\nconst Default$7 = {\n  autofocus: true,\n  trapElement: null // The element to trap focus inside of\n};\nconst DefaultType$7 = {\n  autofocus: 'boolean',\n  trapElement: 'element'\n};\n\n/**\n * Class definition\n */\n\nclass FocusTrap extends Config {\n  constructor(config) {\n    super();\n    this._config = this._getConfig(config);\n    this._isActive = false;\n    this._lastTabNavDirection = null;\n  }\n\n  // Getters\n  static get Default() {\n    return Default$7;\n  }\n  static get DefaultType() {\n    return DefaultType$7;\n  }\n  static get NAME() {\n    return NAME$8;\n  }\n\n  // Public\n  activate() {\n    if (this._isActive) {\n      return;\n    }\n    if (this._config.autofocus) {\n      this._config.trapElement.focus();\n    }\n    EventHandler.off(document, EVENT_KEY$5); // guard against infinite focus loop\n    EventHandler.on(document, EVENT_FOCUSIN$2, event => this._handleFocusin(event));\n    EventHandler.on(document, EVENT_KEYDOWN_TAB, event => this._handleKeydown(event));\n    this._isActive = true;\n  }\n  deactivate() {\n    if (!this._isActive) {\n      return;\n    }\n    this._isActive = false;\n    EventHandler.off(document, EVENT_KEY$5);\n  }\n\n  // Private\n  _handleFocusin(event) {\n    const {\n      trapElement\n    } = this._config;\n    if (event.target === document || event.target === trapElement || trapElement.contains(event.target)) {\n      return;\n    }\n    const elements = SelectorEngine.focusableChildren(trapElement);\n    if (elements.length === 0) {\n      trapElement.focus();\n    } else if (this._lastTabNavDirection === TAB_NAV_BACKWARD) {\n      elements[elements.length - 1].focus();\n    } else {\n      elements[0].focus();\n    }\n  }\n  _handleKeydown(event) {\n    if (event.key !== TAB_KEY) {\n      return;\n    }\n    this._lastTabNavDirection = event.shiftKey ? TAB_NAV_BACKWARD : TAB_NAV_FORWARD;\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/scrollBar.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst SELECTOR_FIXED_CONTENT = '.fixed-top, .fixed-bottom, .is-fixed, .sticky-top';\nconst SELECTOR_STICKY_CONTENT = '.sticky-top';\nconst PROPERTY_PADDING = 'padding-right';\nconst PROPERTY_MARGIN = 'margin-right';\n\n/**\n * Class definition\n */\n\nclass ScrollBarHelper {\n  constructor() {\n    this._element = document.body;\n  }\n\n  // Public\n  getWidth() {\n    // https://developer.mozilla.org/en-US/docs/Web/API/Window/innerWidth#usage_notes\n    const documentWidth = document.documentElement.clientWidth;\n    return Math.abs(window.innerWidth - documentWidth);\n  }\n  hide() {\n    const width = this.getWidth();\n    this._disableOverFlow();\n    // give padding to element to balance the hidden scrollbar width\n    this._setElementAttributes(this._element, PROPERTY_PADDING, calculatedValue => calculatedValue + width);\n    // trick: We adjust positive paddingRight and negative marginRight to sticky-top elements to keep showing fullwidth\n    this._setElementAttributes(SELECTOR_FIXED_CONTENT, PROPERTY_PADDING, calculatedValue => calculatedValue + width);\n    this._setElementAttributes(SELECTOR_STICKY_CONTENT, PROPERTY_MARGIN, calculatedValue => calculatedValue - width);\n  }\n  reset() {\n    this._resetElementAttributes(this._element, 'overflow');\n    this._resetElementAttributes(this._element, PROPERTY_PADDING);\n    this._resetElementAttributes(SELECTOR_FIXED_CONTENT, PROPERTY_PADDING);\n    this._resetElementAttributes(SELECTOR_STICKY_CONTENT, PROPERTY_MARGIN);\n  }\n  isOverflowing() {\n    return this.getWidth() > 0;\n  }\n\n  // Private\n  _disableOverFlow() {\n    this._saveInitialAttribute(this._element, 'overflow');\n    this._element.style.overflow = 'hidden';\n  }\n  _setElementAttributes(selector, styleProperty, callback) {\n    const scrollbarWidth = this.getWidth();\n    const manipulationCallBack = element => {\n      if (element !== this._element && window.innerWidth > element.clientWidth + scrollbarWidth) {\n        return;\n      }\n      this._saveInitialAttribute(element, styleProperty);\n      const calculatedValue = window.getComputedStyle(element).getPropertyValue(styleProperty);\n      element.style.setProperty(styleProperty, `${callback(Number.parseFloat(calculatedValue))}px`);\n    };\n    this._applyManipulationCallback(selector, manipulationCallBack);\n  }\n  _saveInitialAttribute(element, styleProperty) {\n    const actualValue = element.style.getPropertyValue(styleProperty);\n    if (actualValue) {\n      Manipulator.setDataAttribute(element, styleProperty, actualValue);\n    }\n  }\n  _resetElementAttributes(selector, styleProperty) {\n    const manipulationCallBack = element => {\n      const value = Manipulator.getDataAttribute(element, styleProperty);\n      // We only want to remove the property if the value is `null`; the value can also be zero\n      if (value === null) {\n        element.style.removeProperty(styleProperty);\n        return;\n      }\n      Manipulator.removeDataAttribute(element, styleProperty);\n      element.style.setProperty(styleProperty, value);\n    };\n    this._applyManipulationCallback(selector, manipulationCallBack);\n  }\n  _applyManipulationCallback(selector, callBack) {\n    if (isElement(selector)) {\n      callBack(selector);\n      return;\n    }\n    for (const sel of SelectorEngine.find(selector, this._element)) {\n      callBack(sel);\n    }\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap modal.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$7 = 'modal';\nconst DATA_KEY$4 = 'bs.modal';\nconst EVENT_KEY$4 = `.${DATA_KEY$4}`;\nconst DATA_API_KEY$2 = '.data-api';\nconst ESCAPE_KEY$1 = 'Escape';\nconst EVENT_HIDE$4 = `hide${EVENT_KEY$4}`;\nconst EVENT_HIDE_PREVENTED$1 = `hidePrevented${EVENT_KEY$4}`;\nconst EVENT_HIDDEN$4 = `hidden${EVENT_KEY$4}`;\nconst EVENT_SHOW$4 = `show${EVENT_KEY$4}`;\nconst EVENT_SHOWN$4 = `shown${EVENT_KEY$4}`;\nconst EVENT_RESIZE$1 = `resize${EVENT_KEY$4}`;\nconst EVENT_CLICK_DISMISS = `click.dismiss${EVENT_KEY$4}`;\nconst EVENT_MOUSEDOWN_DISMISS = `mousedown.dismiss${EVENT_KEY$4}`;\nconst EVENT_KEYDOWN_DISMISS$1 = `keydown.dismiss${EVENT_KEY$4}`;\nconst EVENT_CLICK_DATA_API$2 = `click${EVENT_KEY$4}${DATA_API_KEY$2}`;\nconst CLASS_NAME_OPEN = 'modal-open';\nconst CLASS_NAME_FADE$3 = 'fade';\nconst CLASS_NAME_SHOW$4 = 'show';\nconst CLASS_NAME_STATIC = 'modal-static';\nconst OPEN_SELECTOR$1 = '.modal.show';\nconst SELECTOR_DIALOG = '.modal-dialog';\nconst SELECTOR_MODAL_BODY = '.modal-body';\nconst SELECTOR_DATA_TOGGLE$2 = '[data-bs-toggle=\"modal\"]';\nconst Default$6 = {\n  backdrop: true,\n  focus: true,\n  keyboard: true\n};\nconst DefaultType$6 = {\n  backdrop: '(boolean|string)',\n  focus: 'boolean',\n  keyboard: 'boolean'\n};\n\n/**\n * Class definition\n */\n\nclass Modal extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._dialog = SelectorEngine.findOne(SELECTOR_DIALOG, this._element);\n    this._backdrop = this._initializeBackDrop();\n    this._focustrap = this._initializeFocusTrap();\n    this._isShown = false;\n    this._isTransitioning = false;\n    this._scrollBar = new ScrollBarHelper();\n    this._addEventListeners();\n  }\n\n  // Getters\n  static get Default() {\n    return Default$6;\n  }\n  static get DefaultType() {\n    return DefaultType$6;\n  }\n  static get NAME() {\n    return NAME$7;\n  }\n\n  // Public\n  toggle(relatedTarget) {\n    return this._isShown ? this.hide() : this.show(relatedTarget);\n  }\n  show(relatedTarget) {\n    if (this._isShown || this._isTransitioning) {\n      return;\n    }\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW$4, {\n      relatedTarget\n    });\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n    this._isShown = true;\n    this._isTransitioning = true;\n    this._scrollBar.hide();\n    document.body.classList.add(CLASS_NAME_OPEN);\n    this._adjustDialog();\n    this._backdrop.show(() => this._showElement(relatedTarget));\n  }\n  hide() {\n    if (!this._isShown || this._isTransitioning) {\n      return;\n    }\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE$4);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    this._isShown = false;\n    this._isTransitioning = true;\n    this._focustrap.deactivate();\n    this._element.classList.remove(CLASS_NAME_SHOW$4);\n    this._queueCallback(() => this._hideModal(), this._element, this._isAnimated());\n  }\n  dispose() {\n    EventHandler.off(window, EVENT_KEY$4);\n    EventHandler.off(this._dialog, EVENT_KEY$4);\n    this._backdrop.dispose();\n    this._focustrap.deactivate();\n    super.dispose();\n  }\n  handleUpdate() {\n    this._adjustDialog();\n  }\n\n  // Private\n  _initializeBackDrop() {\n    return new Backdrop({\n      isVisible: Boolean(this._config.backdrop),\n      // 'static' option will be translated to true, and booleans will keep their value,\n      isAnimated: this._isAnimated()\n    });\n  }\n  _initializeFocusTrap() {\n    return new FocusTrap({\n      trapElement: this._element\n    });\n  }\n  _showElement(relatedTarget) {\n    // try to append dynamic modal\n    if (!document.body.contains(this._element)) {\n      document.body.append(this._element);\n    }\n    this._element.style.display = 'block';\n    this._element.removeAttribute('aria-hidden');\n    this._element.setAttribute('aria-modal', true);\n    this._element.setAttribute('role', 'dialog');\n    this._element.scrollTop = 0;\n    const modalBody = SelectorEngine.findOne(SELECTOR_MODAL_BODY, this._dialog);\n    if (modalBody) {\n      modalBody.scrollTop = 0;\n    }\n    reflow(this._element);\n    this._element.classList.add(CLASS_NAME_SHOW$4);\n    const transitionComplete = () => {\n      if (this._config.focus) {\n        this._focustrap.activate();\n      }\n      this._isTransitioning = false;\n      EventHandler.trigger(this._element, EVENT_SHOWN$4, {\n        relatedTarget\n      });\n    };\n    this._queueCallback(transitionComplete, this._dialog, this._isAnimated());\n  }\n  _addEventListeners() {\n    EventHandler.on(this._element, EVENT_KEYDOWN_DISMISS$1, event => {\n      if (event.key !== ESCAPE_KEY$1) {\n        return;\n      }\n      if (this._config.keyboard) {\n        this.hide();\n        return;\n      }\n      this._triggerBackdropTransition();\n    });\n    EventHandler.on(window, EVENT_RESIZE$1, () => {\n      if (this._isShown && !this._isTransitioning) {\n        this._adjustDialog();\n      }\n    });\n    EventHandler.on(this._element, EVENT_MOUSEDOWN_DISMISS, event => {\n      // a bad trick to segregate clicks that may start inside dialog but end outside, and avoid listen to scrollbar clicks\n      EventHandler.one(this._element, EVENT_CLICK_DISMISS, event2 => {\n        if (this._element !== event.target || this._element !== event2.target) {\n          return;\n        }\n        if (this._config.backdrop === 'static') {\n          this._triggerBackdropTransition();\n          return;\n        }\n        if (this._config.backdrop) {\n          this.hide();\n        }\n      });\n    });\n  }\n  _hideModal() {\n    this._element.style.display = 'none';\n    this._element.setAttribute('aria-hidden', true);\n    this._element.removeAttribute('aria-modal');\n    this._element.removeAttribute('role');\n    this._isTransitioning = false;\n    this._backdrop.hide(() => {\n      document.body.classList.remove(CLASS_NAME_OPEN);\n      this._resetAdjustments();\n      this._scrollBar.reset();\n      EventHandler.trigger(this._element, EVENT_HIDDEN$4);\n    });\n  }\n  _isAnimated() {\n    return this._element.classList.contains(CLASS_NAME_FADE$3);\n  }\n  _triggerBackdropTransition() {\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE_PREVENTED$1);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    const isModalOverflowing = this._element.scrollHeight > document.documentElement.clientHeight;\n    const initialOverflowY = this._element.style.overflowY;\n    // return if the following background transition hasn't yet completed\n    if (initialOverflowY === 'hidden' || this._element.classList.contains(CLASS_NAME_STATIC)) {\n      return;\n    }\n    if (!isModalOverflowing) {\n      this._element.style.overflowY = 'hidden';\n    }\n    this._element.classList.add(CLASS_NAME_STATIC);\n    this._queueCallback(() => {\n      this._element.classList.remove(CLASS_NAME_STATIC);\n      this._queueCallback(() => {\n        this._element.style.overflowY = initialOverflowY;\n      }, this._dialog);\n    }, this._dialog);\n    this._element.focus();\n  }\n\n  /**\n   * The following methods are used to handle overflowing modals\n   */\n\n  _adjustDialog() {\n    const isModalOverflowing = this._element.scrollHeight > document.documentElement.clientHeight;\n    const scrollbarWidth = this._scrollBar.getWidth();\n    const isBodyOverflowing = scrollbarWidth > 0;\n    if (isBodyOverflowing && !isModalOverflowing) {\n      const property = isRTL() ? 'paddingLeft' : 'paddingRight';\n      this._element.style[property] = `${scrollbarWidth}px`;\n    }\n    if (!isBodyOverflowing && isModalOverflowing) {\n      const property = isRTL() ? 'paddingRight' : 'paddingLeft';\n      this._element.style[property] = `${scrollbarWidth}px`;\n    }\n  }\n  _resetAdjustments() {\n    this._element.style.paddingLeft = '';\n    this._element.style.paddingRight = '';\n  }\n\n  // Static\n  static jQueryInterface(config, relatedTarget) {\n    return this.each(function () {\n      const data = Modal.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config](relatedTarget);\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$2, SELECTOR_DATA_TOGGLE$2, function (event) {\n  const target = SelectorEngine.getElementFromSelector(this);\n  if (['A', 'AREA'].includes(this.tagName)) {\n    event.preventDefault();\n  }\n  EventHandler.one(target, EVENT_SHOW$4, showEvent => {\n    if (showEvent.defaultPrevented) {\n      // only register focus restorer if modal will actually get shown\n      return;\n    }\n    EventHandler.one(target, EVENT_HIDDEN$4, () => {\n      if (isVisible(this)) {\n        this.focus();\n      }\n    });\n  });\n\n  // avoid conflict when clicking modal toggler while another one is open\n  const alreadyOpen = SelectorEngine.findOne(OPEN_SELECTOR$1);\n  if (alreadyOpen) {\n    Modal.getInstance(alreadyOpen).hide();\n  }\n  const data = Modal.getOrCreateInstance(target);\n  data.toggle(this);\n});\nenableDismissTrigger(Modal);\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Modal);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap offcanvas.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$6 = 'offcanvas';\nconst DATA_KEY$3 = 'bs.offcanvas';\nconst EVENT_KEY$3 = `.${DATA_KEY$3}`;\nconst DATA_API_KEY$1 = '.data-api';\nconst EVENT_LOAD_DATA_API$2 = `load${EVENT_KEY$3}${DATA_API_KEY$1}`;\nconst ESCAPE_KEY = 'Escape';\nconst CLASS_NAME_SHOW$3 = 'show';\nconst CLASS_NAME_SHOWING$1 = 'showing';\nconst CLASS_NAME_HIDING = 'hiding';\nconst CLASS_NAME_BACKDROP = 'offcanvas-backdrop';\nconst OPEN_SELECTOR = '.offcanvas.show';\nconst EVENT_SHOW$3 = `show${EVENT_KEY$3}`;\nconst EVENT_SHOWN$3 = `shown${EVENT_KEY$3}`;\nconst EVENT_HIDE$3 = `hide${EVENT_KEY$3}`;\nconst EVENT_HIDE_PREVENTED = `hidePrevented${EVENT_KEY$3}`;\nconst EVENT_HIDDEN$3 = `hidden${EVENT_KEY$3}`;\nconst EVENT_RESIZE = `resize${EVENT_KEY$3}`;\nconst EVENT_CLICK_DATA_API$1 = `click${EVENT_KEY$3}${DATA_API_KEY$1}`;\nconst EVENT_KEYDOWN_DISMISS = `keydown.dismiss${EVENT_KEY$3}`;\nconst SELECTOR_DATA_TOGGLE$1 = '[data-bs-toggle=\"offcanvas\"]';\nconst Default$5 = {\n  backdrop: true,\n  keyboard: true,\n  scroll: false\n};\nconst DefaultType$5 = {\n  backdrop: '(boolean|string)',\n  keyboard: 'boolean',\n  scroll: 'boolean'\n};\n\n/**\n * Class definition\n */\n\nclass Offcanvas extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._isShown = false;\n    this._backdrop = this._initializeBackDrop();\n    this._focustrap = this._initializeFocusTrap();\n    this._addEventListeners();\n  }\n\n  // Getters\n  static get Default() {\n    return Default$5;\n  }\n  static get DefaultType() {\n    return DefaultType$5;\n  }\n  static get NAME() {\n    return NAME$6;\n  }\n\n  // Public\n  toggle(relatedTarget) {\n    return this._isShown ? this.hide() : this.show(relatedTarget);\n  }\n  show(relatedTarget) {\n    if (this._isShown) {\n      return;\n    }\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW$3, {\n      relatedTarget\n    });\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n    this._isShown = true;\n    this._backdrop.show();\n    if (!this._config.scroll) {\n      new ScrollBarHelper().hide();\n    }\n    this._element.setAttribute('aria-modal', true);\n    this._element.setAttribute('role', 'dialog');\n    this._element.classList.add(CLASS_NAME_SHOWING$1);\n    const completeCallBack = () => {\n      if (!this._config.scroll || this._config.backdrop) {\n        this._focustrap.activate();\n      }\n      this._element.classList.add(CLASS_NAME_SHOW$3);\n      this._element.classList.remove(CLASS_NAME_SHOWING$1);\n      EventHandler.trigger(this._element, EVENT_SHOWN$3, {\n        relatedTarget\n      });\n    };\n    this._queueCallback(completeCallBack, this._element, true);\n  }\n  hide() {\n    if (!this._isShown) {\n      return;\n    }\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE$3);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    this._focustrap.deactivate();\n    this._element.blur();\n    this._isShown = false;\n    this._element.classList.add(CLASS_NAME_HIDING);\n    this._backdrop.hide();\n    const completeCallback = () => {\n      this._element.classList.remove(CLASS_NAME_SHOW$3, CLASS_NAME_HIDING);\n      this._element.removeAttribute('aria-modal');\n      this._element.removeAttribute('role');\n      if (!this._config.scroll) {\n        new ScrollBarHelper().reset();\n      }\n      EventHandler.trigger(this._element, EVENT_HIDDEN$3);\n    };\n    this._queueCallback(completeCallback, this._element, true);\n  }\n  dispose() {\n    this._backdrop.dispose();\n    this._focustrap.deactivate();\n    super.dispose();\n  }\n\n  // Private\n  _initializeBackDrop() {\n    const clickCallback = () => {\n      if (this._config.backdrop === 'static') {\n        EventHandler.trigger(this._element, EVENT_HIDE_PREVENTED);\n        return;\n      }\n      this.hide();\n    };\n\n    // 'static' option will be translated to true, and booleans will keep their value\n    const isVisible = Boolean(this._config.backdrop);\n    return new Backdrop({\n      className: CLASS_NAME_BACKDROP,\n      isVisible,\n      isAnimated: true,\n      rootElement: this._element.parentNode,\n      clickCallback: isVisible ? clickCallback : null\n    });\n  }\n  _initializeFocusTrap() {\n    return new FocusTrap({\n      trapElement: this._element\n    });\n  }\n  _addEventListeners() {\n    EventHandler.on(this._element, EVENT_KEYDOWN_DISMISS, event => {\n      if (event.key !== ESCAPE_KEY) {\n        return;\n      }\n      if (this._config.keyboard) {\n        this.hide();\n        return;\n      }\n      EventHandler.trigger(this._element, EVENT_HIDE_PREVENTED);\n    });\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Offcanvas.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config](this);\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$1, SELECTOR_DATA_TOGGLE$1, function (event) {\n  const target = SelectorEngine.getElementFromSelector(this);\n  if (['A', 'AREA'].includes(this.tagName)) {\n    event.preventDefault();\n  }\n  if (isDisabled(this)) {\n    return;\n  }\n  EventHandler.one(target, EVENT_HIDDEN$3, () => {\n    // focus on trigger when it is closed\n    if (isVisible(this)) {\n      this.focus();\n    }\n  });\n\n  // avoid conflict when clicking a toggler of an offcanvas, while another is open\n  const alreadyOpen = SelectorEngine.findOne(OPEN_SELECTOR);\n  if (alreadyOpen && alreadyOpen !== target) {\n    Offcanvas.getInstance(alreadyOpen).hide();\n  }\n  const data = Offcanvas.getOrCreateInstance(target);\n  data.toggle(this);\n});\nEventHandler.on(window, EVENT_LOAD_DATA_API$2, () => {\n  for (const selector of SelectorEngine.find(OPEN_SELECTOR)) {\n    Offcanvas.getOrCreateInstance(selector).show();\n  }\n});\nEventHandler.on(window, EVENT_RESIZE, () => {\n  for (const element of SelectorEngine.find('[aria-modal][class*=show][class*=offcanvas-]')) {\n    if (getComputedStyle(element).position !== 'fixed') {\n      Offcanvas.getOrCreateInstance(element).hide();\n    }\n  }\n});\nenableDismissTrigger(Offcanvas);\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Offcanvas);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/sanitizer.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n// js-docs-start allow-list\nconst ARIA_ATTRIBUTE_PATTERN = /^aria-[\\w-]*$/i;\nconst DefaultAllowlist = {\n  // Global attributes allowed on any supplied element below.\n  '*': ['class', 'dir', 'id', 'lang', 'role', ARIA_ATTRIBUTE_PATTERN],\n  a: ['target', 'href', 'title', 'rel'],\n  area: [],\n  b: [],\n  br: [],\n  col: [],\n  code: [],\n  dd: [],\n  div: [],\n  dl: [],\n  dt: [],\n  em: [],\n  hr: [],\n  h1: [],\n  h2: [],\n  h3: [],\n  h4: [],\n  h5: [],\n  h6: [],\n  i: [],\n  img: ['src', 'srcset', 'alt', 'title', 'width', 'height'],\n  li: [],\n  ol: [],\n  p: [],\n  pre: [],\n  s: [],\n  small: [],\n  span: [],\n  sub: [],\n  sup: [],\n  strong: [],\n  u: [],\n  ul: []\n};\n// js-docs-end allow-list\n\nconst uriAttributes = new Set(['background', 'cite', 'href', 'itemtype', 'longdesc', 'poster', 'src', 'xlink:href']);\n\n/**\n * A pattern that recognizes URLs that are safe wrt. XSS in URL navigation\n * contexts.\n *\n * Shout-out to Angular https://github.com/angular/angular/blob/15.2.8/packages/core/src/sanitization/url_sanitizer.ts#L38\n */\n// eslint-disable-next-line unicorn/better-regex\nconst SAFE_URL_PATTERN = /^(?!javascript:)(?:[a-z0-9+.-]+:|[^&:/?#]*(?:[/?#]|$))/i;\nconst allowedAttribute = (attribute, allowedAttributeList) => {\n  const attributeName = attribute.nodeName.toLowerCase();\n  if (allowedAttributeList.includes(attributeName)) {\n    if (uriAttributes.has(attributeName)) {\n      return Boolean(SAFE_URL_PATTERN.test(attribute.nodeValue));\n    }\n    return true;\n  }\n\n  // Check if a regular expression validates the attribute.\n  return allowedAttributeList.filter(attributeRegex => attributeRegex instanceof RegExp).some(regex => regex.test(attributeName));\n};\nfunction sanitizeHtml(unsafeHtml, allowList, sanitizeFunction) {\n  if (!unsafeHtml.length) {\n    return unsafeHtml;\n  }\n  if (sanitizeFunction && typeof sanitizeFunction === 'function') {\n    return sanitizeFunction(unsafeHtml);\n  }\n  const domParser = new window.DOMParser();\n  const createdDocument = domParser.parseFromString(unsafeHtml, 'text/html');\n  const elements = [].concat(...createdDocument.body.querySelectorAll('*'));\n  for (const element of elements) {\n    const elementName = element.nodeName.toLowerCase();\n    if (!Object.keys(allowList).includes(elementName)) {\n      element.remove();\n      continue;\n    }\n    const attributeList = [].concat(...element.attributes);\n    const allowedAttributes = [].concat(allowList['*'] || [], allowList[elementName] || []);\n    for (const attribute of attributeList) {\n      if (!allowedAttribute(attribute, allowedAttributes)) {\n        element.removeAttribute(attribute.nodeName);\n      }\n    }\n  }\n  return createdDocument.body.innerHTML;\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/template-factory.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$5 = 'TemplateFactory';\nconst Default$4 = {\n  allowList: DefaultAllowlist,\n  content: {},\n  // { selector : text ,  selector2 : text2 , }\n  extraClass: '',\n  html: false,\n  sanitize: true,\n  sanitizeFn: null,\n  template: '<div></div>'\n};\nconst DefaultType$4 = {\n  allowList: 'object',\n  content: 'object',\n  extraClass: '(string|function)',\n  html: 'boolean',\n  sanitize: 'boolean',\n  sanitizeFn: '(null|function)',\n  template: 'string'\n};\nconst DefaultContentType = {\n  entry: '(string|element|function|null)',\n  selector: '(string|element)'\n};\n\n/**\n * Class definition\n */\n\nclass TemplateFactory extends Config {\n  constructor(config) {\n    super();\n    this._config = this._getConfig(config);\n  }\n\n  // Getters\n  static get Default() {\n    return Default$4;\n  }\n  static get DefaultType() {\n    return DefaultType$4;\n  }\n  static get NAME() {\n    return NAME$5;\n  }\n\n  // Public\n  getContent() {\n    return Object.values(this._config.content).map(config => this._resolvePossibleFunction(config)).filter(Boolean);\n  }\n  hasContent() {\n    return this.getContent().length > 0;\n  }\n  changeContent(content) {\n    this._checkContent(content);\n    this._config.content = {\n      ...this._config.content,\n      ...content\n    };\n    return this;\n  }\n  toHtml() {\n    const templateWrapper = document.createElement('div');\n    templateWrapper.innerHTML = this._maybeSanitize(this._config.template);\n    for (const [selector, text] of Object.entries(this._config.content)) {\n      this._setContent(templateWrapper, text, selector);\n    }\n    const template = templateWrapper.children[0];\n    const extraClass = this._resolvePossibleFunction(this._config.extraClass);\n    if (extraClass) {\n      template.classList.add(...extraClass.split(' '));\n    }\n    return template;\n  }\n\n  // Private\n  _typeCheckConfig(config) {\n    super._typeCheckConfig(config);\n    this._checkContent(config.content);\n  }\n  _checkContent(arg) {\n    for (const [selector, content] of Object.entries(arg)) {\n      super._typeCheckConfig({\n        selector,\n        entry: content\n      }, DefaultContentType);\n    }\n  }\n  _setContent(template, content, selector) {\n    const templateElement = SelectorEngine.findOne(selector, template);\n    if (!templateElement) {\n      return;\n    }\n    content = this._resolvePossibleFunction(content);\n    if (!content) {\n      templateElement.remove();\n      return;\n    }\n    if (isElement(content)) {\n      this._putElementInTemplate(getElement(content), templateElement);\n      return;\n    }\n    if (this._config.html) {\n      templateElement.innerHTML = this._maybeSanitize(content);\n      return;\n    }\n    templateElement.textContent = content;\n  }\n  _maybeSanitize(arg) {\n    return this._config.sanitize ? sanitizeHtml(arg, this._config.allowList, this._config.sanitizeFn) : arg;\n  }\n  _resolvePossibleFunction(arg) {\n    return execute(arg, [this]);\n  }\n  _putElementInTemplate(element, templateElement) {\n    if (this._config.html) {\n      templateElement.innerHTML = '';\n      templateElement.append(element);\n      return;\n    }\n    templateElement.textContent = element.textContent;\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap tooltip.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$4 = 'tooltip';\nconst DISALLOWED_ATTRIBUTES = new Set(['sanitize', 'allowList', 'sanitizeFn']);\nconst CLASS_NAME_FADE$2 = 'fade';\nconst CLASS_NAME_MODAL = 'modal';\nconst CLASS_NAME_SHOW$2 = 'show';\nconst SELECTOR_TOOLTIP_INNER = '.tooltip-inner';\nconst SELECTOR_MODAL = `.${CLASS_NAME_MODAL}`;\nconst EVENT_MODAL_HIDE = 'hide.bs.modal';\nconst TRIGGER_HOVER = 'hover';\nconst TRIGGER_FOCUS = 'focus';\nconst TRIGGER_CLICK = 'click';\nconst TRIGGER_MANUAL = 'manual';\nconst EVENT_HIDE$2 = 'hide';\nconst EVENT_HIDDEN$2 = 'hidden';\nconst EVENT_SHOW$2 = 'show';\nconst EVENT_SHOWN$2 = 'shown';\nconst EVENT_INSERTED = 'inserted';\nconst EVENT_CLICK$1 = 'click';\nconst EVENT_FOCUSIN$1 = 'focusin';\nconst EVENT_FOCUSOUT$1 = 'focusout';\nconst EVENT_MOUSEENTER = 'mouseenter';\nconst EVENT_MOUSELEAVE = 'mouseleave';\nconst AttachmentMap = {\n  AUTO: 'auto',\n  TOP: 'top',\n  RIGHT: isRTL() ? 'left' : 'right',\n  BOTTOM: 'bottom',\n  LEFT: isRTL() ? 'right' : 'left'\n};\nconst Default$3 = {\n  allowList: DefaultAllowlist,\n  animation: true,\n  boundary: 'clippingParents',\n  container: false,\n  customClass: '',\n  delay: 0,\n  fallbackPlacements: ['top', 'right', 'bottom', 'left'],\n  html: false,\n  offset: [0, 6],\n  placement: 'top',\n  popperConfig: null,\n  sanitize: true,\n  sanitizeFn: null,\n  selector: false,\n  template: '<div class=\"tooltip\" role=\"tooltip\">' + '<div class=\"tooltip-arrow\"></div>' + '<div class=\"tooltip-inner\"></div>' + '</div>',\n  title: '',\n  trigger: 'hover focus'\n};\nconst DefaultType$3 = {\n  allowList: 'object',\n  animation: 'boolean',\n  boundary: '(string|element)',\n  container: '(string|element|boolean)',\n  customClass: '(string|function)',\n  delay: '(number|object)',\n  fallbackPlacements: 'array',\n  html: 'boolean',\n  offset: '(array|string|function)',\n  placement: '(string|function)',\n  popperConfig: '(null|object|function)',\n  sanitize: 'boolean',\n  sanitizeFn: '(null|function)',\n  selector: '(string|boolean)',\n  template: 'string',\n  title: '(string|element|function)',\n  trigger: 'string'\n};\n\n/**\n * Class definition\n */\n\nclass Tooltip extends BaseComponent {\n  constructor(element, config) {\n    if (typeof Popper === 'undefined') {\n      throw new TypeError('Bootstrap\\'s tooltips require Popper (https://popper.js.org)');\n    }\n    super(element, config);\n\n    // Private\n    this._isEnabled = true;\n    this._timeout = 0;\n    this._isHovered = null;\n    this._activeTrigger = {};\n    this._popper = null;\n    this._templateFactory = null;\n    this._newContent = null;\n\n    // Protected\n    this.tip = null;\n    this._setListeners();\n    if (!this._config.selector) {\n      this._fixTitle();\n    }\n  }\n\n  // Getters\n  static get Default() {\n    return Default$3;\n  }\n  static get DefaultType() {\n    return DefaultType$3;\n  }\n  static get NAME() {\n    return NAME$4;\n  }\n\n  // Public\n  enable() {\n    this._isEnabled = true;\n  }\n  disable() {\n    this._isEnabled = false;\n  }\n  toggleEnabled() {\n    this._isEnabled = !this._isEnabled;\n  }\n  toggle() {\n    if (!this._isEnabled) {\n      return;\n    }\n    this._activeTrigger.click = !this._activeTrigger.click;\n    if (this._isShown()) {\n      this._leave();\n      return;\n    }\n    this._enter();\n  }\n  dispose() {\n    clearTimeout(this._timeout);\n    EventHandler.off(this._element.closest(SELECTOR_MODAL), EVENT_MODAL_HIDE, this._hideModalHandler);\n    if (this._element.getAttribute('data-bs-original-title')) {\n      this._element.setAttribute('title', this._element.getAttribute('data-bs-original-title'));\n    }\n    this._disposePopper();\n    super.dispose();\n  }\n  show() {\n    if (this._element.style.display === 'none') {\n      throw new Error('Please use show on visible elements');\n    }\n    if (!(this._isWithContent() && this._isEnabled)) {\n      return;\n    }\n    const showEvent = EventHandler.trigger(this._element, this.constructor.eventName(EVENT_SHOW$2));\n    const shadowRoot = findShadowRoot(this._element);\n    const isInTheDom = (shadowRoot || this._element.ownerDocument.documentElement).contains(this._element);\n    if (showEvent.defaultPrevented || !isInTheDom) {\n      return;\n    }\n\n    // TODO: v6 remove this or make it optional\n    this._disposePopper();\n    const tip = this._getTipElement();\n    this._element.setAttribute('aria-describedby', tip.getAttribute('id'));\n    const {\n      container\n    } = this._config;\n    if (!this._element.ownerDocument.documentElement.contains(this.tip)) {\n      container.append(tip);\n      EventHandler.trigger(this._element, this.constructor.eventName(EVENT_INSERTED));\n    }\n    this._popper = this._createPopper(tip);\n    tip.classList.add(CLASS_NAME_SHOW$2);\n\n    // If this is a touch-enabled device we add extra\n    // empty mouseover listeners to the body's immediate children;\n    // only needed because of broken event delegation on iOS\n    // https://www.quirksmode.org/blog/archives/2014/02/mouse_event_bub.html\n    if ('ontouchstart' in document.documentElement) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.on(element, 'mouseover', noop);\n      }\n    }\n    const complete = () => {\n      EventHandler.trigger(this._element, this.constructor.eventName(EVENT_SHOWN$2));\n      if (this._isHovered === false) {\n        this._leave();\n      }\n      this._isHovered = false;\n    };\n    this._queueCallback(complete, this.tip, this._isAnimated());\n  }\n  hide() {\n    if (!this._isShown()) {\n      return;\n    }\n    const hideEvent = EventHandler.trigger(this._element, this.constructor.eventName(EVENT_HIDE$2));\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    const tip = this._getTipElement();\n    tip.classList.remove(CLASS_NAME_SHOW$2);\n\n    // If this is a touch-enabled device we remove the extra\n    // empty mouseover listeners we added for iOS support\n    if ('ontouchstart' in document.documentElement) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.off(element, 'mouseover', noop);\n      }\n    }\n    this._activeTrigger[TRIGGER_CLICK] = false;\n    this._activeTrigger[TRIGGER_FOCUS] = false;\n    this._activeTrigger[TRIGGER_HOVER] = false;\n    this._isHovered = null; // it is a trick to support manual triggering\n\n    const complete = () => {\n      if (this._isWithActiveTrigger()) {\n        return;\n      }\n      if (!this._isHovered) {\n        this._disposePopper();\n      }\n      this._element.removeAttribute('aria-describedby');\n      EventHandler.trigger(this._element, this.constructor.eventName(EVENT_HIDDEN$2));\n    };\n    this._queueCallback(complete, this.tip, this._isAnimated());\n  }\n  update() {\n    if (this._popper) {\n      this._popper.update();\n    }\n  }\n\n  // Protected\n  _isWithContent() {\n    return Boolean(this._getTitle());\n  }\n  _getTipElement() {\n    if (!this.tip) {\n      this.tip = this._createTipElement(this._newContent || this._getContentForTemplate());\n    }\n    return this.tip;\n  }\n  _createTipElement(content) {\n    const tip = this._getTemplateFactory(content).toHtml();\n\n    // TODO: remove this check in v6\n    if (!tip) {\n      return null;\n    }\n    tip.classList.remove(CLASS_NAME_FADE$2, CLASS_NAME_SHOW$2);\n    // TODO: v6 the following can be achieved with CSS only\n    tip.classList.add(`bs-${this.constructor.NAME}-auto`);\n    const tipId = getUID(this.constructor.NAME).toString();\n    tip.setAttribute('id', tipId);\n    if (this._isAnimated()) {\n      tip.classList.add(CLASS_NAME_FADE$2);\n    }\n    return tip;\n  }\n  setContent(content) {\n    this._newContent = content;\n    if (this._isShown()) {\n      this._disposePopper();\n      this.show();\n    }\n  }\n  _getTemplateFactory(content) {\n    if (this._templateFactory) {\n      this._templateFactory.changeContent(content);\n    } else {\n      this._templateFactory = new TemplateFactory({\n        ...this._config,\n        // the `content` var has to be after `this._config`\n        // to override config.content in case of popover\n        content,\n        extraClass: this._resolvePossibleFunction(this._config.customClass)\n      });\n    }\n    return this._templateFactory;\n  }\n  _getContentForTemplate() {\n    return {\n      [SELECTOR_TOOLTIP_INNER]: this._getTitle()\n    };\n  }\n  _getTitle() {\n    return this._resolvePossibleFunction(this._config.title) || this._element.getAttribute('data-bs-original-title');\n  }\n\n  // Private\n  _initializeOnDelegatedTarget(event) {\n    return this.constructor.getOrCreateInstance(event.delegateTarget, this._getDelegateConfig());\n  }\n  _isAnimated() {\n    return this._config.animation || this.tip && this.tip.classList.contains(CLASS_NAME_FADE$2);\n  }\n  _isShown() {\n    return this.tip && this.tip.classList.contains(CLASS_NAME_SHOW$2);\n  }\n  _createPopper(tip) {\n    const placement = execute(this._config.placement, [this, tip, this._element]);\n    const attachment = AttachmentMap[placement.toUpperCase()];\n    return Popper.createPopper(this._element, tip, this._getPopperConfig(attachment));\n  }\n  _getOffset() {\n    const {\n      offset\n    } = this._config;\n    if (typeof offset === 'string') {\n      return offset.split(',').map(value => Number.parseInt(value, 10));\n    }\n    if (typeof offset === 'function') {\n      return popperData => offset(popperData, this._element);\n    }\n    return offset;\n  }\n  _resolvePossibleFunction(arg) {\n    return execute(arg, [this._element]);\n  }\n  _getPopperConfig(attachment) {\n    const defaultBsPopperConfig = {\n      placement: attachment,\n      modifiers: [{\n        name: 'flip',\n        options: {\n          fallbackPlacements: this._config.fallbackPlacements\n        }\n      }, {\n        name: 'offset',\n        options: {\n          offset: this._getOffset()\n        }\n      }, {\n        name: 'preventOverflow',\n        options: {\n          boundary: this._config.boundary\n        }\n      }, {\n        name: 'arrow',\n        options: {\n          element: `.${this.constructor.NAME}-arrow`\n        }\n      }, {\n        name: 'preSetPlacement',\n        enabled: true,\n        phase: 'beforeMain',\n        fn: data => {\n          // Pre-set Popper's placement attribute in order to read the arrow sizes properly.\n          // Otherwise, Popper mixes up the width and height dimensions since the initial arrow style is for top placement\n          this._getTipElement().setAttribute('data-popper-placement', data.state.placement);\n        }\n      }]\n    };\n    return {\n      ...defaultBsPopperConfig,\n      ...execute(this._config.popperConfig, [defaultBsPopperConfig])\n    };\n  }\n  _setListeners() {\n    const triggers = this._config.trigger.split(' ');\n    for (const trigger of triggers) {\n      if (trigger === 'click') {\n        EventHandler.on(this._element, this.constructor.eventName(EVENT_CLICK$1), this._config.selector, event => {\n          const context = this._initializeOnDelegatedTarget(event);\n          context.toggle();\n        });\n      } else if (trigger !== TRIGGER_MANUAL) {\n        const eventIn = trigger === TRIGGER_HOVER ? this.constructor.eventName(EVENT_MOUSEENTER) : this.constructor.eventName(EVENT_FOCUSIN$1);\n        const eventOut = trigger === TRIGGER_HOVER ? this.constructor.eventName(EVENT_MOUSELEAVE) : this.constructor.eventName(EVENT_FOCUSOUT$1);\n        EventHandler.on(this._element, eventIn, this._config.selector, event => {\n          const context = this._initializeOnDelegatedTarget(event);\n          context._activeTrigger[event.type === 'focusin' ? TRIGGER_FOCUS : TRIGGER_HOVER] = true;\n          context._enter();\n        });\n        EventHandler.on(this._element, eventOut, this._config.selector, event => {\n          const context = this._initializeOnDelegatedTarget(event);\n          context._activeTrigger[event.type === 'focusout' ? TRIGGER_FOCUS : TRIGGER_HOVER] = context._element.contains(event.relatedTarget);\n          context._leave();\n        });\n      }\n    }\n    this._hideModalHandler = () => {\n      if (this._element) {\n        this.hide();\n      }\n    };\n    EventHandler.on(this._element.closest(SELECTOR_MODAL), EVENT_MODAL_HIDE, this._hideModalHandler);\n  }\n  _fixTitle() {\n    const title = this._element.getAttribute('title');\n    if (!title) {\n      return;\n    }\n    if (!this._element.getAttribute('aria-label') && !this._element.textContent.trim()) {\n      this._element.setAttribute('aria-label', title);\n    }\n    this._element.setAttribute('data-bs-original-title', title); // DO NOT USE IT. Is only for backwards compatibility\n    this._element.removeAttribute('title');\n  }\n  _enter() {\n    if (this._isShown() || this._isHovered) {\n      this._isHovered = true;\n      return;\n    }\n    this._isHovered = true;\n    this._setTimeout(() => {\n      if (this._isHovered) {\n        this.show();\n      }\n    }, this._config.delay.show);\n  }\n  _leave() {\n    if (this._isWithActiveTrigger()) {\n      return;\n    }\n    this._isHovered = false;\n    this._setTimeout(() => {\n      if (!this._isHovered) {\n        this.hide();\n      }\n    }, this._config.delay.hide);\n  }\n  _setTimeout(handler, timeout) {\n    clearTimeout(this._timeout);\n    this._timeout = setTimeout(handler, timeout);\n  }\n  _isWithActiveTrigger() {\n    return Object.values(this._activeTrigger).includes(true);\n  }\n  _getConfig(config) {\n    const dataAttributes = Manipulator.getDataAttributes(this._element);\n    for (const dataAttribute of Object.keys(dataAttributes)) {\n      if (DISALLOWED_ATTRIBUTES.has(dataAttribute)) {\n        delete dataAttributes[dataAttribute];\n      }\n    }\n    config = {\n      ...dataAttributes,\n      ...(typeof config === 'object' && config ? config : {})\n    };\n    config = this._mergeConfigObj(config);\n    config = this._configAfterMerge(config);\n    this._typeCheckConfig(config);\n    return config;\n  }\n  _configAfterMerge(config) {\n    config.container = config.container === false ? document.body : getElement(config.container);\n    if (typeof config.delay === 'number') {\n      config.delay = {\n        show: config.delay,\n        hide: config.delay\n      };\n    }\n    if (typeof config.title === 'number') {\n      config.title = config.title.toString();\n    }\n    if (typeof config.content === 'number') {\n      config.content = config.content.toString();\n    }\n    return config;\n  }\n  _getDelegateConfig() {\n    const config = {};\n    for (const [key, value] of Object.entries(this._config)) {\n      if (this.constructor.Default[key] !== value) {\n        config[key] = value;\n      }\n    }\n    config.selector = false;\n    config.trigger = 'manual';\n\n    // In the future can be replaced with:\n    // const keysWithDifferentValues = Object.entries(this._config).filter(entry => this.constructor.Default[entry[0]] !== this._config[entry[0]])\n    // `Object.fromEntries(keysWithDifferentValues)`\n    return config;\n  }\n  _disposePopper() {\n    if (this._popper) {\n      this._popper.destroy();\n      this._popper = null;\n    }\n    if (this.tip) {\n      this.tip.remove();\n      this.tip = null;\n    }\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Tooltip.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n}\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Tooltip);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap popover.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$3 = 'popover';\nconst SELECTOR_TITLE = '.popover-header';\nconst SELECTOR_CONTENT = '.popover-body';\nconst Default$2 = {\n  ...Tooltip.Default,\n  content: '',\n  offset: [0, 8],\n  placement: 'right',\n  template: '<div class=\"popover\" role=\"tooltip\">' + '<div class=\"popover-arrow\"></div>' + '<h3 class=\"popover-header\"></h3>' + '<div class=\"popover-body\"></div>' + '</div>',\n  trigger: 'click'\n};\nconst DefaultType$2 = {\n  ...Tooltip.DefaultType,\n  content: '(null|string|element|function)'\n};\n\n/**\n * Class definition\n */\n\nclass Popover extends Tooltip {\n  // Getters\n  static get Default() {\n    return Default$2;\n  }\n  static get DefaultType() {\n    return DefaultType$2;\n  }\n  static get NAME() {\n    return NAME$3;\n  }\n\n  // Overrides\n  _isWithContent() {\n    return this._getTitle() || this._getContent();\n  }\n\n  // Private\n  _getContentForTemplate() {\n    return {\n      [SELECTOR_TITLE]: this._getTitle(),\n      [SELECTOR_CONTENT]: this._getContent()\n    };\n  }\n  _getContent() {\n    return this._resolvePossibleFunction(this._config.content);\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Popover.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n}\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Popover);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap scrollspy.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$2 = 'scrollspy';\nconst DATA_KEY$2 = 'bs.scrollspy';\nconst EVENT_KEY$2 = `.${DATA_KEY$2}`;\nconst DATA_API_KEY = '.data-api';\nconst EVENT_ACTIVATE = `activate${EVENT_KEY$2}`;\nconst EVENT_CLICK = `click${EVENT_KEY$2}`;\nconst EVENT_LOAD_DATA_API$1 = `load${EVENT_KEY$2}${DATA_API_KEY}`;\nconst CLASS_NAME_DROPDOWN_ITEM = 'dropdown-item';\nconst CLASS_NAME_ACTIVE$1 = 'active';\nconst SELECTOR_DATA_SPY = '[data-bs-spy=\"scroll\"]';\nconst SELECTOR_TARGET_LINKS = '[href]';\nconst SELECTOR_NAV_LIST_GROUP = '.nav, .list-group';\nconst SELECTOR_NAV_LINKS = '.nav-link';\nconst SELECTOR_NAV_ITEMS = '.nav-item';\nconst SELECTOR_LIST_ITEMS = '.list-group-item';\nconst SELECTOR_LINK_ITEMS = `${SELECTOR_NAV_LINKS}, ${SELECTOR_NAV_ITEMS} > ${SELECTOR_NAV_LINKS}, ${SELECTOR_LIST_ITEMS}`;\nconst SELECTOR_DROPDOWN = '.dropdown';\nconst SELECTOR_DROPDOWN_TOGGLE$1 = '.dropdown-toggle';\nconst Default$1 = {\n  offset: null,\n  // TODO: v6 @deprecated, keep it for backwards compatibility reasons\n  rootMargin: '0px 0px -25%',\n  smoothScroll: false,\n  target: null,\n  threshold: [0.1, 0.5, 1]\n};\nconst DefaultType$1 = {\n  offset: '(number|null)',\n  // TODO v6 @deprecated, keep it for backwards compatibility reasons\n  rootMargin: 'string',\n  smoothScroll: 'boolean',\n  target: 'element',\n  threshold: 'array'\n};\n\n/**\n * Class definition\n */\n\nclass ScrollSpy extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n\n    // this._element is the observablesContainer and config.target the menu links wrapper\n    this._targetLinks = new Map();\n    this._observableSections = new Map();\n    this._rootElement = getComputedStyle(this._element).overflowY === 'visible' ? null : this._element;\n    this._activeTarget = null;\n    this._observer = null;\n    this._previousScrollData = {\n      visibleEntryTop: 0,\n      parentScrollTop: 0\n    };\n    this.refresh(); // initialize\n  }\n\n  // Getters\n  static get Default() {\n    return Default$1;\n  }\n  static get DefaultType() {\n    return DefaultType$1;\n  }\n  static get NAME() {\n    return NAME$2;\n  }\n\n  // Public\n  refresh() {\n    this._initializeTargetsAndObservables();\n    this._maybeEnableSmoothScroll();\n    if (this._observer) {\n      this._observer.disconnect();\n    } else {\n      this._observer = this._getNewObserver();\n    }\n    for (const section of this._observableSections.values()) {\n      this._observer.observe(section);\n    }\n  }\n  dispose() {\n    this._observer.disconnect();\n    super.dispose();\n  }\n\n  // Private\n  _configAfterMerge(config) {\n    // TODO: on v6 target should be given explicitly & remove the {target: 'ss-target'} case\n    config.target = getElement(config.target) || document.body;\n\n    // TODO: v6 Only for backwards compatibility reasons. Use rootMargin only\n    config.rootMargin = config.offset ? `${config.offset}px 0px -30%` : config.rootMargin;\n    if (typeof config.threshold === 'string') {\n      config.threshold = config.threshold.split(',').map(value => Number.parseFloat(value));\n    }\n    return config;\n  }\n  _maybeEnableSmoothScroll() {\n    if (!this._config.smoothScroll) {\n      return;\n    }\n\n    // unregister any previous listeners\n    EventHandler.off(this._config.target, EVENT_CLICK);\n    EventHandler.on(this._config.target, EVENT_CLICK, SELECTOR_TARGET_LINKS, event => {\n      const observableSection = this._observableSections.get(event.target.hash);\n      if (observableSection) {\n        event.preventDefault();\n        const root = this._rootElement || window;\n        const height = observableSection.offsetTop - this._element.offsetTop;\n        if (root.scrollTo) {\n          root.scrollTo({\n            top: height,\n            behavior: 'smooth'\n          });\n          return;\n        }\n\n        // Chrome 60 doesn't support `scrollTo`\n        root.scrollTop = height;\n      }\n    });\n  }\n  _getNewObserver() {\n    const options = {\n      root: this._rootElement,\n      threshold: this._config.threshold,\n      rootMargin: this._config.rootMargin\n    };\n    return new IntersectionObserver(entries => this._observerCallback(entries), options);\n  }\n\n  // The logic of selection\n  _observerCallback(entries) {\n    const targetElement = entry => this._targetLinks.get(`#${entry.target.id}`);\n    const activate = entry => {\n      this._previousScrollData.visibleEntryTop = entry.target.offsetTop;\n      this._process(targetElement(entry));\n    };\n    const parentScrollTop = (this._rootElement || document.documentElement).scrollTop;\n    const userScrollsDown = parentScrollTop >= this._previousScrollData.parentScrollTop;\n    this._previousScrollData.parentScrollTop = parentScrollTop;\n    for (const entry of entries) {\n      if (!entry.isIntersecting) {\n        this._activeTarget = null;\n        this._clearActiveClass(targetElement(entry));\n        continue;\n      }\n      const entryIsLowerThanPrevious = entry.target.offsetTop >= this._previousScrollData.visibleEntryTop;\n      // if we are scrolling down, pick the bigger offsetTop\n      if (userScrollsDown && entryIsLowerThanPrevious) {\n        activate(entry);\n        // if parent isn't scrolled, let's keep the first visible item, breaking the iteration\n        if (!parentScrollTop) {\n          return;\n        }\n        continue;\n      }\n\n      // if we are scrolling up, pick the smallest offsetTop\n      if (!userScrollsDown && !entryIsLowerThanPrevious) {\n        activate(entry);\n      }\n    }\n  }\n  _initializeTargetsAndObservables() {\n    this._targetLinks = new Map();\n    this._observableSections = new Map();\n    const targetLinks = SelectorEngine.find(SELECTOR_TARGET_LINKS, this._config.target);\n    for (const anchor of targetLinks) {\n      // ensure that the anchor has an id and is not disabled\n      if (!anchor.hash || isDisabled(anchor)) {\n        continue;\n      }\n      const observableSection = SelectorEngine.findOne(decodeURI(anchor.hash), this._element);\n\n      // ensure that the observableSection exists & is visible\n      if (isVisible(observableSection)) {\n        this._targetLinks.set(decodeURI(anchor.hash), anchor);\n        this._observableSections.set(anchor.hash, observableSection);\n      }\n    }\n  }\n  _process(target) {\n    if (this._activeTarget === target) {\n      return;\n    }\n    this._clearActiveClass(this._config.target);\n    this._activeTarget = target;\n    target.classList.add(CLASS_NAME_ACTIVE$1);\n    this._activateParents(target);\n    EventHandler.trigger(this._element, EVENT_ACTIVATE, {\n      relatedTarget: target\n    });\n  }\n  _activateParents(target) {\n    // Activate dropdown parents\n    if (target.classList.contains(CLASS_NAME_DROPDOWN_ITEM)) {\n      SelectorEngine.findOne(SELECTOR_DROPDOWN_TOGGLE$1, target.closest(SELECTOR_DROPDOWN)).classList.add(CLASS_NAME_ACTIVE$1);\n      return;\n    }\n    for (const listGroup of SelectorEngine.parents(target, SELECTOR_NAV_LIST_GROUP)) {\n      // Set triggered links parents as active\n      // With both <ul> and <nav> markup a parent is the previous sibling of any nav ancestor\n      for (const item of SelectorEngine.prev(listGroup, SELECTOR_LINK_ITEMS)) {\n        item.classList.add(CLASS_NAME_ACTIVE$1);\n      }\n    }\n  }\n  _clearActiveClass(parent) {\n    parent.classList.remove(CLASS_NAME_ACTIVE$1);\n    const activeNodes = SelectorEngine.find(`${SELECTOR_TARGET_LINKS}.${CLASS_NAME_ACTIVE$1}`, parent);\n    for (const node of activeNodes) {\n      node.classList.remove(CLASS_NAME_ACTIVE$1);\n    }\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = ScrollSpy.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(window, EVENT_LOAD_DATA_API$1, () => {\n  for (const spy of SelectorEngine.find(SELECTOR_DATA_SPY)) {\n    ScrollSpy.getOrCreateInstance(spy);\n  }\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(ScrollSpy);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap tab.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$1 = 'tab';\nconst DATA_KEY$1 = 'bs.tab';\nconst EVENT_KEY$1 = `.${DATA_KEY$1}`;\nconst EVENT_HIDE$1 = `hide${EVENT_KEY$1}`;\nconst EVENT_HIDDEN$1 = `hidden${EVENT_KEY$1}`;\nconst EVENT_SHOW$1 = `show${EVENT_KEY$1}`;\nconst EVENT_SHOWN$1 = `shown${EVENT_KEY$1}`;\nconst EVENT_CLICK_DATA_API = `click${EVENT_KEY$1}`;\nconst EVENT_KEYDOWN = `keydown${EVENT_KEY$1}`;\nconst EVENT_LOAD_DATA_API = `load${EVENT_KEY$1}`;\nconst ARROW_LEFT_KEY = 'ArrowLeft';\nconst ARROW_RIGHT_KEY = 'ArrowRight';\nconst ARROW_UP_KEY = 'ArrowUp';\nconst ARROW_DOWN_KEY = 'ArrowDown';\nconst HOME_KEY = 'Home';\nconst END_KEY = 'End';\nconst CLASS_NAME_ACTIVE = 'active';\nconst CLASS_NAME_FADE$1 = 'fade';\nconst CLASS_NAME_SHOW$1 = 'show';\nconst CLASS_DROPDOWN = 'dropdown';\nconst SELECTOR_DROPDOWN_TOGGLE = '.dropdown-toggle';\nconst SELECTOR_DROPDOWN_MENU = '.dropdown-menu';\nconst NOT_SELECTOR_DROPDOWN_TOGGLE = `:not(${SELECTOR_DROPDOWN_TOGGLE})`;\nconst SELECTOR_TAB_PANEL = '.list-group, .nav, [role=\"tablist\"]';\nconst SELECTOR_OUTER = '.nav-item, .list-group-item';\nconst SELECTOR_INNER = `.nav-link${NOT_SELECTOR_DROPDOWN_TOGGLE}, .list-group-item${NOT_SELECTOR_DROPDOWN_TOGGLE}, [role=\"tab\"]${NOT_SELECTOR_DROPDOWN_TOGGLE}`;\nconst SELECTOR_DATA_TOGGLE = '[data-bs-toggle=\"tab\"], [data-bs-toggle=\"pill\"], [data-bs-toggle=\"list\"]'; // TODO: could only be `tab` in v6\nconst SELECTOR_INNER_ELEM = `${SELECTOR_INNER}, ${SELECTOR_DATA_TOGGLE}`;\nconst SELECTOR_DATA_TOGGLE_ACTIVE = `.${CLASS_NAME_ACTIVE}[data-bs-toggle=\"tab\"], .${CLASS_NAME_ACTIVE}[data-bs-toggle=\"pill\"], .${CLASS_NAME_ACTIVE}[data-bs-toggle=\"list\"]`;\n\n/**\n * Class definition\n */\n\nclass Tab extends BaseComponent {\n  constructor(element) {\n    super(element);\n    this._parent = this._element.closest(SELECTOR_TAB_PANEL);\n    if (!this._parent) {\n      return;\n      // TODO: should throw exception in v6\n      // throw new TypeError(`${element.outerHTML} has not a valid parent ${SELECTOR_INNER_ELEM}`)\n    }\n\n    // Set up initial aria attributes\n    this._setInitialAttributes(this._parent, this._getChildren());\n    EventHandler.on(this._element, EVENT_KEYDOWN, event => this._keydown(event));\n  }\n\n  // Getters\n  static get NAME() {\n    return NAME$1;\n  }\n\n  // Public\n  show() {\n    // Shows this elem and deactivate the active sibling if exists\n    const innerElem = this._element;\n    if (this._elemIsActive(innerElem)) {\n      return;\n    }\n\n    // Search for active tab on same parent to deactivate it\n    const active = this._getActiveElem();\n    const hideEvent = active ? EventHandler.trigger(active, EVENT_HIDE$1, {\n      relatedTarget: innerElem\n    }) : null;\n    const showEvent = EventHandler.trigger(innerElem, EVENT_SHOW$1, {\n      relatedTarget: active\n    });\n    if (showEvent.defaultPrevented || hideEvent && hideEvent.defaultPrevented) {\n      return;\n    }\n    this._deactivate(active, innerElem);\n    this._activate(innerElem, active);\n  }\n\n  // Private\n  _activate(element, relatedElem) {\n    if (!element) {\n      return;\n    }\n    element.classList.add(CLASS_NAME_ACTIVE);\n    this._activate(SelectorEngine.getElementFromSelector(element)); // Search and activate/show the proper section\n\n    const complete = () => {\n      if (element.getAttribute('role') !== 'tab') {\n        element.classList.add(CLASS_NAME_SHOW$1);\n        return;\n      }\n      element.removeAttribute('tabindex');\n      element.setAttribute('aria-selected', true);\n      this._toggleDropDown(element, true);\n      EventHandler.trigger(element, EVENT_SHOWN$1, {\n        relatedTarget: relatedElem\n      });\n    };\n    this._queueCallback(complete, element, element.classList.contains(CLASS_NAME_FADE$1));\n  }\n  _deactivate(element, relatedElem) {\n    if (!element) {\n      return;\n    }\n    element.classList.remove(CLASS_NAME_ACTIVE);\n    element.blur();\n    this._deactivate(SelectorEngine.getElementFromSelector(element)); // Search and deactivate the shown section too\n\n    const complete = () => {\n      if (element.getAttribute('role') !== 'tab') {\n        element.classList.remove(CLASS_NAME_SHOW$1);\n        return;\n      }\n      element.setAttribute('aria-selected', false);\n      element.setAttribute('tabindex', '-1');\n      this._toggleDropDown(element, false);\n      EventHandler.trigger(element, EVENT_HIDDEN$1, {\n        relatedTarget: relatedElem\n      });\n    };\n    this._queueCallback(complete, element, element.classList.contains(CLASS_NAME_FADE$1));\n  }\n  _keydown(event) {\n    if (![ARROW_LEFT_KEY, ARROW_RIGHT_KEY, ARROW_UP_KEY, ARROW_DOWN_KEY, HOME_KEY, END_KEY].includes(event.key)) {\n      return;\n    }\n    event.stopPropagation(); // stopPropagation/preventDefault both added to support up/down keys without scrolling the page\n    event.preventDefault();\n    const children = this._getChildren().filter(element => !isDisabled(element));\n    let nextActiveElement;\n    if ([HOME_KEY, END_KEY].includes(event.key)) {\n      nextActiveElement = children[event.key === HOME_KEY ? 0 : children.length - 1];\n    } else {\n      const isNext = [ARROW_RIGHT_KEY, ARROW_DOWN_KEY].includes(event.key);\n      nextActiveElement = getNextActiveElement(children, event.target, isNext, true);\n    }\n    if (nextActiveElement) {\n      nextActiveElement.focus({\n        preventScroll: true\n      });\n      Tab.getOrCreateInstance(nextActiveElement).show();\n    }\n  }\n  _getChildren() {\n    // collection of inner elements\n    return SelectorEngine.find(SELECTOR_INNER_ELEM, this._parent);\n  }\n  _getActiveElem() {\n    return this._getChildren().find(child => this._elemIsActive(child)) || null;\n  }\n  _setInitialAttributes(parent, children) {\n    this._setAttributeIfNotExists(parent, 'role', 'tablist');\n    for (const child of children) {\n      this._setInitialAttributesOnChild(child);\n    }\n  }\n  _setInitialAttributesOnChild(child) {\n    child = this._getInnerElement(child);\n    const isActive = this._elemIsActive(child);\n    const outerElem = this._getOuterElement(child);\n    child.setAttribute('aria-selected', isActive);\n    if (outerElem !== child) {\n      this._setAttributeIfNotExists(outerElem, 'role', 'presentation');\n    }\n    if (!isActive) {\n      child.setAttribute('tabindex', '-1');\n    }\n    this._setAttributeIfNotExists(child, 'role', 'tab');\n\n    // set attributes to the related panel too\n    this._setInitialAttributesOnTargetPanel(child);\n  }\n  _setInitialAttributesOnTargetPanel(child) {\n    const target = SelectorEngine.getElementFromSelector(child);\n    if (!target) {\n      return;\n    }\n    this._setAttributeIfNotExists(target, 'role', 'tabpanel');\n    if (child.id) {\n      this._setAttributeIfNotExists(target, 'aria-labelledby', `${child.id}`);\n    }\n  }\n  _toggleDropDown(element, open) {\n    const outerElem = this._getOuterElement(element);\n    if (!outerElem.classList.contains(CLASS_DROPDOWN)) {\n      return;\n    }\n    const toggle = (selector, className) => {\n      const element = SelectorEngine.findOne(selector, outerElem);\n      if (element) {\n        element.classList.toggle(className, open);\n      }\n    };\n    toggle(SELECTOR_DROPDOWN_TOGGLE, CLASS_NAME_ACTIVE);\n    toggle(SELECTOR_DROPDOWN_MENU, CLASS_NAME_SHOW$1);\n    outerElem.setAttribute('aria-expanded', open);\n  }\n  _setAttributeIfNotExists(element, attribute, value) {\n    if (!element.hasAttribute(attribute)) {\n      element.setAttribute(attribute, value);\n    }\n  }\n  _elemIsActive(elem) {\n    return elem.classList.contains(CLASS_NAME_ACTIVE);\n  }\n\n  // Try to get the inner element (usually the .nav-link)\n  _getInnerElement(elem) {\n    return elem.matches(SELECTOR_INNER_ELEM) ? elem : SelectorEngine.findOne(SELECTOR_INNER_ELEM, elem);\n  }\n\n  // Try to get the outer element (usually the .nav-item)\n  _getOuterElement(elem) {\n    return elem.closest(SELECTOR_OUTER) || elem;\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Tab.getOrCreateInstance(this);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API, SELECTOR_DATA_TOGGLE, function (event) {\n  if (['A', 'AREA'].includes(this.tagName)) {\n    event.preventDefault();\n  }\n  if (isDisabled(this)) {\n    return;\n  }\n  Tab.getOrCreateInstance(this).show();\n});\n\n/**\n * Initialize on focus\n */\nEventHandler.on(window, EVENT_LOAD_DATA_API, () => {\n  for (const element of SelectorEngine.find(SELECTOR_DATA_TOGGLE_ACTIVE)) {\n    Tab.getOrCreateInstance(element);\n  }\n});\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Tab);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap toast.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME = 'toast';\nconst DATA_KEY = 'bs.toast';\nconst EVENT_KEY = `.${DATA_KEY}`;\nconst EVENT_MOUSEOVER = `mouseover${EVENT_KEY}`;\nconst EVENT_MOUSEOUT = `mouseout${EVENT_KEY}`;\nconst EVENT_FOCUSIN = `focusin${EVENT_KEY}`;\nconst EVENT_FOCUSOUT = `focusout${EVENT_KEY}`;\nconst EVENT_HIDE = `hide${EVENT_KEY}`;\nconst EVENT_HIDDEN = `hidden${EVENT_KEY}`;\nconst EVENT_SHOW = `show${EVENT_KEY}`;\nconst EVENT_SHOWN = `shown${EVENT_KEY}`;\nconst CLASS_NAME_FADE = 'fade';\nconst CLASS_NAME_HIDE = 'hide'; // @deprecated - kept here only for backwards compatibility\nconst CLASS_NAME_SHOW = 'show';\nconst CLASS_NAME_SHOWING = 'showing';\nconst DefaultType = {\n  animation: 'boolean',\n  autohide: 'boolean',\n  delay: 'number'\n};\nconst Default = {\n  animation: true,\n  autohide: true,\n  delay: 5000\n};\n\n/**\n * Class definition\n */\n\nclass Toast extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._timeout = null;\n    this._hasMouseInteraction = false;\n    this._hasKeyboardInteraction = false;\n    this._setListeners();\n  }\n\n  // Getters\n  static get Default() {\n    return Default;\n  }\n  static get DefaultType() {\n    return DefaultType;\n  }\n  static get NAME() {\n    return NAME;\n  }\n\n  // Public\n  show() {\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW);\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n    this._clearTimeout();\n    if (this._config.animation) {\n      this._element.classList.add(CLASS_NAME_FADE);\n    }\n    const complete = () => {\n      this._element.classList.remove(CLASS_NAME_SHOWING);\n      EventHandler.trigger(this._element, EVENT_SHOWN);\n      this._maybeScheduleHide();\n    };\n    this._element.classList.remove(CLASS_NAME_HIDE); // @deprecated\n    reflow(this._element);\n    this._element.classList.add(CLASS_NAME_SHOW, CLASS_NAME_SHOWING);\n    this._queueCallback(complete, this._element, this._config.animation);\n  }\n  hide() {\n    if (!this.isShown()) {\n      return;\n    }\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    const complete = () => {\n      this._element.classList.add(CLASS_NAME_HIDE); // @deprecated\n      this._element.classList.remove(CLASS_NAME_SHOWING, CLASS_NAME_SHOW);\n      EventHandler.trigger(this._element, EVENT_HIDDEN);\n    };\n    this._element.classList.add(CLASS_NAME_SHOWING);\n    this._queueCallback(complete, this._element, this._config.animation);\n  }\n  dispose() {\n    this._clearTimeout();\n    if (this.isShown()) {\n      this._element.classList.remove(CLASS_NAME_SHOW);\n    }\n    super.dispose();\n  }\n  isShown() {\n    return this._element.classList.contains(CLASS_NAME_SHOW);\n  }\n\n  // Private\n\n  _maybeScheduleHide() {\n    if (!this._config.autohide) {\n      return;\n    }\n    if (this._hasMouseInteraction || this._hasKeyboardInteraction) {\n      return;\n    }\n    this._timeout = setTimeout(() => {\n      this.hide();\n    }, this._config.delay);\n  }\n  _onInteraction(event, isInteracting) {\n    switch (event.type) {\n      case 'mouseover':\n      case 'mouseout':\n        {\n          this._hasMouseInteraction = isInteracting;\n          break;\n        }\n      case 'focusin':\n      case 'focusout':\n        {\n          this._hasKeyboardInteraction = isInteracting;\n          break;\n        }\n    }\n    if (isInteracting) {\n      this._clearTimeout();\n      return;\n    }\n    const nextElement = event.relatedTarget;\n    if (this._element === nextElement || this._element.contains(nextElement)) {\n      return;\n    }\n    this._maybeScheduleHide();\n  }\n  _setListeners() {\n    EventHandler.on(this._element, EVENT_MOUSEOVER, event => this._onInteraction(event, true));\n    EventHandler.on(this._element, EVENT_MOUSEOUT, event => this._onInteraction(event, false));\n    EventHandler.on(this._element, EVENT_FOCUSIN, event => this._onInteraction(event, true));\n    EventHandler.on(this._element, EVENT_FOCUSOUT, event => this._onInteraction(event, false));\n  }\n  _clearTimeout() {\n    clearTimeout(this._timeout);\n    this._timeout = null;\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Toast.getOrCreateInstance(this, config);\n      if (typeof config === 'string') {\n        if (typeof data[config] === 'undefined') {\n          throw new TypeError(`No method named \"${config}\"`);\n        }\n        data[config](this);\n      }\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nenableDismissTrigger(Toast);\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Toast);\n\nexport { Alert, Button, Carousel, Collapse, Dropdown, Modal, Offcanvas, Popover, ScrollSpy, Tab, Toast, Tooltip };\n//# sourceMappingURL=bootstrap.esm.js.map\n","/* define several functions to replace jQuery methods\n * inspired by https://tobiasahlin.com/blog/move-from-jquery-to-vanilla-javascript/\n */\n\n/**\n * Execute a method if DOM has finished loading\n *\n * @param {function} callback the method to execute\n */\nexport function documentReady(callback) {\n  if 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define several functions to replace jQuery methods\n * inspired by https://tobiasahlin.com/blog/move-from-jquery-to-vanilla-javascript/\n */\n\n/**\n * Execute a method if DOM has finished loading\n *\n * @param {function} callback the method to execute\n */\nexport function documentReady(callback) {\n  if (document.readyState != \"loading\") callback();\n  else document.addEventListener(\"DOMContentLoaded\", callback);\n}\n","/**\n * Compare [semver](https://semver.org/) version strings to find greater, equal or lesser.\n * This library supports the full semver specification, including comparing versions with different number of digits like `1.0.0`, `1.0`, `1`, and pre-release versions like `1.0.0-alpha`.\n * @param v1 - First version to compare\n * @param v2 - Second version to compare\n * @returns Numeric value compatible with the [Array.sort(fn) interface](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/sort#Parameters).\n */\nexport const compareVersions = (v1, v2) => {\n    // validate input and split into segments\n    const n1 = validateAndParse(v1);\n    const n2 = validateAndParse(v2);\n    // pop off the patch\n    const p1 = n1.pop();\n    const p2 = n2.pop();\n    // validate numbers\n    const r = compareSegments(n1, n2);\n    if (r !== 0)\n        return r;\n    // validate pre-release\n    if (p1 && p2) {\n        return compareSegments(p1.split('.'), p2.split('.'));\n    }\n    else if (p1 || p2) {\n        return p1 ? -1 : 1;\n    }\n    return 0;\n};\n/**\n * Validate [semver](https://semver.org/) version strings.\n *\n * @param version Version number to validate\n * @returns `true` if the version number is a valid semver version number, `false` otherwise.\n *\n * @example\n * ```\n * validate('1.0.0-rc.1'); // return true\n * validate('1.0-rc.1'); // return false\n * validate('foo'); // return false\n * ```\n */\nexport const validate = (version) => typeof version === 'string' && /^[v\\d]/.test(version) && semver.test(version);\n/**\n * Compare [semver](https://semver.org/) version strings using the specified operator.\n *\n * @param v1 First version to compare\n * @param v2 Second version to compare\n * @param operator Allowed arithmetic operator to use\n * @returns `true` if the comparison between the firstVersion and the secondVersion satisfies the operator, `false` otherwise.\n *\n * @example\n * ```\n * compare('10.1.8', '10.0.4', '>'); // return true\n * compare('10.0.1', '10.0.1', '='); // return true\n * compare('10.1.1', '10.2.2', '<'); // return true\n * compare('10.1.1', '10.2.2', '<='); // return true\n * compare('10.1.1', '10.2.2', '>='); // return false\n * ```\n */\nexport const compare = (v1, v2, operator) => {\n    // validate input operator\n    assertValidOperator(operator);\n    // since result of compareVersions can only be -1 or 0 or 1\n    // a simple map can be used to replace switch\n    const res = compareVersions(v1, v2);\n    return operatorResMap[operator].includes(res);\n};\n/**\n * Match [npm semver](https://docs.npmjs.com/cli/v6/using-npm/semver) version range.\n *\n * @param version Version number to match\n * @param range Range pattern for version\n * @returns `true` if the version number is within the range, `false` otherwise.\n *\n * @example\n * ```\n * satisfies('1.1.0', '^1.0.0'); // return true\n * satisfies('1.1.0', '~1.0.0'); // return false\n * ```\n */\nexport const satisfies = (version, range) => {\n    // if no range operator then \"=\"\n    const m = range.match(/^([<>=~^]+)/);\n    const op = m ? m[1] : '=';\n    // if gt/lt/eq then operator compare\n    if (op !== '^' && op !== '~')\n        return compare(version, range, op);\n    // else range of either \"~\" or \"^\" is assumed\n    const [v1, v2, v3, , vp] = validateAndParse(version);\n    const [r1, r2, r3, , rp] = validateAndParse(range);\n    const v = [v1, v2, v3];\n    const r = [r1, r2 !== null && r2 !== void 0 ? r2 : 'x', r3 !== null && r3 !== void 0 ? r3 : 'x'];\n    // validate pre-release\n    if (rp) {\n        if (!vp)\n            return false;\n        if (compareSegments(v, r) !== 0)\n            return false;\n        if (compareSegments(vp.split('.'), rp.split('.')) === -1)\n            return false;\n    }\n    // first non-zero number\n    const nonZero = r.findIndex((v) => v !== '0') + 1;\n    // pointer to where segments can be >=\n    const i = op === '~' ? 2 : nonZero > 1 ? nonZero : 1;\n    // before pointer must be equal\n    if (compareSegments(v.slice(0, i), r.slice(0, i)) !== 0)\n        return false;\n    // after pointer must be >=\n    if (compareSegments(v.slice(i), r.slice(i)) === -1)\n        return false;\n    return true;\n};\nconst semver = /^[v^~<>=]*?(\\d+)(?:\\.([x*]|\\d+)(?:\\.([x*]|\\d+)(?:\\.([x*]|\\d+))?(?:-([\\da-z\\-]+(?:\\.[\\da-z\\-]+)*))?(?:\\+[\\da-z\\-]+(?:\\.[\\da-z\\-]+)*)?)?)?$/i;\nconst validateAndParse = (version) => {\n    if (typeof version !== 'string') {\n        throw new TypeError('Invalid argument expected string');\n    }\n    const match = version.match(semver);\n    if (!match) {\n        throw new Error(`Invalid argument not valid semver ('${version}' received)`);\n    }\n    match.shift();\n    return match;\n};\nconst isWildcard = (s) => s === '*' || s === 'x' || s === 'X';\nconst tryParse = (v) => {\n    const n = parseInt(v, 10);\n    return isNaN(n) ? v : n;\n};\nconst forceType = (a, b) => typeof a !== typeof b ? [String(a), String(b)] : [a, b];\nconst compareStrings = (a, b) => {\n    if (isWildcard(a) || isWildcard(b))\n        return 0;\n    const [ap, bp] = forceType(tryParse(a), tryParse(b));\n    if (ap > bp)\n        return 1;\n    if (ap < bp)\n        return -1;\n    return 0;\n};\nconst compareSegments = (a, b) => {\n    for (let i = 0; i < Math.max(a.length, b.length); i++) {\n        const r = compareStrings(a[i] || '0', b[i] || '0');\n        if (r !== 0)\n            return r;\n    }\n    return 0;\n};\nconst operatorResMap = {\n    '>': [1],\n    '>=': [0, 1],\n    '=': [0],\n    '<=': [-1, 0],\n    '<': [-1],\n};\nconst allowedOperators = Object.keys(operatorResMap);\nconst assertValidOperator = (op) => {\n    if (typeof op !== 'string') {\n        throw new TypeError(`Invalid operator type, expected string but got ${typeof op}`);\n    }\n    if (allowedOperators.indexOf(op) === -1) {\n        throw new Error(`Invalid operator, expected one of ${allowedOperators.join('|')}`);\n    }\n};\n//# sourceMappingURL=index.js.map","// Define the custom behavior of the page\nimport { documentReady } from \"./mixin\";\nimport { compare, validate } from \"compare-versions\";\n\nimport \"../styles/pydata-sphinx-theme.scss\";\n\n/*******************************************************************************\n * Theme interaction\n */\n\nvar prefersDark = window.matchMedia(\"(prefers-color-scheme: dark)\");\n\n/**\n * set the the body theme to the one specified by the user browser\n *\n * @param {event} e\n */\nfunction autoTheme(e) {\n  document.documentElement.dataset.theme = prefersDark.matches\n    ? \"dark\"\n    : \"light\";\n}\n\n/**\n * Set the theme using the specified mode.\n * It can be one of [\"auto\", \"dark\", \"light\"]\n *\n * @param {str} mode\n */\nfunction setTheme(mode) {\n  if (mode !== \"light\" && mode !== \"dark\" && mode !== \"auto\") {\n    console.error(`Got invalid theme mode: ${mode}. Resetting to auto.`);\n    mode = \"auto\";\n  }\n\n  // get the theme\n  var colorScheme = prefersDark.matches ? \"dark\" : \"light\";\n  document.documentElement.dataset.mode = mode;\n  var theme = mode == \"auto\" ? colorScheme : mode;\n  document.documentElement.dataset.theme = theme;\n  // TODO: remove this line after Bootstrap upgrade\n  // v5.3 has a colors mode: https://getbootstrap.com/docs/5.3/customize/color-modes/\n  document.querySelectorAll(\".dropdown-menu\").forEach((el) => {\n    if (theme === \"dark\") {\n      el.classList.add(\"dropdown-menu-dark\");\n    } else {\n      el.classList.remove(\"dropdown-menu-dark\");\n    }\n  });\n\n  // save mode and theme\n  localStorage.setItem(\"mode\", mode);\n  localStorage.setItem(\"theme\", theme);\n  console.log(`[PST]: Changed to ${mode} mode using the ${theme} theme.`);\n\n  // add a listener if set on auto\n  prefersDark.onchange = mode == \"auto\" ? autoTheme : \"\";\n}\n\n/**\n * Change the theme option order so that clicking on the btn is always a change\n * from \"auto\"\n */\nfunction cycleMode() {\n  const defaultMode = document.documentElement.dataset.defaultMode || \"auto\";\n  const currentMode = localStorage.getItem(\"mode\") || defaultMode;\n\n  var loopArray = (arr, current) => {\n    var nextPosition = arr.indexOf(current) + 1;\n    if (nextPosition === arr.length) {\n      nextPosition = 0;\n    }\n    return arr[nextPosition];\n  };\n\n  // make sure the next theme after auto is always a change\n  var modeList = prefersDark.matches\n    ? [\"auto\", \"light\", \"dark\"]\n    : [\"auto\", \"dark\", \"light\"];\n  var newMode = loopArray(modeList, currentMode);\n  setTheme(newMode);\n}\n\n/**\n * add the theme listener on the btns of the navbar\n */\nfunction addModeListener() {\n  // the theme was set a first time using the initial mini-script\n  // running setMode will ensure the use of the dark mode if auto is selected\n  setTheme(document.documentElement.dataset.mode);\n\n  // Attach event handlers for toggling themes colors\n  document.querySelectorAll(\".theme-switch-button\").forEach((el) => {\n    el.addEventListener(\"click\", cycleMode);\n  });\n}\n\n/*******************************************************************************\n * TOC interactivity\n */\n\n/**\n * TOC sidebar - add \"active\" class to parent list\n *\n * Bootstrap's scrollspy adds the active class to the <a> link,\n * but for the automatic collapsing we need this on the parent list item.\n *\n * The event is triggered on \"window\" (and not the nav item as documented),\n * see https://github.com/twbs/bootstrap/issues/20086\n */\nfunction addTOCInteractivity() {\n  window.addEventListener(\"activate.bs.scrollspy\", function () {\n    const navLinks = document.querySelectorAll(\".bd-toc-nav a\");\n\n    navLinks.forEach((navLink) => {\n      navLink.parentElement.classList.remove(\"active\");\n    });\n\n    const activeNavLinks = document.querySelectorAll(\".bd-toc-nav a.active\");\n    activeNavLinks.forEach((navLink) => {\n      navLink.parentElement.classList.add(\"active\");\n    });\n  });\n}\n\n/*******************************************************************************\n * Scroll\n */\n\n/**\n * Navigation sidebar scrolling to active page\n */\nfunction scrollToActive() {\n  // If the docs nav doesn't exist, do nothing (e.g., on search page)\n  if (!document.querySelector(\".bd-docs-nav\")) {\n    return;\n  }\n\n  var sidebar = document.querySelector(\"div.bd-sidebar\");\n\n  // Remember the sidebar scroll position between page loads\n  // Inspired on source of revealjs.com\n  let storedScrollTop = parseInt(\n    sessionStorage.getItem(\"sidebar-scroll-top\"),\n    10,\n  );\n\n  if (!isNaN(storedScrollTop)) {\n    // If we've got a saved scroll position, just use that\n    sidebar.scrollTop = storedScrollTop;\n    console.log(\"[PST]: Scrolled sidebar using stored browser position...\");\n  } else {\n    // Otherwise, calculate a position to scroll to based on the lowest `active` link\n    var sidebarNav = document.querySelector(\".bd-docs-nav\");\n    var active_pages = sidebarNav.querySelectorAll(\".active\");\n    if (active_pages.length > 0) {\n      // Use the last active page as the offset since it's the page we're on\n      var latest_active = active_pages[active_pages.length - 1];\n      var offset =\n        latest_active.getBoundingClientRect().y -\n        sidebar.getBoundingClientRect().y;\n      // Only scroll the navbar if the active link is lower than 50% of the page\n      if (latest_active.getBoundingClientRect().y > window.innerHeight * 0.5) {\n        let buffer = 0.25; // Buffer so we have some space above the scrolled item\n        sidebar.scrollTop = offset - sidebar.clientHeight * buffer;\n        console.log(\"[PST]: Scrolled sidebar using last active link...\");\n      }\n    }\n  }\n\n  // Store the sidebar scroll position\n  window.addEventListener(\"beforeunload\", () => {\n    sessionStorage.setItem(\"sidebar-scroll-top\", sidebar.scrollTop);\n  });\n}\n\n/*******************************************************************************\n * Search\n */\n\n/**\n * Find any search forms on the page and return their input element\n */\nvar findSearchInput = () => {\n  let forms = document.querySelectorAll(\"form.bd-search\");\n  if (!forms.length) {\n    // no search form found\n    return;\n  } else {\n    var form;\n    if (forms.length == 1) {\n      // there is exactly one search form (persistent or hidden)\n      form = forms[0];\n    } else {\n      // must be at least one persistent form, use the first persistent one\n      form = document.querySelector(\n        \"div:not(.search-button__search-container) > form.bd-search\",\n      );\n    }\n    return form.querySelector(\"input\");\n  }\n};\n\n/**\n * Activate the search field on the page.\n * - If there is a search field already visible it will be activated.\n * - If not, then a search field will pop up.\n */\nvar toggleSearchField = () => {\n  // Find the search input to highlight\n  let input = findSearchInput();\n\n  // if the input field is the hidden one (the one associated with the\n  // search button) then toggle the button state (to show/hide the field)\n  let searchPopupWrapper = document.querySelector(\".search-button__wrapper\");\n  let hiddenInput = searchPopupWrapper.querySelector(\"input\");\n  if (input === hiddenInput) {\n    searchPopupWrapper.classList.toggle(\"show\");\n  }\n  // when toggling off the search field, remove its focus\n  if (document.activeElement === input) {\n    input.blur();\n  } else {\n    input.focus();\n    input.select();\n    input.scrollIntoView({ block: \"center\" });\n  }\n};\n\n/**\n * Add an event listener for toggleSearchField() for Ctrl/Cmd + K\n */\nvar addEventListenerForSearchKeyboard = () => {\n  window.addEventListener(\n    \"keydown\",\n    (event) => {\n      let input = findSearchInput();\n      // toggle on Ctrl+k or ⌘+k\n      if (\n        // Ignore if shift or alt are pressed\n        !event.shiftKey &&\n        !event.altKey &&\n        // On Mac use ⌘, all other OS use Ctrl\n        (useCommandKey\n          ? event.metaKey && !event.ctrlKey\n          : !event.metaKey && event.ctrlKey) &&\n        // Case-insensitive so the shortcut still works with caps lock\n        /^k$/i.test(event.key)\n      ) {\n        event.preventDefault();\n        toggleSearchField();\n      }\n      // also allow Escape key to hide (but not show) the dynamic search field\n      else if (document.activeElement === input && /Escape/i.test(event.key)) {\n        toggleSearchField();\n      }\n    },\n    true,\n  );\n};\n\n/**\n * If the user is on a Mac, use command (⌘) instead of control (ctrl) key\n *\n * Note: `navigator.platform` is deprecated; however MDN still recommends using\n * it for the one specific use case of detecting whether a keyboard shortcut\n * should use control or command:\n * https://developer.mozilla.org/en-US/docs/Web/API/Navigator/platform#examples\n */\nvar useCommandKey =\n  navigator.platform.indexOf(\"Mac\") === 0 || navigator.platform === \"iPhone\";\n\n/**\n * Change the search hint to `meta key` if we are a Mac\n */\n\nvar changeSearchShortcutKey = () => {\n  let shortcuts = document.querySelectorAll(\".search-button__kbd-shortcut\");\n  if (useCommandKey) {\n    shortcuts.forEach(\n      (f) => (f.querySelector(\"kbd.kbd-shortcut__modifier\").innerText = \"⌘\"),\n    );\n  }\n};\n\n/**\n * Activate callbacks for search button popup\n */\nvar setupSearchButtons = () => {\n  changeSearchShortcutKey();\n  addEventListenerForSearchKeyboard();\n\n  // Add the search button trigger event callback\n  document.querySelectorAll(\".search-button__button\").forEach((btn) => {\n    btn.onclick = toggleSearchField;\n  });\n\n  // Add the search button overlay event callback\n  let overlay = document.querySelector(\".search-button__overlay\");\n  if (overlay) {\n    overlay.onclick = toggleSearchField;\n  }\n};\n\n/*******************************************************************************\n * Version Switcher\n * Note that this depends on two variables existing that are defined in\n * and `html-page-context` hook:\n *\n * - DOCUMENTATION_OPTIONS.pagename\n * - DOCUMENTATION_OPTIONS.theme_switcher_url\n */\n\n/**\n * path component of URL\n */\nvar getCurrentUrlPath = () => {\n  if (DOCUMENTATION_OPTIONS.BUILDER == \"dirhtml\") {\n    return DOCUMENTATION_OPTIONS.pagename == \"index\"\n      ? `/`\n      : `${DOCUMENTATION_OPTIONS.pagename}/`;\n  }\n  return `${DOCUMENTATION_OPTIONS.pagename}.html`;\n};\n\n/**\n * Allow user to dismiss the warning banner about the docs version being dev / old.\n * We store the dismissal date and version, to give us flexibility about making the\n * dismissal last for longer than one browser session, if we decide to do that.\n *\n * @param {event} event the event that trigger the check\n */\nasync function DismissBannerAndStorePref(event) {\n  const banner = document.querySelector(\"#bd-header-version-warning\");\n  banner.remove();\n  const version = DOCUMENTATION_OPTIONS.VERSION;\n  const now = new Date();\n  const banner_pref = JSON.parse(\n    localStorage.getItem(\"pst_banner_pref\") || \"{}\",\n  );\n  console.debug(\n    `[PST] Dismissing the version warning banner on ${version} starting ${now}.`,\n  );\n  banner_pref[version] = now;\n  localStorage.setItem(\"pst_banner_pref\", JSON.stringify(banner_pref));\n}\n\n/**\n * Check if corresponding page path exists in other version of docs\n * and, if so, go there instead of the homepage of the other docs version\n *\n * @param {event} event the event that trigger the check\n */\nasync function checkPageExistsAndRedirect(event) {\n  // ensure we don't follow the initial link\n  event.preventDefault();\n  const currentFilePath = getCurrentUrlPath();\n  let tryUrl = event.currentTarget.getAttribute(\"href\");\n  let otherDocsHomepage = tryUrl.replace(currentFilePath, \"\");\n  try {\n    let head = await fetch(tryUrl, { method: \"HEAD\" });\n    if (head.ok) {\n      location.href = tryUrl; // the page exists, go there\n    } else {\n      location.href = otherDocsHomepage;\n    }\n  } catch (err) {\n    // something went wrong, probably CORS restriction, fallback to other docs homepage\n    location.href = otherDocsHomepage;\n  }\n}\n\n/**\n * Load and parse the version switcher JSON file from an absolute or relative URL.\n *\n * @param {string} url The URL to load version switcher entries from.\n */\nasync function fetchVersionSwitcherJSON(url) {\n  // first check if it's a valid URL\n  try {\n    var result = new URL(url);\n  } catch (err) {\n    if (err instanceof TypeError) {\n      if (!window.location.origin) {\n        // window.location.origin is null for local static sites\n        // (ie. window.location.protocol == 'file:')\n        //\n        // TODO: Fix this to return the static version switcher by working out\n        // how to get the correct path to the switcher JSON file on local static builds\n        return null;\n      }\n      // assume we got a relative path, and fix accordingly. But first, we need to\n      // use `fetch()` to follow redirects so we get the correct final base URL\n      const origin = await fetch(window.location.origin, { method: \"HEAD\" });\n      result = new URL(url, origin.url);\n    } else {\n      // something unexpected happened\n      throw err;\n    }\n  }\n  // load and return the JSON\n  const response = await fetch(result);\n  const data = await response.json();\n  return data;\n}\n\n// Populate the version switcher from the JSON data\nfunction populateVersionSwitcher(data, versionSwitcherBtns) {\n  const currentFilePath = getCurrentUrlPath();\n  versionSwitcherBtns.forEach((btn) => {\n    // Set empty strings by default so that these attributes exist and can be used in CSS selectors\n    btn.dataset[\"activeVersionName\"] = \"\";\n    btn.dataset[\"activeVersion\"] = \"\";\n  });\n  // in case there are multiple entries with the same version string, this helps us\n  // decide which entry's `name` to put on the button itself. Without this, it would\n  // always be the *last* version-matching entry; now it will be either the\n  // version-matching entry that is also marked as `\"preferred\": true`, or if that\n  // doesn't exist: the *first* version-matching entry.\n  data = data.map((entry) => {\n    // does this entry match the version that we're currently building/viewing?\n    entry.match =\n      entry.version == DOCUMENTATION_OPTIONS.theme_switcher_version_match;\n    entry.preferred = entry.preferred || false;\n    // if no custom name specified (e.g., \"latest\"), use version string\n    if (!(\"name\" in entry)) {\n      entry.name = entry.version;\n    }\n    return entry;\n  });\n  const hasMatchingPreferredEntry = data\n    .map((entry) => entry.preferred && entry.match)\n    .some(Boolean);\n  var foundMatch = false;\n  // create links to the corresponding page in the other docs versions\n  data.forEach((entry) => {\n    // create the node\n    const anchor = document.createElement(\"a\");\n    anchor.setAttribute(\n      \"class\",\n      \"dropdown-item list-group-item list-group-item-action py-1\",\n    );\n    anchor.setAttribute(\"href\", `${entry.url}${currentFilePath}`);\n    anchor.setAttribute(\"role\", \"option\");\n    const span = document.createElement(\"span\");\n    span.textContent = `${entry.name}`;\n    anchor.appendChild(span);\n    // Add dataset values for the version and name in case people want\n    // to apply CSS styling based on this information.\n    anchor.dataset[\"versionName\"] = entry.name;\n    anchor.dataset[\"version\"] = entry.version;\n    // replace dropdown button text with the preferred display name of the\n    // currently-viewed version, rather than using sphinx's {{ version }} variable.\n    // also highlight the dropdown entry for the currently-viewed version's entry\n    let matchesAndIsPreferred = hasMatchingPreferredEntry && entry.preferred;\n    let matchesAndIsFirst =\n      !hasMatchingPreferredEntry && !foundMatch && entry.match;\n    if (matchesAndIsPreferred || matchesAndIsFirst) {\n      anchor.classList.add(\"active\");\n      versionSwitcherBtns.forEach((btn) => {\n        btn.innerText = entry.name;\n        btn.dataset[\"activeVersionName\"] = entry.name;\n        btn.dataset[\"activeVersion\"] = entry.version;\n      });\n      foundMatch = true;\n    }\n    // There may be multiple version-switcher elements, e.g. one\n    // in a slide-over panel displayed on smaller screens.\n    document.querySelectorAll(\".version-switcher__menu\").forEach((menu) => {\n      // we need to clone the node for each menu, but onclick attributes are not\n      // preserved by `.cloneNode()` so we add onclick here after cloning.\n      let node = anchor.cloneNode(true);\n      node.onclick = checkPageExistsAndRedirect;\n      // on click, AJAX calls will check if the linked page exists before\n      // trying to redirect, and if not, will redirect to the homepage\n      // for that version of the docs.\n      menu.append(node);\n    });\n  });\n}\n\n/*******************************************************************************\n * Warning banner when viewing non-stable version of the docs.\n */\n\n/**\n * Show a warning banner when viewing a non-stable version of the docs.\n *\n * adapted 2023-06 from https://mne.tools/versionwarning.js, which was\n * originally adapted 2020-05 from https://scikit-learn.org/versionwarning.js\n *\n * @param {Array} data The version data used to populate the switcher menu.\n */\nfunction showVersionWarningBanner(data) {\n  var version = DOCUMENTATION_OPTIONS.VERSION;\n  // figure out what latest stable version is\n  var preferredEntries = data.filter((entry) => entry.preferred);\n  if (preferredEntries.length !== 1) {\n    const howMany = preferredEntries.length == 0 ? \"No\" : \"Multiple\";\n    console.log(\n      `[PST] ${howMany} versions marked \"preferred\" found in versions JSON, ignoring.`,\n    );\n    return;\n  }\n  const preferredVersion = preferredEntries[0].version;\n  const preferredURL = preferredEntries[0].url;\n  // if already on preferred version, nothing to do\n  const versionsAreComparable = validate(version) && validate(preferredVersion);\n  if (versionsAreComparable && compare(version, preferredVersion, \"=\")) {\n    console.log(\n      \"This is the prefered version of the docs, not showing the warning banner.\",\n    );\n    return;\n  }\n  // check if banner has been dismissed recently\n  const dismiss_date_str = JSON.parse(\n    localStorage.getItem(\"pst_banner_pref\") || \"{}\",\n  )[version];\n  if (dismiss_date_str != null) {\n    const dismiss_date = new Date(dismiss_date_str);\n    const now = new Date();\n    const milliseconds_in_a_day = 24 * 60 * 60 * 1000;\n    const days_passed = (now - dismiss_date) / milliseconds_in_a_day;\n    const timeout_in_days = 14;\n    if (days_passed < timeout_in_days) {\n      console.info(\n        `[PST] Suppressing version warning banner; was dismissed ${Math.floor(days_passed)} day(s) ago`,\n      );\n      return;\n    }\n  }\n\n  // now construct the warning banner\n  const banner = document.querySelector(\"#bd-header-version-warning\");\n  const middle = document.createElement(\"div\");\n  const inner = document.createElement(\"div\");\n  const bold = document.createElement(\"strong\");\n  const button = document.createElement(\"a\");\n  const close_btn = document.createElement(\"a\");\n  // these classes exist since pydata-sphinx-theme v0.10.0\n  // the init class is used for animation\n  middle.classList = \"bd-header-announcement__content  ms-auto me-auto\";\n  inner.classList = \"sidebar-message\";\n  button.classList =\n    \"btn text-wrap font-weight-bold ms-3 my-1 align-baseline pst-button-link-to-stable-version\";\n  button.href = `${preferredURL}${getCurrentUrlPath()}`;\n  button.innerText = \"Switch to stable version\";\n  button.onclick = checkPageExistsAndRedirect;\n  close_btn.classList = \"ms-3 my-1 align-baseline\";\n  const close_x = document.createElement(\"i\");\n  close_btn.append(close_x);\n  close_x.classList = \"fa-solid fa-xmark\";\n  close_btn.onclick = DismissBannerAndStorePref;\n  // add the version-dependent text\n  inner.innerText = \"This is documentation for \";\n  const isDev =\n    version.includes(\"dev\") ||\n    version.includes(\"rc\") ||\n    version.includes(\"pre\");\n  const newerThanPreferred =\n    versionsAreComparable && compare(version, preferredVersion, \">\");\n  if (isDev || newerThanPreferred) {\n    bold.innerText = \"an unstable development version\";\n  } else if (versionsAreComparable && compare(version, preferredVersion, \"<\")) {\n    bold.innerText = `an old version (${version})`;\n  } else if (!version) {\n    bold.innerText = \"an unknown version\"; // e.g., an empty string\n  } else {\n    bold.innerText = `version ${version}`;\n  }\n  banner.appendChild(middle);\n  banner.append(close_btn);\n  middle.appendChild(inner);\n  inner.appendChild(bold);\n  inner.appendChild(document.createTextNode(\".\"));\n  inner.appendChild(button);\n  banner.classList.remove(\"d-none\");\n}\n\n/*******************************************************************************\n * MutationObserver to move the ReadTheDocs button\n */\n\n/**\n * intercept the RTD flyout and place it in the rtd-footer-container if existing\n * if not it stays where on top of the page\n */\nfunction initRTDObserver() {\n  const mutatedCallback = (mutationList, observer) => {\n    mutationList.forEach((mutation) => {\n      // Check whether the mutation is for RTD, which will have a specific structure\n      if (mutation.addedNodes.length === 0) {\n        return;\n      }\n      if (mutation.addedNodes[0].data === undefined) {\n        return;\n      }\n      if (mutation.addedNodes[0].data.search(\"Inserted RTD Footer\") != -1) {\n        mutation.addedNodes.forEach((node) => {\n          document.getElementById(\"rtd-footer-container\").append(node);\n        });\n      }\n    });\n  };\n\n  const observer = new MutationObserver(mutatedCallback);\n  const config = { childList: true };\n  observer.observe(document.body, config);\n}\n\nasync function fetchAndUseVersions() {\n  // fetch the JSON version data (only once), then use it to populate the version\n  // switcher and maybe show the version warning bar\n  var versionSwitcherBtns = document.querySelectorAll(\n    \".version-switcher__button\",\n  );\n  const hasSwitcherMenu = versionSwitcherBtns.length > 0;\n  const hasVersionsJSON = DOCUMENTATION_OPTIONS.hasOwnProperty(\n    \"theme_switcher_json_url\",\n  );\n  const wantsWarningBanner = DOCUMENTATION_OPTIONS.show_version_warning_banner;\n\n  if (hasVersionsJSON && (hasSwitcherMenu || wantsWarningBanner)) {\n    const data = await fetchVersionSwitcherJSON(\n      DOCUMENTATION_OPTIONS.theme_switcher_json_url,\n    );\n    // TODO: remove the `if(data)` once the `return null` is fixed within fetchVersionSwitcherJSON.\n    // We don't really want the switcher and warning bar to silently not work.\n    if (data) {\n      populateVersionSwitcher(data, versionSwitcherBtns);\n      if (wantsWarningBanner) {\n        showVersionWarningBanner(data);\n      }\n    }\n  }\n}\n\n/*******************************************************************************\n * Add keyboard functionality to mobile sidebars.\n *\n * Wire up the hamburger-style buttons using the click event which (on buttons)\n * handles both mouse clicks and the space and enter keys.\n */\nfunction setupMobileSidebarKeyboardHandlers() {\n  // These are hidden checkboxes at the top of the page whose :checked property\n  // allows the mobile sidebars to be hidden or revealed via CSS.\n  const primaryToggle = document.getElementById(\"pst-primary-sidebar-checkbox\");\n  const secondaryToggle = document.getElementById(\n    \"pst-secondary-sidebar-checkbox\",\n  );\n  const primarySidebar = document.querySelector(\".bd-sidebar-primary\");\n  const secondarySidebar = document.querySelector(\".bd-sidebar-secondary\");\n\n  // Toggle buttons -\n  //\n  // These are the hamburger-style buttons in the header nav bar. 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(!pstAnnouncementUrl) {\n    return;\n  }\n\n  try {\n    const response = await fetch(pstAnnouncementUrl);\n    if (!response.ok) {\n      throw new Error(\n        `[PST]: HTTP response status not ok: ${response.status} ${response.statusText}`,\n      );\n    }\n    const data = await response.text();\n    if (data.length === 0) {\n      console.log(`[PST]: Empty announcement at: ${pstAnnouncementUrl}`);\n      return;\n    }\n    banner.innerHTML = `<div class=\"bd-header-announcement__content\">${data}</div>`;\n    banner.classList.remove(\"d-none\");\n  } catch (_error) {\n    console.log(`[PST]: Failed to load announcement at: ${pstAnnouncementUrl}`);\n    console.error(_error);\n  }\n}\n\n/*******************************************************************************\n * Reveal (and animate) the banners (version warning, announcement) together\n */\nasync function fetchRevealBannersTogether() {\n  // Wait until finished fetching and loading banners\n  await 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diff --git a/docs/html/_static/scripts/sphinx-book-theme.js b/docs/html/_static/scripts/sphinx-book-theme.js
new file mode 100644
index 0000000..481ba0c
--- /dev/null
+++ b/docs/html/_static/scripts/sphinx-book-theme.js
@@ -0,0 +1,2 @@
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diff --git a/docs/html/_static/scripts/sphinx-book-theme.js.map b/docs/html/_static/scripts/sphinx-book-theme.js.map
new file mode 100644
index 0000000..2cc145a
--- /dev/null
+++ b/docs/html/_static/scripts/sphinx-book-theme.js.map
@@ -0,0 +1 @@
+{"version":3,"file":"scripts/sphinx-book-theme.js","mappings":"mBASA,IAAIA,EAAsBC,IACG,WAAvBC,SAASC,WACXF,IACSC,SAASE,iBAClBF,SAASE,iBAAiB,mBAAoBH,GAE9CC,SAASG,YAAY,sBAAsB,WACd,YAAvBH,SAASC,YAA0BF,GACzC,GACF,EAkKFK,OAAOC,aAtCY,KACjB,IAAIC,EAAQN,SAASO,cAAc,cACrBD,EAAME,mBAGPC,UAAUC,SAAS,wBAC9BJ,EAAMK,mBACJ,WACA,iDAIJC,WAAW,EA2BbR,OAAOS,iBAzJgB,KACrB,IAAIC,EACDd,SAASe,mBAAoD,OAA/Bf,SAASe,mBACvCf,SAASgB,yBAC6B,OAArChB,SAASgB,wBACb,IAAIC,EAASjB,SAASkB,gBACjBJ,GAQHK,QAAQC,IAAI,8BACRpB,SAASqB,eACXrB,SAASqB,iBACArB,SAASsB,sBAClBtB,SAASsB,yBAXXH,QAAQC,IAAI,+BACRH,EAAOM,kBACTN,EAAOM,oBACEN,EAAOO,yBAChBP,EAAOO,0BASX,EA0IF1B,GAzHkB,KAChB,IAAI2B,EAAgB,GACpB,IAwCIC,EAAc,IAAIC,sBAxCA,CAACC,EAASC,KAE9BD,EAAQE,SAASC,IACf,GAAIA,EAAMC,eAERP,EAAcQ,KAAKF,EAAMG,aAGzB,IAAK,IAAIC,EAAK,EAAGA,EAAKV,EAAcW,OAAQD,IAC1C,GAAIV,EAAcU,KAAQJ,EAAMG,OAAQ,CACtCT,EAAcY,OAAOF,EAAI,GACzB,KACF,CAEJ,IAIEV,EAAcW,OAAS,EACzBpC,SAASO,cAAc,4BAA4BE,UAAU6B,IAAI,QAEjEtC,SACGO,cAAc,4BACdE,UAAU8B,OAAO,OACtB,GAYY,CAEZC,WAAY,qBAaVC,EAAiB,GATG,CACtB,aACA,WACA,SACA,iBACA,aACA,UACA,UAGcX,SAASK,IAEvBM,EAAeR,KAEX,IAAIE,IACJ,QAAQA,IACR,IAAIA,EAAGO,QAAQ,IAAK,OACpB,QAAQP,EAAGO,QAAQ,IAAK,OAE3B,IAEH1C,SAAS2C,iBAAiBF,EAAeG,KAAK,OAAOd,SAASK,IAC5DT,EAAYmB,QAAQV,EAAG,IAIJ,IAAIR,sBA1CO,CAACC,EAASC,KAEpCD,EAAQ,GAAGkB,mBAAmBC,EAAI,EACpC/C,SAASgD,KAAKvC,UAAU6B,IAAI,YAE5BtC,SAASgD,KAAKvC,UAAU8B,OAAO,WACjC,IAqCaM,QAAQ7C,SAASO,cAAc,4BAA4B,IAmD5ET,GA1BA,WACE,IAAImD,EAAkB,CACpB,0BACA,aACA,qBACA,sBACA,wBACA,qBACA,qBACA,cACAL,KAAK,KACP5C,SAAS2C,iBAAiBM,GAAiBnB,SAASK,IAClDA,EAAG1B,UAAU6B,IAAI,UAAU,GAE/B,G","sources":["webpack://sphinx_book_theme/./src/sphinx_book_theme/assets/scripts/index.js"],"sourcesContent":["// Import CSS variables\n// ref: https://css-tricks.com/getting-javascript-to-talk-to-css-and-sass/\nimport \"../styles/index.scss\";\n\n/**\n * A helper function to load scripts when the DOM is loaded.\n * This waits for everything to be on the page first before running, since\n * some functionality doesn't behave properly until everything is ready.\n */\nvar sbRunWhenDOMLoaded = (cb) => {\n  if (document.readyState != \"loading\") {\n    cb();\n  } else if (document.addEventListener) {\n    document.addEventListener(\"DOMContentLoaded\", cb);\n  } else {\n    document.attachEvent(\"onreadystatechange\", function () {\n      if (document.readyState == \"complete\") cb();\n    });\n  }\n};\n\n/**\n * Toggle full-screen with button\n *\n * There are some browser-specific hacks in here:\n * - Safari requires a `webkit` prefix, so this uses conditionals to check for that\n *   ref: https://developer.mozilla.org/en-US/docs/Web/API/Fullscreen_API\n */\nvar toggleFullScreen = () => {\n  var isInFullScreen =\n    (document.fullscreenElement && document.fullscreenElement !== null) ||\n    (document.webkitFullscreenElement &&\n      document.webkitFullscreenElement !== null);\n  let docElm = document.documentElement;\n  if (!isInFullScreen) {\n    console.log(\"[SBT]: Entering full screen\");\n    if (docElm.requestFullscreen) {\n      docElm.requestFullscreen();\n    } else if (docElm.webkitRequestFullscreen) {\n      docElm.webkitRequestFullscreen();\n    }\n  } else {\n    console.log(\"[SBT]: Exiting full screen\");\n    if (document.exitFullscreen) {\n      document.exitFullscreen();\n    } else if (document.webkitExitFullscreen) {\n      document.webkitExitFullscreen();\n    }\n  }\n};\n\n/**\n * Manage scrolling behavior. This is primarily two things:\n *\n * 1. Hide the Table of Contents any time sidebar content is on the screen.\n *\n * This will be triggered any time a sidebar item enters or exits the screen.\n * It adds/removes items from an array if they have entered the screen, and\n * removes them when they exit the screen. It hides the TOC if anything is\n * on-screen.\n *\n * ref: https://developer.mozilla.org/en-US/docs/Web/API/Intersection_Observer_API\n *\n * 2. Add a `scrolled` class to <body> to trigger CSS changes.\n */\nvar initTocHide = () => {\n  var onScreenItems = [];\n  let hideTocCallback = (entries, observer) => {\n    // Check whether any sidebar item is displayed\n    entries.forEach((entry) => {\n      if (entry.isIntersecting) {\n        // If an element just came on screen, add it our list\n        onScreenItems.push(entry.target);\n      } else {\n        // Otherwise, if it's in our list then remove it\n        for (let ii = 0; ii < onScreenItems.length; ii++) {\n          if (onScreenItems[ii] === entry.target) {\n            onScreenItems.splice(ii, 1);\n            break;\n          }\n        }\n      }\n    });\n\n    // Hide the TOC if any margin content is displayed on the screen\n    if (onScreenItems.length > 0) {\n      document.querySelector(\"div.bd-sidebar-secondary\").classList.add(\"hide\");\n    } else {\n      document\n        .querySelector(\"div.bd-sidebar-secondary\")\n        .classList.remove(\"hide\");\n    }\n  };\n  let manageScrolledClassOnBody = (entries, observer) => {\n    // The pixel is at the top, so if we're < 0 that it means we've scrolled\n    if (entries[0].boundingClientRect.y < 0) {\n      document.body.classList.add(\"scrolled\");\n    } else {\n      document.body.classList.remove(\"scrolled\");\n    }\n  };\n\n  // Set up the intersection observer to watch all margin content\n  let options = {\n    // Trigger callback when the top of a margin item is 1/3 up the screen\n    rootMargin: \"0px 0px -33% 0px\",\n  };\n  let tocObserver = new IntersectionObserver(hideTocCallback, options);\n  // TODO: deprecate popout after v0.5.0\n  const selectorClasses = [\n    \"marginnote\",\n    \"sidenote\",\n    \"margin\",\n    \"margin-caption\",\n    \"full-width\",\n    \"sidebar\",\n    \"popout\",\n  ];\n  let marginSelector = [];\n  selectorClasses.forEach((ii) => {\n    // Use three permutations of each class name because `tag_` and `_` used to be supported\n    marginSelector.push(\n      ...[\n        `.${ii}`,\n        `.tag_${ii}`,\n        `.${ii.replace(\"-\", \"_\")}`,\n        `.tag_${ii.replace(\"-\", \"_\")}`,\n      ],\n    );\n  });\n  document.querySelectorAll(marginSelector.join(\", \")).forEach((ii) => {\n    tocObserver.observe(ii);\n  });\n\n  // Set up the observer to check if we've scrolled from top of page\n  let scrollObserver = new IntersectionObserver(manageScrolledClassOnBody);\n  scrollObserver.observe(document.querySelector(\".sbt-scroll-pixel-helper\"));\n};\n\n/**\n * Activate Thebe with a custom button click.\n */\nvar initThebeSBT = () => {\n  var title = document.querySelector(\"section h1\");\n  var sibling = title.nextElementSibling;\n  // If the next element after the title isn't a thebe button, add one now.\n  // That way it is initiatlized when thebe is first-clicked and isn't re-added after.\n  if (!sibling.classList.contains(\"thebe-launch-button\")) {\n    title.insertAdjacentHTML(\n      \"afterend\",\n      \"<button class='thebe-launch-button'></button>\",\n    );\n  }\n  // This function is provided by sphinx-thebe\n  initThebe();\n};\n\n/**\n * Add no print class to certain DOM elements\n */\n\nfunction addNoPrint() {\n  var noPrintSelector = [\n    \".bd-header-announcement\",\n    \".bd-header\",\n    \".bd-header-article\",\n    \".bd-sidebar-primary\",\n    \".bd-sidebar-secondary\",\n    \".bd-footer-article\",\n    \".bd-footer-content\",\n    \".bd-footer\",\n  ].join(\",\");\n  document.querySelectorAll(noPrintSelector).forEach((ii) => {\n    ii.classList.add(\"noprint\");\n  });\n}\n\n/**\n * Set up callback functions for UI click actions\n */\nwindow.initThebeSBT = initThebeSBT;\nwindow.toggleFullScreen = toggleFullScreen;\n\n/**\n * Set up functions to load when the DOM is ready\n */\nsbRunWhenDOMLoaded(initTocHide);\nsbRunWhenDOMLoaded(addNoPrint);\n"],"names":["sbRunWhenDOMLoaded","cb","document","readyState","addEventListener","attachEvent","window","initThebeSBT","title","querySelector","nextElementSibling","classList","contains","insertAdjacentHTML","initThebe","toggleFullScreen","isInFullScreen","fullscreenElement","webkitFullscreenElement","docElm","documentElement","console","log","exitFullscreen","webkitExitFullscreen","requestFullscreen","webkitRequestFullscreen","onScreenItems","tocObserver","IntersectionObserver","entries","observer","forEach","entry","isIntersecting","push","target","ii","length","splice","add","remove","rootMargin","marginSelector","replace","querySelectorAll","join","observe","boundingClientRect","y","body","noPrintSelector"],"sourceRoot":""}
\ No newline at end of file
diff --git a/docs/html/_static/searchtools.js b/docs/html/_static/searchtools.js
new file mode 100644
index 0000000..92da3f8
--- /dev/null
+++ b/docs/html/_static/searchtools.js
@@ -0,0 +1,619 @@
+/*
+ * searchtools.js
+ * ~~~~~~~~~~~~~~~~
+ *
+ * Sphinx JavaScript utilities for the full-text search.
+ *
+ * :copyright: Copyright 2007-2024 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+"use strict";
+
+/**
+ * Simple result scoring code.
+ */
+if (typeof Scorer === "undefined") {
+  var Scorer = {
+    // Implement the following function to further tweak the score for each result
+    // The function takes a result array [docname, title, anchor, descr, score, filename]
+    // and returns the new score.
+    /*
+    score: result => {
+      const [docname, title, anchor, descr, score, filename] = result
+      return score
+    },
+    */
+
+    // query matches the full name of an object
+    objNameMatch: 11,
+    // or matches in the last dotted part of the object name
+    objPartialMatch: 6,
+    // Additive scores depending on the priority of the object
+    objPrio: {
+      0: 15, // used to be importantResults
+      1: 5, // used to be objectResults
+      2: -5, // used to be unimportantResults
+    },
+    //  Used when the priority is not in the mapping.
+    objPrioDefault: 0,
+
+    // query found in title
+    title: 15,
+    partialTitle: 7,
+    // query found in terms
+    term: 5,
+    partialTerm: 2,
+  };
+}
+
+const _removeChildren = (element) => {
+  while (element && element.lastChild) element.removeChild(element.lastChild);
+};
+
+/**
+ * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping
+ */
+const _escapeRegExp = (string) =>
+  string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string
+
+const _displayItem = (item, searchTerms, highlightTerms) => {
+  const docBuilder = DOCUMENTATION_OPTIONS.BUILDER;
+  const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX;
+  const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX;
+  const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY;
+  const contentRoot = document.documentElement.dataset.content_root;
+
+  const [docName, title, anchor, descr, score, _filename] = item;
+
+  let listItem = document.createElement("li");
+  let requestUrl;
+  let linkUrl;
+  if (docBuilder === "dirhtml") {
+    // dirhtml builder
+    let dirname = docName + "/";
+    if (dirname.match(/\/index\/$/))
+      dirname = dirname.substring(0, dirname.length - 6);
+    else if (dirname === "index/") dirname = "";
+    requestUrl = contentRoot + dirname;
+    linkUrl = requestUrl;
+  } else {
+    // normal html builders
+    requestUrl = contentRoot + docName + docFileSuffix;
+    linkUrl = docName + docLinkSuffix;
+  }
+  let linkEl = listItem.appendChild(document.createElement("a"));
+  linkEl.href = linkUrl + anchor;
+  linkEl.dataset.score = score;
+  linkEl.innerHTML = title;
+  if (descr) {
+    listItem.appendChild(document.createElement("span")).innerHTML =
+      " (" + descr + ")";
+    // highlight search terms in the description
+    if (SPHINX_HIGHLIGHT_ENABLED)  // set in sphinx_highlight.js
+      highlightTerms.forEach((term) => _highlightText(listItem, term, "highlighted"));
+  }
+  else if (showSearchSummary)
+    fetch(requestUrl)
+      .then((responseData) => responseData.text())
+      .then((data) => {
+        if (data)
+          listItem.appendChild(
+            Search.makeSearchSummary(data, searchTerms, anchor)
+          );
+        // highlight search terms in the summary
+        if (SPHINX_HIGHLIGHT_ENABLED)  // set in sphinx_highlight.js
+          highlightTerms.forEach((term) => _highlightText(listItem, term, "highlighted"));
+      });
+  Search.output.appendChild(listItem);
+};
+const _finishSearch = (resultCount) => {
+  Search.stopPulse();
+  Search.title.innerText = _("Search Results");
+  if (!resultCount)
+    Search.status.innerText = Documentation.gettext(
+      "Your search did not match any documents. Please make sure that all words are spelled correctly and that you've selected enough categories."
+    );
+  else
+    Search.status.innerText = _(
+      "Search finished, found ${resultCount} page(s) matching the search query."
+    ).replace('${resultCount}', resultCount);
+};
+const _displayNextItem = (
+  results,
+  resultCount,
+  searchTerms,
+  highlightTerms,
+) => {
+  // results left, load the summary and display it
+  // this is intended to be dynamic (don't sub resultsCount)
+  if (results.length) {
+    _displayItem(results.pop(), searchTerms, highlightTerms);
+    setTimeout(
+      () => _displayNextItem(results, resultCount, searchTerms, highlightTerms),
+      5
+    );
+  }
+  // search finished, update title and status message
+  else _finishSearch(resultCount);
+};
+// Helper function used by query() to order search results.
+// Each input is an array of [docname, title, anchor, descr, score, filename].
+// Order the results by score (in opposite order of appearance, since the
+// `_displayNextItem` function uses pop() to retrieve items) and then alphabetically.
+const _orderResultsByScoreThenName = (a, b) => {
+  const leftScore = a[4];
+  const rightScore = b[4];
+  if (leftScore === rightScore) {
+    // same score: sort alphabetically
+    const leftTitle = a[1].toLowerCase();
+    const rightTitle = b[1].toLowerCase();
+    if (leftTitle === rightTitle) return 0;
+    return leftTitle > rightTitle ? -1 : 1; // inverted is intentional
+  }
+  return leftScore > rightScore ? 1 : -1;
+};
+
+/**
+ * Default splitQuery function. Can be overridden in ``sphinx.search`` with a
+ * custom function per language.
+ *
+ * The regular expression works by splitting the string on consecutive characters
+ * that are not Unicode letters, numbers, underscores, or emoji characters.
+ * This is the same as ``\W+`` in Python, preserving the surrogate pair area.
+ */
+if (typeof splitQuery === "undefined") {
+  var splitQuery = (query) => query
+      .split(/[^\p{Letter}\p{Number}_\p{Emoji_Presentation}]+/gu)
+      .filter(term => term)  // remove remaining empty strings
+}
+
+/**
+ * Search Module
+ */
+const Search = {
+  _index: null,
+  _queued_query: null,
+  _pulse_status: -1,
+
+  htmlToText: (htmlString, anchor) => {
+    const htmlElement = new DOMParser().parseFromString(htmlString, 'text/html');
+    for (const removalQuery of [".headerlinks", "script", "style"]) {
+      htmlElement.querySelectorAll(removalQuery).forEach((el) => { el.remove() });
+    }
+    if (anchor) {
+      const anchorContent = htmlElement.querySelector(`[role="main"] ${anchor}`);
+      if (anchorContent) return anchorContent.textContent;
+
+      console.warn(
+        `Anchored content block not found. Sphinx search tries to obtain it via DOM query '[role=main] ${anchor}'. Check your theme or template.`
+      );
+    }
+
+    // if anchor not specified or not found, fall back to main content
+    const docContent = htmlElement.querySelector('[role="main"]');
+    if (docContent) return docContent.textContent;
+
+    console.warn(
+      "Content block not found. Sphinx search tries to obtain it via DOM query '[role=main]'. Check your theme or template."
+    );
+    return "";
+  },
+
+  init: () => {
+    const query = new URLSearchParams(window.location.search).get("q");
+    document
+      .querySelectorAll('input[name="q"]')
+      .forEach((el) => (el.value = query));
+    if (query) Search.performSearch(query);
+  },
+
+  loadIndex: (url) =>
+    (document.body.appendChild(document.createElement("script")).src = url),
+
+  setIndex: (index) => {
+    Search._index = index;
+    if (Search._queued_query !== null) {
+      const query = Search._queued_query;
+      Search._queued_query = null;
+      Search.query(query);
+    }
+  },
+
+  hasIndex: () => Search._index !== null,
+
+  deferQuery: (query) => (Search._queued_query = query),
+
+  stopPulse: () => (Search._pulse_status = -1),
+
+  startPulse: () => {
+    if (Search._pulse_status >= 0) return;
+
+    const pulse = () => {
+      Search._pulse_status = (Search._pulse_status + 1) % 4;
+      Search.dots.innerText = ".".repeat(Search._pulse_status);
+      if (Search._pulse_status >= 0) window.setTimeout(pulse, 500);
+    };
+    pulse();
+  },
+
+  /**
+   * perform a search for something (or wait until index is loaded)
+   */
+  performSearch: (query) => {
+    // create the required interface elements
+    const searchText = document.createElement("h2");
+    searchText.textContent = _("Searching");
+    const searchSummary = document.createElement("p");
+    searchSummary.classList.add("search-summary");
+    searchSummary.innerText = "";
+    const searchList = document.createElement("ul");
+    searchList.classList.add("search");
+
+    const out = document.getElementById("search-results");
+    Search.title = out.appendChild(searchText);
+    Search.dots = Search.title.appendChild(document.createElement("span"));
+    Search.status = out.appendChild(searchSummary);
+    Search.output = out.appendChild(searchList);
+
+    const searchProgress = document.getElementById("search-progress");
+    // Some themes don't use the search progress node
+    if (searchProgress) {
+      searchProgress.innerText = _("Preparing search...");
+    }
+    Search.startPulse();
+
+    // index already loaded, the browser was quick!
+    if (Search.hasIndex()) Search.query(query);
+    else Search.deferQuery(query);
+  },
+
+  _parseQuery: (query) => {
+    // stem the search terms and add them to the correct list
+    const stemmer = new Stemmer();
+    const searchTerms = new Set();
+    const excludedTerms = new Set();
+    const highlightTerms = new Set();
+    const objectTerms = new Set(splitQuery(query.toLowerCase().trim()));
+    splitQuery(query.trim()).forEach((queryTerm) => {
+      const queryTermLower = queryTerm.toLowerCase();
+
+      // maybe skip this "word"
+      // stopwords array is from language_data.js
+      if (
+        stopwords.indexOf(queryTermLower) !== -1 ||
+        queryTerm.match(/^\d+$/)
+      )
+        return;
+
+      // stem the word
+      let word = stemmer.stemWord(queryTermLower);
+      // select the correct list
+      if (word[0] === "-") excludedTerms.add(word.substr(1));
+      else {
+        searchTerms.add(word);
+        highlightTerms.add(queryTermLower);
+      }
+    });
+
+    if (SPHINX_HIGHLIGHT_ENABLED) {  // set in sphinx_highlight.js
+      localStorage.setItem("sphinx_highlight_terms", [...highlightTerms].join(" "))
+    }
+
+    // console.debug("SEARCH: searching for:");
+    // console.info("required: ", [...searchTerms]);
+    // console.info("excluded: ", [...excludedTerms]);
+
+    return [query, searchTerms, excludedTerms, highlightTerms, objectTerms];
+  },
+
+  /**
+   * execute search (requires search index to be loaded)
+   */
+  _performSearch: (query, searchTerms, excludedTerms, highlightTerms, objectTerms) => {
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const titles = Search._index.titles;
+    const allTitles = Search._index.alltitles;
+    const indexEntries = Search._index.indexentries;
+
+    // Collect multiple result groups to be sorted separately and then ordered.
+    // Each is an array of [docname, title, anchor, descr, score, filename].
+    const normalResults = [];
+    const nonMainIndexResults = [];
+
+    _removeChildren(document.getElementById("search-progress"));
+
+    const queryLower = query.toLowerCase().trim();
+    for (const [title, foundTitles] of Object.entries(allTitles)) {
+      if (title.toLowerCase().trim().includes(queryLower) && (queryLower.length >= title.length/2)) {
+        for (const [file, id] of foundTitles) {
+          let score = Math.round(100 * queryLower.length / title.length)
+          normalResults.push([
+            docNames[file],
+            titles[file] !== title ? `${titles[file]} > ${title}` : title,
+            id !== null ? "#" + id : "",
+            null,
+            score,
+            filenames[file],
+          ]);
+        }
+      }
+    }
+
+    // search for explicit entries in index directives
+    for (const [entry, foundEntries] of Object.entries(indexEntries)) {
+      if (entry.includes(queryLower) && (queryLower.length >= entry.length/2)) {
+        for (const [file, id, isMain] of foundEntries) {
+          const score = Math.round(100 * queryLower.length / entry.length);
+          const result = [
+            docNames[file],
+            titles[file],
+            id ? "#" + id : "",
+            null,
+            score,
+            filenames[file],
+          ];
+          if (isMain) {
+            normalResults.push(result);
+          } else {
+            nonMainIndexResults.push(result);
+          }
+        }
+      }
+    }
+
+    // lookup as object
+    objectTerms.forEach((term) =>
+      normalResults.push(...Search.performObjectSearch(term, objectTerms))
+    );
+
+    // lookup as search terms in fulltext
+    normalResults.push(...Search.performTermsSearch(searchTerms, excludedTerms));
+
+    // let the scorer override scores with a custom scoring function
+    if (Scorer.score) {
+      normalResults.forEach((item) => (item[4] = Scorer.score(item)));
+      nonMainIndexResults.forEach((item) => (item[4] = Scorer.score(item)));
+    }
+
+    // Sort each group of results by score and then alphabetically by name.
+    normalResults.sort(_orderResultsByScoreThenName);
+    nonMainIndexResults.sort(_orderResultsByScoreThenName);
+
+    // Combine the result groups in (reverse) order.
+    // Non-main index entries are typically arbitrary cross-references,
+    // so display them after other results.
+    let results = [...nonMainIndexResults, ...normalResults];
+
+    // remove duplicate search results
+    // note the reversing of results, so that in the case of duplicates, the highest-scoring entry is kept
+    let seen = new Set();
+    results = results.reverse().reduce((acc, result) => {
+      let resultStr = result.slice(0, 4).concat([result[5]]).map(v => String(v)).join(',');
+      if (!seen.has(resultStr)) {
+        acc.push(result);
+        seen.add(resultStr);
+      }
+      return acc;
+    }, []);
+
+    return results.reverse();
+  },
+
+  query: (query) => {
+    const [searchQuery, searchTerms, excludedTerms, highlightTerms, objectTerms] = Search._parseQuery(query);
+    const results = Search._performSearch(searchQuery, searchTerms, excludedTerms, highlightTerms, objectTerms);
+
+    // for debugging
+    //Search.lastresults = results.slice();  // a copy
+    // console.info("search results:", Search.lastresults);
+
+    // print the results
+    _displayNextItem(results, results.length, searchTerms, highlightTerms);
+  },
+
+  /**
+   * search for object names
+   */
+  performObjectSearch: (object, objectTerms) => {
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const objects = Search._index.objects;
+    const objNames = Search._index.objnames;
+    const titles = Search._index.titles;
+
+    const results = [];
+
+    const objectSearchCallback = (prefix, match) => {
+      const name = match[4]
+      const fullname = (prefix ? prefix + "." : "") + name;
+      const fullnameLower = fullname.toLowerCase();
+      if (fullnameLower.indexOf(object) < 0) return;
+
+      let score = 0;
+      const parts = fullnameLower.split(".");
+
+      // check for different match types: exact matches of full name or
+      // "last name" (i.e. last dotted part)
+      if (fullnameLower === object || parts.slice(-1)[0] === object)
+        score += Scorer.objNameMatch;
+      else if (parts.slice(-1)[0].indexOf(object) > -1)
+        score += Scorer.objPartialMatch; // matches in last name
+
+      const objName = objNames[match[1]][2];
+      const title = titles[match[0]];
+
+      // If more than one term searched for, we require other words to be
+      // found in the name/title/description
+      const otherTerms = new Set(objectTerms);
+      otherTerms.delete(object);
+      if (otherTerms.size > 0) {
+        const haystack = `${prefix} ${name} ${objName} ${title}`.toLowerCase();
+        if (
+          [...otherTerms].some((otherTerm) => haystack.indexOf(otherTerm) < 0)
+        )
+          return;
+      }
+
+      let anchor = match[3];
+      if (anchor === "") anchor = fullname;
+      else if (anchor === "-") anchor = objNames[match[1]][1] + "-" + fullname;
+
+      const descr = objName + _(", in ") + title;
+
+      // add custom score for some objects according to scorer
+      if (Scorer.objPrio.hasOwnProperty(match[2]))
+        score += Scorer.objPrio[match[2]];
+      else score += Scorer.objPrioDefault;
+
+      results.push([
+        docNames[match[0]],
+        fullname,
+        "#" + anchor,
+        descr,
+        score,
+        filenames[match[0]],
+      ]);
+    };
+    Object.keys(objects).forEach((prefix) =>
+      objects[prefix].forEach((array) =>
+        objectSearchCallback(prefix, array)
+      )
+    );
+    return results;
+  },
+
+  /**
+   * search for full-text terms in the index
+   */
+  performTermsSearch: (searchTerms, excludedTerms) => {
+    // prepare search
+    const terms = Search._index.terms;
+    const titleTerms = Search._index.titleterms;
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const titles = Search._index.titles;
+
+    const scoreMap = new Map();
+    const fileMap = new Map();
+
+    // perform the search on the required terms
+    searchTerms.forEach((word) => {
+      const files = [];
+      const arr = [
+        { files: terms[word], score: Scorer.term },
+        { files: titleTerms[word], score: Scorer.title },
+      ];
+      // add support for partial matches
+      if (word.length > 2) {
+        const escapedWord = _escapeRegExp(word);
+        if (!terms.hasOwnProperty(word)) {
+          Object.keys(terms).forEach((term) => {
+            if (term.match(escapedWord))
+              arr.push({ files: terms[term], score: Scorer.partialTerm });
+          });
+        }
+        if (!titleTerms.hasOwnProperty(word)) {
+          Object.keys(titleTerms).forEach((term) => {
+            if (term.match(escapedWord))
+              arr.push({ files: titleTerms[term], score: Scorer.partialTitle });
+          });
+        }
+      }
+
+      // no match but word was a required one
+      if (arr.every((record) => record.files === undefined)) return;
+
+      // found search word in contents
+      arr.forEach((record) => {
+        if (record.files === undefined) return;
+
+        let recordFiles = record.files;
+        if (recordFiles.length === undefined) recordFiles = [recordFiles];
+        files.push(...recordFiles);
+
+        // set score for the word in each file
+        recordFiles.forEach((file) => {
+          if (!scoreMap.has(file)) scoreMap.set(file, {});
+          scoreMap.get(file)[word] = record.score;
+        });
+      });
+
+      // create the mapping
+      files.forEach((file) => {
+        if (!fileMap.has(file)) fileMap.set(file, [word]);
+        else if (fileMap.get(file).indexOf(word) === -1) fileMap.get(file).push(word);
+      });
+    });
+
+    // now check if the files don't contain excluded terms
+    const results = [];
+    for (const [file, wordList] of fileMap) {
+      // check if all requirements are matched
+
+      // as search terms with length < 3 are discarded
+      const filteredTermCount = [...searchTerms].filter(
+        (term) => term.length > 2
+      ).length;
+      if (
+        wordList.length !== searchTerms.size &&
+        wordList.length !== filteredTermCount
+      )
+        continue;
+
+      // ensure that none of the excluded terms is in the search result
+      if (
+        [...excludedTerms].some(
+          (term) =>
+            terms[term] === file ||
+            titleTerms[term] === file ||
+            (terms[term] || []).includes(file) ||
+            (titleTerms[term] || []).includes(file)
+        )
+      )
+        break;
+
+      // select one (max) score for the file.
+      const score = Math.max(...wordList.map((w) => scoreMap.get(file)[w]));
+      // add result to the result list
+      results.push([
+        docNames[file],
+        titles[file],
+        "",
+        null,
+        score,
+        filenames[file],
+      ]);
+    }
+    return results;
+  },
+
+  /**
+   * helper function to return a node containing the
+   * search summary for a given text. keywords is a list
+   * of stemmed words.
+   */
+  makeSearchSummary: (htmlText, keywords, anchor) => {
+    const text = Search.htmlToText(htmlText, anchor);
+    if (text === "") return null;
+
+    const textLower = text.toLowerCase();
+    const actualStartPosition = [...keywords]
+      .map((k) => textLower.indexOf(k.toLowerCase()))
+      .filter((i) => i > -1)
+      .slice(-1)[0];
+    const startWithContext = Math.max(actualStartPosition - 120, 0);
+
+    const top = startWithContext === 0 ? "" : "...";
+    const tail = startWithContext + 240 < text.length ? "..." : "";
+
+    let summary = document.createElement("p");
+    summary.classList.add("context");
+    summary.textContent = top + text.substr(startWithContext, 240).trim() + tail;
+
+    return summary;
+  },
+};
+
+_ready(Search.init);
diff --git a/docs/html/_static/sphinx_highlight.js b/docs/html/_static/sphinx_highlight.js
new file mode 100644
index 0000000..8a96c69
--- /dev/null
+++ b/docs/html/_static/sphinx_highlight.js
@@ -0,0 +1,154 @@
+/* Highlighting utilities for Sphinx HTML documentation. */
+"use strict";
+
+const SPHINX_HIGHLIGHT_ENABLED = true
+
+/**
+ * highlight a given string on a node by wrapping it in
+ * span elements with the given class name.
+ */
+const _highlight = (node, addItems, text, className) => {
+  if (node.nodeType === Node.TEXT_NODE) {
+    const val = node.nodeValue;
+    const parent = node.parentNode;
+    const pos = val.toLowerCase().indexOf(text);
+    if (
+      pos >= 0 &&
+      !parent.classList.contains(className) &&
+      !parent.classList.contains("nohighlight")
+    ) {
+      let span;
+
+      const closestNode = parent.closest("body, svg, foreignObject");
+      const isInSVG = closestNode && closestNode.matches("svg");
+      if (isInSVG) {
+        span = document.createElementNS("http://www.w3.org/2000/svg", "tspan");
+      } else {
+        span = document.createElement("span");
+        span.classList.add(className);
+      }
+
+      span.appendChild(document.createTextNode(val.substr(pos, text.length)));
+      const rest = document.createTextNode(val.substr(pos + text.length));
+      parent.insertBefore(
+        span,
+        parent.insertBefore(
+          rest,
+          node.nextSibling
+        )
+      );
+      node.nodeValue = val.substr(0, pos);
+      /* There may be more occurrences of search term in this node. So call this
+       * function recursively on the remaining fragment.
+       */
+      _highlight(rest, addItems, text, className);
+
+      if (isInSVG) {
+        const rect = document.createElementNS(
+          "http://www.w3.org/2000/svg",
+          "rect"
+        );
+        const bbox = parent.getBBox();
+        rect.x.baseVal.value = bbox.x;
+        rect.y.baseVal.value = bbox.y;
+        rect.width.baseVal.value = bbox.width;
+        rect.height.baseVal.value = bbox.height;
+        rect.setAttribute("class", className);
+        addItems.push({ parent: parent, target: rect });
+      }
+    }
+  } else if (node.matches && !node.matches("button, select, textarea")) {
+    node.childNodes.forEach((el) => _highlight(el, addItems, text, className));
+  }
+};
+const _highlightText = (thisNode, text, className) => {
+  let addItems = [];
+  _highlight(thisNode, addItems, text, className);
+  addItems.forEach((obj) =>
+    obj.parent.insertAdjacentElement("beforebegin", obj.target)
+  );
+};
+
+/**
+ * Small JavaScript module for the documentation.
+ */
+const SphinxHighlight = {
+
+  /**
+   * highlight the search words provided in localstorage in the text
+   */
+  highlightSearchWords: () => {
+    if (!SPHINX_HIGHLIGHT_ENABLED) return;  // bail if no highlight
+
+    // get and clear terms from localstorage
+    const url = new URL(window.location);
+    const highlight =
+        localStorage.getItem("sphinx_highlight_terms")
+        || url.searchParams.get("highlight")
+        || "";
+    localStorage.removeItem("sphinx_highlight_terms")
+    url.searchParams.delete("highlight");
+    window.history.replaceState({}, "", url);
+
+    // get individual terms from highlight string
+    const terms = highlight.toLowerCase().split(/\s+/).filter(x => x);
+    if (terms.length === 0) return; // nothing to do
+
+    // There should never be more than one element matching "div.body"
+    const divBody = document.querySelectorAll("div.body");
+    const body = divBody.length ? divBody[0] : document.querySelector("body");
+    window.setTimeout(() => {
+      terms.forEach((term) => _highlightText(body, term, "highlighted"));
+    }, 10);
+
+    const searchBox = document.getElementById("searchbox");
+    if (searchBox === null) return;
+    searchBox.appendChild(
+      document
+        .createRange()
+        .createContextualFragment(
+          '<p class="highlight-link">' +
+            '<a href="javascript:SphinxHighlight.hideSearchWords()">' +
+            _("Hide Search Matches") +
+            "</a></p>"
+        )
+    );
+  },
+
+  /**
+   * helper function to hide the search marks again
+   */
+  hideSearchWords: () => {
+    document
+      .querySelectorAll("#searchbox .highlight-link")
+      .forEach((el) => el.remove());
+    document
+      .querySelectorAll("span.highlighted")
+      .forEach((el) => el.classList.remove("highlighted"));
+    localStorage.removeItem("sphinx_highlight_terms")
+  },
+
+  initEscapeListener: () => {
+    // only install a listener if it is really needed
+    if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) return;
+
+    document.addEventListener("keydown", (event) => {
+      // bail for input elements
+      if (BLACKLISTED_KEY_CONTROL_ELEMENTS.has(document.activeElement.tagName)) return;
+      // bail with special keys
+      if (event.shiftKey || event.altKey || event.ctrlKey || event.metaKey) return;
+      if (DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS && (event.key === "Escape")) {
+        SphinxHighlight.hideSearchWords();
+        event.preventDefault();
+      }
+    });
+  },
+};
+
+_ready(() => {
+  /* Do not call highlightSearchWords() when we are on the search page.
+   * It will highlight words from the *previous* search query.
+   */
+  if (typeof Search === "undefined") SphinxHighlight.highlightSearchWords();
+  SphinxHighlight.initEscapeListener();
+});
diff --git a/docs/html/_static/stationary_synthetic_FDC.png b/docs/html/_static/stationary_synthetic_FDC.png
new file mode 100644
index 0000000..822cfef
Binary files /dev/null and b/docs/html/_static/stationary_synthetic_FDC.png differ
diff --git a/docs/html/_static/styles/bootstrap.css b/docs/html/_static/styles/bootstrap.css
new file mode 100644
index 0000000..5ac709f
--- /dev/null
+++ b/docs/html/_static/styles/bootstrap.css
@@ -0,0 +1,6 @@
+/*!
+ * Bootstrap  v5.3.3 (https://getbootstrap.com/)
+ * Copyright 2011-2024 The Bootstrap Authors
+ * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)
+ */:root,[data-bs-theme=light]{--bs-blue:#0d6efd;--bs-indigo:#6610f2;--bs-purple:#6f42c1;--bs-pink:#d63384;--bs-red:#dc3545;--bs-orange:#fd7e14;--bs-yellow:#ffc107;--bs-green:#198754;--bs-teal:#20c997;--bs-cyan:#0dcaf0;--bs-black:#000;--bs-white:#fff;--bs-gray:#6c757d;--bs-gray-dark:#343a40;--bs-gray-100:#f8f9fa;--bs-gray-200:#e9ecef;--bs-gray-300:#dee2e6;--bs-gray-400:#ced4da;--bs-gray-500:#adb5bd;--bs-gray-600:#6c757d;--bs-gray-700:#495057;--bs-gray-800:#343a40;--bs-gray-900:#212529;--bs-primary:#0d6efd;--bs-secondary:#6c757d;--bs-success:#198754;--bs-info:#0dcaf0;--bs-warning:#ffc107;--bs-danger:#dc3545;--bs-light:#f8f9fa;--bs-dark:#212529;--bs-primary-rgb:13,110,253;--bs-secondary-rgb:108,117,125;--bs-success-rgb:25,135,84;--bs-info-rgb:13,202,240;--bs-warning-rgb:255,193,7;--bs-danger-rgb:220,53,69;--bs-light-rgb:248,249,250;--bs-dark-rgb:33,37,41;--bs-primary-text-emphasis:#052c65;--bs-secondary-text-emphasis:#2b2f32;--bs-success-text-emphasis:#0a3622;--bs-info-text-emphasis:#055160;--bs-warning-text-emphasis:#664d03;--bs-danger-text-emphasis:#58151c;--bs-light-text-emphasis:#495057;--bs-dark-text-emphasis:#495057;--bs-primary-bg-subtle:#cfe2ff;--bs-secondary-bg-subtle:#e2e3e5;--bs-success-bg-subtle:#d1e7dd;--bs-info-bg-subtle:#cff4fc;--bs-warning-bg-subtle:#fff3cd;--bs-danger-bg-subtle:#f8d7da;--bs-light-bg-subtle:#fcfcfd;--bs-dark-bg-subtle:#ced4da;--bs-primary-border-subtle:#9ec5fe;--bs-secondary-border-subtle:#c4c8cb;--bs-success-border-subtle:#a3cfbb;--bs-info-border-subtle:#9eeaf9;--bs-warning-border-subtle:#ffe69c;--bs-danger-border-subtle:#f1aeb5;--bs-light-border-subtle:#e9ecef;--bs-dark-border-subtle:#adb5bd;--bs-white-rgb:255,255,255;--bs-black-rgb:0,0,0;--bs-font-sans-serif:system-ui,-apple-system,"Segoe UI",Roboto,"Helvetica Neue","Noto Sans","Liberation Sans",Arial,sans-serif,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol","Noto Color Emoji";--bs-font-monospace:SFMono-Regular,Menlo,Monaco,Consolas,"Liberation Mono","Courier 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.offcanvas{background-color:transparent!important;border:0!important;flex-grow:1;height:auto!important;position:static;transform:none!important;transition:none;visibility:visible!important;width:auto!important;z-index:auto}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;flex-grow:0;overflow-y:visible;padding:0}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color:hsla(0,0%,100%,.55);--bs-navbar-hover-color:hsla(0,0%,100%,.75);--bs-navbar-disabled-color:hsla(0,0%,100%,.25);--bs-navbar-active-color:#fff;--bs-navbar-brand-color:#fff;--bs-navbar-brand-hover-color:#fff;--bs-navbar-toggler-border-color:hsla(0,0%,100%,.1)}.navbar-dark,.navbar[data-bs-theme=dark],[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg:url("data:image/svg+xml;charset=utf-8,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3E%3Cpath stroke='rgba(255, 255, 255, 0.55)' stroke-linecap='round' stroke-miterlimit='10' 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.nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-left:calc(var(--bs-card-cap-padding-x)*-.5);margin-right:calc(var(--bs-card-cap-padding-x)*-.5)}.card-img-overlay{border-radius:var(--bs-card-inner-border-radius);bottom:0;left:0;padding:var(--bs-card-img-overlay-padding);position:absolute;right:0;top:0}.card-img,.card-img-bottom,.card-img-top{width:100%}.card-img,.card-img-top{border-top-left-radius:var(--bs-card-inner-border-radius);border-top-right-radius:var(--bs-card-inner-border-radius)}.card-img,.card-img-bottom{border-bottom-left-radius:var(--bs-card-inner-border-radius);border-bottom-right-radius:var(--bs-card-inner-border-radius)}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width:540px){.card-group{display:flex;flex-flow:row wrap}.card-group>.card{flex:1 0 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var(--bs-accordion-border-color);color:var(--bs-accordion-active-color)}.accordion-button:not(.collapsed):after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button:after{background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);content:"";flex-shrink:0;height:var(--bs-accordion-btn-icon-width);margin-left:auto;transition:var(--bs-accordion-btn-icon-transition);width:var(--bs-accordion-btn-icon-width)}@media(prefers-reduced-motion:reduce){.accordion-button:after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{box-shadow:var(--bs-accordion-btn-focus-box-shadow);outline:0;z-index:3}.accordion-header{margin-bottom:0}.accordion-item{background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color);color:var(--bs-accordion-color)}.accordion-item:first-of-type{border-top-left-radius:var(--bs-accordion-border-radius);border-top-right-radius:var(--bs-accordion-border-radius)}.accordion-item:first-of-type>.accordion-header .accordion-button{border-top-left-radius:var(--bs-accordion-inner-border-radius);border-top-right-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:not(:first-of-type){border-top:0}.accordion-item:last-of-type{border-bottom-left-radius:var(--bs-accordion-border-radius);border-bottom-right-radius:var(--bs-accordion-border-radius)}.accordion-item:last-of-type>.accordion-header .accordion-button.collapsed{border-bottom-left-radius:var(--bs-accordion-inner-border-radius);border-bottom-right-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:last-of-type>.accordion-collapse{border-bottom-left-radius:var(--bs-accordion-border-radius);border-bottom-right-radius:var(--bs-accordion-border-radius)}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush>.accordion-item{border-left:0;border-radius:0;border-right:0}.accordion-flush>.accordion-item:first-child{border-top:0}.accordion-flush>.accordion-item:last-child{border-bottom:0}.accordion-flush>.accordion-item>.accordion-collapse,.accordion-flush>.accordion-item>.accordion-header .accordion-button,.accordion-flush>.accordion-item>.accordion-header .accordion-button.collapsed{border-radius:0}[data-bs-theme=dark] .accordion-button:after{--bs-accordion-btn-icon:url("data:image/svg+xml;charset=utf-8,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3E%3Cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3E%3C/svg%3E");--bs-accordion-btn-active-icon:url("data:image/svg+xml;charset=utf-8,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3E%3Cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3E%3C/svg%3E")}.breadcrumb{--bs-breadcrumb-padding-x:0;--bs-breadcrumb-padding-y:0;--bs-breadcrumb-margin-bottom:1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color:var(--bs-secondary-color);--bs-breadcrumb-item-padding-x:0.5rem;--bs-breadcrumb-item-active-color:var(--bs-secondary-color);background-color:var(--bs-breadcrumb-bg);border-radius:var(--bs-breadcrumb-border-radius);display:flex;flex-wrap:wrap;font-size:var(--bs-breadcrumb-font-size);list-style:none;margin-bottom:var(--bs-breadcrumb-margin-bottom);padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item:before{color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider,"/");float:left;padding-right:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x:0.75rem;--bs-pagination-padding-y:0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color:var(--bs-link-color);--bs-pagination-bg:var(--bs-body-bg);--bs-pagination-border-width:var(--bs-border-width);--bs-pagination-border-color:var(--bs-border-color);--bs-pagination-border-radius:var(--bs-border-radius);--bs-pagination-hover-color:var(--bs-link-hover-color);--bs-pagination-hover-bg:var(--bs-tertiary-bg);--bs-pagination-hover-border-color:var(--bs-border-color);--bs-pagination-focus-color:var(--bs-link-hover-color);--bs-pagination-focus-bg:var(--bs-secondary-bg);--bs-pagination-focus-box-shadow:0 0 0 0.1875rem var(--pst-color-accent);--bs-pagination-active-color:#fff;--bs-pagination-active-bg:#0d6efd;--bs-pagination-active-border-color:#0d6efd;--bs-pagination-disabled-color:var(--bs-secondary-color);--bs-pagination-disabled-bg:var(--bs-secondary-bg);--bs-pagination-disabled-border-color:var(--bs-border-color);display:flex;list-style:none;padding-left:0}.page-link{background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);color:var(--bs-pagination-color);display:block;font-size:var(--bs-pagination-font-size);padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);position:relative;text-decoration:none;transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion:reduce){.page-link{transition:none}}.page-link:hover{background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color);color:var(--bs-pagination-hover-color);z-index:2}.page-link:focus{background-color:var(--bs-pagination-focus-bg);box-shadow:var(--bs-pagination-focus-box-shadow);color:var(--bs-pagination-focus-color);outline:0;z-index:3}.active>.page-link,.page-link.active{background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color);color:var(--bs-pagination-active-color);z-index:3}.disabled>.page-link,.page-link.disabled{background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color);color:var(--bs-pagination-disabled-color);pointer-events:none}.page-item:not(:first-child) .page-link{margin-left:calc(var(--bs-border-width)*-1)}.page-item:first-child .page-link{border-bottom-left-radius:var(--bs-pagination-border-radius);border-top-left-radius:var(--bs-pagination-border-radius)}.page-item:last-child .page-link{border-bottom-right-radius:var(--bs-pagination-border-radius);border-top-right-radius:var(--bs-pagination-border-radius)}.pagination-lg{--bs-pagination-padding-x:1.5rem;--bs-pagination-padding-y:0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius:var(--bs-border-radius-lg)}.pagination-sm{--bs-pagination-padding-x:0.5rem;--bs-pagination-padding-y:0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius:var(--bs-border-radius-sm)}.badge{--bs-badge-padding-x:0.65em;--bs-badge-padding-y:0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight:700;--bs-badge-color:#fff;--bs-badge-border-radius:var(--bs-border-radius);border-radius:var(--bs-badge-border-radius);color:var(--bs-badge-color);display:inline-block;font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);text-align:center;vertical-align:baseline;white-space:nowrap}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg:transparent;--bs-alert-padding-x:1rem;--bs-alert-padding-y:1rem;--bs-alert-margin-bottom:1rem;--bs-alert-color:inherit;--bs-alert-border-color:transparent;--bs-alert-border:var(--bs-border-width) solid var(--bs-alert-border-color);--bs-alert-border-radius:var(--bs-border-radius);--bs-alert-link-color:inherit;background-color:var(--bs-alert-bg);border:var(--bs-alert-border);border-radius:var(--bs-alert-border-radius);color:var(--bs-alert-color);margin-bottom:var(--bs-alert-margin-bottom);padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);position:relative}.alert-heading{color:inherit}.alert-link{color:var(--bs-alert-link-color);font-weight:700}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{padding:1.25rem 1rem;position:absolute;right:0;top:0;z-index:2}.alert-primary{--bs-alert-color:var(--bs-primary-text-emphasis);--bs-alert-bg:var(--bs-primary-bg-subtle);--bs-alert-border-color:var(--bs-primary-border-subtle);--bs-alert-link-color:var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color:var(--bs-secondary-text-emphasis);--bs-alert-bg:var(--bs-secondary-bg-subtle);--bs-alert-border-color:var(--bs-secondary-border-subtle);--bs-alert-link-color:var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color:var(--bs-success-text-emphasis);--bs-alert-bg:var(--bs-success-bg-subtle);--bs-alert-border-color:var(--bs-success-border-subtle);--bs-alert-link-color:var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color:var(--bs-info-text-emphasis);--bs-alert-bg:var(--bs-info-bg-subtle);--bs-alert-border-color:var(--bs-info-border-subtle);--bs-alert-link-color:var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color:var(--bs-warning-text-emphasis);--bs-alert-bg:var(--bs-warning-bg-subtle);--bs-alert-border-color:var(--bs-warning-border-subtle);--bs-alert-link-color:var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color:var(--bs-danger-text-emphasis);--bs-alert-bg:var(--bs-danger-bg-subtle);--bs-alert-border-color:var(--bs-danger-border-subtle);--bs-alert-link-color:var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color:var(--bs-light-text-emphasis);--bs-alert-bg:var(--bs-light-bg-subtle);--bs-alert-border-color:var(--bs-light-border-subtle);--bs-alert-link-color:var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color:var(--bs-dark-text-emphasis);--bs-alert-bg:var(--bs-dark-bg-subtle);--bs-alert-border-color:var(--bs-dark-border-subtle);--bs-alert-link-color:var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:1rem}}.progress,.progress-stacked{--bs-progress-height:1rem;--bs-progress-font-size:0.75rem;--bs-progress-bg:var(--bs-secondary-bg);--bs-progress-border-radius:var(--bs-border-radius);--bs-progress-box-shadow:var(--bs-box-shadow-inset);--bs-progress-bar-color:#fff;--bs-progress-bar-bg:#0d6efd;--bs-progress-bar-transition:width 0.6s ease;background-color:var(--bs-progress-bg);border-radius:var(--bs-progress-border-radius);display:flex;font-size:var(--bs-progress-font-size);height:var(--bs-progress-height);overflow:hidden}.progress-bar{background-color:var(--bs-progress-bar-bg);color:var(--bs-progress-bar-color);display:flex;flex-direction:column;justify-content:center;overflow:hidden;text-align:center;transition:var(--bs-progress-bar-transition);white-space:nowrap}@media(prefers-reduced-motion:reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg,hsla(0,0%,100%,.15) 25%,transparent 0,transparent 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";counter-increment:section}.list-group-item-action{color:var(--bs-list-group-action-color);text-align:inherit;width:100%}.list-group-item-action:focus,.list-group-item-action:hover{background-color:var(--bs-list-group-action-hover-bg);color:var(--bs-list-group-action-hover-color);text-decoration:none;z-index:1}.list-group-item-action:active{background-color:var(--bs-list-group-action-active-bg);color:var(--bs-list-group-action-active-color)}.list-group-item{background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color);color:var(--bs-list-group-color);display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);position:relative;text-decoration:none}.list-group-item:first-child{border-top-left-radius:inherit;border-top-right-radius:inherit}.list-group-item:last-child{border-bottom-left-radius:inherit;border-bottom-right-radius:inherit}.list-group-item.disabled,.list-group-item:disabled{background-color:var(--bs-list-group-disabled-bg);color:var(--bs-list-group-disabled-color);pointer-events:none}.list-group-item.active{background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color);color:var(--bs-list-group-active-color);z-index:2}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{border-top-width:var(--bs-list-group-border-width);margin-top:calc(var(--bs-list-group-border-width)*-1)}.list-group-horizontal{flex-direction:row}.list-group-horizontal>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}@media(min-width:540px){.list-group-horizontal-sm{flex-direction:row}.list-group-horizontal-sm>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-sm>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}}@media(min-width:720px){.list-group-horizontal-md{flex-direction:row}.list-group-horizontal-md>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-md>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal-md>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}}@media(min-width:960px){.list-group-horizontal-lg{flex-direction:row}.list-group-horizontal-lg>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-lg>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}}@media(min-width:1200px){.list-group-horizontal-xl{flex-direction:row}.list-group-horizontal-xl>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-xl>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}}.list-group-flush{border-radius:0}.list-group-flush>.list-group-item{border-width:0 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var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-primary{--bs-list-group-color:var(--bs-primary-text-emphasis);--bs-list-group-bg:var(--bs-primary-bg-subtle);--bs-list-group-border-color:var(--bs-primary-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-primary-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-primary-border-subtle);--bs-list-group-active-color:var(--bs-primary-bg-subtle);--bs-list-group-active-bg:var(--bs-primary-text-emphasis);--bs-list-group-active-border-color:var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color:var(--bs-secondary-text-emphasis);--bs-list-group-bg:var(--bs-secondary-bg-subtle);--bs-list-group-border-color:var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-secondary-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-secondary-border-subtle);--bs-list-group-active-color:var(--bs-secondary-bg-subtle);--bs-list-group-active-bg:var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color:var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color:var(--bs-success-text-emphasis);--bs-list-group-bg:var(--bs-success-bg-subtle);--bs-list-group-border-color:var(--bs-success-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-success-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-success-border-subtle);--bs-list-group-active-color:var(--bs-success-bg-subtle);--bs-list-group-active-bg:var(--bs-success-text-emphasis);--bs-list-group-active-border-color:var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color:var(--bs-info-text-emphasis);--bs-list-group-bg:var(--bs-info-bg-subtle);--bs-list-group-border-color:var(--bs-info-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-info-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-info-border-subtle);--bs-list-group-active-color:var(--bs-info-bg-subtle);--bs-list-group-active-bg:var(--bs-info-text-emphasis);--bs-list-group-active-border-color:var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color:var(--bs-warning-text-emphasis);--bs-list-group-bg:var(--bs-warning-bg-subtle);--bs-list-group-border-color:var(--bs-warning-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-warning-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-warning-border-subtle);--bs-list-group-active-color:var(--bs-warning-bg-subtle);--bs-list-group-active-bg:var(--bs-warning-text-emphasis);--bs-list-group-active-border-color:var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color:var(--bs-danger-text-emphasis);--bs-list-group-bg:var(--bs-danger-bg-subtle);--bs-list-group-border-color:var(--bs-danger-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-danger-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-danger-border-subtle);--bs-list-group-active-color:var(--bs-danger-bg-subtle);--bs-list-group-active-bg:var(--bs-danger-text-emphasis);--bs-list-group-active-border-color:var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color:var(--bs-light-text-emphasis);--bs-list-group-bg:var(--bs-light-bg-subtle);--bs-list-group-border-color:var(--bs-light-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-light-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-light-border-subtle);--bs-list-group-active-color:var(--bs-light-bg-subtle);--bs-list-group-active-bg:var(--bs-light-text-emphasis);--bs-list-group-active-border-color:var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color:var(--bs-dark-text-emphasis);--bs-list-group-bg:var(--bs-dark-bg-subtle);--bs-list-group-border-color:var(--bs-dark-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-dark-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-dark-border-subtle);--bs-list-group-active-color:var(--bs-dark-bg-subtle);--bs-list-group-active-bg:var(--bs-dark-text-emphasis);--bs-list-group-active-border-color:var(--bs-dark-text-emphasis)}.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0)}.dropdown-item:hover,.dropdown-item:focus{color:var(--bs-dropdown-link-hover-color);background-color:var(--bs-dropdown-link-hover-bg)}.dropdown-item.active,.dropdown-item:active{color:var(--bs-dropdown-link-active-color);text-decoration:none;background-color:var(--bs-dropdown-link-active-bg)}.dropdown-item.disabled,.dropdown-item:disabled{color:var(--bs-dropdown-link-disabled-color);pointer-events:none;background-color:rgba(0,0,0,0)}.dropdown-menu.show{display:block}.dropdown-header{display:block;padding:var(--bs-dropdown-header-padding-y) var(--bs-dropdown-header-padding-x);margin-bottom:0;font-size:0.875rem;color:var(--bs-dropdown-header-color);white-space:nowrap}.dropdown-item-text{display:block;padding:var(--bs-dropdown-item-padding-y) var(--bs-dropdown-item-padding-x);color:var(--bs-dropdown-link-color)}.dropdown-menu-dark{--bs-dropdown-color: #dee2e6;--bs-dropdown-bg: #343a40;--bs-dropdown-border-color: var(--bs-border-color-translucent);--bs-dropdown-box-shadow: ;--bs-dropdown-link-color: #dee2e6;--bs-dropdown-link-hover-color: #fff;--bs-dropdown-divider-bg: var(--bs-border-color-translucent);--bs-dropdown-link-hover-bg: var(--pst-color-surface);--bs-dropdown-link-active-color: #fff;--bs-dropdown-link-active-bg: var(--pst-color-surface);--bs-dropdown-link-disabled-color: #adb5bd;--bs-dropdown-header-color: #adb5bd}.btn-group,.btn-group-vertical{position:relative;display:inline-flex;vertical-align:middle}.btn-group>.btn,.btn-group-vertical>.btn{position:relative;flex:1 1 auto}.btn-group>.btn-check:checked+.btn,.btn-group>.btn-check:focus+.btn,.btn-group>.btn:hover,.btn-group>.btn:focus,.btn-group>.btn:active,.btn-group>.btn.active,.btn-group-vertical>.btn-check:checked+.btn,.btn-group-vertical>.btn-check:focus+.btn,.btn-group-vertical>.btn:hover,.btn-group-vertical>.btn:focus,.btn-group-vertical>.btn:active,.btn-group-vertical>.btn.active{z-index:1}.btn-toolbar{display:flex;flex-wrap:wrap;justify-content:flex-start}.btn-toolbar .input-group{width:auto}.btn-group{border-radius:var(--bs-border-radius)}.btn-group>:not(.btn-check:first-child)+.btn,.btn-group>.btn-group:not(:first-child){margin-left:calc(var(--bs-border-width)*-1)}.btn-group>.btn:not(:last-child):not(.dropdown-toggle),.btn-group>.btn.dropdown-toggle-split:first-child,.btn-group>.btn-group:not(:last-child)>.btn{border-top-right-radius:0;border-bottom-right-radius:0}.btn-group>.btn:nth-child(n+3),.btn-group>:not(.btn-check)+.btn,.btn-group>.btn-group:not(:first-child)>.btn{border-top-left-radius:0;border-bottom-left-radius:0}.dropdown-toggle-split{padding-right:.5625rem;padding-left:.5625rem}.dropdown-toggle-split::after,.dropup .dropdown-toggle-split::after,.dropend .dropdown-toggle-split::after{margin-left:0}.dropstart .dropdown-toggle-split::before{margin-right:0}.btn-sm+.dropdown-toggle-split,.btn-group-sm>.btn+.dropdown-toggle-split{padding-right:.375rem;padding-left:.375rem}.btn-lg+.dropdown-toggle-split,.btn-group-lg>.btn+.dropdown-toggle-split{padding-right:.75rem;padding-left:.75rem}.btn-group-vertical{flex-direction:column;align-items:flex-start;justify-content:center}.btn-group-vertical>.btn,.btn-group-vertical>.btn-group{width:100%}.btn-group-vertical>.btn:not(:first-child),.btn-group-vertical>.btn-group:not(:first-child){margin-top:calc(var(--bs-border-width)*-1)}.btn-group-vertical>.btn:not(:last-child):not(.dropdown-toggle),.btn-group-vertical>.btn-group:not(:last-child)>.btn{border-bottom-right-radius:0;border-bottom-left-radius:0}.btn-group-vertical>.btn~.btn,.btn-group-vertical>.btn-group:not(:first-child)>.btn{border-top-left-radius:0;border-top-right-radius:0}.nav{--bs-nav-link-padding-x: 1rem;--bs-nav-link-padding-y: 0.5rem;--bs-nav-link-font-weight: ;--bs-nav-link-color: 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1.25rem;--bs-navbar-brand-color: rgba(var(--bs-emphasis-color-rgb), 1);--bs-navbar-brand-hover-color: rgba(var(--bs-emphasis-color-rgb), 1);--bs-navbar-nav-link-padding-x: 0.5rem;--bs-navbar-toggler-padding-y: 0.25rem;--bs-navbar-toggler-padding-x: 0.75rem;--bs-navbar-toggler-font-size: 1.25rem;--bs-navbar-toggler-icon-bg: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%2833, 37, 41, 0.75%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e\");--bs-navbar-toggler-border-color: rgba(var(--bs-emphasis-color-rgb), 0.15);--bs-navbar-toggler-border-radius: var(--bs-border-radius);--bs-navbar-toggler-focus-width: 0.1875rem;--bs-navbar-toggler-transition: box-shadow 0.15s ease-in-out;position:relative;display:flex;flex-wrap:wrap;align-items:center;justify-content:space-between;padding:var(--bs-navbar-padding-y) 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.offcanvas-body{display:flex;flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;flex-basis:auto}.navbar-expand .navbar-toggler{display:none}.navbar-expand .offcanvas{position:static;z-index:auto;flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;flex-grow:0;padding:0;overflow-y:visible}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color: rgba(255, 255, 255, 0.55);--bs-navbar-hover-color: rgba(255, 255, 255, 0.75);--bs-navbar-disabled-color: rgba(255, 255, 255, 0.25);--bs-navbar-active-color: #fff;--bs-navbar-brand-color: #fff;--bs-navbar-brand-hover-color: #fff;--bs-navbar-toggler-border-color: rgba(255, 255, 255, 0.1);--bs-navbar-toggler-icon-bg: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%28255, 255, 255, 0.55%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e\")}[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%28255, 255, 255, 0.55%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e\")}.card{--bs-card-spacer-y: 1rem;--bs-card-spacer-x: 1rem;--bs-card-title-spacer-y: 0.5rem;--bs-card-title-color: ;--bs-card-subtitle-color: ;--bs-card-border-width: var(--bs-border-width);--bs-card-border-color: var(--bs-border-color-translucent);--bs-card-border-radius: var(--bs-border-radius);--bs-card-box-shadow: ;--bs-card-inner-border-radius: calc(var(--bs-border-radius) - (var(--bs-border-width)));--bs-card-cap-padding-y: 0.5rem;--bs-card-cap-padding-x: 1rem;--bs-card-cap-bg: rgba(var(--bs-body-color-rgb), 0.03);--bs-card-cap-color: ;--bs-card-height: ;--bs-card-color: ;--bs-card-bg: var(--bs-body-bg);--bs-card-img-overlay-padding: 1rem;--bs-card-group-margin: 0.75rem;position:relative;display:flex;flex-direction:column;min-width:0;height:var(--bs-card-height);color:var(--bs-body-color);word-wrap:break-word;background-color:var(--bs-card-bg);background-clip:border-box;border:var(--bs-card-border-width) solid var(--bs-card-border-color);border-radius:var(--bs-card-border-radius)}.card>hr{margin-right:0;margin-left:0}.card>.list-group{border-top:inherit;border-bottom:inherit}.card>.list-group:first-child{border-top-width:0;border-top-left-radius:var(--bs-card-inner-border-radius);border-top-right-radius:var(--bs-card-inner-border-radius)}.card>.list-group:last-child{border-bottom-width:0;border-bottom-right-radius:var(--bs-card-inner-border-radius);border-bottom-left-radius:var(--bs-card-inner-border-radius)}.card>.card-header+.list-group,.card>.list-group+.card-footer{border-top:0}.card-body{flex:1 1 auto;padding:var(--bs-card-spacer-y) var(--bs-card-spacer-x);color:var(--bs-card-color)}.card-title{margin-bottom:var(--bs-card-title-spacer-y);color:var(--bs-card-title-color)}.card-subtitle{margin-top:calc(-0.5*var(--bs-card-title-spacer-y));margin-bottom:0;color:var(--bs-card-subtitle-color)}.card-text:last-child{margin-bottom:0}.card-link+.card-link{margin-left:var(--bs-card-spacer-x)}.card-header{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);margin-bottom:0;color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-bottom:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-header:first-child{border-radius:var(--bs-card-inner-border-radius) var(--bs-card-inner-border-radius) 0 0}.card-footer{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-top:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-footer:last-child{border-radius:0 0 var(--bs-card-inner-border-radius) var(--bs-card-inner-border-radius)}.card-header-tabs{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-bottom:calc(-1*var(--bs-card-cap-padding-y));margin-left:calc(-0.5*var(--bs-card-cap-padding-x));border-bottom:0}.card-header-tabs .nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-left:calc(-0.5*var(--bs-card-cap-padding-x))}.card-img-overlay{position:absolute;top:0;right:0;bottom:0;left:0;padding:var(--bs-card-img-overlay-padding);border-radius:var(--bs-card-inner-border-radius)}.card-img,.card-img-top,.card-img-bottom{width:100%}.card-img,.card-img-top{border-top-left-radius:var(--bs-card-inner-border-radius);border-top-right-radius:var(--bs-card-inner-border-radius)}.card-img,.card-img-bottom{border-bottom-right-radius:var(--bs-card-inner-border-radius);border-bottom-left-radius:var(--bs-card-inner-border-radius)}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width: 540px){.card-group{display:flex;flex-flow:row wrap}.card-group>.card{flex:1 0 0%;margin-bottom:0}.card-group>.card+.card{margin-left:0;border-left:0}.card-group>.card:not(:last-child){border-top-right-radius:0;border-bottom-right-radius:0}.card-group>.card:not(:last-child) .card-img-top,.card-group>.card:not(:last-child) .card-header{border-top-right-radius:0}.card-group>.card:not(:last-child) .card-img-bottom,.card-group>.card:not(:last-child) .card-footer{border-bottom-right-radius:0}.card-group>.card:not(:first-child){border-top-left-radius:0;border-bottom-left-radius:0}.card-group>.card:not(:first-child) .card-img-top,.card-group>.card:not(:first-child) .card-header{border-top-left-radius:0}.card-group>.card:not(:first-child) .card-img-bottom,.card-group>.card:not(:first-child) .card-footer{border-bottom-left-radius:0}}.accordion{--bs-accordion-color: var(--bs-body-color);--bs-accordion-bg: var(--bs-body-bg);--bs-accordion-transition: color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, border-radius 0.15s ease;--bs-accordion-border-color: var(--bs-border-color);--bs-accordion-border-width: var(--bs-border-width);--bs-accordion-border-radius: var(--bs-border-radius);--bs-accordion-inner-border-radius: calc(var(--bs-border-radius) - (var(--bs-border-width)));--bs-accordion-btn-padding-x: 1.25rem;--bs-accordion-btn-padding-y: 1rem;--bs-accordion-btn-color: var(--bs-body-color);--bs-accordion-btn-bg: var(--bs-accordion-bg);--bs-accordion-btn-icon: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='none' stroke='%23212529' stroke-linecap='round' stroke-linejoin='round'%3e%3cpath d='M2 5L8 11L14 5'/%3e%3c/svg%3e\");--bs-accordion-btn-icon-width: 1.25rem;--bs-accordion-btn-icon-transform: rotate(-180deg);--bs-accordion-btn-icon-transition: transform 0.2s ease-in-out;--bs-accordion-btn-active-icon: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='none' stroke='%23052c65' stroke-linecap='round' stroke-linejoin='round'%3e%3cpath d='M2 5L8 11L14 5'/%3e%3c/svg%3e\");--bs-accordion-btn-focus-box-shadow: 0 0 0 0.1875rem var(--pst-color-accent);--bs-accordion-body-padding-x: 1.25rem;--bs-accordion-body-padding-y: 1rem;--bs-accordion-active-color: var(--bs-primary-text-emphasis);--bs-accordion-active-bg: var(--bs-primary-bg-subtle)}.accordion-button{position:relative;display:flex;align-items:center;width:100%;padding:var(--bs-accordion-btn-padding-y) var(--bs-accordion-btn-padding-x);font-size:1rem;color:var(--bs-accordion-btn-color);text-align:left;background-color:var(--bs-accordion-btn-bg);border:0;border-radius:0;overflow-anchor:none;transition:var(--bs-accordion-transition)}@media(prefers-reduced-motion: reduce){.accordion-button{transition:none}}.accordion-button:not(.collapsed){color:var(--bs-accordion-active-color);background-color:var(--bs-accordion-active-bg);box-shadow:inset 0 calc(-1*var(--bs-accordion-border-width)) 0 var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:\"\";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:first-of-type{border-top-left-radius:var(--bs-accordion-border-radius);border-top-right-radius:var(--bs-accordion-border-radius)}.accordion-item:first-of-type>.accordion-header .accordion-button{border-top-left-radius:var(--bs-accordion-inner-border-radius);border-top-right-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:not(:first-of-type){border-top:0}.accordion-item:last-of-type{border-bottom-right-radius:var(--bs-accordion-border-radius);border-bottom-left-radius:var(--bs-accordion-border-radius)}.accordion-item:last-of-type>.accordion-header .accordion-button.collapsed{border-bottom-right-radius:var(--bs-accordion-inner-border-radius);border-bottom-left-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:last-of-type>.accordion-collapse{border-bottom-right-radius:var(--bs-accordion-border-radius);border-bottom-left-radius:var(--bs-accordion-border-radius)}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush>.accordion-item{border-right:0;border-left:0;border-radius:0}.accordion-flush>.accordion-item:first-child{border-top:0}.accordion-flush>.accordion-item:last-child{border-bottom:0}.accordion-flush>.accordion-item>.accordion-header .accordion-button,.accordion-flush>.accordion-item>.accordion-header .accordion-button.collapsed{border-radius:0}.accordion-flush>.accordion-item>.accordion-collapse{border-radius:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e\");--bs-accordion-btn-active-icon: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e\")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: var(--bs-secondary-color);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: var(--bs-secondary-color);display:flex;flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg);border-radius:var(--bs-breadcrumb-border-radius)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, \"/\") /* rtl: var(--bs-breadcrumb-divider, \"/\") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: var(--bs-link-color);--bs-pagination-bg: var(--bs-body-bg);--bs-pagination-border-width: var(--bs-border-width);--bs-pagination-border-color: var(--bs-border-color);--bs-pagination-border-radius: var(--bs-border-radius);--bs-pagination-hover-color: var(--bs-link-hover-color);--bs-pagination-hover-bg: var(--bs-tertiary-bg);--bs-pagination-hover-border-color: var(--bs-border-color);--bs-pagination-focus-color: var(--bs-link-hover-color);--bs-pagination-focus-bg: var(--bs-secondary-bg);--bs-pagination-focus-box-shadow: 0 0 0 0.1875rem var(--pst-color-accent);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #0d6efd;--bs-pagination-active-border-color: #0d6efd;--bs-pagination-disabled-color: var(--bs-secondary-color);--bs-pagination-disabled-bg: var(--bs-secondary-bg);--bs-pagination-disabled-border-color: var(--bs-border-color);display:flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(var(--bs-border-width)*-1)}.page-item:first-child .page-link{border-top-left-radius:var(--bs-pagination-border-radius);border-bottom-left-radius:var(--bs-pagination-border-radius)}.page-item:last-child .page-link{border-top-right-radius:var(--bs-pagination-border-radius);border-bottom-right-radius:var(--bs-pagination-border-radius)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: var(--bs-border-radius-lg)}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: var(--bs-border-radius-sm)}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: var(--bs-border-radius);display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline;border-radius:var(--bs-badge-border-radius)}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: var(--bs-border-width) solid var(--bs-alert-border-color);--bs-alert-border-radius: var(--bs-border-radius);--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border);border-radius:var(--bs-alert-border-radius)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:1rem}}.progress,.progress-stacked{--bs-progress-height: 1rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: var(--bs-secondary-bg);--bs-progress-border-radius: var(--bs-border-radius);--bs-progress-box-shadow: var(--bs-box-shadow-inset);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #0d6efd;--bs-progress-bar-transition: width 0.6s ease;display:flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg);border-radius:var(--bs-progress-border-radius)}.progress-bar{display:flex;flex-direction:column;justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: var(--bs-body-color);--bs-list-group-bg: var(--bs-body-bg);--bs-list-group-border-color: var(--bs-border-color);--bs-list-group-border-width: var(--bs-border-width);--bs-list-group-border-radius: var(--bs-border-radius);--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: var(--bs-secondary-color);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-tertiary-bg);--bs-list-group-action-active-color: var(--bs-body-color);--bs-list-group-action-active-bg: var(--bs-secondary-bg);--bs-list-group-disabled-color: var(--bs-secondary-color);--bs-list-group-disabled-bg: var(--bs-body-bg);--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #0d6efd;--bs-list-group-active-border-color: 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var(--bs-list-group-border-color)}.list-group-item:first-child{border-top-left-radius:inherit;border-top-right-radius:inherit}.list-group-item:last-child{border-bottom-right-radius:inherit;border-bottom-left-radius:inherit}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row}.list-group-horizontal>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 540px){.list-group-horizontal-sm{flex-direction:row}.list-group-horizontal-sm>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-sm>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 720px){.list-group-horizontal-md{flex-direction:row}.list-group-horizontal-md>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-md>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 960px){.list-group-horizontal-lg{flex-direction:row}.list-group-horizontal-lg>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-lg>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row}.list-group-horizontal-xl>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-xl>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush{border-radius:0}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e\");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.1875rem var(--pst-color-accent);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;border-radius:.375rem;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(var(--bs-body-bg-rgb), 0.85);--bs-toast-border-width: var(--bs-border-width);--bs-toast-border-color: var(--bs-border-color-translucent);--bs-toast-border-radius: var(--bs-border-radius);--bs-toast-box-shadow: var(--bs-box-shadow);--bs-toast-header-color: var(--bs-secondary-color);--bs-toast-header-bg: rgba(var(--bs-body-bg-rgb), 0.85);--bs-toast-header-border-color: var(--bs-border-color-translucent);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow);border-radius:var(--bs-toast-border-radius)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 1090;position:absolute;z-index:var(--bs-toast-zindex);width:max-content;max-width:100%;pointer-events:none}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{display:flex;align-items:center;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x);color:var(--bs-toast-header-color);background-color:var(--bs-toast-header-bg);background-clip:padding-box;border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color);border-top-left-radius:calc(var(--bs-toast-border-radius) - var(--bs-toast-border-width));border-top-right-radius:calc(var(--bs-toast-border-radius) - var(--bs-toast-border-width))}.toast-header .btn-close{margin-right:calc(-0.5*var(--bs-toast-padding-x));margin-left:var(--bs-toast-padding-x)}.toast-body{padding:var(--bs-toast-padding-x);word-wrap:break-word}.modal{--bs-modal-zindex: 1055;--bs-modal-width: 500px;--bs-modal-padding: 1rem;--bs-modal-margin: 0.5rem;--bs-modal-color: ;--bs-modal-bg: var(--bs-body-bg);--bs-modal-border-color: var(--bs-border-color-translucent);--bs-modal-border-width: var(--bs-border-width);--bs-modal-border-radius: var(--bs-border-radius-lg);--bs-modal-box-shadow: var(--bs-box-shadow-sm);--bs-modal-inner-border-radius: calc(var(--bs-border-radius-lg) - (var(--bs-border-width)));--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: var(--bs-border-color);--bs-modal-header-border-width: var(--bs-border-width);--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: var(--bs-border-color);--bs-modal-footer-border-width: var(--bs-border-width);position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static 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0.5;position:fixed;top:0;left:0;z-index:var(--bs-backdrop-zindex);width:100vw;height:100vh;background-color:var(--bs-backdrop-bg)}.modal-backdrop.fade{opacity:0}.modal-backdrop.show{opacity:var(--bs-backdrop-opacity)}.modal-header{display:flex;flex-shrink:0;align-items:center;padding:var(--bs-modal-header-padding);border-bottom:var(--bs-modal-header-border-width) solid var(--bs-modal-header-border-color);border-top-left-radius:var(--bs-modal-inner-border-radius);border-top-right-radius:var(--bs-modal-inner-border-radius)}.modal-header .btn-close{padding:calc(var(--bs-modal-header-padding-y)*.5) calc(var(--bs-modal-header-padding-x)*.5);margin:calc(-0.5*var(--bs-modal-header-padding-y)) calc(-0.5*var(--bs-modal-header-padding-x)) calc(-0.5*var(--bs-modal-header-padding-y)) auto}.modal-title{margin-bottom:0;line-height:var(--bs-modal-title-line-height)}.modal-body{position:relative;flex:1 1 auto;padding:var(--bs-modal-padding)}.modal-footer{display:flex;flex-shrink:0;flex-wrap:wrap;align-items:center;justify-content:flex-end;padding:calc(var(--bs-modal-padding) - var(--bs-modal-footer-gap)*.5);background-color:var(--bs-modal-footer-bg);border-top:var(--bs-modal-footer-border-width) solid var(--bs-modal-footer-border-color);border-bottom-right-radius:var(--bs-modal-inner-border-radius);border-bottom-left-radius:var(--bs-modal-inner-border-radius)}.modal-footer>*{margin:calc(var(--bs-modal-footer-gap)*.5)}@media(min-width: 540px){.modal{--bs-modal-margin: 1.75rem;--bs-modal-box-shadow: var(--bs-box-shadow)}.modal-dialog{max-width:var(--bs-modal-width);margin-right:auto;margin-left:auto}.modal-sm{--bs-modal-width: 300px}}@media(min-width: 960px){.modal-lg,.modal-xl{--bs-modal-width: 800px}}@media(min-width: 1200px){.modal-xl{--bs-modal-width: 1140px}}.modal-fullscreen{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen 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!important}.p-lg-5{padding:3rem !important}.px-lg-0{padding-right:0 !important;padding-left:0 !important}.px-lg-1{padding-right:.25rem !important;padding-left:.25rem !important}.px-lg-2{padding-right:.5rem !important;padding-left:.5rem !important}.px-lg-3{padding-right:1rem !important;padding-left:1rem !important}.px-lg-4{padding-right:1.5rem !important;padding-left:1.5rem !important}.px-lg-5{padding-right:3rem !important;padding-left:3rem !important}.py-lg-0{padding-top:0 !important;padding-bottom:0 !important}.py-lg-1{padding-top:.25rem !important;padding-bottom:.25rem !important}.py-lg-2{padding-top:.5rem !important;padding-bottom:.5rem !important}.py-lg-3{padding-top:1rem !important;padding-bottom:1rem !important}.py-lg-4{padding-top:1.5rem !important;padding-bottom:1.5rem !important}.py-lg-5{padding-top:3rem !important;padding-bottom:3rem !important}.pt-lg-0{padding-top:0 !important}.pt-lg-1{padding-top:.25rem !important}.pt-lg-2{padding-top:.5rem !important}.pt-lg-3{padding-top:1rem !important}.pt-lg-4{padding-top:1.5rem !important}.pt-lg-5{padding-top:3rem !important}.pe-lg-0{padding-right:0 !important}.pe-lg-1{padding-right:.25rem !important}.pe-lg-2{padding-right:.5rem !important}.pe-lg-3{padding-right:1rem !important}.pe-lg-4{padding-right:1.5rem !important}.pe-lg-5{padding-right:3rem !important}.pb-lg-0{padding-bottom:0 !important}.pb-lg-1{padding-bottom:.25rem !important}.pb-lg-2{padding-bottom:.5rem !important}.pb-lg-3{padding-bottom:1rem !important}.pb-lg-4{padding-bottom:1.5rem !important}.pb-lg-5{padding-bottom:3rem !important}.ps-lg-0{padding-left:0 !important}.ps-lg-1{padding-left:.25rem !important}.ps-lg-2{padding-left:.5rem !important}.ps-lg-3{padding-left:1rem !important}.ps-lg-4{padding-left:1.5rem !important}.ps-lg-5{padding-left:3rem !important}.gap-lg-0{gap:0 !important}.gap-lg-1{gap:.25rem !important}.gap-lg-2{gap:.5rem !important}.gap-lg-3{gap:1rem !important}.gap-lg-4{gap:1.5rem !important}.gap-lg-5{gap:3rem !important}.row-gap-lg-0{row-gap:0 !important}.row-gap-lg-1{row-gap:.25rem !important}.row-gap-lg-2{row-gap:.5rem !important}.row-gap-lg-3{row-gap:1rem !important}.row-gap-lg-4{row-gap:1.5rem !important}.row-gap-lg-5{row-gap:3rem !important}.column-gap-lg-0{column-gap:0 !important}.column-gap-lg-1{column-gap:.25rem !important}.column-gap-lg-2{column-gap:.5rem !important}.column-gap-lg-3{column-gap:1rem !important}.column-gap-lg-4{column-gap:1.5rem !important}.column-gap-lg-5{column-gap:3rem !important}.text-lg-start{text-align:left !important}.text-lg-end{text-align:right !important}.text-lg-center{text-align:center !important}}@media(min-width: 1200px){.float-xl-start{float:left !important}.float-xl-end{float:right !important}.float-xl-none{float:none !important}.object-fit-xl-contain{object-fit:contain !important}.object-fit-xl-cover{object-fit:cover !important}.object-fit-xl-fill{object-fit:fill !important}.object-fit-xl-scale{object-fit:scale-down !important}.object-fit-xl-none{object-fit:none !important}.d-xl-inline{display:inline !important}.d-xl-inline-block{display:inline-block !important}.d-xl-block{display:block !important}.d-xl-grid{display:grid !important}.d-xl-inline-grid{display:inline-grid !important}.d-xl-table{display:table !important}.d-xl-table-row{display:table-row !important}.d-xl-table-cell{display:table-cell !important}.d-xl-flex{display:flex !important}.d-xl-inline-flex{display:inline-flex !important}.d-xl-none{display:none !important}.flex-xl-fill{flex:1 1 auto !important}.flex-xl-row{flex-direction:row !important}.flex-xl-column{flex-direction:column !important}.flex-xl-row-reverse{flex-direction:row-reverse !important}.flex-xl-column-reverse{flex-direction:column-reverse !important}.flex-xl-grow-0{flex-grow:0 !important}.flex-xl-grow-1{flex-grow:1 !important}.flex-xl-shrink-0{flex-shrink:0 !important}.flex-xl-shrink-1{flex-shrink:1 !important}.flex-xl-wrap{flex-wrap:wrap !important}.flex-xl-nowrap{flex-wrap:nowrap !important}.flex-xl-wrap-reverse{flex-wrap:wrap-reverse !important}.justify-content-xl-start{justify-content:flex-start !important}.justify-content-xl-end{justify-content:flex-end !important}.justify-content-xl-center{justify-content:center !important}.justify-content-xl-between{justify-content:space-between !important}.justify-content-xl-around{justify-content:space-around !important}.justify-content-xl-evenly{justify-content:space-evenly !important}.align-items-xl-start{align-items:flex-start !important}.align-items-xl-end{align-items:flex-end !important}.align-items-xl-center{align-items:center !important}.align-items-xl-baseline{align-items:baseline !important}.align-items-xl-stretch{align-items:stretch !important}.align-content-xl-start{align-content:flex-start !important}.align-content-xl-end{align-content:flex-end !important}.align-content-xl-center{align-content:center !important}.align-content-xl-between{align-content:space-between !important}.align-content-xl-around{align-content:space-around !important}.align-content-xl-stretch{align-content:stretch !important}.align-self-xl-auto{align-self:auto !important}.align-self-xl-start{align-self:flex-start !important}.align-self-xl-end{align-self:flex-end !important}.align-self-xl-center{align-self:center !important}.align-self-xl-baseline{align-self:baseline !important}.align-self-xl-stretch{align-self:stretch !important}.order-xl-first{order:-1 !important}.order-xl-0{order:0 !important}.order-xl-1{order:1 !important}.order-xl-2{order:2 !important}.order-xl-3{order:3 !important}.order-xl-4{order:4 !important}.order-xl-5{order:5 !important}.order-xl-last{order:6 !important}.m-xl-0{margin:0 !important}.m-xl-1{margin:.25rem !important}.m-xl-2{margin:.5rem !important}.m-xl-3{margin:1rem !important}.m-xl-4{margin:1.5rem !important}.m-xl-5{margin:3rem !important}.m-xl-auto{margin:auto !important}.mx-xl-0{margin-right:0 !important;margin-left:0 !important}.mx-xl-1{margin-right:.25rem !important;margin-left:.25rem !important}.mx-xl-2{margin-right:.5rem !important;margin-left:.5rem !important}.mx-xl-3{margin-right:1rem !important;margin-left:1rem !important}.mx-xl-4{margin-right:1.5rem !important;margin-left:1.5rem !important}.mx-xl-5{margin-right:3rem !important;margin-left:3rem !important}.mx-xl-auto{margin-right:auto !important;margin-left:auto !important}.my-xl-0{margin-top:0 !important;margin-bottom:0 !important}.my-xl-1{margin-top:.25rem !important;margin-bottom:.25rem !important}.my-xl-2{margin-top:.5rem !important;margin-bottom:.5rem !important}.my-xl-3{margin-top:1rem !important;margin-bottom:1rem !important}.my-xl-4{margin-top:1.5rem !important;margin-bottom:1.5rem !important}.my-xl-5{margin-top:3rem !important;margin-bottom:3rem !important}.my-xl-auto{margin-top:auto !important;margin-bottom:auto !important}.mt-xl-0{margin-top:0 !important}.mt-xl-1{margin-top:.25rem !important}.mt-xl-2{margin-top:.5rem !important}.mt-xl-3{margin-top:1rem !important}.mt-xl-4{margin-top:1.5rem !important}.mt-xl-5{margin-top:3rem !important}.mt-xl-auto{margin-top:auto !important}.me-xl-0{margin-right:0 !important}.me-xl-1{margin-right:.25rem !important}.me-xl-2{margin-right:.5rem !important}.me-xl-3{margin-right:1rem !important}.me-xl-4{margin-right:1.5rem !important}.me-xl-5{margin-right:3rem !important}.me-xl-auto{margin-right:auto !important}.mb-xl-0{margin-bottom:0 !important}.mb-xl-1{margin-bottom:.25rem !important}.mb-xl-2{margin-bottom:.5rem !important}.mb-xl-3{margin-bottom:1rem !important}.mb-xl-4{margin-bottom:1.5rem !important}.mb-xl-5{margin-bottom:3rem !important}.mb-xl-auto{margin-bottom:auto !important}.ms-xl-0{margin-left:0 !important}.ms-xl-1{margin-left:.25rem !important}.ms-xl-2{margin-left:.5rem !important}.ms-xl-3{margin-left:1rem !important}.ms-xl-4{margin-left:1.5rem !important}.ms-xl-5{margin-left:3rem !important}.ms-xl-auto{margin-left:auto !important}.p-xl-0{padding:0 !important}.p-xl-1{padding:.25rem !important}.p-xl-2{padding:.5rem !important}.p-xl-3{padding:1rem !important}.p-xl-4{padding:1.5rem !important}.p-xl-5{padding:3rem !important}.px-xl-0{padding-right:0 !important;padding-left:0 !important}.px-xl-1{padding-right:.25rem !important;padding-left:.25rem !important}.px-xl-2{padding-right:.5rem !important;padding-left:.5rem !important}.px-xl-3{padding-right:1rem !important;padding-left:1rem !important}.px-xl-4{padding-right:1.5rem !important;padding-left:1.5rem !important}.px-xl-5{padding-right:3rem !important;padding-left:3rem !important}.py-xl-0{padding-top:0 !important;padding-bottom:0 !important}.py-xl-1{padding-top:.25rem !important;padding-bottom:.25rem !important}.py-xl-2{padding-top:.5rem !important;padding-bottom:.5rem !important}.py-xl-3{padding-top:1rem !important;padding-bottom:1rem !important}.py-xl-4{padding-top:1.5rem !important;padding-bottom:1.5rem !important}.py-xl-5{padding-top:3rem !important;padding-bottom:3rem !important}.pt-xl-0{padding-top:0 !important}.pt-xl-1{padding-top:.25rem !important}.pt-xl-2{padding-top:.5rem !important}.pt-xl-3{padding-top:1rem !important}.pt-xl-4{padding-top:1.5rem !important}.pt-xl-5{padding-top:3rem !important}.pe-xl-0{padding-right:0 !important}.pe-xl-1{padding-right:.25rem !important}.pe-xl-2{padding-right:.5rem !important}.pe-xl-3{padding-right:1rem !important}.pe-xl-4{padding-right:1.5rem !important}.pe-xl-5{padding-right:3rem !important}.pb-xl-0{padding-bottom:0 !important}.pb-xl-1{padding-bottom:.25rem !important}.pb-xl-2{padding-bottom:.5rem !important}.pb-xl-3{padding-bottom:1rem !important}.pb-xl-4{padding-bottom:1.5rem !important}.pb-xl-5{padding-bottom:3rem !important}.ps-xl-0{padding-left:0 !important}.ps-xl-1{padding-left:.25rem !important}.ps-xl-2{padding-left:.5rem !important}.ps-xl-3{padding-left:1rem !important}.ps-xl-4{padding-left:1.5rem !important}.ps-xl-5{padding-left:3rem !important}.gap-xl-0{gap:0 !important}.gap-xl-1{gap:.25rem !important}.gap-xl-2{gap:.5rem !important}.gap-xl-3{gap:1rem !important}.gap-xl-4{gap:1.5rem !important}.gap-xl-5{gap:3rem !important}.row-gap-xl-0{row-gap:0 !important}.row-gap-xl-1{row-gap:.25rem !important}.row-gap-xl-2{row-gap:.5rem !important}.row-gap-xl-3{row-gap:1rem !important}.row-gap-xl-4{row-gap:1.5rem !important}.row-gap-xl-5{row-gap:3rem !important}.column-gap-xl-0{column-gap:0 !important}.column-gap-xl-1{column-gap:.25rem !important}.column-gap-xl-2{column-gap:.5rem !important}.column-gap-xl-3{column-gap:1rem !important}.column-gap-xl-4{column-gap:1.5rem !important}.column-gap-xl-5{column-gap:3rem !important}.text-xl-start{text-align:left !important}.text-xl-end{text-align:right !important}.text-xl-center{text-align:center !important}}@media(min-width: 1200px){.fs-1{font-size:2.5rem !important}.fs-2{font-size:2rem !important}.fs-3{font-size:1.75rem !important}.fs-4{font-size:1.5rem !important}}@media print{.d-print-inline{display:inline !important}.d-print-inline-block{display:inline-block !important}.d-print-block{display:block !important}.d-print-grid{display:grid !important}.d-print-inline-grid{display:inline-grid !important}.d-print-table{display:table !important}.d-print-table-row{display:table-row !important}.d-print-table-cell{display:table-cell !important}.d-print-flex{display:flex !important}.d-print-inline-flex{display:inline-flex !important}.d-print-none{display:none !important}}","@mixin bsBanner($file) {\n  /*!\n   * Bootstrap #{$file} v5.3.3 (https://getbootstrap.com/)\n   * Copyright 2011-2024 The Bootstrap Authors\n   * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n   */\n}\n",":root,\n[data-bs-theme=\"light\"] {\n  // Note: Custom variable values only support SassScript inside `#{}`.\n\n  // Colors\n  //\n  // Generate palettes for full colors, grays, and theme colors.\n\n  @each $color, $value in $colors {\n    --#{$prefix}#{$color}: #{$value};\n  }\n\n  @each $color, $value in $grays {\n    --#{$prefix}gray-#{$color}: #{$value};\n  }\n\n  @each $color, $value in $theme-colors {\n    --#{$prefix}#{$color}: #{$value};\n  }\n\n  @each $color, $value in $theme-colors-rgb {\n    --#{$prefix}#{$color}-rgb: #{$value};\n  }\n\n  @each $color, $value in $theme-colors-text {\n    --#{$prefix}#{$color}-text-emphasis: #{$value};\n  }\n\n  @each $color, $value in $theme-colors-bg-subtle {\n    --#{$prefix}#{$color}-bg-subtle: #{$value};\n  }\n\n  @each $color, $value in $theme-colors-border-subtle {\n    --#{$prefix}#{$color}-border-subtle: #{$value};\n  }\n\n  --#{$prefix}white-rgb: #{to-rgb($white)};\n  --#{$prefix}black-rgb: #{to-rgb($black)};\n\n  // Fonts\n\n  // Note: Use `inspect` for lists so that quoted items keep the quotes.\n  // See https://github.com/sass/sass/issues/2383#issuecomment-336349172\n  --#{$prefix}font-sans-serif: #{inspect($font-family-sans-serif)};\n  --#{$prefix}font-monospace: #{inspect($font-family-monospace)};\n  --#{$prefix}gradient: #{$gradient};\n\n  // Root and body\n  // scss-docs-start root-body-variables\n  @if $font-size-root != null {\n    --#{$prefix}root-font-size: #{$font-size-root};\n  }\n  --#{$prefix}body-font-family: #{inspect($font-family-base)};\n  @include rfs($font-size-base, --#{$prefix}body-font-size);\n  --#{$prefix}body-font-weight: #{$font-weight-base};\n  --#{$prefix}body-line-height: #{$line-height-base};\n  @if $body-text-align != null {\n    --#{$prefix}body-text-align: #{$body-text-align};\n  }\n\n  --#{$prefix}body-color: #{$body-color};\n  --#{$prefix}body-color-rgb: #{to-rgb($body-color)};\n  --#{$prefix}body-bg: #{$body-bg};\n  --#{$prefix}body-bg-rgb: #{to-rgb($body-bg)};\n\n  --#{$prefix}emphasis-color: #{$body-emphasis-color};\n  --#{$prefix}emphasis-color-rgb: #{to-rgb($body-emphasis-color)};\n\n  --#{$prefix}secondary-color: #{$body-secondary-color};\n  --#{$prefix}secondary-color-rgb: #{to-rgb($body-secondary-color)};\n  --#{$prefix}secondary-bg: #{$body-secondary-bg};\n  --#{$prefix}secondary-bg-rgb: #{to-rgb($body-secondary-bg)};\n\n  --#{$prefix}tertiary-color: #{$body-tertiary-color};\n  --#{$prefix}tertiary-color-rgb: #{to-rgb($body-tertiary-color)};\n  --#{$prefix}tertiary-bg: #{$body-tertiary-bg};\n  --#{$prefix}tertiary-bg-rgb: #{to-rgb($body-tertiary-bg)};\n  // scss-docs-end root-body-variables\n\n  --#{$prefix}heading-color: #{$headings-color};\n\n  --#{$prefix}link-color: #{$link-color};\n  --#{$prefix}link-color-rgb: #{to-rgb($link-color)};\n  --#{$prefix}link-decoration: #{$link-decoration};\n\n  --#{$prefix}link-hover-color: #{$link-hover-color};\n  --#{$prefix}link-hover-color-rgb: #{to-rgb($link-hover-color)};\n\n  @if $link-hover-decoration != null {\n    --#{$prefix}link-hover-decoration: #{$link-hover-decoration};\n  }\n\n  --#{$prefix}code-color: #{$code-color};\n  --#{$prefix}highlight-color: #{$mark-color};\n  --#{$prefix}highlight-bg: #{$mark-bg};\n\n  // scss-docs-start root-border-var\n  --#{$prefix}border-width: #{$border-width};\n  --#{$prefix}border-style: #{$border-style};\n  --#{$prefix}border-color: #{$border-color};\n  --#{$prefix}border-color-translucent: #{$border-color-translucent};\n\n  --#{$prefix}border-radius: #{$border-radius};\n  --#{$prefix}border-radius-sm: #{$border-radius-sm};\n  --#{$prefix}border-radius-lg: #{$border-radius-lg};\n  --#{$prefix}border-radius-xl: #{$border-radius-xl};\n  --#{$prefix}border-radius-xxl: #{$border-radius-xxl};\n  --#{$prefix}border-radius-2xl: var(--#{$prefix}border-radius-xxl); // Deprecated in v5.3.0 for consistency\n  --#{$prefix}border-radius-pill: #{$border-radius-pill};\n  // scss-docs-end root-border-var\n\n  --#{$prefix}box-shadow: #{$box-shadow};\n  --#{$prefix}box-shadow-sm: #{$box-shadow-sm};\n  --#{$prefix}box-shadow-lg: #{$box-shadow-lg};\n  --#{$prefix}box-shadow-inset: #{$box-shadow-inset};\n\n  // Focus styles\n  // scss-docs-start root-focus-variables\n  --#{$prefix}focus-ring-width: #{$focus-ring-width};\n  --#{$prefix}focus-ring-opacity: #{$focus-ring-opacity};\n  --#{$prefix}focus-ring-color: #{$focus-ring-color};\n  // scss-docs-end root-focus-variables\n\n  // scss-docs-start root-form-validation-variables\n  --#{$prefix}form-valid-color: #{$form-valid-color};\n  --#{$prefix}form-valid-border-color: #{$form-valid-border-color};\n  --#{$prefix}form-invalid-color: #{$form-invalid-color};\n  --#{$prefix}form-invalid-border-color: #{$form-invalid-border-color};\n  // scss-docs-end root-form-validation-variables\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark, true) {\n    color-scheme: dark;\n\n    // scss-docs-start root-dark-mode-vars\n    --#{$prefix}body-color: #{$body-color-dark};\n    --#{$prefix}body-color-rgb: #{to-rgb($body-color-dark)};\n    --#{$prefix}body-bg: #{$body-bg-dark};\n    --#{$prefix}body-bg-rgb: #{to-rgb($body-bg-dark)};\n\n    --#{$prefix}emphasis-color: #{$body-emphasis-color-dark};\n    --#{$prefix}emphasis-color-rgb: #{to-rgb($body-emphasis-color-dark)};\n\n    --#{$prefix}secondary-color: #{$body-secondary-color-dark};\n    --#{$prefix}secondary-color-rgb: #{to-rgb($body-secondary-color-dark)};\n    --#{$prefix}secondary-bg: #{$body-secondary-bg-dark};\n    --#{$prefix}secondary-bg-rgb: #{to-rgb($body-secondary-bg-dark)};\n\n    --#{$prefix}tertiary-color: #{$body-tertiary-color-dark};\n    --#{$prefix}tertiary-color-rgb: #{to-rgb($body-tertiary-color-dark)};\n    --#{$prefix}tertiary-bg: #{$body-tertiary-bg-dark};\n    --#{$prefix}tertiary-bg-rgb: #{to-rgb($body-tertiary-bg-dark)};\n\n    @each $color, $value in $theme-colors-text-dark {\n      --#{$prefix}#{$color}-text-emphasis: #{$value};\n    }\n\n    @each $color, $value in $theme-colors-bg-subtle-dark {\n      --#{$prefix}#{$color}-bg-subtle: #{$value};\n    }\n\n    @each $color, $value in $theme-colors-border-subtle-dark {\n      --#{$prefix}#{$color}-border-subtle: #{$value};\n    }\n\n    --#{$prefix}heading-color: #{$headings-color-dark};\n\n    --#{$prefix}link-color: #{$link-color-dark};\n    --#{$prefix}link-hover-color: #{$link-hover-color-dark};\n    --#{$prefix}link-color-rgb: #{to-rgb($link-color-dark)};\n    --#{$prefix}link-hover-color-rgb: #{to-rgb($link-hover-color-dark)};\n\n    --#{$prefix}code-color: #{$code-color-dark};\n    --#{$prefix}highlight-color: #{$mark-color-dark};\n    --#{$prefix}highlight-bg: #{$mark-bg-dark};\n\n    --#{$prefix}border-color: #{$border-color-dark};\n    --#{$prefix}border-color-translucent: #{$border-color-translucent-dark};\n\n    --#{$prefix}form-valid-color: #{$form-valid-color-dark};\n    --#{$prefix}form-valid-border-color: #{$form-valid-border-color-dark};\n    --#{$prefix}form-invalid-color: #{$form-invalid-color-dark};\n    --#{$prefix}form-invalid-border-color: #{$form-invalid-border-color-dark};\n    // scss-docs-end root-dark-mode-vars\n  }\n}\n","// stylelint-disable scss/dimension-no-non-numeric-values\n\n// SCSS RFS mixin\n//\n// Automated responsive values for font sizes, paddings, margins and much more\n//\n// Licensed under MIT (https://github.com/twbs/rfs/blob/main/LICENSE)\n\n// Configuration\n\n// Base value\n$rfs-base-value: 1.25rem !default;\n$rfs-unit: rem !default;\n\n@if $rfs-unit != rem and $rfs-unit != px {\n  @error \"`#{$rfs-unit}` is not a valid unit for $rfs-unit. Use `px` or `rem`.\";\n}\n\n// Breakpoint at where values start decreasing if screen width is smaller\n$rfs-breakpoint: 1200px !default;\n$rfs-breakpoint-unit: px !default;\n\n@if $rfs-breakpoint-unit != px and $rfs-breakpoint-unit != em and $rfs-breakpoint-unit != rem {\n  @error \"`#{$rfs-breakpoint-unit}` is not a valid unit for $rfs-breakpoint-unit. Use `px`, `em` or `rem`.\";\n}\n\n// Resize values based on screen height and width\n$rfs-two-dimensional: false !default;\n\n// Factor of decrease\n$rfs-factor: 10 !default;\n\n@if type-of($rfs-factor) != number or $rfs-factor <= 1 {\n  @error \"`#{$rfs-factor}` is not a valid  $rfs-factor, it must be greater than 1.\";\n}\n\n// Mode. Possibilities: \"min-media-query\", \"max-media-query\"\n$rfs-mode: min-media-query !default;\n\n// Generate enable or disable classes. Possibilities: false, \"enable\" or \"disable\"\n$rfs-class: false !default;\n\n// 1 rem = $rfs-rem-value px\n$rfs-rem-value: 16 !default;\n\n// Safari iframe resize bug: https://github.com/twbs/rfs/issues/14\n$rfs-safari-iframe-resize-bug-fix: false !default;\n\n// Disable RFS by setting $enable-rfs to false\n$enable-rfs: true !default;\n\n// Cache $rfs-base-value unit\n$rfs-base-value-unit: unit($rfs-base-value);\n\n@function divide($dividend, $divisor, $precision: 10) {\n  $sign: if($dividend > 0 and $divisor > 0 or $dividend < 0 and $divisor < 0, 1, -1);\n  $dividend: abs($dividend);\n  $divisor: abs($divisor);\n  @if $dividend == 0 {\n    @return 0;\n  }\n  @if $divisor == 0 {\n    @error \"Cannot divide by 0\";\n  }\n  $remainder: $dividend;\n  $result: 0;\n  $factor: 10;\n  @while ($remainder > 0 and $precision >= 0) {\n    $quotient: 0;\n    @while ($remainder >= $divisor) {\n      $remainder: $remainder - $divisor;\n      $quotient: $quotient + 1;\n    }\n    $result: $result * 10 + $quotient;\n    $factor: $factor * .1;\n    $remainder: $remainder * 10;\n    $precision: $precision - 1;\n    @if ($precision < 0 and $remainder >= $divisor * 5) {\n      $result: $result + 1;\n    }\n  }\n  $result: $result * $factor * $sign;\n  $dividend-unit: unit($dividend);\n  $divisor-unit: unit($divisor);\n  $unit-map: (\n    \"px\": 1px,\n    \"rem\": 1rem,\n    \"em\": 1em,\n    \"%\": 1%\n  );\n  @if ($dividend-unit != $divisor-unit and map-has-key($unit-map, $dividend-unit)) {\n    $result: $result * map-get($unit-map, $dividend-unit);\n  }\n  @return $result;\n}\n\n// Remove px-unit from $rfs-base-value for calculations\n@if $rfs-base-value-unit == px {\n  $rfs-base-value: divide($rfs-base-value, $rfs-base-value * 0 + 1);\n}\n@else if $rfs-base-value-unit == rem {\n  $rfs-base-value: divide($rfs-base-value, divide($rfs-base-value * 0 + 1, $rfs-rem-value));\n}\n\n// Cache $rfs-breakpoint unit to prevent multiple calls\n$rfs-breakpoint-unit-cache: unit($rfs-breakpoint);\n\n// Remove unit from $rfs-breakpoint for calculations\n@if $rfs-breakpoint-unit-cache == px {\n  $rfs-breakpoint: divide($rfs-breakpoint, $rfs-breakpoint * 0 + 1);\n}\n@else if $rfs-breakpoint-unit-cache == rem or $rfs-breakpoint-unit-cache == \"em\" {\n  $rfs-breakpoint: divide($rfs-breakpoint, divide($rfs-breakpoint * 0 + 1, $rfs-rem-value));\n}\n\n// Calculate the media query value\n$rfs-mq-value: if($rfs-breakpoint-unit == px, #{$rfs-breakpoint}px, #{divide($rfs-breakpoint, $rfs-rem-value)}#{$rfs-breakpoint-unit});\n$rfs-mq-property-width: if($rfs-mode == max-media-query, max-width, min-width);\n$rfs-mq-property-height: if($rfs-mode == max-media-query, max-height, min-height);\n\n// Internal mixin used to determine which media query needs to be used\n@mixin _rfs-media-query {\n  @if $rfs-two-dimensional {\n    @if $rfs-mode == max-media-query {\n      @media (#{$rfs-mq-property-width}: #{$rfs-mq-value}), (#{$rfs-mq-property-height}: #{$rfs-mq-value}) {\n        @content;\n      }\n    }\n    @else {\n      @media (#{$rfs-mq-property-width}: #{$rfs-mq-value}) and (#{$rfs-mq-property-height}: #{$rfs-mq-value}) {\n        @content;\n      }\n    }\n  }\n  @else {\n    @media (#{$rfs-mq-property-width}: #{$rfs-mq-value}) {\n      @content;\n    }\n  }\n}\n\n// Internal mixin that adds disable classes to the selector if needed.\n@mixin _rfs-rule {\n  @if $rfs-class == disable and $rfs-mode == max-media-query {\n    // Adding an extra class increases specificity, which prevents the media query to override the property\n    &,\n    .disable-rfs &,\n    &.disable-rfs {\n      @content;\n    }\n  }\n  @else if $rfs-class == enable and $rfs-mode == min-media-query {\n    .enable-rfs &,\n    &.enable-rfs {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Internal mixin that adds enable classes to the selector if needed.\n@mixin _rfs-media-query-rule {\n\n  @if $rfs-class == enable {\n    @if $rfs-mode == min-media-query {\n      @content;\n    }\n\n    @include _rfs-media-query () {\n      .enable-rfs &,\n      &.enable-rfs {\n        @content;\n      }\n    }\n  }\n  @else {\n    @if $rfs-class == disable and $rfs-mode == min-media-query {\n      .disable-rfs &,\n      &.disable-rfs {\n        @content;\n      }\n    }\n    @include _rfs-media-query () {\n      @content;\n    }\n  }\n}\n\n// Helper function to get the formatted non-responsive value\n@function rfs-value($values) {\n  // Convert to list\n  $values: if(type-of($values) != list, ($values,), $values);\n\n  $val: \"\";\n\n  // Loop over each value and calculate value\n  @each $value in $values {\n    @if $value == 0 {\n      $val: $val + \" 0\";\n    }\n    @else {\n      // Cache $value unit\n      $unit: if(type-of($value) == \"number\", unit($value), false);\n\n      @if $unit == px {\n        // Convert to rem if needed\n        $val: $val + \" \" + if($rfs-unit == rem, #{divide($value, $value * 0 + $rfs-rem-value)}rem, $value);\n      }\n      @else if $unit == rem {\n        // Convert to px if needed\n        $val: $val + \" \" + if($rfs-unit == px, #{divide($value, $value * 0 + 1) * $rfs-rem-value}px, $value);\n      } @else {\n        // If $value isn't a number (like inherit) or $value has a unit (not px or rem, like 1.5em) or $ is 0, just print the value\n        $val: $val + \" \" + $value;\n      }\n    }\n  }\n\n  // Remove first space\n  @return unquote(str-slice($val, 2));\n}\n\n// Helper function to get the responsive value calculated by RFS\n@function rfs-fluid-value($values) {\n  // Convert to list\n  $values: if(type-of($values) != list, ($values,), $values);\n\n  $val: \"\";\n\n  // Loop over each value and calculate value\n  @each $value in $values {\n    @if $value == 0 {\n      $val: $val + \" 0\";\n    } @else {\n      // Cache $value unit\n      $unit: if(type-of($value) == \"number\", unit($value), false);\n\n      // If $value isn't a number (like inherit) or $value has a unit (not px or rem, like 1.5em) or $ is 0, just print the value\n      @if not $unit or $unit != px and $unit != rem {\n        $val: $val + \" \" + $value;\n      } @else {\n        // Remove unit from $value for calculations\n        $value: divide($value, $value * 0 + if($unit == px, 1, divide(1, $rfs-rem-value)));\n\n        // Only add the media query if the value is greater than the minimum value\n        @if abs($value) <= $rfs-base-value or not $enable-rfs {\n          $val: $val + \" \" + if($rfs-unit == rem, #{divide($value, $rfs-rem-value)}rem, #{$value}px);\n        }\n        @else {\n          // Calculate the minimum value\n          $value-min: $rfs-base-value + divide(abs($value) - $rfs-base-value, $rfs-factor);\n\n          // Calculate difference between $value and the minimum value\n          $value-diff: abs($value) - $value-min;\n\n          // Base value formatting\n          $min-width: if($rfs-unit == rem, #{divide($value-min, $rfs-rem-value)}rem, #{$value-min}px);\n\n          // Use negative value if needed\n          $min-width: if($value < 0, -$min-width, $min-width);\n\n          // Use `vmin` if two-dimensional is enabled\n          $variable-unit: if($rfs-two-dimensional, vmin, vw);\n\n          // Calculate the variable width between 0 and $rfs-breakpoint\n          $variable-width: #{divide($value-diff * 100, $rfs-breakpoint)}#{$variable-unit};\n\n          // Return the calculated value\n          $val: $val + \" calc(\" + $min-width + if($value < 0, \" - \", \" + \") + $variable-width + \")\";\n        }\n      }\n    }\n  }\n\n  // Remove first space\n  @return unquote(str-slice($val, 2));\n}\n\n// RFS mixin\n@mixin rfs($values, $property: font-size) {\n  @if $values != null {\n    $val: rfs-value($values);\n    $fluid-val: rfs-fluid-value($values);\n\n    // Do not print the media query if responsive & non-responsive values are the same\n    @if $val == $fluid-val {\n      #{$property}: $val;\n    }\n    @else {\n      @include _rfs-rule () {\n        #{$property}: if($rfs-mode == max-media-query, $val, $fluid-val);\n\n        // Include safari iframe resize fix if needed\n        min-width: if($rfs-safari-iframe-resize-bug-fix, (0 * 1vw), null);\n      }\n\n      @include _rfs-media-query-rule () {\n        #{$property}: if($rfs-mode == max-media-query, $fluid-val, $val);\n      }\n    }\n  }\n}\n\n// Shorthand helper mixins\n@mixin font-size($value) {\n  @include rfs($value);\n}\n\n@mixin padding($value) {\n  @include rfs($value, padding);\n}\n\n@mixin padding-top($value) {\n  @include rfs($value, padding-top);\n}\n\n@mixin padding-right($value) {\n  @include rfs($value, padding-right);\n}\n\n@mixin padding-bottom($value) {\n  @include rfs($value, padding-bottom);\n}\n\n@mixin padding-left($value) {\n  @include rfs($value, padding-left);\n}\n\n@mixin margin($value) {\n  @include rfs($value, margin);\n}\n\n@mixin margin-top($value) {\n  @include rfs($value, margin-top);\n}\n\n@mixin margin-right($value) {\n  @include rfs($value, margin-right);\n}\n\n@mixin margin-bottom($value) {\n  @include rfs($value, margin-bottom);\n}\n\n@mixin margin-left($value) {\n  @include rfs($value, margin-left);\n}\n","// scss-docs-start color-mode-mixin\n@mixin color-mode($mode: light, $root: false) {\n  @if $color-mode-type == \"media-query\" {\n    @if $root == true {\n      @media (prefers-color-scheme: $mode) {\n        :root {\n          @content;\n        }\n      }\n    } @else {\n      @media (prefers-color-scheme: $mode) {\n        @content;\n      }\n    }\n  } @else {\n    [data-bs-theme=\"#{$mode}\"] {\n      @content;\n    }\n  }\n}\n// scss-docs-end color-mode-mixin\n","// stylelint-disable declaration-no-important, selector-no-qualifying-type, property-no-vendor-prefix\n\n\n// Reboot\n//\n// Normalization of HTML elements, manually forked from Normalize.css to remove\n// styles targeting irrelevant browsers while applying new styles.\n//\n// Normalize is licensed MIT. https://github.com/necolas/normalize.css\n\n\n// Document\n//\n// Change from `box-sizing: content-box` so that `width` is not affected by `padding` or `border`.\n\n*,\n*::before,\n*::after {\n  box-sizing: border-box;\n}\n\n\n// Root\n//\n// Ability to the value of the root font sizes, affecting the value of `rem`.\n// null by default, thus nothing is generated.\n\n:root {\n  @if $font-size-root != null {\n    @include font-size(var(--#{$prefix}root-font-size));\n  }\n\n  @if $enable-smooth-scroll {\n    @media (prefers-reduced-motion: no-preference) {\n      scroll-behavior: smooth;\n    }\n  }\n}\n\n\n// Body\n//\n// 1. Remove the margin in all browsers.\n// 2. As a best practice, apply a default `background-color`.\n// 3. Prevent adjustments of font size after orientation changes in iOS.\n// 4. Change the default tap highlight to be completely transparent in iOS.\n\n// scss-docs-start reboot-body-rules\nbody {\n  margin: 0; // 1\n  font-family: var(--#{$prefix}body-font-family);\n  @include font-size(var(--#{$prefix}body-font-size));\n  font-weight: var(--#{$prefix}body-font-weight);\n  line-height: var(--#{$prefix}body-line-height);\n  color: var(--#{$prefix}body-color);\n  text-align: var(--#{$prefix}body-text-align);\n  background-color: var(--#{$prefix}body-bg); // 2\n  -webkit-text-size-adjust: 100%; // 3\n  -webkit-tap-highlight-color: rgba($black, 0); // 4\n}\n// scss-docs-end reboot-body-rules\n\n\n// Content grouping\n//\n// 1. Reset Firefox's gray color\n\nhr {\n  margin: $hr-margin-y 0;\n  color: $hr-color; // 1\n  border: 0;\n  border-top: $hr-border-width solid $hr-border-color;\n  opacity: $hr-opacity;\n}\n\n\n// Typography\n//\n// 1. Remove top margins from headings\n//    By default, `<h1>`-`<h6>` all receive top and bottom margins. We nuke the top\n//    margin for easier control within type scales as it avoids margin collapsing.\n\n%heading {\n  margin-top: 0; // 1\n  margin-bottom: $headings-margin-bottom;\n  font-family: $headings-font-family;\n  font-style: $headings-font-style;\n  font-weight: $headings-font-weight;\n  line-height: $headings-line-height;\n  color: var(--#{$prefix}heading-color);\n}\n\nh1 {\n  @extend %heading;\n  @include font-size($h1-font-size);\n}\n\nh2 {\n  @extend %heading;\n  @include font-size($h2-font-size);\n}\n\nh3 {\n  @extend %heading;\n  @include font-size($h3-font-size);\n}\n\nh4 {\n  @extend %heading;\n  @include font-size($h4-font-size);\n}\n\nh5 {\n  @extend %heading;\n  @include font-size($h5-font-size);\n}\n\nh6 {\n  @extend %heading;\n  @include font-size($h6-font-size);\n}\n\n\n// Reset margins on paragraphs\n//\n// Similarly, the top margin on `<p>`s get reset. However, we also reset the\n// bottom margin to use `rem` units instead of `em`.\n\np {\n  margin-top: 0;\n  margin-bottom: $paragraph-margin-bottom;\n}\n\n\n// Abbreviations\n//\n// 1. Add the correct text decoration in Chrome, Edge, Opera, and Safari.\n// 2. Add explicit cursor to indicate changed behavior.\n// 3. Prevent the text-decoration to be skipped.\n\nabbr[title] {\n  text-decoration: underline dotted; // 1\n  cursor: help; // 2\n  text-decoration-skip-ink: none; // 3\n}\n\n\n// Address\n\naddress {\n  margin-bottom: 1rem;\n  font-style: normal;\n  line-height: inherit;\n}\n\n\n// Lists\n\nol,\nul {\n  padding-left: 2rem;\n}\n\nol,\nul,\ndl {\n  margin-top: 0;\n  margin-bottom: 1rem;\n}\n\nol ol,\nul ul,\nol ul,\nul ol {\n  margin-bottom: 0;\n}\n\ndt {\n  font-weight: $dt-font-weight;\n}\n\n// 1. Undo browser default\n\ndd {\n  margin-bottom: .5rem;\n  margin-left: 0; // 1\n}\n\n\n// Blockquote\n\nblockquote {\n  margin: 0 0 1rem;\n}\n\n\n// Strong\n//\n// Add the correct font weight in Chrome, Edge, and Safari\n\nb,\nstrong {\n  font-weight: $font-weight-bolder;\n}\n\n\n// Small\n//\n// Add the correct font size in all browsers\n\nsmall {\n  @include font-size($small-font-size);\n}\n\n\n// Mark\n\nmark {\n  padding: $mark-padding;\n  color: var(--#{$prefix}highlight-color);\n  background-color: var(--#{$prefix}highlight-bg);\n}\n\n\n// Sub and Sup\n//\n// Prevent `sub` and `sup` elements from affecting the line height in\n// all browsers.\n\nsub,\nsup {\n  position: relative;\n  @include font-size($sub-sup-font-size);\n  line-height: 0;\n  vertical-align: baseline;\n}\n\nsub { bottom: -.25em; }\nsup { top: -.5em; }\n\n\n// Links\n\na {\n  color: rgba(var(--#{$prefix}link-color-rgb), var(--#{$prefix}link-opacity, 1));\n  text-decoration: $link-decoration;\n\n  &:hover {\n    --#{$prefix}link-color-rgb: var(--#{$prefix}link-hover-color-rgb);\n    text-decoration: $link-hover-decoration;\n  }\n}\n\n// And undo these styles for placeholder links/named anchors (without href).\n// It would be more straightforward to just use a[href] in previous block, but that\n// causes specificity issues in many other styles that are too complex to fix.\n// See https://github.com/twbs/bootstrap/issues/19402\n\na:not([href]):not([class]) {\n  &,\n  &:hover {\n    color: inherit;\n    text-decoration: none;\n  }\n}\n\n\n// Code\n\npre,\ncode,\nkbd,\nsamp {\n  font-family: $font-family-code;\n  @include font-size(1em); // Correct the odd `em` font sizing in all browsers.\n}\n\n// 1. Remove browser default top margin\n// 2. Reset browser default of `1em` to use `rem`s\n// 3. Don't allow content to break outside\n\npre {\n  display: block;\n  margin-top: 0; // 1\n  margin-bottom: 1rem; // 2\n  overflow: auto; // 3\n  @include font-size($code-font-size);\n  color: $pre-color;\n\n  // Account for some code outputs that place code tags in pre tags\n  code {\n    @include font-size(inherit);\n    color: inherit;\n    word-break: normal;\n  }\n}\n\ncode {\n  @include font-size($code-font-size);\n  color: var(--#{$prefix}code-color);\n  word-wrap: break-word;\n\n  // Streamline the style when inside anchors to avoid broken underline and more\n  a > & {\n    color: inherit;\n  }\n}\n\nkbd {\n  padding: $kbd-padding-y $kbd-padding-x;\n  @include font-size($kbd-font-size);\n  color: $kbd-color;\n  background-color: $kbd-bg;\n  @include border-radius($border-radius-sm);\n\n  kbd {\n    padding: 0;\n    @include font-size(1em);\n    font-weight: $nested-kbd-font-weight;\n  }\n}\n\n\n// Figures\n//\n// Apply a consistent margin strategy (matches our type styles).\n\nfigure {\n  margin: 0 0 1rem;\n}\n\n\n// Images and content\n\nimg,\nsvg {\n  vertical-align: middle;\n}\n\n\n// Tables\n//\n// Prevent double borders\n\ntable {\n  caption-side: bottom;\n  border-collapse: collapse;\n}\n\ncaption {\n  padding-top: $table-cell-padding-y;\n  padding-bottom: $table-cell-padding-y;\n  color: $table-caption-color;\n  text-align: left;\n}\n\n// 1. Removes font-weight bold by inheriting\n// 2. Matches default `<td>` alignment by inheriting `text-align`.\n// 3. Fix alignment for Safari\n\nth {\n  font-weight: $table-th-font-weight; // 1\n  text-align: inherit; // 2\n  text-align: -webkit-match-parent; // 3\n}\n\nthead,\ntbody,\ntfoot,\ntr,\ntd,\nth {\n  border-color: inherit;\n  border-style: solid;\n  border-width: 0;\n}\n\n\n// Forms\n//\n// 1. Allow labels to use `margin` for spacing.\n\nlabel {\n  display: inline-block; // 1\n}\n\n// Remove the default `border-radius` that macOS Chrome adds.\n// See https://github.com/twbs/bootstrap/issues/24093\n\nbutton {\n  // stylelint-disable-next-line property-disallowed-list\n  border-radius: 0;\n}\n\n// Explicitly remove focus outline in Chromium when it shouldn't be\n// visible (e.g. as result of mouse click or touch tap). It already\n// should be doing this automatically, but seems to currently be\n// confused and applies its very visible two-tone outline anyway.\n\nbutton:focus:not(:focus-visible) {\n  outline: 0;\n}\n\n// 1. Remove the margin in Firefox and Safari\n\ninput,\nbutton,\nselect,\noptgroup,\ntextarea {\n  margin: 0; // 1\n  font-family: inherit;\n  @include font-size(inherit);\n  line-height: inherit;\n}\n\n// Remove the inheritance of text transform in Firefox\nbutton,\nselect {\n  text-transform: none;\n}\n// Set the cursor for non-`<button>` buttons\n//\n// Details at https://github.com/twbs/bootstrap/pull/30562\n[role=\"button\"] {\n  cursor: pointer;\n}\n\nselect {\n  // Remove the inheritance of word-wrap in Safari.\n  // See https://github.com/twbs/bootstrap/issues/24990\n  word-wrap: normal;\n\n  // Undo the opacity change from Chrome\n  &:disabled {\n    opacity: 1;\n  }\n}\n\n// Remove the dropdown arrow only from text type inputs built with datalists in Chrome.\n// See https://stackoverflow.com/a/54997118\n\n[list]:not([type=\"date\"]):not([type=\"datetime-local\"]):not([type=\"month\"]):not([type=\"week\"]):not([type=\"time\"])::-webkit-calendar-picker-indicator {\n  display: none !important;\n}\n\n// 1. Prevent a WebKit bug where (2) destroys native `audio` and `video`\n//    controls in Android 4.\n// 2. Correct the inability to style clickable types in iOS and Safari.\n// 3. Opinionated: add \"hand\" cursor to non-disabled button elements.\n\nbutton,\n[type=\"button\"], // 1\n[type=\"reset\"],\n[type=\"submit\"] {\n  -webkit-appearance: button; // 2\n\n  @if $enable-button-pointers {\n    &:not(:disabled) {\n      cursor: pointer; // 3\n    }\n  }\n}\n\n// Remove inner border and padding from Firefox, but don't restore the outline like Normalize.\n\n::-moz-focus-inner {\n  padding: 0;\n  border-style: none;\n}\n\n// 1. Textareas should really only resize vertically so they don't break their (horizontal) containers.\n\ntextarea {\n  resize: vertical; // 1\n}\n\n// 1. Browsers set a default `min-width: min-content;` on fieldsets,\n//    unlike e.g. `<div>`s, which have `min-width: 0;` by default.\n//    So we reset that to ensure fieldsets behave more like a standard block element.\n//    See https://github.com/twbs/bootstrap/issues/12359\n//    and https://html.spec.whatwg.org/multipage/#the-fieldset-and-legend-elements\n// 2. Reset the default outline behavior of fieldsets so they don't affect page layout.\n\nfieldset {\n  min-width: 0; // 1\n  padding: 0; // 2\n  margin: 0; // 2\n  border: 0; // 2\n}\n\n// 1. By using `float: left`, the legend will behave like a block element.\n//    This way the border of a fieldset wraps around the legend if present.\n// 2. Fix wrapping bug.\n//    See https://github.com/twbs/bootstrap/issues/29712\n\nlegend {\n  float: left; // 1\n  width: 100%;\n  padding: 0;\n  margin-bottom: $legend-margin-bottom;\n  @include font-size($legend-font-size);\n  font-weight: $legend-font-weight;\n  line-height: inherit;\n\n  + * {\n    clear: left; // 2\n  }\n}\n\n// Fix height of inputs with a type of datetime-local, date, month, week, or time\n// See https://github.com/twbs/bootstrap/issues/18842\n\n::-webkit-datetime-edit-fields-wrapper,\n::-webkit-datetime-edit-text,\n::-webkit-datetime-edit-minute,\n::-webkit-datetime-edit-hour-field,\n::-webkit-datetime-edit-day-field,\n::-webkit-datetime-edit-month-field,\n::-webkit-datetime-edit-year-field {\n  padding: 0;\n}\n\n::-webkit-inner-spin-button {\n  height: auto;\n}\n\n// 1. This overrides the extra rounded corners on search inputs in iOS so that our\n//    `.form-control` class can properly style them. Note that this cannot simply\n//    be added to `.form-control` as it's not specific enough. For details, see\n//    https://github.com/twbs/bootstrap/issues/11586.\n// 2. Correct the outline style in Safari.\n\n[type=\"search\"] {\n  -webkit-appearance: textfield; // 1\n  outline-offset: -2px; // 2\n}\n\n// 1. A few input types should stay LTR\n// See https://rtlstyling.com/posts/rtl-styling#form-inputs\n// 2. RTL only output\n// See https://rtlcss.com/learn/usage-guide/control-directives/#raw\n\n/* rtl:raw:\n[type=\"tel\"],\n[type=\"url\"],\n[type=\"email\"],\n[type=\"number\"] {\n  direction: ltr;\n}\n*/\n\n// Remove the inner padding in Chrome and Safari on macOS.\n\n::-webkit-search-decoration {\n  -webkit-appearance: none;\n}\n\n// Remove padding around color pickers in webkit browsers\n\n::-webkit-color-swatch-wrapper {\n  padding: 0;\n}\n\n\n// 1. Inherit font family and line height for file input buttons\n// 2. Correct the inability to style clickable types in iOS and Safari.\n\n::file-selector-button {\n  font: inherit; // 1\n  -webkit-appearance: button; // 2\n}\n\n// Correct element displays\n\noutput {\n  display: inline-block;\n}\n\n// Remove border from iframe\n\niframe {\n  border: 0;\n}\n\n// Summary\n//\n// 1. Add the correct display in all browsers\n\nsummary {\n  display: list-item; // 1\n  cursor: pointer;\n}\n\n\n// Progress\n//\n// Add the correct vertical alignment in Chrome, Firefox, and Opera.\n\nprogress {\n  vertical-align: baseline;\n}\n\n\n// Hidden attribute\n//\n// Always hide an element with the `hidden` HTML attribute.\n\n[hidden] {\n  display: none !important;\n}\n","// Variables\n//\n// Variables should follow the `$component-state-property-size` formula for\n// consistent naming. Ex: $nav-link-disabled-color and $modal-content-box-shadow-xs.\n\n// Color system\n\n// scss-docs-start gray-color-variables\n$white:    #fff !default;\n$gray-100: #f8f9fa !default;\n$gray-200: #e9ecef !default;\n$gray-300: #dee2e6 !default;\n$gray-400: #ced4da !default;\n$gray-500: #adb5bd !default;\n$gray-600: #6c757d !default;\n$gray-700: #495057 !default;\n$gray-800: #343a40 !default;\n$gray-900: #212529 !default;\n$black:    #000 !default;\n// scss-docs-end gray-color-variables\n\n// fusv-disable\n// scss-docs-start gray-colors-map\n$grays: (\n  \"100\": $gray-100,\n  \"200\": $gray-200,\n  \"300\": $gray-300,\n  \"400\": $gray-400,\n  \"500\": $gray-500,\n  \"600\": $gray-600,\n  \"700\": $gray-700,\n  \"800\": $gray-800,\n  \"900\": $gray-900\n) !default;\n// scss-docs-end gray-colors-map\n// fusv-enable\n\n// scss-docs-start color-variables\n$blue:    #0d6efd !default;\n$indigo:  #6610f2 !default;\n$purple:  #6f42c1 !default;\n$pink:    #d63384 !default;\n$red:     #dc3545 !default;\n$orange:  #fd7e14 !default;\n$yellow:  #ffc107 !default;\n$green:   #198754 !default;\n$teal:    #20c997 !default;\n$cyan:    #0dcaf0 !default;\n// scss-docs-end color-variables\n\n// scss-docs-start colors-map\n$colors: (\n  \"blue\":       $blue,\n  \"indigo\":     $indigo,\n  \"purple\":     $purple,\n  \"pink\":       $pink,\n  \"red\":        $red,\n  \"orange\":     $orange,\n  \"yellow\":     $yellow,\n  \"green\":      $green,\n  \"teal\":       $teal,\n  \"cyan\":       $cyan,\n  \"black\":      $black,\n  \"white\":      $white,\n  \"gray\":       $gray-600,\n  \"gray-dark\":  $gray-800\n) !default;\n// scss-docs-end colors-map\n\n// The contrast ratio to reach against white, to determine if color changes from \"light\" to \"dark\". Acceptable values for WCAG 2.0 are 3, 4.5 and 7.\n// See https://www.w3.org/TR/WCAG20/#visual-audio-contrast-contrast\n$min-contrast-ratio:   4.5 !default;\n\n// Customize the light and dark text colors for use in our color contrast function.\n$color-contrast-dark:      $black !default;\n$color-contrast-light:     $white !default;\n\n// fusv-disable\n$blue-100: tint-color($blue, 80%) !default;\n$blue-200: tint-color($blue, 60%) !default;\n$blue-300: tint-color($blue, 40%) !default;\n$blue-400: tint-color($blue, 20%) !default;\n$blue-500: $blue !default;\n$blue-600: shade-color($blue, 20%) !default;\n$blue-700: shade-color($blue, 40%) !default;\n$blue-800: shade-color($blue, 60%) !default;\n$blue-900: shade-color($blue, 80%) !default;\n\n$indigo-100: tint-color($indigo, 80%) !default;\n$indigo-200: tint-color($indigo, 60%) !default;\n$indigo-300: tint-color($indigo, 40%) !default;\n$indigo-400: tint-color($indigo, 20%) !default;\n$indigo-500: $indigo !default;\n$indigo-600: shade-color($indigo, 20%) !default;\n$indigo-700: shade-color($indigo, 40%) !default;\n$indigo-800: shade-color($indigo, 60%) !default;\n$indigo-900: shade-color($indigo, 80%) !default;\n\n$purple-100: tint-color($purple, 80%) !default;\n$purple-200: tint-color($purple, 60%) !default;\n$purple-300: tint-color($purple, 40%) !default;\n$purple-400: tint-color($purple, 20%) !default;\n$purple-500: $purple !default;\n$purple-600: shade-color($purple, 20%) !default;\n$purple-700: shade-color($purple, 40%) !default;\n$purple-800: shade-color($purple, 60%) !default;\n$purple-900: shade-color($purple, 80%) !default;\n\n$pink-100: tint-color($pink, 80%) !default;\n$pink-200: tint-color($pink, 60%) !default;\n$pink-300: tint-color($pink, 40%) !default;\n$pink-400: tint-color($pink, 20%) !default;\n$pink-500: $pink !default;\n$pink-600: shade-color($pink, 20%) !default;\n$pink-700: shade-color($pink, 40%) !default;\n$pink-800: shade-color($pink, 60%) !default;\n$pink-900: shade-color($pink, 80%) !default;\n\n$red-100: tint-color($red, 80%) !default;\n$red-200: tint-color($red, 60%) !default;\n$red-300: tint-color($red, 40%) !default;\n$red-400: tint-color($red, 20%) !default;\n$red-500: $red !default;\n$red-600: shade-color($red, 20%) !default;\n$red-700: shade-color($red, 40%) !default;\n$red-800: shade-color($red, 60%) !default;\n$red-900: shade-color($red, 80%) !default;\n\n$orange-100: tint-color($orange, 80%) !default;\n$orange-200: tint-color($orange, 60%) !default;\n$orange-300: tint-color($orange, 40%) !default;\n$orange-400: tint-color($orange, 20%) !default;\n$orange-500: $orange !default;\n$orange-600: shade-color($orange, 20%) !default;\n$orange-700: shade-color($orange, 40%) !default;\n$orange-800: shade-color($orange, 60%) !default;\n$orange-900: shade-color($orange, 80%) !default;\n\n$yellow-100: tint-color($yellow, 80%) !default;\n$yellow-200: tint-color($yellow, 60%) !default;\n$yellow-300: tint-color($yellow, 40%) !default;\n$yellow-400: tint-color($yellow, 20%) !default;\n$yellow-500: $yellow !default;\n$yellow-600: shade-color($yellow, 20%) !default;\n$yellow-700: shade-color($yellow, 40%) !default;\n$yellow-800: shade-color($yellow, 60%) !default;\n$yellow-900: shade-color($yellow, 80%) !default;\n\n$green-100: tint-color($green, 80%) !default;\n$green-200: tint-color($green, 60%) !default;\n$green-300: tint-color($green, 40%) !default;\n$green-400: tint-color($green, 20%) !default;\n$green-500: $green !default;\n$green-600: shade-color($green, 20%) !default;\n$green-700: shade-color($green, 40%) !default;\n$green-800: shade-color($green, 60%) !default;\n$green-900: shade-color($green, 80%) !default;\n\n$teal-100: tint-color($teal, 80%) !default;\n$teal-200: tint-color($teal, 60%) !default;\n$teal-300: tint-color($teal, 40%) !default;\n$teal-400: tint-color($teal, 20%) !default;\n$teal-500: $teal !default;\n$teal-600: shade-color($teal, 20%) !default;\n$teal-700: shade-color($teal, 40%) !default;\n$teal-800: shade-color($teal, 60%) !default;\n$teal-900: shade-color($teal, 80%) !default;\n\n$cyan-100: tint-color($cyan, 80%) !default;\n$cyan-200: tint-color($cyan, 60%) !default;\n$cyan-300: tint-color($cyan, 40%) !default;\n$cyan-400: tint-color($cyan, 20%) !default;\n$cyan-500: $cyan !default;\n$cyan-600: shade-color($cyan, 20%) !default;\n$cyan-700: shade-color($cyan, 40%) !default;\n$cyan-800: shade-color($cyan, 60%) !default;\n$cyan-900: shade-color($cyan, 80%) !default;\n\n$blues: (\n  \"blue-100\": $blue-100,\n  \"blue-200\": $blue-200,\n  \"blue-300\": $blue-300,\n  \"blue-400\": $blue-400,\n  \"blue-500\": $blue-500,\n  \"blue-600\": $blue-600,\n  \"blue-700\": $blue-700,\n  \"blue-800\": $blue-800,\n  \"blue-900\": $blue-900\n) !default;\n\n$indigos: (\n  \"indigo-100\": $indigo-100,\n  \"indigo-200\": $indigo-200,\n  \"indigo-300\": $indigo-300,\n  \"indigo-400\": $indigo-400,\n  \"indigo-500\": $indigo-500,\n  \"indigo-600\": $indigo-600,\n  \"indigo-700\": $indigo-700,\n  \"indigo-800\": $indigo-800,\n  \"indigo-900\": $indigo-900\n) !default;\n\n$purples: (\n  \"purple-100\": $purple-100,\n  \"purple-200\": $purple-200,\n  \"purple-300\": $purple-300,\n  \"purple-400\": $purple-400,\n  \"purple-500\": $purple-500,\n  \"purple-600\": $purple-600,\n  \"purple-700\": $purple-700,\n  \"purple-800\": $purple-800,\n  \"purple-900\": $purple-900\n) !default;\n\n$pinks: (\n  \"pink-100\": $pink-100,\n  \"pink-200\": $pink-200,\n  \"pink-300\": $pink-300,\n  \"pink-400\": $pink-400,\n  \"pink-500\": $pink-500,\n  \"pink-600\": $pink-600,\n  \"pink-700\": $pink-700,\n  \"pink-800\": $pink-800,\n  \"pink-900\": $pink-900\n) !default;\n\n$reds: (\n  \"red-100\": $red-100,\n  \"red-200\": $red-200,\n  \"red-300\": $red-300,\n  \"red-400\": $red-400,\n  \"red-500\": $red-500,\n  \"red-600\": $red-600,\n  \"red-700\": $red-700,\n  \"red-800\": $red-800,\n  \"red-900\": $red-900\n) !default;\n\n$oranges: (\n  \"orange-100\": $orange-100,\n  \"orange-200\": $orange-200,\n  \"orange-300\": $orange-300,\n  \"orange-400\": $orange-400,\n  \"orange-500\": $orange-500,\n  \"orange-600\": $orange-600,\n  \"orange-700\": $orange-700,\n  \"orange-800\": $orange-800,\n  \"orange-900\": $orange-900\n) !default;\n\n$yellows: (\n  \"yellow-100\": $yellow-100,\n  \"yellow-200\": $yellow-200,\n  \"yellow-300\": $yellow-300,\n  \"yellow-400\": $yellow-400,\n  \"yellow-500\": $yellow-500,\n  \"yellow-600\": $yellow-600,\n  \"yellow-700\": $yellow-700,\n  \"yellow-800\": $yellow-800,\n  \"yellow-900\": $yellow-900\n) !default;\n\n$greens: (\n  \"green-100\": $green-100,\n  \"green-200\": $green-200,\n  \"green-300\": $green-300,\n  \"green-400\": $green-400,\n  \"green-500\": $green-500,\n  \"green-600\": $green-600,\n  \"green-700\": $green-700,\n  \"green-800\": $green-800,\n  \"green-900\": $green-900\n) !default;\n\n$teals: (\n  \"teal-100\": $teal-100,\n  \"teal-200\": $teal-200,\n  \"teal-300\": $teal-300,\n  \"teal-400\": $teal-400,\n  \"teal-500\": $teal-500,\n  \"teal-600\": $teal-600,\n  \"teal-700\": $teal-700,\n  \"teal-800\": $teal-800,\n  \"teal-900\": $teal-900\n) !default;\n\n$cyans: (\n  \"cyan-100\": $cyan-100,\n  \"cyan-200\": $cyan-200,\n  \"cyan-300\": $cyan-300,\n  \"cyan-400\": $cyan-400,\n  \"cyan-500\": $cyan-500,\n  \"cyan-600\": $cyan-600,\n  \"cyan-700\": $cyan-700,\n  \"cyan-800\": $cyan-800,\n  \"cyan-900\": $cyan-900\n) !default;\n// fusv-enable\n\n// scss-docs-start theme-color-variables\n$primary:       $blue !default;\n$secondary:     $gray-600 !default;\n$success:       $green !default;\n$info:          $cyan !default;\n$warning:       $yellow !default;\n$danger:        $red !default;\n$light:         $gray-100 !default;\n$dark:          $gray-900 !default;\n// scss-docs-end theme-color-variables\n\n// scss-docs-start theme-colors-map\n$theme-colors: (\n  \"primary\":    $primary,\n  \"secondary\":  $secondary,\n  \"success\":    $success,\n  \"info\":       $info,\n  \"warning\":    $warning,\n  \"danger\":     $danger,\n  \"light\":      $light,\n  \"dark\":       $dark\n) !default;\n// scss-docs-end theme-colors-map\n\n// scss-docs-start theme-text-variables\n$primary-text-emphasis:   shade-color($primary, 60%) !default;\n$secondary-text-emphasis: shade-color($secondary, 60%) !default;\n$success-text-emphasis:   shade-color($success, 60%) !default;\n$info-text-emphasis:      shade-color($info, 60%) !default;\n$warning-text-emphasis:   shade-color($warning, 60%) !default;\n$danger-text-emphasis:    shade-color($danger, 60%) !default;\n$light-text-emphasis:     $gray-700 !default;\n$dark-text-emphasis:      $gray-700 !default;\n// scss-docs-end theme-text-variables\n\n// scss-docs-start theme-bg-subtle-variables\n$primary-bg-subtle:       tint-color($primary, 80%) !default;\n$secondary-bg-subtle:     tint-color($secondary, 80%) !default;\n$success-bg-subtle:       tint-color($success, 80%) !default;\n$info-bg-subtle:          tint-color($info, 80%) !default;\n$warning-bg-subtle:       tint-color($warning, 80%) !default;\n$danger-bg-subtle:        tint-color($danger, 80%) !default;\n$light-bg-subtle:         mix($gray-100, $white) !default;\n$dark-bg-subtle:          $gray-400 !default;\n// scss-docs-end theme-bg-subtle-variables\n\n// scss-docs-start theme-border-subtle-variables\n$primary-border-subtle:   tint-color($primary, 60%) !default;\n$secondary-border-subtle: tint-color($secondary, 60%) !default;\n$success-border-subtle:   tint-color($success, 60%) !default;\n$info-border-subtle:      tint-color($info, 60%) !default;\n$warning-border-subtle:   tint-color($warning, 60%) !default;\n$danger-border-subtle:    tint-color($danger, 60%) !default;\n$light-border-subtle:     $gray-200 !default;\n$dark-border-subtle:      $gray-500 !default;\n// scss-docs-end theme-border-subtle-variables\n\n// Characters which are escaped by the escape-svg function\n$escaped-characters: (\n  (\"<\", \"%3c\"),\n  (\">\", \"%3e\"),\n  (\"#\", \"%23\"),\n  (\"(\", \"%28\"),\n  (\")\", \"%29\"),\n) !default;\n\n// Options\n//\n// Quickly modify global styling by enabling or disabling optional features.\n\n$enable-caret:                true !default;\n$enable-rounded:              true !default;\n$enable-shadows:              false !default;\n$enable-gradients:            false !default;\n$enable-transitions:          true !default;\n$enable-reduced-motion:       true !default;\n$enable-smooth-scroll:        true !default;\n$enable-grid-classes:         true !default;\n$enable-container-classes:    true !default;\n$enable-cssgrid:              false !default;\n$enable-button-pointers:      true !default;\n$enable-rfs:                  true !default;\n$enable-validation-icons:     true !default;\n$enable-negative-margins:     false !default;\n$enable-deprecation-messages: true !default;\n$enable-important-utilities:  true !default;\n\n$enable-dark-mode:            true !default;\n$color-mode-type:             data !default; // `data` or `media-query`\n\n// Prefix for :root CSS variables\n\n$variable-prefix:             bs- !default; // Deprecated in v5.2.0 for the shorter `$prefix`\n$prefix:                      $variable-prefix !default;\n\n// Gradient\n//\n// The gradient which is added to components if `$enable-gradients` is `true`\n// This gradient is also added to elements with `.bg-gradient`\n// scss-docs-start variable-gradient\n$gradient: linear-gradient(180deg, rgba($white, .15), rgba($white, 0)) !default;\n// scss-docs-end variable-gradient\n\n// Spacing\n//\n// Control the default styling of most Bootstrap elements by modifying these\n// variables. Mostly focused on spacing.\n// You can add more entries to the $spacers map, should you need more variation.\n\n// scss-docs-start spacer-variables-maps\n$spacer: 1rem !default;\n$spacers: (\n  0: 0,\n  1: $spacer * .25,\n  2: $spacer * .5,\n  3: $spacer,\n  4: $spacer * 1.5,\n  5: $spacer * 3,\n) !default;\n// scss-docs-end spacer-variables-maps\n\n// Position\n//\n// Define the edge positioning anchors of the position utilities.\n\n// scss-docs-start position-map\n$position-values: (\n  0: 0,\n  50: 50%,\n  100: 100%\n) !default;\n// scss-docs-end position-map\n\n// Body\n//\n// Settings for the `<body>` element.\n\n$body-text-align:           null !default;\n$body-color:                $gray-900 !default;\n$body-bg:                   $white !default;\n\n$body-secondary-color:      rgba($body-color, .75) !default;\n$body-secondary-bg:         $gray-200 !default;\n\n$body-tertiary-color:       rgba($body-color, .5) !default;\n$body-tertiary-bg:          $gray-100 !default;\n\n$body-emphasis-color:       $black !default;\n\n// Links\n//\n// Style anchor elements.\n\n$link-color:                              $primary !default;\n$link-decoration:                         underline !default;\n$link-shade-percentage:                   20% !default;\n$link-hover-color:                        shift-color($link-color, $link-shade-percentage) !default;\n$link-hover-decoration:                   null !default;\n\n$stretched-link-pseudo-element:           after !default;\n$stretched-link-z-index:                  1 !default;\n\n// Icon links\n// scss-docs-start icon-link-variables\n$icon-link-gap:               .375rem !default;\n$icon-link-underline-offset:  .25em !default;\n$icon-link-icon-size:         1em !default;\n$icon-link-icon-transition:   .2s ease-in-out transform !default;\n$icon-link-icon-transform:    translate3d(.25em, 0, 0) !default;\n// scss-docs-end icon-link-variables\n\n// Paragraphs\n//\n// Style p element.\n\n$paragraph-margin-bottom:   1rem !default;\n\n\n// Grid breakpoints\n//\n// Define the minimum dimensions at which your layout will change,\n// adapting to different screen sizes, for use in media queries.\n\n// scss-docs-start grid-breakpoints\n$grid-breakpoints: (\n  xs: 0,\n  sm: 576px,\n  md: 768px,\n  lg: 992px,\n  xl: 1200px,\n  xxl: 1400px\n) !default;\n// scss-docs-end grid-breakpoints\n\n@include _assert-ascending($grid-breakpoints, \"$grid-breakpoints\");\n@include _assert-starts-at-zero($grid-breakpoints, \"$grid-breakpoints\");\n\n\n// Grid containers\n//\n// Define the maximum width of `.container` for different screen sizes.\n\n// scss-docs-start container-max-widths\n$container-max-widths: (\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1140px,\n  xxl: 1320px\n) !default;\n// scss-docs-end container-max-widths\n\n@include _assert-ascending($container-max-widths, \"$container-max-widths\");\n\n\n// Grid columns\n//\n// Set the number of columns and specify the width of the gutters.\n\n$grid-columns:                12 !default;\n$grid-gutter-width:           1.5rem !default;\n$grid-row-columns:            6 !default;\n\n// Container padding\n\n$container-padding-x: $grid-gutter-width !default;\n\n\n// Components\n//\n// Define common padding and border radius sizes and more.\n\n// scss-docs-start border-variables\n$border-width:                1px !default;\n$border-widths: (\n  1: 1px,\n  2: 2px,\n  3: 3px,\n  4: 4px,\n  5: 5px\n) !default;\n$border-style:                solid !default;\n$border-color:                $gray-300 !default;\n$border-color-translucent:    rgba($black, .175) !default;\n// scss-docs-end border-variables\n\n// scss-docs-start border-radius-variables\n$border-radius:               .375rem !default;\n$border-radius-sm:            .25rem !default;\n$border-radius-lg:            .5rem !default;\n$border-radius-xl:            1rem !default;\n$border-radius-xxl:           2rem !default;\n$border-radius-pill:          50rem !default;\n// scss-docs-end border-radius-variables\n// fusv-disable\n$border-radius-2xl:           $border-radius-xxl !default; // Deprecated in v5.3.0\n// fusv-enable\n\n// scss-docs-start box-shadow-variables\n$box-shadow:                  0 .5rem 1rem rgba($black, .15) !default;\n$box-shadow-sm:               0 .125rem .25rem rgba($black, .075) !default;\n$box-shadow-lg:               0 1rem 3rem rgba($black, .175) !default;\n$box-shadow-inset:            inset 0 1px 2px rgba($black, .075) !default;\n// scss-docs-end box-shadow-variables\n\n$component-active-color:      $white !default;\n$component-active-bg:         $primary !default;\n\n// scss-docs-start focus-ring-variables\n$focus-ring-width:      .25rem !default;\n$focus-ring-opacity:    .25 !default;\n$focus-ring-color:      rgba($primary, $focus-ring-opacity) !default;\n$focus-ring-blur:       0 !default;\n$focus-ring-box-shadow: 0 0 $focus-ring-blur $focus-ring-width $focus-ring-color !default;\n// scss-docs-end focus-ring-variables\n\n// scss-docs-start caret-variables\n$caret-width:                 .3em !default;\n$caret-vertical-align:        $caret-width * .85 !default;\n$caret-spacing:               $caret-width * .85 !default;\n// scss-docs-end caret-variables\n\n$transition-base:             all .2s ease-in-out !default;\n$transition-fade:             opacity .15s linear !default;\n// scss-docs-start collapse-transition\n$transition-collapse:         height .35s ease !default;\n$transition-collapse-width:   width .35s ease !default;\n// scss-docs-end collapse-transition\n\n// stylelint-disable function-disallowed-list\n// scss-docs-start aspect-ratios\n$aspect-ratios: (\n  \"1x1\": 100%,\n  \"4x3\": calc(3 / 4 * 100%),\n  \"16x9\": calc(9 / 16 * 100%),\n  \"21x9\": calc(9 / 21 * 100%)\n) !default;\n// scss-docs-end aspect-ratios\n// stylelint-enable function-disallowed-list\n\n// Typography\n//\n// Font, line-height, and color for body text, headings, and more.\n\n// scss-docs-start font-variables\n// stylelint-disable value-keyword-case\n$font-family-sans-serif:      system-ui, -apple-system, \"Segoe UI\", Roboto, \"Helvetica Neue\", \"Noto Sans\", \"Liberation Sans\", Arial, sans-serif, \"Apple Color Emoji\", \"Segoe UI Emoji\", \"Segoe UI Symbol\", \"Noto Color Emoji\" !default;\n$font-family-monospace:       SFMono-Regular, Menlo, Monaco, Consolas, \"Liberation Mono\", \"Courier New\", monospace !default;\n// stylelint-enable value-keyword-case\n$font-family-base:            var(--#{$prefix}font-sans-serif) !default;\n$font-family-code:            var(--#{$prefix}font-monospace) !default;\n\n// $font-size-root affects the value of `rem`, which is used for as well font sizes, paddings, and margins\n// $font-size-base affects the font size of the body text\n$font-size-root:              null !default;\n$font-size-base:              1rem !default; // Assumes the browser default, typically `16px`\n$font-size-sm:                $font-size-base * .875 !default;\n$font-size-lg:                $font-size-base * 1.25 !default;\n\n$font-weight-lighter:         lighter !default;\n$font-weight-light:           300 !default;\n$font-weight-normal:          400 !default;\n$font-weight-medium:          500 !default;\n$font-weight-semibold:        600 !default;\n$font-weight-bold:            700 !default;\n$font-weight-bolder:          bolder !default;\n\n$font-weight-base:            $font-weight-normal !default;\n\n$line-height-base:            1.5 !default;\n$line-height-sm:              1.25 !default;\n$line-height-lg:              2 !default;\n\n$h1-font-size:                $font-size-base * 2.5 !default;\n$h2-font-size:                $font-size-base * 2 !default;\n$h3-font-size:                $font-size-base * 1.75 !default;\n$h4-font-size:                $font-size-base * 1.5 !default;\n$h5-font-size:                $font-size-base * 1.25 !default;\n$h6-font-size:                $font-size-base !default;\n// scss-docs-end font-variables\n\n// scss-docs-start font-sizes\n$font-sizes: (\n  1: $h1-font-size,\n  2: $h2-font-size,\n  3: $h3-font-size,\n  4: $h4-font-size,\n  5: $h5-font-size,\n  6: $h6-font-size\n) !default;\n// scss-docs-end font-sizes\n\n// scss-docs-start headings-variables\n$headings-margin-bottom:      $spacer * .5 !default;\n$headings-font-family:        null !default;\n$headings-font-style:         null !default;\n$headings-font-weight:        500 !default;\n$headings-line-height:        1.2 !default;\n$headings-color:              inherit !default;\n// scss-docs-end headings-variables\n\n// scss-docs-start display-headings\n$display-font-sizes: (\n  1: 5rem,\n  2: 4.5rem,\n  3: 4rem,\n  4: 3.5rem,\n  5: 3rem,\n  6: 2.5rem\n) !default;\n\n$display-font-family: null !default;\n$display-font-style:  null !default;\n$display-font-weight: 300 !default;\n$display-line-height: $headings-line-height !default;\n// scss-docs-end display-headings\n\n// scss-docs-start type-variables\n$lead-font-size:              $font-size-base * 1.25 !default;\n$lead-font-weight:            300 !default;\n\n$small-font-size:             .875em !default;\n\n$sub-sup-font-size:           .75em !default;\n\n// fusv-disable\n$text-muted:                  var(--#{$prefix}secondary-color) !default; // Deprecated in 5.3.0\n// fusv-enable\n\n$initialism-font-size:        $small-font-size !default;\n\n$blockquote-margin-y:         $spacer !default;\n$blockquote-font-size:        $font-size-base * 1.25 !default;\n$blockquote-footer-color:     $gray-600 !default;\n$blockquote-footer-font-size: $small-font-size !default;\n\n$hr-margin-y:                 $spacer !default;\n$hr-color:                    inherit !default;\n\n// fusv-disable\n$hr-bg-color:                 null !default; // Deprecated in v5.2.0\n$hr-height:                   null !default; // Deprecated in v5.2.0\n// fusv-enable\n\n$hr-border-color:             null !default; // Allows for inherited colors\n$hr-border-width:             var(--#{$prefix}border-width) !default;\n$hr-opacity:                  .25 !default;\n\n// scss-docs-start vr-variables\n$vr-border-width:             var(--#{$prefix}border-width) !default;\n// scss-docs-end vr-variables\n\n$legend-margin-bottom:        .5rem !default;\n$legend-font-size:            1.5rem !default;\n$legend-font-weight:          null !default;\n\n$dt-font-weight:              $font-weight-bold !default;\n\n$list-inline-padding:         .5rem !default;\n\n$mark-padding:                .1875em !default;\n$mark-color:                  $body-color !default;\n$mark-bg:                     $yellow-100 !default;\n// scss-docs-end type-variables\n\n\n// Tables\n//\n// Customizes the `.table` component with basic values, each used across all table variations.\n\n// scss-docs-start table-variables\n$table-cell-padding-y:        .5rem !default;\n$table-cell-padding-x:        .5rem !default;\n$table-cell-padding-y-sm:     .25rem !default;\n$table-cell-padding-x-sm:     .25rem !default;\n\n$table-cell-vertical-align:   top !default;\n\n$table-color:                 var(--#{$prefix}emphasis-color) !default;\n$table-bg:                    var(--#{$prefix}body-bg) !default;\n$table-accent-bg:             transparent !default;\n\n$table-th-font-weight:        null !default;\n\n$table-striped-color:         $table-color !default;\n$table-striped-bg-factor:     .05 !default;\n$table-striped-bg:            rgba(var(--#{$prefix}emphasis-color-rgb), $table-striped-bg-factor) !default;\n\n$table-active-color:          $table-color !default;\n$table-active-bg-factor:      .1 !default;\n$table-active-bg:             rgba(var(--#{$prefix}emphasis-color-rgb), $table-active-bg-factor) !default;\n\n$table-hover-color:           $table-color !default;\n$table-hover-bg-factor:       .075 !default;\n$table-hover-bg:              rgba(var(--#{$prefix}emphasis-color-rgb), $table-hover-bg-factor) !default;\n\n$table-border-factor:         .2 !default;\n$table-border-width:          var(--#{$prefix}border-width) !default;\n$table-border-color:          var(--#{$prefix}border-color) !default;\n\n$table-striped-order:         odd !default;\n$table-striped-columns-order: even !default;\n\n$table-group-separator-color: currentcolor !default;\n\n$table-caption-color:         var(--#{$prefix}secondary-color) !default;\n\n$table-bg-scale:              -80% !default;\n// scss-docs-end table-variables\n\n// scss-docs-start table-loop\n$table-variants: (\n  \"primary\":    shift-color($primary, $table-bg-scale),\n  \"secondary\":  shift-color($secondary, $table-bg-scale),\n  \"success\":    shift-color($success, $table-bg-scale),\n  \"info\":       shift-color($info, $table-bg-scale),\n  \"warning\":    shift-color($warning, $table-bg-scale),\n  \"danger\":     shift-color($danger, $table-bg-scale),\n  \"light\":      $light,\n  \"dark\":       $dark,\n) !default;\n// scss-docs-end table-loop\n\n\n// Buttons + Forms\n//\n// Shared variables that are reassigned to `$input-` and `$btn-` specific variables.\n\n// scss-docs-start input-btn-variables\n$input-btn-padding-y:         .375rem !default;\n$input-btn-padding-x:         .75rem !default;\n$input-btn-font-family:       null !default;\n$input-btn-font-size:         $font-size-base !default;\n$input-btn-line-height:       $line-height-base !default;\n\n$input-btn-focus-width:         $focus-ring-width !default;\n$input-btn-focus-color-opacity: $focus-ring-opacity !default;\n$input-btn-focus-color:         $focus-ring-color !default;\n$input-btn-focus-blur:          $focus-ring-blur !default;\n$input-btn-focus-box-shadow:    $focus-ring-box-shadow !default;\n\n$input-btn-padding-y-sm:      .25rem !default;\n$input-btn-padding-x-sm:      .5rem !default;\n$input-btn-font-size-sm:      $font-size-sm !default;\n\n$input-btn-padding-y-lg:      .5rem !default;\n$input-btn-padding-x-lg:      1rem !default;\n$input-btn-font-size-lg:      $font-size-lg !default;\n\n$input-btn-border-width:      var(--#{$prefix}border-width) !default;\n// scss-docs-end input-btn-variables\n\n\n// Buttons\n//\n// For each of Bootstrap's buttons, define text, background, and border color.\n\n// scss-docs-start btn-variables\n$btn-color:                   var(--#{$prefix}body-color) !default;\n$btn-padding-y:               $input-btn-padding-y !default;\n$btn-padding-x:               $input-btn-padding-x !default;\n$btn-font-family:             $input-btn-font-family !default;\n$btn-font-size:               $input-btn-font-size !default;\n$btn-line-height:             $input-btn-line-height !default;\n$btn-white-space:             null !default; // Set to `nowrap` to prevent text wrapping\n\n$btn-padding-y-sm:            $input-btn-padding-y-sm !default;\n$btn-padding-x-sm:            $input-btn-padding-x-sm !default;\n$btn-font-size-sm:            $input-btn-font-size-sm !default;\n\n$btn-padding-y-lg:            $input-btn-padding-y-lg !default;\n$btn-padding-x-lg:            $input-btn-padding-x-lg !default;\n$btn-font-size-lg:            $input-btn-font-size-lg !default;\n\n$btn-border-width:            $input-btn-border-width !default;\n\n$btn-font-weight:             $font-weight-normal !default;\n$btn-box-shadow:              inset 0 1px 0 rgba($white, .15), 0 1px 1px rgba($black, .075) !default;\n$btn-focus-width:             $input-btn-focus-width !default;\n$btn-focus-box-shadow:        $input-btn-focus-box-shadow !default;\n$btn-disabled-opacity:        .65 !default;\n$btn-active-box-shadow:       inset 0 3px 5px rgba($black, .125) !default;\n\n$btn-link-color:              var(--#{$prefix}link-color) !default;\n$btn-link-hover-color:        var(--#{$prefix}link-hover-color) !default;\n$btn-link-disabled-color:     $gray-600 !default;\n$btn-link-focus-shadow-rgb:   to-rgb(mix(color-contrast($link-color), $link-color, 15%)) !default;\n\n// Allows for customizing button radius independently from global border radius\n$btn-border-radius:           var(--#{$prefix}border-radius) !default;\n$btn-border-radius-sm:        var(--#{$prefix}border-radius-sm) !default;\n$btn-border-radius-lg:        var(--#{$prefix}border-radius-lg) !default;\n\n$btn-transition:              color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$btn-hover-bg-shade-amount:       15% !default;\n$btn-hover-bg-tint-amount:        15% !default;\n$btn-hover-border-shade-amount:   20% !default;\n$btn-hover-border-tint-amount:    10% !default;\n$btn-active-bg-shade-amount:      20% !default;\n$btn-active-bg-tint-amount:       20% !default;\n$btn-active-border-shade-amount:  25% !default;\n$btn-active-border-tint-amount:   10% !default;\n// scss-docs-end btn-variables\n\n\n// Forms\n\n// scss-docs-start form-text-variables\n$form-text-margin-top:                  .25rem !default;\n$form-text-font-size:                   $small-font-size !default;\n$form-text-font-style:                  null !default;\n$form-text-font-weight:                 null !default;\n$form-text-color:                       var(--#{$prefix}secondary-color) !default;\n// scss-docs-end form-text-variables\n\n// scss-docs-start form-label-variables\n$form-label-margin-bottom:              .5rem !default;\n$form-label-font-size:                  null !default;\n$form-label-font-style:                 null !default;\n$form-label-font-weight:                null !default;\n$form-label-color:                      null !default;\n// scss-docs-end form-label-variables\n\n// scss-docs-start form-input-variables\n$input-padding-y:                       $input-btn-padding-y !default;\n$input-padding-x:                       $input-btn-padding-x !default;\n$input-font-family:                     $input-btn-font-family !default;\n$input-font-size:                       $input-btn-font-size !default;\n$input-font-weight:                     $font-weight-base !default;\n$input-line-height:                     $input-btn-line-height !default;\n\n$input-padding-y-sm:                    $input-btn-padding-y-sm !default;\n$input-padding-x-sm:                    $input-btn-padding-x-sm !default;\n$input-font-size-sm:                    $input-btn-font-size-sm !default;\n\n$input-padding-y-lg:                    $input-btn-padding-y-lg !default;\n$input-padding-x-lg:                    $input-btn-padding-x-lg !default;\n$input-font-size-lg:                    $input-btn-font-size-lg !default;\n\n$input-bg:                              var(--#{$prefix}body-bg) !default;\n$input-disabled-color:                  null !default;\n$input-disabled-bg:                     var(--#{$prefix}secondary-bg) !default;\n$input-disabled-border-color:           null !default;\n\n$input-color:                           var(--#{$prefix}body-color) !default;\n$input-border-color:                    var(--#{$prefix}border-color) !default;\n$input-border-width:                    $input-btn-border-width !default;\n$input-box-shadow:                      var(--#{$prefix}box-shadow-inset) !default;\n\n$input-border-radius:                   var(--#{$prefix}border-radius) !default;\n$input-border-radius-sm:                var(--#{$prefix}border-radius-sm) !default;\n$input-border-radius-lg:                var(--#{$prefix}border-radius-lg) !default;\n\n$input-focus-bg:                        $input-bg !default;\n$input-focus-border-color:              tint-color($component-active-bg, 50%) !default;\n$input-focus-color:                     $input-color !default;\n$input-focus-width:                     $input-btn-focus-width !default;\n$input-focus-box-shadow:                $input-btn-focus-box-shadow !default;\n\n$input-placeholder-color:               var(--#{$prefix}secondary-color) !default;\n$input-plaintext-color:                 var(--#{$prefix}body-color) !default;\n\n$input-height-border:                   calc(#{$input-border-width} * 2) !default; // stylelint-disable-line function-disallowed-list\n\n$input-height-inner:                    add($input-line-height * 1em, $input-padding-y * 2) !default;\n$input-height-inner-half:               add($input-line-height * .5em, $input-padding-y) !default;\n$input-height-inner-quarter:            add($input-line-height * .25em, $input-padding-y * .5) !default;\n\n$input-height:                          add($input-line-height * 1em, add($input-padding-y * 2, $input-height-border, false)) !default;\n$input-height-sm:                       add($input-line-height * 1em, add($input-padding-y-sm * 2, $input-height-border, false)) !default;\n$input-height-lg:                       add($input-line-height * 1em, add($input-padding-y-lg * 2, $input-height-border, false)) !default;\n\n$input-transition:                      border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$form-color-width:                      3rem !default;\n// scss-docs-end form-input-variables\n\n// scss-docs-start form-check-variables\n$form-check-input-width:                  1em !default;\n$form-check-min-height:                   $font-size-base * $line-height-base !default;\n$form-check-padding-start:                $form-check-input-width + .5em !default;\n$form-check-margin-bottom:                .125rem !default;\n$form-check-label-color:                  null !default;\n$form-check-label-cursor:                 null !default;\n$form-check-transition:                   null !default;\n\n$form-check-input-active-filter:          brightness(90%) !default;\n\n$form-check-input-bg:                     $input-bg !default;\n$form-check-input-border:                 var(--#{$prefix}border-width) solid var(--#{$prefix}border-color) !default;\n$form-check-input-border-radius:          .25em !default;\n$form-check-radio-border-radius:          50% !default;\n$form-check-input-focus-border:           $input-focus-border-color !default;\n$form-check-input-focus-box-shadow:       $focus-ring-box-shadow !default;\n\n$form-check-input-checked-color:          $component-active-color !default;\n$form-check-input-checked-bg-color:       $component-active-bg !default;\n$form-check-input-checked-border-color:   $form-check-input-checked-bg-color !default;\n$form-check-input-checked-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 20 20'><path fill='none' stroke='#{$form-check-input-checked-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='3' d='m6 10 3 3 6-6'/></svg>\") !default;\n$form-check-radio-checked-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='2' fill='#{$form-check-input-checked-color}'/></svg>\") !default;\n\n$form-check-input-indeterminate-color:          $component-active-color !default;\n$form-check-input-indeterminate-bg-color:       $component-active-bg !default;\n$form-check-input-indeterminate-border-color:   $form-check-input-indeterminate-bg-color !default;\n$form-check-input-indeterminate-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 20 20'><path fill='none' stroke='#{$form-check-input-indeterminate-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='3' d='M6 10h8'/></svg>\") !default;\n\n$form-check-input-disabled-opacity:        .5 !default;\n$form-check-label-disabled-opacity:        $form-check-input-disabled-opacity !default;\n$form-check-btn-check-disabled-opacity:    $btn-disabled-opacity !default;\n\n$form-check-inline-margin-end:    1rem !default;\n// scss-docs-end form-check-variables\n\n// scss-docs-start form-switch-variables\n$form-switch-color:               rgba($black, .25) !default;\n$form-switch-width:               2em !default;\n$form-switch-padding-start:       $form-switch-width + .5em !default;\n$form-switch-bg-image:            url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-color}'/></svg>\") !default;\n$form-switch-border-radius:       $form-switch-width !default;\n$form-switch-transition:          background-position .15s ease-in-out !default;\n\n$form-switch-focus-color:         $input-focus-border-color !default;\n$form-switch-focus-bg-image:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-focus-color}'/></svg>\") !default;\n\n$form-switch-checked-color:       $component-active-color !default;\n$form-switch-checked-bg-image:    url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-checked-color}'/></svg>\") !default;\n$form-switch-checked-bg-position: right center !default;\n// scss-docs-end form-switch-variables\n\n// scss-docs-start input-group-variables\n$input-group-addon-padding-y:           $input-padding-y !default;\n$input-group-addon-padding-x:           $input-padding-x !default;\n$input-group-addon-font-weight:         $input-font-weight !default;\n$input-group-addon-color:               $input-color !default;\n$input-group-addon-bg:                  var(--#{$prefix}tertiary-bg) !default;\n$input-group-addon-border-color:        $input-border-color !default;\n// scss-docs-end input-group-variables\n\n// scss-docs-start form-select-variables\n$form-select-padding-y:             $input-padding-y !default;\n$form-select-padding-x:             $input-padding-x !default;\n$form-select-font-family:           $input-font-family !default;\n$form-select-font-size:             $input-font-size !default;\n$form-select-indicator-padding:     $form-select-padding-x * 3 !default; // Extra padding for background-image\n$form-select-font-weight:           $input-font-weight !default;\n$form-select-line-height:           $input-line-height !default;\n$form-select-color:                 $input-color !default;\n$form-select-bg:                    $input-bg !default;\n$form-select-disabled-color:        null !default;\n$form-select-disabled-bg:           $input-disabled-bg !default;\n$form-select-disabled-border-color: $input-disabled-border-color !default;\n$form-select-bg-position:           right $form-select-padding-x center !default;\n$form-select-bg-size:               16px 12px !default; // In pixels because image dimensions\n$form-select-indicator-color:       $gray-800 !default;\n$form-select-indicator:             url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16'><path fill='none' stroke='#{$form-select-indicator-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='2' d='m2 5 6 6 6-6'/></svg>\") !default;\n\n$form-select-feedback-icon-padding-end: $form-select-padding-x * 2.5 + $form-select-indicator-padding !default;\n$form-select-feedback-icon-position:    center right $form-select-indicator-padding !default;\n$form-select-feedback-icon-size:        $input-height-inner-half $input-height-inner-half !default;\n\n$form-select-border-width:        $input-border-width !default;\n$form-select-border-color:        $input-border-color !default;\n$form-select-border-radius:       $input-border-radius !default;\n$form-select-box-shadow:          var(--#{$prefix}box-shadow-inset) !default;\n\n$form-select-focus-border-color:  $input-focus-border-color !default;\n$form-select-focus-width:         $input-focus-width !default;\n$form-select-focus-box-shadow:    0 0 0 $form-select-focus-width $input-btn-focus-color !default;\n\n$form-select-padding-y-sm:        $input-padding-y-sm !default;\n$form-select-padding-x-sm:        $input-padding-x-sm !default;\n$form-select-font-size-sm:        $input-font-size-sm !default;\n$form-select-border-radius-sm:    $input-border-radius-sm !default;\n\n$form-select-padding-y-lg:        $input-padding-y-lg !default;\n$form-select-padding-x-lg:        $input-padding-x-lg !default;\n$form-select-font-size-lg:        $input-font-size-lg !default;\n$form-select-border-radius-lg:    $input-border-radius-lg !default;\n\n$form-select-transition:          $input-transition !default;\n// scss-docs-end form-select-variables\n\n// scss-docs-start form-range-variables\n$form-range-track-width:          100% !default;\n$form-range-track-height:         .5rem !default;\n$form-range-track-cursor:         pointer !default;\n$form-range-track-bg:             var(--#{$prefix}secondary-bg) !default;\n$form-range-track-border-radius:  1rem !default;\n$form-range-track-box-shadow:     var(--#{$prefix}box-shadow-inset) !default;\n\n$form-range-thumb-width:                   1rem !default;\n$form-range-thumb-height:                  $form-range-thumb-width !default;\n$form-range-thumb-bg:                      $component-active-bg !default;\n$form-range-thumb-border:                  0 !default;\n$form-range-thumb-border-radius:           1rem !default;\n$form-range-thumb-box-shadow:              0 .1rem .25rem rgba($black, .1) !default;\n$form-range-thumb-focus-box-shadow:        0 0 0 1px $body-bg, $input-focus-box-shadow !default;\n$form-range-thumb-focus-box-shadow-width:  $input-focus-width !default; // For focus box shadow issue in Edge\n$form-range-thumb-active-bg:               tint-color($component-active-bg, 70%) !default;\n$form-range-thumb-disabled-bg:             var(--#{$prefix}secondary-color) !default;\n$form-range-thumb-transition:              background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n// scss-docs-end form-range-variables\n\n// scss-docs-start form-file-variables\n$form-file-button-color:          $input-color !default;\n$form-file-button-bg:             var(--#{$prefix}tertiary-bg) !default;\n$form-file-button-hover-bg:       var(--#{$prefix}secondary-bg) !default;\n// scss-docs-end form-file-variables\n\n// scss-docs-start form-floating-variables\n$form-floating-height:                  add(3.5rem, $input-height-border) !default;\n$form-floating-line-height:             1.25 !default;\n$form-floating-padding-x:               $input-padding-x !default;\n$form-floating-padding-y:               1rem !default;\n$form-floating-input-padding-t:         1.625rem !default;\n$form-floating-input-padding-b:         .625rem !default;\n$form-floating-label-height:            1.5em !default;\n$form-floating-label-opacity:           .65 !default;\n$form-floating-label-transform:         scale(.85) translateY(-.5rem) translateX(.15rem) !default;\n$form-floating-label-disabled-color:    $gray-600 !default;\n$form-floating-transition:              opacity .1s ease-in-out, transform .1s ease-in-out !default;\n// scss-docs-end form-floating-variables\n\n// Form validation\n\n// scss-docs-start form-feedback-variables\n$form-feedback-margin-top:          $form-text-margin-top !default;\n$form-feedback-font-size:           $form-text-font-size !default;\n$form-feedback-font-style:          $form-text-font-style !default;\n$form-feedback-valid-color:         $success !default;\n$form-feedback-invalid-color:       $danger !default;\n\n$form-feedback-icon-valid-color:    $form-feedback-valid-color !default;\n$form-feedback-icon-valid:          url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 8 8'><path fill='#{$form-feedback-icon-valid-color}' d='M2.3 6.73.6 4.53c-.4-1.04.46-1.4 1.1-.8l1.1 1.4 3.4-3.8c.6-.63 1.6-.27 1.2.7l-4 4.6c-.43.5-.8.4-1.1.1z'/></svg>\") !default;\n$form-feedback-icon-invalid-color:  $form-feedback-invalid-color !default;\n$form-feedback-icon-invalid:        url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 12 12' width='12' height='12' fill='none' stroke='#{$form-feedback-icon-invalid-color}'><circle cx='6' cy='6' r='4.5'/><path stroke-linejoin='round' d='M5.8 3.6h.4L6 6.5z'/><circle cx='6' cy='8.2' r='.6' fill='#{$form-feedback-icon-invalid-color}' stroke='none'/></svg>\") !default;\n// scss-docs-end form-feedback-variables\n\n// scss-docs-start form-validation-colors\n$form-valid-color:                  $form-feedback-valid-color !default;\n$form-valid-border-color:           $form-feedback-valid-color !default;\n$form-invalid-color:                $form-feedback-invalid-color !default;\n$form-invalid-border-color:         $form-feedback-invalid-color !default;\n// scss-docs-end form-validation-colors\n\n// scss-docs-start form-validation-states\n$form-validation-states: (\n  \"valid\": (\n    \"color\": var(--#{$prefix}form-valid-color),\n    \"icon\": $form-feedback-icon-valid,\n    \"tooltip-color\": #fff,\n    \"tooltip-bg-color\": var(--#{$prefix}success),\n    \"focus-box-shadow\": 0 0 $input-btn-focus-blur $input-focus-width rgba(var(--#{$prefix}success-rgb), $input-btn-focus-color-opacity),\n    \"border-color\": var(--#{$prefix}form-valid-border-color),\n  ),\n  \"invalid\": (\n    \"color\": var(--#{$prefix}form-invalid-color),\n    \"icon\": $form-feedback-icon-invalid,\n    \"tooltip-color\": #fff,\n    \"tooltip-bg-color\": var(--#{$prefix}danger),\n    \"focus-box-shadow\": 0 0 $input-btn-focus-blur $input-focus-width rgba(var(--#{$prefix}danger-rgb), $input-btn-focus-color-opacity),\n    \"border-color\": var(--#{$prefix}form-invalid-border-color),\n  )\n) !default;\n// scss-docs-end form-validation-states\n\n// Z-index master list\n//\n// Warning: Avoid customizing these values. They're used for a bird's eye view\n// of components dependent on the z-axis and are designed to all work together.\n\n// scss-docs-start zindex-stack\n$zindex-dropdown:                   1000 !default;\n$zindex-sticky:                     1020 !default;\n$zindex-fixed:                      1030 !default;\n$zindex-offcanvas-backdrop:         1040 !default;\n$zindex-offcanvas:                  1045 !default;\n$zindex-modal-backdrop:             1050 !default;\n$zindex-modal:                      1055 !default;\n$zindex-popover:                    1070 !default;\n$zindex-tooltip:                    1080 !default;\n$zindex-toast:                      1090 !default;\n// scss-docs-end zindex-stack\n\n// scss-docs-start zindex-levels-map\n$zindex-levels: (\n  n1: -1,\n  0: 0,\n  1: 1,\n  2: 2,\n  3: 3\n) !default;\n// scss-docs-end zindex-levels-map\n\n\n// Navs\n\n// scss-docs-start nav-variables\n$nav-link-padding-y:                .5rem !default;\n$nav-link-padding-x:                1rem !default;\n$nav-link-font-size:                null !default;\n$nav-link-font-weight:              null !default;\n$nav-link-color:                    var(--#{$prefix}link-color) !default;\n$nav-link-hover-color:              var(--#{$prefix}link-hover-color) !default;\n$nav-link-transition:               color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out !default;\n$nav-link-disabled-color:           var(--#{$prefix}secondary-color) !default;\n$nav-link-focus-box-shadow:         $focus-ring-box-shadow !default;\n\n$nav-tabs-border-color:             var(--#{$prefix}border-color) !default;\n$nav-tabs-border-width:             var(--#{$prefix}border-width) !default;\n$nav-tabs-border-radius:            var(--#{$prefix}border-radius) !default;\n$nav-tabs-link-hover-border-color:  var(--#{$prefix}secondary-bg) var(--#{$prefix}secondary-bg) $nav-tabs-border-color !default;\n$nav-tabs-link-active-color:        var(--#{$prefix}emphasis-color) !default;\n$nav-tabs-link-active-bg:           var(--#{$prefix}body-bg) !default;\n$nav-tabs-link-active-border-color: var(--#{$prefix}border-color) var(--#{$prefix}border-color) $nav-tabs-link-active-bg !default;\n\n$nav-pills-border-radius:           var(--#{$prefix}border-radius) !default;\n$nav-pills-link-active-color:       $component-active-color !default;\n$nav-pills-link-active-bg:          $component-active-bg !default;\n\n$nav-underline-gap:                 1rem !default;\n$nav-underline-border-width:        .125rem !default;\n$nav-underline-link-active-color:   var(--#{$prefix}emphasis-color) !default;\n// scss-docs-end nav-variables\n\n\n// Navbar\n\n// scss-docs-start navbar-variables\n$navbar-padding-y:                  $spacer * .5 !default;\n$navbar-padding-x:                  null !default;\n\n$navbar-nav-link-padding-x:         .5rem !default;\n\n$navbar-brand-font-size:            $font-size-lg !default;\n// Compute the navbar-brand padding-y so the navbar-brand will have the same height as navbar-text and nav-link\n$nav-link-height:                   $font-size-base * $line-height-base + $nav-link-padding-y * 2 !default;\n$navbar-brand-height:               $navbar-brand-font-size * $line-height-base !default;\n$navbar-brand-padding-y:            ($nav-link-height - $navbar-brand-height) * .5 !default;\n$navbar-brand-margin-end:           1rem !default;\n\n$navbar-toggler-padding-y:          .25rem !default;\n$navbar-toggler-padding-x:          .75rem !default;\n$navbar-toggler-font-size:          $font-size-lg !default;\n$navbar-toggler-border-radius:      $btn-border-radius !default;\n$navbar-toggler-focus-width:        $btn-focus-width !default;\n$navbar-toggler-transition:         box-shadow .15s ease-in-out !default;\n\n$navbar-light-color:                rgba(var(--#{$prefix}emphasis-color-rgb), .65) !default;\n$navbar-light-hover-color:          rgba(var(--#{$prefix}emphasis-color-rgb), .8) !default;\n$navbar-light-active-color:         rgba(var(--#{$prefix}emphasis-color-rgb), 1) !default;\n$navbar-light-disabled-color:       rgba(var(--#{$prefix}emphasis-color-rgb), .3) !default;\n$navbar-light-icon-color:           rgba($body-color, .75) !default;\n$navbar-light-toggler-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'><path stroke='#{$navbar-light-icon-color}' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/></svg>\") !default;\n$navbar-light-toggler-border-color: rgba(var(--#{$prefix}emphasis-color-rgb), .15) !default;\n$navbar-light-brand-color:          $navbar-light-active-color !default;\n$navbar-light-brand-hover-color:    $navbar-light-active-color !default;\n// scss-docs-end navbar-variables\n\n// scss-docs-start navbar-dark-variables\n$navbar-dark-color:                 rgba($white, .55) !default;\n$navbar-dark-hover-color:           rgba($white, .75) !default;\n$navbar-dark-active-color:          $white !default;\n$navbar-dark-disabled-color:        rgba($white, .25) !default;\n$navbar-dark-icon-color:            $navbar-dark-color !default;\n$navbar-dark-toggler-icon-bg:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'><path stroke='#{$navbar-dark-icon-color}' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/></svg>\") !default;\n$navbar-dark-toggler-border-color:  rgba($white, .1) !default;\n$navbar-dark-brand-color:           $navbar-dark-active-color !default;\n$navbar-dark-brand-hover-color:     $navbar-dark-active-color !default;\n// scss-docs-end navbar-dark-variables\n\n\n// Dropdowns\n//\n// Dropdown menu container and contents.\n\n// scss-docs-start dropdown-variables\n$dropdown-min-width:                10rem !default;\n$dropdown-padding-x:                0 !default;\n$dropdown-padding-y:                .5rem !default;\n$dropdown-spacer:                   .125rem !default;\n$dropdown-font-size:                $font-size-base !default;\n$dropdown-color:                    var(--#{$prefix}body-color) !default;\n$dropdown-bg:                       var(--#{$prefix}body-bg) !default;\n$dropdown-border-color:             var(--#{$prefix}border-color-translucent) !default;\n$dropdown-border-radius:            var(--#{$prefix}border-radius) !default;\n$dropdown-border-width:             var(--#{$prefix}border-width) !default;\n$dropdown-inner-border-radius:      calc(#{$dropdown-border-radius} - #{$dropdown-border-width}) !default; // stylelint-disable-line function-disallowed-list\n$dropdown-divider-bg:               $dropdown-border-color !default;\n$dropdown-divider-margin-y:         $spacer * .5 !default;\n$dropdown-box-shadow:               var(--#{$prefix}box-shadow) !default;\n\n$dropdown-link-color:               var(--#{$prefix}body-color) !default;\n$dropdown-link-hover-color:         $dropdown-link-color !default;\n$dropdown-link-hover-bg:            var(--#{$prefix}tertiary-bg) !default;\n\n$dropdown-link-active-color:        $component-active-color !default;\n$dropdown-link-active-bg:           $component-active-bg !default;\n\n$dropdown-link-disabled-color:      var(--#{$prefix}tertiary-color) !default;\n\n$dropdown-item-padding-y:           $spacer * .25 !default;\n$dropdown-item-padding-x:           $spacer !default;\n\n$dropdown-header-color:             $gray-600 !default;\n$dropdown-header-padding-x:         $dropdown-item-padding-x !default;\n$dropdown-header-padding-y:         $dropdown-padding-y !default;\n// fusv-disable\n$dropdown-header-padding:           $dropdown-header-padding-y $dropdown-header-padding-x !default; // Deprecated in v5.2.0\n// fusv-enable\n// scss-docs-end dropdown-variables\n\n// scss-docs-start dropdown-dark-variables\n$dropdown-dark-color:               $gray-300 !default;\n$dropdown-dark-bg:                  $gray-800 !default;\n$dropdown-dark-border-color:        $dropdown-border-color !default;\n$dropdown-dark-divider-bg:          $dropdown-divider-bg !default;\n$dropdown-dark-box-shadow:          null !default;\n$dropdown-dark-link-color:          $dropdown-dark-color !default;\n$dropdown-dark-link-hover-color:    $white !default;\n$dropdown-dark-link-hover-bg:       rgba($white, .15) !default;\n$dropdown-dark-link-active-color:   $dropdown-link-active-color !default;\n$dropdown-dark-link-active-bg:      $dropdown-link-active-bg !default;\n$dropdown-dark-link-disabled-color: $gray-500 !default;\n$dropdown-dark-header-color:        $gray-500 !default;\n// scss-docs-end dropdown-dark-variables\n\n\n// Pagination\n\n// scss-docs-start pagination-variables\n$pagination-padding-y:              .375rem !default;\n$pagination-padding-x:              .75rem !default;\n$pagination-padding-y-sm:           .25rem !default;\n$pagination-padding-x-sm:           .5rem !default;\n$pagination-padding-y-lg:           .75rem !default;\n$pagination-padding-x-lg:           1.5rem !default;\n\n$pagination-font-size:              $font-size-base !default;\n\n$pagination-color:                  var(--#{$prefix}link-color) !default;\n$pagination-bg:                     var(--#{$prefix}body-bg) !default;\n$pagination-border-radius:          var(--#{$prefix}border-radius) !default;\n$pagination-border-width:           var(--#{$prefix}border-width) !default;\n$pagination-margin-start:           calc(#{$pagination-border-width} * -1) !default; // stylelint-disable-line function-disallowed-list\n$pagination-border-color:           var(--#{$prefix}border-color) !default;\n\n$pagination-focus-color:            var(--#{$prefix}link-hover-color) !default;\n$pagination-focus-bg:               var(--#{$prefix}secondary-bg) !default;\n$pagination-focus-box-shadow:       $focus-ring-box-shadow !default;\n$pagination-focus-outline:          0 !default;\n\n$pagination-hover-color:            var(--#{$prefix}link-hover-color) !default;\n$pagination-hover-bg:               var(--#{$prefix}tertiary-bg) !default;\n$pagination-hover-border-color:     var(--#{$prefix}border-color) !default; // Todo in v6: remove this?\n\n$pagination-active-color:           $component-active-color !default;\n$pagination-active-bg:              $component-active-bg !default;\n$pagination-active-border-color:    $component-active-bg !default;\n\n$pagination-disabled-color:         var(--#{$prefix}secondary-color) !default;\n$pagination-disabled-bg:            var(--#{$prefix}secondary-bg) !default;\n$pagination-disabled-border-color:  var(--#{$prefix}border-color) !default;\n\n$pagination-transition:              color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$pagination-border-radius-sm:       var(--#{$prefix}border-radius-sm) !default;\n$pagination-border-radius-lg:       var(--#{$prefix}border-radius-lg) !default;\n// scss-docs-end pagination-variables\n\n\n// Placeholders\n\n// scss-docs-start placeholders\n$placeholder-opacity-max:           .5 !default;\n$placeholder-opacity-min:           .2 !default;\n// scss-docs-end placeholders\n\n// Cards\n\n// scss-docs-start card-variables\n$card-spacer-y:                     $spacer !default;\n$card-spacer-x:                     $spacer !default;\n$card-title-spacer-y:               $spacer * .5 !default;\n$card-title-color:                  null !default;\n$card-subtitle-color:               null !default;\n$card-border-width:                 var(--#{$prefix}border-width) !default;\n$card-border-color:                 var(--#{$prefix}border-color-translucent) !default;\n$card-border-radius:                var(--#{$prefix}border-radius) !default;\n$card-box-shadow:                   null !default;\n$card-inner-border-radius:          subtract($card-border-radius, $card-border-width) !default;\n$card-cap-padding-y:                $card-spacer-y * .5 !default;\n$card-cap-padding-x:                $card-spacer-x !default;\n$card-cap-bg:                       rgba(var(--#{$prefix}body-color-rgb), .03) !default;\n$card-cap-color:                    null !default;\n$card-height:                       null !default;\n$card-color:                        null !default;\n$card-bg:                           var(--#{$prefix}body-bg) !default;\n$card-img-overlay-padding:          $spacer !default;\n$card-group-margin:                 $grid-gutter-width * .5 !default;\n// scss-docs-end card-variables\n\n// Accordion\n\n// scss-docs-start accordion-variables\n$accordion-padding-y:                     1rem !default;\n$accordion-padding-x:                     1.25rem !default;\n$accordion-color:                         var(--#{$prefix}body-color) !default;\n$accordion-bg:                            var(--#{$prefix}body-bg) !default;\n$accordion-border-width:                  var(--#{$prefix}border-width) !default;\n$accordion-border-color:                  var(--#{$prefix}border-color) !default;\n$accordion-border-radius:                 var(--#{$prefix}border-radius) !default;\n$accordion-inner-border-radius:           subtract($accordion-border-radius, $accordion-border-width) !default;\n\n$accordion-body-padding-y:                $accordion-padding-y !default;\n$accordion-body-padding-x:                $accordion-padding-x !default;\n\n$accordion-button-padding-y:              $accordion-padding-y !default;\n$accordion-button-padding-x:              $accordion-padding-x !default;\n$accordion-button-color:                  var(--#{$prefix}body-color) !default;\n$accordion-button-bg:                     var(--#{$prefix}accordion-bg) !default;\n$accordion-transition:                    $btn-transition, border-radius .15s ease !default;\n$accordion-button-active-bg:              var(--#{$prefix}primary-bg-subtle) !default;\n$accordion-button-active-color:           var(--#{$prefix}primary-text-emphasis) !default;\n\n// fusv-disable\n$accordion-button-focus-border-color:     $input-focus-border-color !default; // Deprecated in v5.3.3\n// fusv-enable\n$accordion-button-focus-box-shadow:       $btn-focus-box-shadow !default;\n\n$accordion-icon-width:                    1.25rem !default;\n$accordion-icon-color:                    $body-color !default;\n$accordion-icon-active-color:             $primary-text-emphasis !default;\n$accordion-icon-transition:               transform .2s ease-in-out !default;\n$accordion-icon-transform:                rotate(-180deg) !default;\n\n$accordion-button-icon:         url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='none' stroke='#{$accordion-icon-color}' stroke-linecap='round' stroke-linejoin='round'><path d='M2 5L8 11L14 5'/></svg>\") !default;\n$accordion-button-active-icon:  url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='none' stroke='#{$accordion-icon-active-color}' stroke-linecap='round' stroke-linejoin='round'><path d='M2 5L8 11L14 5'/></svg>\") !default;\n// scss-docs-end accordion-variables\n\n// Tooltips\n\n// scss-docs-start tooltip-variables\n$tooltip-font-size:                 $font-size-sm !default;\n$tooltip-max-width:                 200px !default;\n$tooltip-color:                     var(--#{$prefix}body-bg) !default;\n$tooltip-bg:                        var(--#{$prefix}emphasis-color) !default;\n$tooltip-border-radius:             var(--#{$prefix}border-radius) !default;\n$tooltip-opacity:                   .9 !default;\n$tooltip-padding-y:                 $spacer * .25 !default;\n$tooltip-padding-x:                 $spacer * .5 !default;\n$tooltip-margin:                    null !default; // TODO: remove this in v6\n\n$tooltip-arrow-width:               .8rem !default;\n$tooltip-arrow-height:              .4rem !default;\n// fusv-disable\n$tooltip-arrow-color:               null !default; // Deprecated in Bootstrap 5.2.0 for CSS variables\n// fusv-enable\n// scss-docs-end tooltip-variables\n\n// Form tooltips must come after regular tooltips\n// scss-docs-start tooltip-feedback-variables\n$form-feedback-tooltip-padding-y:     $tooltip-padding-y !default;\n$form-feedback-tooltip-padding-x:     $tooltip-padding-x !default;\n$form-feedback-tooltip-font-size:     $tooltip-font-size !default;\n$form-feedback-tooltip-line-height:   null !default;\n$form-feedback-tooltip-opacity:       $tooltip-opacity !default;\n$form-feedback-tooltip-border-radius: $tooltip-border-radius !default;\n// scss-docs-end tooltip-feedback-variables\n\n\n// Popovers\n\n// scss-docs-start popover-variables\n$popover-font-size:                 $font-size-sm !default;\n$popover-bg:                        var(--#{$prefix}body-bg) !default;\n$popover-max-width:                 276px !default;\n$popover-border-width:              var(--#{$prefix}border-width) !default;\n$popover-border-color:              var(--#{$prefix}border-color-translucent) !default;\n$popover-border-radius:             var(--#{$prefix}border-radius-lg) !default;\n$popover-inner-border-radius:       calc(#{$popover-border-radius} - #{$popover-border-width}) !default; // stylelint-disable-line function-disallowed-list\n$popover-box-shadow:                var(--#{$prefix}box-shadow) !default;\n\n$popover-header-font-size:          $font-size-base !default;\n$popover-header-bg:                 var(--#{$prefix}secondary-bg) !default;\n$popover-header-color:              $headings-color !default;\n$popover-header-padding-y:          .5rem !default;\n$popover-header-padding-x:          $spacer !default;\n\n$popover-body-color:                var(--#{$prefix}body-color) !default;\n$popover-body-padding-y:            $spacer !default;\n$popover-body-padding-x:            $spacer !default;\n\n$popover-arrow-width:               1rem !default;\n$popover-arrow-height:              .5rem !default;\n// scss-docs-end popover-variables\n\n// fusv-disable\n// Deprecated in Bootstrap 5.2.0 for CSS variables\n$popover-arrow-color:               $popover-bg !default;\n$popover-arrow-outer-color:         var(--#{$prefix}border-color-translucent) !default;\n// fusv-enable\n\n\n// Toasts\n\n// scss-docs-start toast-variables\n$toast-max-width:                   350px !default;\n$toast-padding-x:                   .75rem !default;\n$toast-padding-y:                   .5rem !default;\n$toast-font-size:                   .875rem !default;\n$toast-color:                       null !default;\n$toast-background-color:            rgba(var(--#{$prefix}body-bg-rgb), .85) !default;\n$toast-border-width:                var(--#{$prefix}border-width) !default;\n$toast-border-color:                var(--#{$prefix}border-color-translucent) !default;\n$toast-border-radius:               var(--#{$prefix}border-radius) !default;\n$toast-box-shadow:                  var(--#{$prefix}box-shadow) !default;\n$toast-spacing:                     $container-padding-x !default;\n\n$toast-header-color:                var(--#{$prefix}secondary-color) !default;\n$toast-header-background-color:     rgba(var(--#{$prefix}body-bg-rgb), .85) !default;\n$toast-header-border-color:         $toast-border-color !default;\n// scss-docs-end toast-variables\n\n\n// Badges\n\n// scss-docs-start badge-variables\n$badge-font-size:                   .75em !default;\n$badge-font-weight:                 $font-weight-bold !default;\n$badge-color:                       $white !default;\n$badge-padding-y:                   .35em !default;\n$badge-padding-x:                   .65em !default;\n$badge-border-radius:               var(--#{$prefix}border-radius) !default;\n// scss-docs-end badge-variables\n\n\n// Modals\n\n// scss-docs-start modal-variables\n$modal-inner-padding:               $spacer !default;\n\n$modal-footer-margin-between:       .5rem !default;\n\n$modal-dialog-margin:               .5rem !default;\n$modal-dialog-margin-y-sm-up:       1.75rem !default;\n\n$modal-title-line-height:           $line-height-base !default;\n\n$modal-content-color:               null !default;\n$modal-content-bg:                  var(--#{$prefix}body-bg) !default;\n$modal-content-border-color:        var(--#{$prefix}border-color-translucent) !default;\n$modal-content-border-width:        var(--#{$prefix}border-width) !default;\n$modal-content-border-radius:       var(--#{$prefix}border-radius-lg) !default;\n$modal-content-inner-border-radius: subtract($modal-content-border-radius, $modal-content-border-width) !default;\n$modal-content-box-shadow-xs:       var(--#{$prefix}box-shadow-sm) !default;\n$modal-content-box-shadow-sm-up:    var(--#{$prefix}box-shadow) !default;\n\n$modal-backdrop-bg:                 $black !default;\n$modal-backdrop-opacity:            .5 !default;\n\n$modal-header-border-color:         var(--#{$prefix}border-color) !default;\n$modal-header-border-width:         $modal-content-border-width !default;\n$modal-header-padding-y:            $modal-inner-padding !default;\n$modal-header-padding-x:            $modal-inner-padding !default;\n$modal-header-padding:              $modal-header-padding-y $modal-header-padding-x !default; // Keep this for backwards compatibility\n\n$modal-footer-bg:                   null !default;\n$modal-footer-border-color:         $modal-header-border-color !default;\n$modal-footer-border-width:         $modal-header-border-width !default;\n\n$modal-sm:                          300px !default;\n$modal-md:                          500px !default;\n$modal-lg:                          800px !default;\n$modal-xl:                          1140px !default;\n\n$modal-fade-transform:              translate(0, -50px) !default;\n$modal-show-transform:              none !default;\n$modal-transition:                  transform .3s ease-out !default;\n$modal-scale-transform:             scale(1.02) !default;\n// scss-docs-end modal-variables\n\n\n// Alerts\n//\n// Define alert colors, border radius, and padding.\n\n// scss-docs-start alert-variables\n$alert-padding-y:               $spacer !default;\n$alert-padding-x:               $spacer !default;\n$alert-margin-bottom:           1rem !default;\n$alert-border-radius:           var(--#{$prefix}border-radius) !default;\n$alert-link-font-weight:        $font-weight-bold !default;\n$alert-border-width:            var(--#{$prefix}border-width) !default;\n$alert-dismissible-padding-r:   $alert-padding-x * 3 !default; // 3x covers width of x plus default padding on either side\n// scss-docs-end alert-variables\n\n// fusv-disable\n$alert-bg-scale:                -80% !default; // Deprecated in v5.2.0, to be removed in v6\n$alert-border-scale:            -70% !default; // Deprecated in v5.2.0, to be removed in v6\n$alert-color-scale:             40% !default; // Deprecated in v5.2.0, to be removed in v6\n// fusv-enable\n\n// Progress bars\n\n// scss-docs-start progress-variables\n$progress-height:                   1rem !default;\n$progress-font-size:                $font-size-base * .75 !default;\n$progress-bg:                       var(--#{$prefix}secondary-bg) !default;\n$progress-border-radius:            var(--#{$prefix}border-radius) !default;\n$progress-box-shadow:               var(--#{$prefix}box-shadow-inset) !default;\n$progress-bar-color:                $white !default;\n$progress-bar-bg:                   $primary !default;\n$progress-bar-animation-timing:     1s linear infinite !default;\n$progress-bar-transition:           width .6s ease !default;\n// scss-docs-end progress-variables\n\n\n// List group\n\n// scss-docs-start list-group-variables\n$list-group-color:                  var(--#{$prefix}body-color) !default;\n$list-group-bg:                     var(--#{$prefix}body-bg) !default;\n$list-group-border-color:           var(--#{$prefix}border-color) !default;\n$list-group-border-width:           var(--#{$prefix}border-width) !default;\n$list-group-border-radius:          var(--#{$prefix}border-radius) !default;\n\n$list-group-item-padding-y:         $spacer * .5 !default;\n$list-group-item-padding-x:         $spacer !default;\n// fusv-disable\n$list-group-item-bg-scale:          -80% !default; // Deprecated in v5.3.0\n$list-group-item-color-scale:       40% !default; // Deprecated in v5.3.0\n// fusv-enable\n\n$list-group-hover-bg:               var(--#{$prefix}tertiary-bg) !default;\n$list-group-active-color:           $component-active-color !default;\n$list-group-active-bg:              $component-active-bg !default;\n$list-group-active-border-color:    $list-group-active-bg !default;\n\n$list-group-disabled-color:         var(--#{$prefix}secondary-color) !default;\n$list-group-disabled-bg:            $list-group-bg !default;\n\n$list-group-action-color:           var(--#{$prefix}secondary-color) !default;\n$list-group-action-hover-color:     var(--#{$prefix}emphasis-color) !default;\n\n$list-group-action-active-color:    var(--#{$prefix}body-color) !default;\n$list-group-action-active-bg:       var(--#{$prefix}secondary-bg) !default;\n// scss-docs-end list-group-variables\n\n\n// Image thumbnails\n\n// scss-docs-start thumbnail-variables\n$thumbnail-padding:                 .25rem !default;\n$thumbnail-bg:                      var(--#{$prefix}body-bg) !default;\n$thumbnail-border-width:            var(--#{$prefix}border-width) !default;\n$thumbnail-border-color:            var(--#{$prefix}border-color) !default;\n$thumbnail-border-radius:           var(--#{$prefix}border-radius) !default;\n$thumbnail-box-shadow:              var(--#{$prefix}box-shadow-sm) !default;\n// scss-docs-end thumbnail-variables\n\n\n// Figures\n\n// scss-docs-start figure-variables\n$figure-caption-font-size:          $small-font-size !default;\n$figure-caption-color:              var(--#{$prefix}secondary-color) !default;\n// scss-docs-end figure-variables\n\n\n// Breadcrumbs\n\n// scss-docs-start breadcrumb-variables\n$breadcrumb-font-size:              null !default;\n$breadcrumb-padding-y:              0 !default;\n$breadcrumb-padding-x:              0 !default;\n$breadcrumb-item-padding-x:         .5rem !default;\n$breadcrumb-margin-bottom:          1rem !default;\n$breadcrumb-bg:                     null !default;\n$breadcrumb-divider-color:          var(--#{$prefix}secondary-color) !default;\n$breadcrumb-active-color:           var(--#{$prefix}secondary-color) !default;\n$breadcrumb-divider:                quote(\"/\") !default;\n$breadcrumb-divider-flipped:        $breadcrumb-divider !default;\n$breadcrumb-border-radius:          null !default;\n// scss-docs-end breadcrumb-variables\n\n// Carousel\n\n// scss-docs-start carousel-variables\n$carousel-control-color:             $white !default;\n$carousel-control-width:             15% !default;\n$carousel-control-opacity:           .5 !default;\n$carousel-control-hover-opacity:     .9 !default;\n$carousel-control-transition:        opacity .15s ease !default;\n\n$carousel-indicator-width:           30px !default;\n$carousel-indicator-height:          3px !default;\n$carousel-indicator-hit-area-height: 10px !default;\n$carousel-indicator-spacer:          3px !default;\n$carousel-indicator-opacity:         .5 !default;\n$carousel-indicator-active-bg:       $white !default;\n$carousel-indicator-active-opacity:  1 !default;\n$carousel-indicator-transition:      opacity .6s ease !default;\n\n$carousel-caption-width:             70% !default;\n$carousel-caption-color:             $white !default;\n$carousel-caption-padding-y:         1.25rem !default;\n$carousel-caption-spacer:            1.25rem !default;\n\n$carousel-control-icon-width:        2rem !default;\n\n$carousel-control-prev-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$carousel-control-color}'><path d='M11.354 1.646a.5.5 0 0 1 0 .708L5.707 8l5.647 5.646a.5.5 0 0 1-.708.708l-6-6a.5.5 0 0 1 0-.708l6-6a.5.5 0 0 1 .708 0z'/></svg>\") !default;\n$carousel-control-next-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$carousel-control-color}'><path d='M4.646 1.646a.5.5 0 0 1 .708 0l6 6a.5.5 0 0 1 0 .708l-6 6a.5.5 0 0 1-.708-.708L10.293 8 4.646 2.354a.5.5 0 0 1 0-.708z'/></svg>\") !default;\n\n$carousel-transition-duration:       .6s !default;\n$carousel-transition:                transform $carousel-transition-duration ease-in-out !default; // Define transform transition first if using multiple transitions (e.g., `transform 2s ease, opacity .5s ease-out`)\n// scss-docs-end carousel-variables\n\n// scss-docs-start carousel-dark-variables\n$carousel-dark-indicator-active-bg:  $black !default;\n$carousel-dark-caption-color:        $black !default;\n$carousel-dark-control-icon-filter:  invert(1) grayscale(100) !default;\n// scss-docs-end carousel-dark-variables\n\n\n// Spinners\n\n// scss-docs-start spinner-variables\n$spinner-width:           2rem !default;\n$spinner-height:          $spinner-width !default;\n$spinner-vertical-align:  -.125em !default;\n$spinner-border-width:    .25em !default;\n$spinner-animation-speed: .75s !default;\n\n$spinner-width-sm:        1rem !default;\n$spinner-height-sm:       $spinner-width-sm !default;\n$spinner-border-width-sm: .2em !default;\n// scss-docs-end spinner-variables\n\n\n// Close\n\n// scss-docs-start close-variables\n$btn-close-width:            1em !default;\n$btn-close-height:           $btn-close-width !default;\n$btn-close-padding-x:        .25em !default;\n$btn-close-padding-y:        $btn-close-padding-x !default;\n$btn-close-color:            $black !default;\n$btn-close-bg:               url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$btn-close-color}'><path d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/></svg>\") !default;\n$btn-close-focus-shadow:     $focus-ring-box-shadow !default;\n$btn-close-opacity:          .5 !default;\n$btn-close-hover-opacity:    .75 !default;\n$btn-close-focus-opacity:    1 !default;\n$btn-close-disabled-opacity: .25 !default;\n$btn-close-white-filter:     invert(1) grayscale(100%) brightness(200%) !default;\n// scss-docs-end close-variables\n\n\n// Offcanvas\n\n// scss-docs-start offcanvas-variables\n$offcanvas-padding-y:               $modal-inner-padding !default;\n$offcanvas-padding-x:               $modal-inner-padding !default;\n$offcanvas-horizontal-width:        400px !default;\n$offcanvas-vertical-height:         30vh !default;\n$offcanvas-transition-duration:     .3s !default;\n$offcanvas-border-color:            $modal-content-border-color !default;\n$offcanvas-border-width:            $modal-content-border-width !default;\n$offcanvas-title-line-height:       $modal-title-line-height !default;\n$offcanvas-bg-color:                var(--#{$prefix}body-bg) !default;\n$offcanvas-color:                   var(--#{$prefix}body-color) !default;\n$offcanvas-box-shadow:              $modal-content-box-shadow-xs !default;\n$offcanvas-backdrop-bg:             $modal-backdrop-bg !default;\n$offcanvas-backdrop-opacity:        $modal-backdrop-opacity !default;\n// scss-docs-end offcanvas-variables\n\n// Code\n\n$code-font-size:                    $small-font-size !default;\n$code-color:                        $pink !default;\n\n$kbd-padding-y:                     .1875rem !default;\n$kbd-padding-x:                     .375rem !default;\n$kbd-font-size:                     $code-font-size !default;\n$kbd-color:                         var(--#{$prefix}body-bg) !default;\n$kbd-bg:                            var(--#{$prefix}body-color) !default;\n$nested-kbd-font-weight:            null !default; // Deprecated in v5.2.0, removing in v6\n\n$pre-color:                         null !default;\n\n@import \"variables-dark\"; // TODO: can be removed safely in v6, only here to avoid breaking changes in v5.3\n","// stylelint-disable property-disallowed-list\n// Single side border-radius\n\n// Helper function to replace negative values with 0\n@function valid-radius($radius) {\n  $return: ();\n  @each $value in $radius {\n    @if type-of($value) == number {\n      $return: append($return, max($value, 0));\n    } @else {\n      $return: append($return, $value);\n    }\n  }\n  @return $return;\n}\n\n// scss-docs-start border-radius-mixins\n@mixin border-radius($radius: $border-radius, $fallback-border-radius: false) {\n  @if $enable-rounded {\n    border-radius: valid-radius($radius);\n  }\n  @else if $fallback-border-radius != false {\n    border-radius: $fallback-border-radius;\n  }\n}\n\n@mixin border-top-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-left-radius: valid-radius($radius);\n    border-top-right-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-end-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-right-radius: valid-radius($radius);\n    border-bottom-right-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-bottom-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-bottom-right-radius: valid-radius($radius);\n    border-bottom-left-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-start-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-left-radius: valid-radius($radius);\n    border-bottom-left-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-top-start-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-left-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-top-end-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-right-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-bottom-end-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-bottom-right-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-bottom-start-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-bottom-left-radius: valid-radius($radius);\n  }\n}\n// scss-docs-end border-radius-mixins\n","//\n// Headings\n//\n.h1 {\n  @extend h1;\n}\n\n.h2 {\n  @extend h2;\n}\n\n.h3 {\n  @extend h3;\n}\n\n.h4 {\n  @extend h4;\n}\n\n.h5 {\n  @extend h5;\n}\n\n.h6 {\n  @extend h6;\n}\n\n\n.lead {\n  @include font-size($lead-font-size);\n  font-weight: $lead-font-weight;\n}\n\n// Type display classes\n@each $display, $font-size in $display-font-sizes {\n  .display-#{$display} {\n    @include font-size($font-size);\n    font-family: $display-font-family;\n    font-style: $display-font-style;\n    font-weight: $display-font-weight;\n    line-height: $display-line-height;\n  }\n}\n\n//\n// Emphasis\n//\n.small {\n  @extend small;\n}\n\n.mark {\n  @extend mark;\n}\n\n//\n// Lists\n//\n\n.list-unstyled {\n  @include list-unstyled();\n}\n\n// Inline turns list items into inline-block\n.list-inline {\n  @include list-unstyled();\n}\n.list-inline-item {\n  display: inline-block;\n\n  &:not(:last-child) {\n    margin-right: $list-inline-padding;\n  }\n}\n\n\n//\n// Misc\n//\n\n// Builds on `abbr`\n.initialism {\n  @include font-size($initialism-font-size);\n  text-transform: uppercase;\n}\n\n// Blockquotes\n.blockquote {\n  margin-bottom: $blockquote-margin-y;\n  @include font-size($blockquote-font-size);\n\n  > :last-child {\n    margin-bottom: 0;\n  }\n}\n\n.blockquote-footer {\n  margin-top: -$blockquote-margin-y;\n  margin-bottom: $blockquote-margin-y;\n  @include font-size($blockquote-footer-font-size);\n  color: $blockquote-footer-color;\n\n  &::before {\n    content: \"\\2014\\00A0\"; // em dash, nbsp\n  }\n}\n","// Lists\n\n// Unstyled keeps list items block level, just removes default browser padding and list-style\n@mixin list-unstyled {\n  padding-left: 0;\n  list-style: none;\n}\n","// Override bootstrap variables\n$spacer: 1rem;\n$container-max-widths: (\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1400px,\n);\n$grid-breakpoints: (\n  xs: 0,\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1200px,\n);\n$dropdown-link-hover-bg: var(--pst-color-surface);\n\n// --pst-color-surface can also be assigned to the dark variant because it is\n// scoped to different values depending on light/dark theme\n$dropdown-dark-link-hover-bg: var(--pst-color-surface);\n$dropdown-link-active-bg: var(--pst-color-surface);\n$dropdown-dark-link-active-bg: var(--pst-color-surface);\n$focus-ring-width: 0.1875rem; // 3px at 100% zoom (0.1875 * 16px base font = 3px)\n$focus-ring-opacity: 1;\n$focus-ring-color: var(--pst-color-accent);\n$focus-ring-blur: 0;\n$focus-ring-box-shadow: 0 0 $focus-ring-blur $focus-ring-width $focus-ring-color;\n\n// outline creates the same style of focus ring, it just uses CSS outline instead of box shadow\n$focus-ring-outline: $focus-ring-color solid $focus-ring-width;\n$btn-focus-box-shadow: $focus-ring-box-shadow;\n","// Responsive images (ensure images don't scale beyond their parents)\n//\n// This is purposefully opt-in via an explicit class rather than being the default for all `<img>`s.\n// We previously tried the \"images are responsive by default\" approach in Bootstrap v2,\n// and abandoned it in Bootstrap v3 because it breaks lots of third-party widgets (including Google Maps)\n// which weren't expecting the images within themselves to be involuntarily resized.\n// See also https://github.com/twbs/bootstrap/issues/18178\n.img-fluid {\n  @include img-fluid();\n}\n\n\n// Image thumbnails\n.img-thumbnail {\n  padding: $thumbnail-padding;\n  background-color: $thumbnail-bg;\n  border: $thumbnail-border-width solid $thumbnail-border-color;\n  @include border-radius($thumbnail-border-radius);\n  @include box-shadow($thumbnail-box-shadow);\n\n  // Keep them at most 100% wide\n  @include img-fluid();\n}\n\n//\n// Figures\n//\n\n.figure {\n  // Ensures the caption's text aligns with the image.\n  display: inline-block;\n}\n\n.figure-img {\n  margin-bottom: $spacer * .5;\n  line-height: 1;\n}\n\n.figure-caption {\n  @include font-size($figure-caption-font-size);\n  color: $figure-caption-color;\n}\n","// Image Mixins\n// - Responsive image\n// - Retina image\n\n\n// Responsive image\n//\n// Keep images from scaling beyond the width of their parents.\n\n@mixin img-fluid {\n  // Part 1: Set a maximum relative to the parent\n  max-width: 100%;\n  // Part 2: Override the height to auto, otherwise images will be stretched\n  // when setting a width and height attribute on the img element.\n  height: auto;\n}\n","// Container widths\n//\n// Set the container width, and override it for fixed navbars in media queries.\n\n@if $enable-container-classes {\n  // Single container class with breakpoint max-widths\n  .container,\n  // 100% wide container at all breakpoints\n  .container-fluid {\n    @include make-container();\n  }\n\n  // Responsive containers that are 100% wide until a breakpoint\n  @each $breakpoint, $container-max-width in $container-max-widths {\n    .container-#{$breakpoint} {\n      @extend .container-fluid;\n    }\n\n    @include media-breakpoint-up($breakpoint, $grid-breakpoints) {\n      %responsive-container-#{$breakpoint} {\n        max-width: $container-max-width;\n      }\n\n      // Extend each breakpoint which is smaller or equal to the current breakpoint\n      $extend-breakpoint: true;\n\n      @each $name, $width in $grid-breakpoints {\n        @if ($extend-breakpoint) {\n          .container#{breakpoint-infix($name, $grid-breakpoints)} {\n            @extend %responsive-container-#{$breakpoint};\n          }\n\n          // Once the current breakpoint is reached, stop extending\n          @if ($breakpoint == $name) {\n            $extend-breakpoint: false;\n          }\n        }\n      }\n    }\n  }\n}\n","// Container mixins\n\n@mixin make-container($gutter: $container-padding-x) {\n  --#{$prefix}gutter-x: #{$gutter};\n  --#{$prefix}gutter-y: 0;\n  width: 100%;\n  padding-right: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  padding-left: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  margin-right: auto;\n  margin-left: auto;\n}\n","// Breakpoint viewport sizes and media queries.\n//\n// Breakpoints are defined as a map of (name: minimum width), order from small to large:\n//\n//    (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px)\n//\n// The map defined in the `$grid-breakpoints` global variable is used as the `$breakpoints` argument by default.\n\n// Name of the next breakpoint, or null for the last breakpoint.\n//\n//    >> breakpoint-next(sm)\n//    md\n//    >> breakpoint-next(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    md\n//    >> breakpoint-next(sm, $breakpoint-names: (xs sm md lg xl xxl))\n//    md\n@function breakpoint-next($name, $breakpoints: $grid-breakpoints, $breakpoint-names: map-keys($breakpoints)) {\n  $n: index($breakpoint-names, $name);\n  @if not $n {\n    @error \"breakpoint `#{$name}` not found in `#{$breakpoints}`\";\n  }\n  @return if($n < length($breakpoint-names), nth($breakpoint-names, $n + 1), null);\n}\n\n// Minimum breakpoint width. Null for the smallest (first) breakpoint.\n//\n//    >> breakpoint-min(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    576px\n@function breakpoint-min($name, $breakpoints: $grid-breakpoints) {\n  $min: map-get($breakpoints, $name);\n  @return if($min != 0, $min, null);\n}\n\n// Maximum breakpoint width.\n// The maximum value is reduced by 0.02px to work around the limitations of\n// `min-` and `max-` prefixes and viewports with fractional widths.\n// See https://www.w3.org/TR/mediaqueries-4/#mq-min-max\n// Uses 0.02px rather than 0.01px to work around a current rounding bug in Safari.\n// See https://bugs.webkit.org/show_bug.cgi?id=178261\n//\n//    >> breakpoint-max(md, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    767.98px\n@function breakpoint-max($name, $breakpoints: $grid-breakpoints) {\n  $max: map-get($breakpoints, $name);\n  @return if($max and $max > 0, $max - .02, null);\n}\n\n// Returns a blank string if smallest breakpoint, otherwise returns the name with a dash in front.\n// Useful for making responsive utilities.\n//\n//    >> breakpoint-infix(xs, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    \"\"  (Returns a blank string)\n//    >> breakpoint-infix(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    \"-sm\"\n@function breakpoint-infix($name, $breakpoints: $grid-breakpoints) {\n  @return if(breakpoint-min($name, $breakpoints) == null, \"\", \"-#{$name}\");\n}\n\n// Media of at least the minimum breakpoint width. No query for the smallest breakpoint.\n// Makes the @content apply to the given breakpoint and wider.\n@mixin media-breakpoint-up($name, $breakpoints: $grid-breakpoints) {\n  $min: breakpoint-min($name, $breakpoints);\n  @if $min {\n    @media (min-width: $min) {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Media of at most the maximum breakpoint width. No query for the largest breakpoint.\n// Makes the @content apply to the given breakpoint and narrower.\n@mixin media-breakpoint-down($name, $breakpoints: $grid-breakpoints) {\n  $max: breakpoint-max($name, $breakpoints);\n  @if $max {\n    @media (max-width: $max) {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Media that spans multiple breakpoint widths.\n// Makes the @content apply between the min and max breakpoints\n@mixin media-breakpoint-between($lower, $upper, $breakpoints: $grid-breakpoints) {\n  $min: breakpoint-min($lower, $breakpoints);\n  $max: breakpoint-max($upper, $breakpoints);\n\n  @if $min != null and $max != null {\n    @media (min-width: $min) and (max-width: $max) {\n      @content;\n    }\n  } @else if $max == null {\n    @include media-breakpoint-up($lower, $breakpoints) {\n      @content;\n    }\n  } @else if $min == null {\n    @include media-breakpoint-down($upper, $breakpoints) {\n      @content;\n    }\n  }\n}\n\n// Media between the breakpoint's minimum and maximum widths.\n// No minimum for the smallest breakpoint, and no maximum for the largest one.\n// Makes the @content apply only to the given breakpoint, not viewports any wider or narrower.\n@mixin media-breakpoint-only($name, $breakpoints: $grid-breakpoints) {\n  $min:  breakpoint-min($name, $breakpoints);\n  $next: breakpoint-next($name, $breakpoints);\n  $max:  breakpoint-max($next, $breakpoints);\n\n  @if $min != null and $max != null {\n    @media (min-width: $min) and (max-width: $max) {\n      @content;\n    }\n  } @else if $max == null {\n    @include media-breakpoint-up($name, $breakpoints) {\n      @content;\n    }\n  } @else if $min == null {\n    @include media-breakpoint-down($next, $breakpoints) {\n      @content;\n    }\n  }\n}\n","// Row\n//\n// Rows contain your columns.\n\n:root {\n  @each $name, $value in $grid-breakpoints {\n    --#{$prefix}breakpoint-#{$name}: #{$value};\n  }\n}\n\n@if $enable-grid-classes {\n  .row {\n    @include make-row();\n\n    > * {\n      @include make-col-ready();\n    }\n  }\n}\n\n@if $enable-cssgrid {\n  .grid {\n    display: grid;\n    grid-template-rows: repeat(var(--#{$prefix}rows, 1), 1fr);\n    grid-template-columns: repeat(var(--#{$prefix}columns, #{$grid-columns}), 1fr);\n    gap: var(--#{$prefix}gap, #{$grid-gutter-width});\n\n    @include make-cssgrid();\n  }\n}\n\n\n// Columns\n//\n// Common styles for small and large grid columns\n\n@if $enable-grid-classes {\n  @include make-grid-columns();\n}\n","// Grid system\n//\n// Generate semantic grid columns with these mixins.\n\n@mixin make-row($gutter: $grid-gutter-width) {\n  --#{$prefix}gutter-x: #{$gutter};\n  --#{$prefix}gutter-y: 0;\n  display: flex;\n  flex-wrap: wrap;\n  // TODO: Revisit calc order after https://github.com/react-bootstrap/react-bootstrap/issues/6039 is fixed\n  margin-top: calc(-1 * var(--#{$prefix}gutter-y)); // stylelint-disable-line function-disallowed-list\n  margin-right: calc(-.5 * var(--#{$prefix}gutter-x)); // stylelint-disable-line function-disallowed-list\n  margin-left: calc(-.5 * var(--#{$prefix}gutter-x)); // stylelint-disable-line function-disallowed-list\n}\n\n@mixin make-col-ready() {\n  // Add box sizing if only the grid is loaded\n  box-sizing: if(variable-exists(include-column-box-sizing) and $include-column-box-sizing, border-box, null);\n  // Prevent columns from becoming too narrow when at smaller grid tiers by\n  // always setting `width: 100%;`. This works because we set the width\n  // later on to override this initial width.\n  flex-shrink: 0;\n  width: 100%;\n  max-width: 100%; // Prevent `.col-auto`, `.col` (& responsive variants) from breaking out the grid\n  padding-right: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  padding-left: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  margin-top: var(--#{$prefix}gutter-y);\n}\n\n@mixin make-col($size: false, $columns: $grid-columns) {\n  @if $size {\n    flex: 0 0 auto;\n    width: percentage(divide($size, $columns));\n\n  } @else {\n    flex: 1 1 0;\n    max-width: 100%;\n  }\n}\n\n@mixin make-col-auto() {\n  flex: 0 0 auto;\n  width: auto;\n}\n\n@mixin make-col-offset($size, $columns: $grid-columns) {\n  $num: divide($size, $columns);\n  margin-left: if($num == 0, 0, percentage($num));\n}\n\n// Row columns\n//\n// Specify on a parent element(e.g., .row) to force immediate children into NN\n// number of columns. Supports wrapping to new lines, but does not do a Masonry\n// style grid.\n@mixin row-cols($count) {\n  > * {\n    flex: 0 0 auto;\n    width: percentage(divide(1, $count));\n  }\n}\n\n// Framework grid generation\n//\n// Used only by Bootstrap to generate the correct number of grid classes given\n// any value of `$grid-columns`.\n\n@mixin make-grid-columns($columns: $grid-columns, $gutter: $grid-gutter-width, $breakpoints: $grid-breakpoints) {\n  @each $breakpoint in map-keys($breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $breakpoints);\n\n    @include media-breakpoint-up($breakpoint, $breakpoints) {\n      // Provide basic `.col-{bp}` classes for equal-width flexbox columns\n      .col#{$infix} {\n        flex: 1 0 0%; // Flexbugs #4: https://github.com/philipwalton/flexbugs#flexbug-4\n      }\n\n      .row-cols#{$infix}-auto > * {\n        @include make-col-auto();\n      }\n\n      @if $grid-row-columns > 0 {\n        @for $i from 1 through $grid-row-columns {\n          .row-cols#{$infix}-#{$i} {\n            @include row-cols($i);\n          }\n        }\n      }\n\n      .col#{$infix}-auto {\n        @include make-col-auto();\n      }\n\n      @if $columns > 0 {\n        @for $i from 1 through $columns {\n          .col#{$infix}-#{$i} {\n            @include make-col($i, $columns);\n          }\n        }\n\n        // `$columns - 1` because offsetting by the width of an entire row isn't possible\n        @for $i from 0 through ($columns - 1) {\n          @if not ($infix == \"\" and $i == 0) { // Avoid emitting useless .offset-0\n            .offset#{$infix}-#{$i} {\n              @include make-col-offset($i, $columns);\n            }\n          }\n        }\n      }\n\n      // Gutters\n      //\n      // Make use of `.g-*`, `.gx-*` or `.gy-*` utilities to change spacing between the columns.\n      @each $key, $value in $gutters {\n        .g#{$infix}-#{$key},\n        .gx#{$infix}-#{$key} {\n          --#{$prefix}gutter-x: #{$value};\n        }\n\n        .g#{$infix}-#{$key},\n        .gy#{$infix}-#{$key} {\n          --#{$prefix}gutter-y: #{$value};\n        }\n      }\n    }\n  }\n}\n\n@mixin make-cssgrid($columns: $grid-columns, $breakpoints: $grid-breakpoints) {\n  @each $breakpoint in map-keys($breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $breakpoints);\n\n    @include media-breakpoint-up($breakpoint, $breakpoints) {\n      @if $columns > 0 {\n        @for $i from 1 through $columns {\n          .g-col#{$infix}-#{$i} {\n            grid-column: auto / span $i;\n          }\n        }\n\n        // Start with `1` because `0` is an invalid value.\n        // Ends with `$columns - 1` because offsetting by the width of an entire row isn't possible.\n        @for $i from 1 through ($columns - 1) {\n          .g-start#{$infix}-#{$i} {\n            grid-column-start: $i;\n          }\n        }\n      }\n    }\n  }\n}\n","//\n// Basic Bootstrap table\n//\n\n.table {\n  // Reset needed for nesting tables\n  --#{$prefix}table-color-type: initial;\n  --#{$prefix}table-bg-type: initial;\n  --#{$prefix}table-color-state: initial;\n  --#{$prefix}table-bg-state: initial;\n  // End of reset\n  --#{$prefix}table-color: #{$table-color};\n  --#{$prefix}table-bg: #{$table-bg};\n  --#{$prefix}table-border-color: #{$table-border-color};\n  --#{$prefix}table-accent-bg: #{$table-accent-bg};\n  --#{$prefix}table-striped-color: #{$table-striped-color};\n  --#{$prefix}table-striped-bg: #{$table-striped-bg};\n  --#{$prefix}table-active-color: #{$table-active-color};\n  --#{$prefix}table-active-bg: #{$table-active-bg};\n  --#{$prefix}table-hover-color: #{$table-hover-color};\n  --#{$prefix}table-hover-bg: #{$table-hover-bg};\n\n  width: 100%;\n  margin-bottom: $spacer;\n  vertical-align: $table-cell-vertical-align;\n  border-color: var(--#{$prefix}table-border-color);\n\n  // Target th & td\n  // We need the child combinator to prevent styles leaking to nested tables which doesn't have a `.table` class.\n  // We use the universal selectors here to simplify the selector (else we would need 6 different selectors).\n  // Another advantage is that this generates less code and makes the selector less specific making it easier to override.\n  // stylelint-disable-next-line selector-max-universal\n  > :not(caption) > * > * {\n    padding: $table-cell-padding-y $table-cell-padding-x;\n    // Following the precept of cascades: https://codepen.io/miriamsuzanne/full/vYNgodb\n    color: var(--#{$prefix}table-color-state, var(--#{$prefix}table-color-type, var(--#{$prefix}table-color)));\n    background-color: var(--#{$prefix}table-bg);\n    border-bottom-width: $table-border-width;\n    box-shadow: inset 0 0 0 9999px var(--#{$prefix}table-bg-state, var(--#{$prefix}table-bg-type, var(--#{$prefix}table-accent-bg)));\n  }\n\n  > tbody {\n    vertical-align: inherit;\n  }\n\n  > thead {\n    vertical-align: bottom;\n  }\n}\n\n.table-group-divider {\n  border-top: calc(#{$table-border-width} * 2) solid $table-group-separator-color; // stylelint-disable-line function-disallowed-list\n}\n\n//\n// Change placement of captions with a class\n//\n\n.caption-top {\n  caption-side: top;\n}\n\n\n//\n// Condensed table w/ half padding\n//\n\n.table-sm {\n  // stylelint-disable-next-line selector-max-universal\n  > :not(caption) > * > * {\n    padding: $table-cell-padding-y-sm $table-cell-padding-x-sm;\n  }\n}\n\n\n// Border versions\n//\n// Add or remove borders all around the table and between all the columns.\n//\n// When borders are added on all sides of the cells, the corners can render odd when\n// these borders do not have the same color or if they are semi-transparent.\n// Therefore we add top and border bottoms to the `tr`s and left and right borders\n// to the `td`s or `th`s\n\n.table-bordered {\n  > :not(caption) > * {\n    border-width: $table-border-width 0;\n\n    // stylelint-disable-next-line selector-max-universal\n    > * {\n      border-width: 0 $table-border-width;\n    }\n  }\n}\n\n.table-borderless {\n  // stylelint-disable-next-line selector-max-universal\n  > :not(caption) > * > * {\n    border-bottom-width: 0;\n  }\n\n  > :not(:first-child) {\n    border-top-width: 0;\n  }\n}\n\n// Zebra-striping\n//\n// Default zebra-stripe styles (alternating gray and transparent backgrounds)\n\n// For rows\n.table-striped {\n  > tbody > tr:nth-of-type(#{$table-striped-order}) > * {\n    --#{$prefix}table-color-type: var(--#{$prefix}table-striped-color);\n    --#{$prefix}table-bg-type: var(--#{$prefix}table-striped-bg);\n  }\n}\n\n// For columns\n.table-striped-columns {\n  > :not(caption) > tr > :nth-child(#{$table-striped-columns-order}) {\n    --#{$prefix}table-color-type: var(--#{$prefix}table-striped-color);\n    --#{$prefix}table-bg-type: var(--#{$prefix}table-striped-bg);\n  }\n}\n\n// Active table\n//\n// The `.table-active` class can be added to highlight rows or cells\n\n.table-active {\n  --#{$prefix}table-color-state: var(--#{$prefix}table-active-color);\n  --#{$prefix}table-bg-state: var(--#{$prefix}table-active-bg);\n}\n\n// Hover effect\n//\n// Placed here since it has to come after the potential zebra striping\n\n.table-hover {\n  > tbody > tr:hover > * {\n    --#{$prefix}table-color-state: var(--#{$prefix}table-hover-color);\n    --#{$prefix}table-bg-state: var(--#{$prefix}table-hover-bg);\n  }\n}\n\n\n// Table variants\n//\n// Table variants set the table cell backgrounds, border colors\n// and the colors of the striped, hovered & active tables\n\n@each $color, $value in $table-variants {\n  @include table-variant($color, $value);\n}\n\n// Responsive tables\n//\n// Generate series of `.table-responsive-*` classes for configuring the screen\n// size of where your table will overflow.\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n  @include media-breakpoint-down($breakpoint) {\n    .table-responsive#{$infix} {\n      overflow-x: auto;\n      -webkit-overflow-scrolling: touch;\n    }\n  }\n}\n","// scss-docs-start table-variant\n@mixin table-variant($state, $background) {\n  .table-#{$state} {\n    $color: color-contrast(opaque($body-bg, $background));\n    $hover-bg: mix($color, $background, percentage($table-hover-bg-factor));\n    $striped-bg: mix($color, $background, percentage($table-striped-bg-factor));\n    $active-bg: mix($color, $background, percentage($table-active-bg-factor));\n    $table-border-color: mix($color, $background, percentage($table-border-factor));\n\n    --#{$prefix}table-color: #{$color};\n    --#{$prefix}table-bg: #{$background};\n    --#{$prefix}table-border-color: #{$table-border-color};\n    --#{$prefix}table-striped-bg: #{$striped-bg};\n    --#{$prefix}table-striped-color: #{color-contrast($striped-bg)};\n    --#{$prefix}table-active-bg: #{$active-bg};\n    --#{$prefix}table-active-color: #{color-contrast($active-bg)};\n    --#{$prefix}table-hover-bg: #{$hover-bg};\n    --#{$prefix}table-hover-color: #{color-contrast($hover-bg)};\n\n    color: var(--#{$prefix}table-color);\n    border-color: var(--#{$prefix}table-border-color);\n  }\n}\n// scss-docs-end table-variant\n","//\n// Labels\n//\n\n.form-label {\n  margin-bottom: $form-label-margin-bottom;\n  @include font-size($form-label-font-size);\n  font-style: $form-label-font-style;\n  font-weight: $form-label-font-weight;\n  color: $form-label-color;\n}\n\n// For use with horizontal and inline forms, when you need the label (or legend)\n// text to align with the form controls.\n.col-form-label {\n  padding-top: add($input-padding-y, $input-border-width);\n  padding-bottom: add($input-padding-y, $input-border-width);\n  margin-bottom: 0; // Override the `<legend>` default\n  @include font-size(inherit); // Override the `<legend>` default\n  font-style: $form-label-font-style;\n  font-weight: $form-label-font-weight;\n  line-height: $input-line-height;\n  color: $form-label-color;\n}\n\n.col-form-label-lg {\n  padding-top: add($input-padding-y-lg, $input-border-width);\n  padding-bottom: add($input-padding-y-lg, $input-border-width);\n  @include font-size($input-font-size-lg);\n}\n\n.col-form-label-sm {\n  padding-top: add($input-padding-y-sm, $input-border-width);\n  padding-bottom: add($input-padding-y-sm, $input-border-width);\n  @include font-size($input-font-size-sm);\n}\n","//\n// Form text\n//\n\n.form-text {\n  margin-top: $form-text-margin-top;\n  @include font-size($form-text-font-size);\n  font-style: $form-text-font-style;\n  font-weight: $form-text-font-weight;\n  color: $form-text-color;\n}\n","//\n// General form controls (plus a few specific high-level interventions)\n//\n\n.form-control {\n  display: block;\n  width: 100%;\n  padding: $input-padding-y $input-padding-x;\n  font-family: $input-font-family;\n  @include font-size($input-font-size);\n  font-weight: $input-font-weight;\n  line-height: $input-line-height;\n  color: $input-color;\n  appearance: none; // Fix appearance for date inputs in Safari\n  background-color: $input-bg;\n  background-clip: padding-box;\n  border: $input-border-width solid $input-border-color;\n\n  // Note: This has no effect on <select>s in some browsers, due to the limited stylability of `<select>`s in CSS.\n  @include border-radius($input-border-radius, 0);\n\n  @include box-shadow($input-box-shadow);\n  @include transition($input-transition);\n\n  &[type=\"file\"] {\n    overflow: hidden; // prevent pseudo element button overlap\n\n    &:not(:disabled):not([readonly]) {\n      cursor: pointer;\n    }\n  }\n\n  // Customize the `:focus` state to imitate native WebKit styles.\n  &:focus {\n    color: $input-focus-color;\n    background-color: $input-focus-bg;\n    border-color: $input-focus-border-color;\n    outline: 0;\n    @if $enable-shadows {\n      @include box-shadow($input-box-shadow, $input-focus-box-shadow);\n    } @else {\n      // Avoid using mixin so we can pass custom focus shadow properly\n      box-shadow: $input-focus-box-shadow;\n    }\n  }\n\n  &::-webkit-date-and-time-value {\n    // On Android Chrome, form-control's \"width: 100%\" makes the input width too small\n    // Tested under Android 11 / Chrome 89, Android 12 / Chrome 100, Android 13 / Chrome 109\n    //\n    // On iOS Safari, form-control's \"appearance: none\" + \"width: 100%\" makes the input width too small\n    // Tested under iOS 16.2 / Safari 16.2\n    min-width: 85px; // Seems to be a good minimum safe width\n\n    // Add some height to date inputs on iOS\n    // https://github.com/twbs/bootstrap/issues/23307\n    // TODO: we can remove this workaround once https://bugs.webkit.org/show_bug.cgi?id=198959 is resolved\n    // Multiply line-height by 1em if it has no unit\n    height: if(unit($input-line-height) == \"\", $input-line-height * 1em, $input-line-height);\n\n    // Android Chrome type=\"date\" is taller than the other inputs\n    // because of \"margin: 1px 24px 1px 4px\" inside the shadow DOM\n    // Tested under Android 11 / Chrome 89, Android 12 / Chrome 100, Android 13 / Chrome 109\n    margin: 0;\n  }\n\n  // Prevent excessive date input height in Webkit\n  // https://github.com/twbs/bootstrap/issues/34433\n  &::-webkit-datetime-edit {\n    display: block;\n    padding: 0;\n  }\n\n  // Placeholder\n  &::placeholder {\n    color: $input-placeholder-color;\n    // Override Firefox's unusual default opacity; see https://github.com/twbs/bootstrap/pull/11526.\n    opacity: 1;\n  }\n\n  // Disabled inputs\n  //\n  // HTML5 says that controls under a fieldset > legend:first-child won't be\n  // disabled if the fieldset is disabled. Due to implementation difficulty, we\n  // don't honor that edge case; we style them as disabled anyway.\n  &:disabled {\n    color: $input-disabled-color;\n    background-color: $input-disabled-bg;\n    border-color: $input-disabled-border-color;\n    // iOS fix for unreadable disabled content; see https://github.com/twbs/bootstrap/issues/11655.\n    opacity: 1;\n  }\n\n  // File input buttons theming\n  &::file-selector-button {\n    padding: $input-padding-y $input-padding-x;\n    margin: (-$input-padding-y) (-$input-padding-x);\n    margin-inline-end: $input-padding-x;\n    color: $form-file-button-color;\n    @include gradient-bg($form-file-button-bg);\n    pointer-events: none;\n    border-color: inherit;\n    border-style: solid;\n    border-width: 0;\n    border-inline-end-width: $input-border-width;\n    border-radius: 0; // stylelint-disable-line property-disallowed-list\n    @include transition($btn-transition);\n  }\n\n  &:hover:not(:disabled):not([readonly])::file-selector-button {\n    background-color: $form-file-button-hover-bg;\n  }\n}\n\n// Readonly controls as plain text\n//\n// Apply class to a readonly input to make it appear like regular plain\n// text (without any border, background color, focus indicator)\n\n.form-control-plaintext {\n  display: block;\n  width: 100%;\n  padding: $input-padding-y 0;\n  margin-bottom: 0; // match inputs if this class comes on inputs with default margins\n  line-height: $input-line-height;\n  color: $input-plaintext-color;\n  background-color: transparent;\n  border: solid transparent;\n  border-width: $input-border-width 0;\n\n  &:focus {\n    outline: 0;\n  }\n\n  &.form-control-sm,\n  &.form-control-lg {\n    padding-right: 0;\n    padding-left: 0;\n  }\n}\n\n// Form control sizing\n//\n// Build on `.form-control` with modifier classes to decrease or increase the\n// height and font-size of form controls.\n//\n// Repeated in `_input_group.scss` to avoid Sass extend issues.\n\n.form-control-sm {\n  min-height: $input-height-sm;\n  padding: $input-padding-y-sm $input-padding-x-sm;\n  @include font-size($input-font-size-sm);\n  @include border-radius($input-border-radius-sm);\n\n  &::file-selector-button {\n    padding: $input-padding-y-sm $input-padding-x-sm;\n    margin: (-$input-padding-y-sm) (-$input-padding-x-sm);\n    margin-inline-end: $input-padding-x-sm;\n  }\n}\n\n.form-control-lg {\n  min-height: $input-height-lg;\n  padding: $input-padding-y-lg $input-padding-x-lg;\n  @include font-size($input-font-size-lg);\n  @include border-radius($input-border-radius-lg);\n\n  &::file-selector-button {\n    padding: $input-padding-y-lg $input-padding-x-lg;\n    margin: (-$input-padding-y-lg) (-$input-padding-x-lg);\n    margin-inline-end: $input-padding-x-lg;\n  }\n}\n\n// Make sure textareas don't shrink too much when resized\n// https://github.com/twbs/bootstrap/pull/29124\n// stylelint-disable selector-no-qualifying-type\ntextarea {\n  &.form-control {\n    min-height: $input-height;\n  }\n\n  &.form-control-sm {\n    min-height: $input-height-sm;\n  }\n\n  &.form-control-lg {\n    min-height: $input-height-lg;\n  }\n}\n// stylelint-enable selector-no-qualifying-type\n\n.form-control-color {\n  width: $form-color-width;\n  height: $input-height;\n  padding: $input-padding-y;\n\n  &:not(:disabled):not([readonly]) {\n    cursor: pointer;\n  }\n\n  &::-moz-color-swatch {\n    border: 0 !important; // stylelint-disable-line declaration-no-important\n    @include border-radius($input-border-radius);\n  }\n\n  &::-webkit-color-swatch {\n    border: 0 !important; // stylelint-disable-line declaration-no-important\n    @include border-radius($input-border-radius);\n  }\n\n  &.form-control-sm { height: $input-height-sm; }\n  &.form-control-lg { height: $input-height-lg; }\n}\n","// stylelint-disable property-disallowed-list\n@mixin transition($transition...) {\n  @if length($transition) == 0 {\n    $transition: $transition-base;\n  }\n\n  @if length($transition) > 1 {\n    @each $value in $transition {\n      @if $value == null or $value == none {\n        @warn \"The keyword 'none' or 'null' must be used as a single argument.\";\n      }\n    }\n  }\n\n  @if $enable-transitions {\n    @if nth($transition, 1) != null {\n      transition: $transition;\n    }\n\n    @if $enable-reduced-motion and nth($transition, 1) != null and nth($transition, 1) != none {\n      @media (prefers-reduced-motion: reduce) {\n        transition: none;\n      }\n    }\n  }\n}\n","// Gradients\n\n// scss-docs-start gradient-bg-mixin\n@mixin gradient-bg($color: null) {\n  background-color: $color;\n\n  @if $enable-gradients {\n    background-image: var(--#{$prefix}gradient);\n  }\n}\n// scss-docs-end gradient-bg-mixin\n\n// scss-docs-start gradient-mixins\n// Horizontal gradient, from left to right\n//\n// Creates two color stops, start and end, by specifying a color and position for each color stop.\n@mixin gradient-x($start-color: $gray-700, $end-color: $gray-800, $start-percent: 0%, $end-percent: 100%) {\n  background-image: linear-gradient(to right, $start-color $start-percent, $end-color $end-percent);\n}\n\n// Vertical gradient, from top to bottom\n//\n// Creates two color stops, start and end, by specifying a color and position for each color stop.\n@mixin gradient-y($start-color: $gray-700, $end-color: $gray-800, $start-percent: null, $end-percent: null) {\n  background-image: linear-gradient(to bottom, $start-color $start-percent, $end-color $end-percent);\n}\n\n@mixin gradient-directional($start-color: $gray-700, $end-color: $gray-800, $deg: 45deg) {\n  background-image: linear-gradient($deg, $start-color, $end-color);\n}\n\n@mixin gradient-x-three-colors($start-color: $blue, $mid-color: $purple, $color-stop: 50%, $end-color: $red) {\n  background-image: linear-gradient(to right, $start-color, $mid-color $color-stop, $end-color);\n}\n\n@mixin gradient-y-three-colors($start-color: $blue, $mid-color: $purple, $color-stop: 50%, $end-color: $red) {\n  background-image: linear-gradient($start-color, $mid-color $color-stop, $end-color);\n}\n\n@mixin gradient-radial($inner-color: $gray-700, $outer-color: $gray-800) {\n  background-image: radial-gradient(circle, $inner-color, $outer-color);\n}\n\n@mixin gradient-striped($color: rgba($white, .15), $angle: 45deg) {\n  background-image: linear-gradient($angle, $color 25%, transparent 25%, transparent 50%, $color 50%, $color 75%, transparent 75%, transparent);\n}\n// scss-docs-end gradient-mixins\n","// Select\n//\n// Replaces the browser default select with a custom one, mostly pulled from\n// https://primer.github.io/.\n\n.form-select {\n  --#{$prefix}form-select-bg-img: #{escape-svg($form-select-indicator)};\n\n  display: block;\n  width: 100%;\n  padding: $form-select-padding-y $form-select-indicator-padding $form-select-padding-y $form-select-padding-x;\n  font-family: $form-select-font-family;\n  @include font-size($form-select-font-size);\n  font-weight: $form-select-font-weight;\n  line-height: $form-select-line-height;\n  color: $form-select-color;\n  appearance: none;\n  background-color: $form-select-bg;\n  background-image: var(--#{$prefix}form-select-bg-img), var(--#{$prefix}form-select-bg-icon, none);\n  background-repeat: no-repeat;\n  background-position: $form-select-bg-position;\n  background-size: $form-select-bg-size;\n  border: $form-select-border-width solid $form-select-border-color;\n  @include border-radius($form-select-border-radius, 0);\n  @include box-shadow($form-select-box-shadow);\n  @include transition($form-select-transition);\n\n  &:focus {\n    border-color: $form-select-focus-border-color;\n    outline: 0;\n    @if $enable-shadows {\n      @include box-shadow($form-select-box-shadow, $form-select-focus-box-shadow);\n    } @else {\n      // Avoid using mixin so we can pass custom focus shadow properly\n      box-shadow: $form-select-focus-box-shadow;\n    }\n  }\n\n  &[multiple],\n  &[size]:not([size=\"1\"]) {\n    padding-right: $form-select-padding-x;\n    background-image: none;\n  }\n\n  &:disabled {\n    color: $form-select-disabled-color;\n    background-color: $form-select-disabled-bg;\n    border-color: $form-select-disabled-border-color;\n  }\n\n  // Remove outline from select box in FF\n  &:-moz-focusring {\n    color: transparent;\n    text-shadow: 0 0 0 $form-select-color;\n  }\n}\n\n.form-select-sm {\n  padding-top: $form-select-padding-y-sm;\n  padding-bottom: $form-select-padding-y-sm;\n  padding-left: $form-select-padding-x-sm;\n  @include font-size($form-select-font-size-sm);\n  @include border-radius($form-select-border-radius-sm);\n}\n\n.form-select-lg {\n  padding-top: $form-select-padding-y-lg;\n  padding-bottom: $form-select-padding-y-lg;\n  padding-left: $form-select-padding-x-lg;\n  @include font-size($form-select-font-size-lg);\n  @include border-radius($form-select-border-radius-lg);\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .form-select {\n      --#{$prefix}form-select-bg-img: #{escape-svg($form-select-indicator-dark)};\n    }\n  }\n}\n","//\n// Check/radio\n//\n\n.form-check {\n  display: block;\n  min-height: $form-check-min-height;\n  padding-left: $form-check-padding-start;\n  margin-bottom: $form-check-margin-bottom;\n\n  .form-check-input {\n    float: left;\n    margin-left: $form-check-padding-start * -1;\n  }\n}\n\n.form-check-reverse {\n  padding-right: $form-check-padding-start;\n  padding-left: 0;\n  text-align: right;\n\n  .form-check-input {\n    float: right;\n    margin-right: $form-check-padding-start * -1;\n    margin-left: 0;\n  }\n}\n\n.form-check-input {\n  --#{$prefix}form-check-bg: #{$form-check-input-bg};\n\n  flex-shrink: 0;\n  width: $form-check-input-width;\n  height: $form-check-input-width;\n  margin-top: ($line-height-base - $form-check-input-width) * .5; // line-height minus check height\n  vertical-align: top;\n  appearance: none;\n  background-color: var(--#{$prefix}form-check-bg);\n  background-image: var(--#{$prefix}form-check-bg-image);\n  background-repeat: no-repeat;\n  background-position: center;\n  background-size: contain;\n  border: $form-check-input-border;\n  print-color-adjust: exact; // Keep themed appearance for print\n  @include transition($form-check-transition);\n\n  &[type=\"checkbox\"] {\n    @include border-radius($form-check-input-border-radius);\n  }\n\n  &[type=\"radio\"] {\n    // stylelint-disable-next-line property-disallowed-list\n    border-radius: $form-check-radio-border-radius;\n  }\n\n  &:active {\n    filter: $form-check-input-active-filter;\n  }\n\n  &:focus {\n    border-color: $form-check-input-focus-border;\n    outline: 0;\n    box-shadow: $form-check-input-focus-box-shadow;\n  }\n\n  &:checked {\n    background-color: $form-check-input-checked-bg-color;\n    border-color: $form-check-input-checked-border-color;\n\n    &[type=\"checkbox\"] {\n      @if $enable-gradients {\n        --#{$prefix}form-check-bg-image: #{escape-svg($form-check-input-checked-bg-image)}, var(--#{$prefix}gradient);\n      } @else {\n        --#{$prefix}form-check-bg-image: #{escape-svg($form-check-input-checked-bg-image)};\n      }\n    }\n\n    &[type=\"radio\"] {\n      @if $enable-gradients {\n        --#{$prefix}form-check-bg-image: #{escape-svg($form-check-radio-checked-bg-image)}, var(--#{$prefix}gradient);\n      } @else {\n        --#{$prefix}form-check-bg-image: #{escape-svg($form-check-radio-checked-bg-image)};\n      }\n    }\n  }\n\n  &[type=\"checkbox\"]:indeterminate {\n    background-color: $form-check-input-indeterminate-bg-color;\n    border-color: $form-check-input-indeterminate-border-color;\n\n    @if $enable-gradients {\n      --#{$prefix}form-check-bg-image: #{escape-svg($form-check-input-indeterminate-bg-image)}, var(--#{$prefix}gradient);\n    } @else {\n      --#{$prefix}form-check-bg-image: #{escape-svg($form-check-input-indeterminate-bg-image)};\n    }\n  }\n\n  &:disabled {\n    pointer-events: none;\n    filter: none;\n    opacity: $form-check-input-disabled-opacity;\n  }\n\n  // Use disabled attribute in addition of :disabled pseudo-class\n  // See: https://github.com/twbs/bootstrap/issues/28247\n  &[disabled],\n  &:disabled {\n    ~ .form-check-label {\n      cursor: default;\n      opacity: $form-check-label-disabled-opacity;\n    }\n  }\n}\n\n.form-check-label {\n  color: $form-check-label-color;\n  cursor: $form-check-label-cursor;\n}\n\n//\n// Switch\n//\n\n.form-switch {\n  padding-left: $form-switch-padding-start;\n\n  .form-check-input {\n    --#{$prefix}form-switch-bg: #{escape-svg($form-switch-bg-image)};\n\n    width: $form-switch-width;\n    margin-left: $form-switch-padding-start * -1;\n    background-image: var(--#{$prefix}form-switch-bg);\n    background-position: left center;\n    @include border-radius($form-switch-border-radius, 0);\n    @include transition($form-switch-transition);\n\n    &:focus {\n      --#{$prefix}form-switch-bg: #{escape-svg($form-switch-focus-bg-image)};\n    }\n\n    &:checked {\n      background-position: $form-switch-checked-bg-position;\n\n      @if $enable-gradients {\n        --#{$prefix}form-switch-bg: #{escape-svg($form-switch-checked-bg-image)}, var(--#{$prefix}gradient);\n      } @else {\n        --#{$prefix}form-switch-bg: #{escape-svg($form-switch-checked-bg-image)};\n      }\n    }\n  }\n\n  &.form-check-reverse {\n    padding-right: $form-switch-padding-start;\n    padding-left: 0;\n\n    .form-check-input {\n      margin-right: $form-switch-padding-start * -1;\n      margin-left: 0;\n    }\n  }\n}\n\n.form-check-inline {\n  display: inline-block;\n  margin-right: $form-check-inline-margin-end;\n}\n\n.btn-check {\n  position: absolute;\n  clip: rect(0, 0, 0, 0);\n  pointer-events: none;\n\n  &[disabled],\n  &:disabled {\n    + .btn {\n      pointer-events: none;\n      filter: none;\n      opacity: $form-check-btn-check-disabled-opacity;\n    }\n  }\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .form-switch .form-check-input:not(:checked):not(:focus) {\n      --#{$prefix}form-switch-bg: #{escape-svg($form-switch-bg-image-dark)};\n    }\n  }\n}\n","// Range\n//\n// Style range inputs the same across browsers. Vendor-specific rules for pseudo\n// elements cannot be mixed. As such, there are no shared styles for focus or\n// active states on prefixed selectors.\n\n.form-range {\n  width: 100%;\n  height: add($form-range-thumb-height, $form-range-thumb-focus-box-shadow-width * 2);\n  padding: 0; // Need to reset padding\n  appearance: none;\n  background-color: transparent;\n\n  &:focus {\n    outline: 0;\n\n    // Pseudo-elements must be split across multiple rulesets to have an effect.\n    // No box-shadow() mixin for focus accessibility.\n    &::-webkit-slider-thumb { box-shadow: $form-range-thumb-focus-box-shadow; }\n    &::-moz-range-thumb     { box-shadow: $form-range-thumb-focus-box-shadow; }\n  }\n\n  &::-moz-focus-outer {\n    border: 0;\n  }\n\n  &::-webkit-slider-thumb {\n    width: $form-range-thumb-width;\n    height: $form-range-thumb-height;\n    margin-top: ($form-range-track-height - $form-range-thumb-height) * .5; // Webkit specific\n    appearance: none;\n    @include gradient-bg($form-range-thumb-bg);\n    border: $form-range-thumb-border;\n    @include border-radius($form-range-thumb-border-radius);\n    @include box-shadow($form-range-thumb-box-shadow);\n    @include transition($form-range-thumb-transition);\n\n    &:active {\n      @include gradient-bg($form-range-thumb-active-bg);\n    }\n  }\n\n  &::-webkit-slider-runnable-track {\n    width: $form-range-track-width;\n    height: $form-range-track-height;\n    color: transparent; // Why?\n    cursor: $form-range-track-cursor;\n    background-color: $form-range-track-bg;\n    border-color: transparent;\n    @include border-radius($form-range-track-border-radius);\n    @include box-shadow($form-range-track-box-shadow);\n  }\n\n  &::-moz-range-thumb {\n    width: $form-range-thumb-width;\n    height: $form-range-thumb-height;\n    appearance: none;\n    @include gradient-bg($form-range-thumb-bg);\n    border: $form-range-thumb-border;\n    @include border-radius($form-range-thumb-border-radius);\n    @include box-shadow($form-range-thumb-box-shadow);\n    @include transition($form-range-thumb-transition);\n\n    &:active {\n      @include gradient-bg($form-range-thumb-active-bg);\n    }\n  }\n\n  &::-moz-range-track {\n    width: $form-range-track-width;\n    height: $form-range-track-height;\n    color: transparent;\n    cursor: $form-range-track-cursor;\n    background-color: $form-range-track-bg;\n    border-color: transparent; // Firefox specific?\n    @include border-radius($form-range-track-border-radius);\n    @include box-shadow($form-range-track-box-shadow);\n  }\n\n  &:disabled {\n    pointer-events: none;\n\n    &::-webkit-slider-thumb {\n      background-color: $form-range-thumb-disabled-bg;\n    }\n\n    &::-moz-range-thumb {\n      background-color: $form-range-thumb-disabled-bg;\n    }\n  }\n}\n",".form-floating {\n  position: relative;\n\n  > .form-control,\n  > .form-control-plaintext,\n  > .form-select {\n    height: $form-floating-height;\n    min-height: $form-floating-height;\n    line-height: $form-floating-line-height;\n  }\n\n  > label {\n    position: absolute;\n    top: 0;\n    left: 0;\n    z-index: 2;\n    height: 100%; // allow textareas\n    padding: $form-floating-padding-y $form-floating-padding-x;\n    overflow: hidden;\n    text-align: start;\n    text-overflow: ellipsis;\n    white-space: nowrap;\n    pointer-events: none;\n    border: $input-border-width solid transparent; // Required for aligning label's text with the input as it affects inner box model\n    transform-origin: 0 0;\n    @include transition($form-floating-transition);\n  }\n\n  > .form-control,\n  > .form-control-plaintext {\n    padding: $form-floating-padding-y $form-floating-padding-x;\n\n    &::placeholder {\n      color: transparent;\n    }\n\n    &:focus,\n    &:not(:placeholder-shown) {\n      padding-top: $form-floating-input-padding-t;\n      padding-bottom: $form-floating-input-padding-b;\n    }\n    // Duplicated because `:-webkit-autofill` invalidates other selectors when grouped\n    &:-webkit-autofill {\n      padding-top: $form-floating-input-padding-t;\n      padding-bottom: $form-floating-input-padding-b;\n    }\n  }\n\n  > .form-select {\n    padding-top: $form-floating-input-padding-t;\n    padding-bottom: $form-floating-input-padding-b;\n  }\n\n  > .form-control:focus,\n  > .form-control:not(:placeholder-shown),\n  > .form-control-plaintext,\n  > .form-select {\n    ~ label {\n      color: rgba(var(--#{$prefix}body-color-rgb), #{$form-floating-label-opacity});\n      transform: $form-floating-label-transform;\n\n      &::after {\n        position: absolute;\n        inset: $form-floating-padding-y ($form-floating-padding-x * .5);\n        z-index: -1;\n        height: $form-floating-label-height;\n        content: \"\";\n        background-color: $input-bg;\n        @include border-radius($input-border-radius);\n      }\n    }\n  }\n  // Duplicated because `:-webkit-autofill` invalidates other selectors when grouped\n  > .form-control:-webkit-autofill {\n    ~ label {\n      color: rgba(var(--#{$prefix}body-color-rgb), #{$form-floating-label-opacity});\n      transform: $form-floating-label-transform;\n    }\n  }\n\n  > .form-control-plaintext {\n    ~ label {\n      border-width: $input-border-width 0; // Required to properly position label text - as explained above\n    }\n  }\n\n  > :disabled ~ label,\n  > .form-control:disabled ~ label { // Required for `.form-control`s because of specificity\n    color: $form-floating-label-disabled-color;\n\n    &::after {\n      background-color: $input-disabled-bg;\n    }\n  }\n}\n","//\n// Base styles\n//\n\n.input-group {\n  position: relative;\n  display: flex;\n  flex-wrap: wrap; // For form validation feedback\n  align-items: stretch;\n  width: 100%;\n\n  > .form-control,\n  > .form-select,\n  > .form-floating {\n    position: relative; // For focus state's z-index\n    flex: 1 1 auto;\n    width: 1%;\n    min-width: 0; // https://stackoverflow.com/questions/36247140/why-dont-flex-items-shrink-past-content-size\n  }\n\n  // Bring the \"active\" form control to the top of surrounding elements\n  > .form-control:focus,\n  > .form-select:focus,\n  > .form-floating:focus-within {\n    z-index: 5;\n  }\n\n  // Ensure buttons are always above inputs for more visually pleasing borders.\n  // This isn't needed for `.input-group-text` since it shares the same border-color\n  // as our inputs.\n  .btn {\n    position: relative;\n    z-index: 2;\n\n    &:focus {\n      z-index: 5;\n    }\n  }\n}\n\n\n// Textual addons\n//\n// Serves as a catch-all element for any text or radio/checkbox input you wish\n// to prepend or append to an input.\n\n.input-group-text {\n  display: flex;\n  align-items: center;\n  padding: $input-group-addon-padding-y $input-group-addon-padding-x;\n  @include font-size($input-font-size); // Match inputs\n  font-weight: $input-group-addon-font-weight;\n  line-height: $input-line-height;\n  color: $input-group-addon-color;\n  text-align: center;\n  white-space: nowrap;\n  background-color: $input-group-addon-bg;\n  border: $input-border-width solid $input-group-addon-border-color;\n  @include border-radius($input-border-radius);\n}\n\n\n// Sizing\n//\n// Remix the default form control sizing classes into new ones for easier\n// manipulation.\n\n.input-group-lg > .form-control,\n.input-group-lg > .form-select,\n.input-group-lg > .input-group-text,\n.input-group-lg > .btn {\n  padding: $input-padding-y-lg $input-padding-x-lg;\n  @include font-size($input-font-size-lg);\n  @include border-radius($input-border-radius-lg);\n}\n\n.input-group-sm > .form-control,\n.input-group-sm > .form-select,\n.input-group-sm > .input-group-text,\n.input-group-sm > .btn {\n  padding: $input-padding-y-sm $input-padding-x-sm;\n  @include font-size($input-font-size-sm);\n  @include border-radius($input-border-radius-sm);\n}\n\n.input-group-lg > .form-select,\n.input-group-sm > .form-select {\n  padding-right: $form-select-padding-x + $form-select-indicator-padding;\n}\n\n\n// Rounded corners\n//\n// These rulesets must come after the sizing ones to properly override sm and lg\n// border-radius values when extending. They're more specific than we'd like\n// with the `.input-group >` part, but without it, we cannot override the sizing.\n\n// stylelint-disable-next-line no-duplicate-selectors\n.input-group {\n  &:not(.has-validation) {\n    > :not(:last-child):not(.dropdown-toggle):not(.dropdown-menu):not(.form-floating),\n    > .dropdown-toggle:nth-last-child(n + 3),\n    > .form-floating:not(:last-child) > .form-control,\n    > .form-floating:not(:last-child) > .form-select {\n      @include border-end-radius(0);\n    }\n  }\n\n  &.has-validation {\n    > :nth-last-child(n + 3):not(.dropdown-toggle):not(.dropdown-menu):not(.form-floating),\n    > .dropdown-toggle:nth-last-child(n + 4),\n    > .form-floating:nth-last-child(n + 3) > .form-control,\n    > .form-floating:nth-last-child(n + 3) > .form-select {\n      @include border-end-radius(0);\n    }\n  }\n\n  $validation-messages: \"\";\n  @each $state in map-keys($form-validation-states) {\n    $validation-messages: $validation-messages + \":not(.\" + unquote($state) + \"-tooltip)\" + \":not(.\" + unquote($state) + \"-feedback)\";\n  }\n\n  > :not(:first-child):not(.dropdown-menu)#{$validation-messages} {\n    margin-left: calc(#{$input-border-width} * -1); // stylelint-disable-line function-disallowed-list\n    @include border-start-radius(0);\n  }\n\n  > .form-floating:not(:first-child) > .form-control,\n  > .form-floating:not(:first-child) > .form-select {\n    @include border-start-radius(0);\n  }\n}\n","// This mixin uses an `if()` technique to be compatible with Dart Sass\n// See https://github.com/sass/sass/issues/1873#issuecomment-152293725 for more details\n\n// scss-docs-start form-validation-mixins\n@mixin form-validation-state-selector($state) {\n  @if ($state == \"valid\" or $state == \"invalid\") {\n    .was-validated #{if(&, \"&\", \"\")}:#{$state},\n    #{if(&, \"&\", \"\")}.is-#{$state} {\n      @content;\n    }\n  } @else {\n    #{if(&, \"&\", \"\")}.is-#{$state} {\n      @content;\n    }\n  }\n}\n\n@mixin form-validation-state(\n  $state,\n  $color,\n  $icon,\n  $tooltip-color: color-contrast($color),\n  $tooltip-bg-color: rgba($color, $form-feedback-tooltip-opacity),\n  $focus-box-shadow: 0 0 $input-btn-focus-blur $input-focus-width rgba($color, $input-btn-focus-color-opacity),\n  $border-color: $color\n) {\n  .#{$state}-feedback {\n    display: none;\n    width: 100%;\n    margin-top: $form-feedback-margin-top;\n    @include font-size($form-feedback-font-size);\n    font-style: $form-feedback-font-style;\n    color: $color;\n  }\n\n  .#{$state}-tooltip {\n    position: absolute;\n    top: 100%;\n    z-index: 5;\n    display: none;\n    max-width: 100%; // Contain to parent when possible\n    padding: $form-feedback-tooltip-padding-y $form-feedback-tooltip-padding-x;\n    margin-top: .1rem;\n    @include font-size($form-feedback-tooltip-font-size);\n    line-height: $form-feedback-tooltip-line-height;\n    color: $tooltip-color;\n    background-color: $tooltip-bg-color;\n    @include border-radius($form-feedback-tooltip-border-radius);\n  }\n\n  @include form-validation-state-selector($state) {\n    ~ .#{$state}-feedback,\n    ~ .#{$state}-tooltip {\n      display: block;\n    }\n  }\n\n  .form-control {\n    @include form-validation-state-selector($state) {\n      border-color: $border-color;\n\n      @if $enable-validation-icons {\n        padding-right: $input-height-inner;\n        background-image: escape-svg($icon);\n        background-repeat: no-repeat;\n        background-position: right $input-height-inner-quarter center;\n        background-size: $input-height-inner-half $input-height-inner-half;\n      }\n\n      &:focus {\n        border-color: $border-color;\n        @if $enable-shadows {\n          @include box-shadow($input-box-shadow, $focus-box-shadow);\n        } @else {\n          // Avoid using mixin so we can pass custom focus shadow properly\n          box-shadow: $focus-box-shadow;\n        }\n      }\n    }\n  }\n\n  // stylelint-disable-next-line selector-no-qualifying-type\n  textarea.form-control {\n    @include form-validation-state-selector($state) {\n      @if $enable-validation-icons {\n        padding-right: $input-height-inner;\n        background-position: top $input-height-inner-quarter right $input-height-inner-quarter;\n      }\n    }\n  }\n\n  .form-select {\n    @include form-validation-state-selector($state) {\n      border-color: $border-color;\n\n      @if $enable-validation-icons {\n        &:not([multiple]):not([size]),\n        &:not([multiple])[size=\"1\"] {\n          --#{$prefix}form-select-bg-icon: #{escape-svg($icon)};\n          padding-right: $form-select-feedback-icon-padding-end;\n          background-position: $form-select-bg-position, $form-select-feedback-icon-position;\n          background-size: $form-select-bg-size, $form-select-feedback-icon-size;\n        }\n      }\n\n      &:focus {\n        border-color: $border-color;\n        @if $enable-shadows {\n          @include box-shadow($form-select-box-shadow, $focus-box-shadow);\n        } @else {\n          // Avoid using mixin so we can pass custom focus shadow properly\n          box-shadow: $focus-box-shadow;\n        }\n      }\n    }\n  }\n\n  .form-control-color {\n    @include form-validation-state-selector($state) {\n      @if $enable-validation-icons {\n        width: add($form-color-width, $input-height-inner);\n      }\n    }\n  }\n\n  .form-check-input {\n    @include form-validation-state-selector($state) {\n      border-color: $border-color;\n\n      &:checked {\n        background-color: $color;\n      }\n\n      &:focus {\n        box-shadow: $focus-box-shadow;\n      }\n\n      ~ .form-check-label {\n        color: $color;\n      }\n    }\n  }\n  .form-check-inline .form-check-input {\n    ~ .#{$state}-feedback {\n      margin-left: .5em;\n    }\n  }\n\n  .input-group {\n    > .form-control:not(:focus),\n    > .form-select:not(:focus),\n    > .form-floating:not(:focus-within) {\n      @include form-validation-state-selector($state) {\n        @if $state == \"valid\" {\n          z-index: 3;\n        } @else if $state == \"invalid\" {\n          z-index: 4;\n        }\n      }\n    }\n  }\n}\n// scss-docs-end form-validation-mixins\n","//\n// Base styles\n//\n\n.btn {\n  // scss-docs-start btn-css-vars\n  --#{$prefix}btn-padding-x: #{$btn-padding-x};\n  --#{$prefix}btn-padding-y: #{$btn-padding-y};\n  --#{$prefix}btn-font-family: #{$btn-font-family};\n  @include rfs($btn-font-size, --#{$prefix}btn-font-size);\n  --#{$prefix}btn-font-weight: #{$btn-font-weight};\n  --#{$prefix}btn-line-height: #{$btn-line-height};\n  --#{$prefix}btn-color: #{$btn-color};\n  --#{$prefix}btn-bg: transparent;\n  --#{$prefix}btn-border-width: #{$btn-border-width};\n  --#{$prefix}btn-border-color: transparent;\n  --#{$prefix}btn-border-radius: #{$btn-border-radius};\n  --#{$prefix}btn-hover-border-color: transparent;\n  --#{$prefix}btn-box-shadow: #{$btn-box-shadow};\n  --#{$prefix}btn-disabled-opacity: #{$btn-disabled-opacity};\n  --#{$prefix}btn-focus-box-shadow: 0 0 0 #{$btn-focus-width} rgba(var(--#{$prefix}btn-focus-shadow-rgb), .5);\n  // scss-docs-end btn-css-vars\n\n  display: inline-block;\n  padding: var(--#{$prefix}btn-padding-y) var(--#{$prefix}btn-padding-x);\n  font-family: var(--#{$prefix}btn-font-family);\n  @include font-size(var(--#{$prefix}btn-font-size));\n  font-weight: var(--#{$prefix}btn-font-weight);\n  line-height: var(--#{$prefix}btn-line-height);\n  color: var(--#{$prefix}btn-color);\n  text-align: center;\n  text-decoration: if($link-decoration == none, null, none);\n  white-space: $btn-white-space;\n  vertical-align: middle;\n  cursor: if($enable-button-pointers, pointer, null);\n  user-select: none;\n  border: var(--#{$prefix}btn-border-width) solid var(--#{$prefix}btn-border-color);\n  @include border-radius(var(--#{$prefix}btn-border-radius));\n  @include gradient-bg(var(--#{$prefix}btn-bg));\n  @include box-shadow(var(--#{$prefix}btn-box-shadow));\n  @include transition($btn-transition);\n\n  &:hover {\n    color: var(--#{$prefix}btn-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n    background-color: var(--#{$prefix}btn-hover-bg);\n    border-color: var(--#{$prefix}btn-hover-border-color);\n  }\n\n  .btn-check + &:hover {\n    // override for the checkbox/radio buttons\n    color: var(--#{$prefix}btn-color);\n    background-color: var(--#{$prefix}btn-bg);\n    border-color: var(--#{$prefix}btn-border-color);\n  }\n\n  &:focus-visible {\n    color: var(--#{$prefix}btn-hover-color);\n    @include gradient-bg(var(--#{$prefix}btn-hover-bg));\n    border-color: var(--#{$prefix}btn-hover-border-color);\n    outline: 0;\n    // Avoid using mixin so we can pass custom focus shadow properly\n    @if $enable-shadows {\n      box-shadow: var(--#{$prefix}btn-box-shadow), var(--#{$prefix}btn-focus-box-shadow);\n    } @else {\n      box-shadow: var(--#{$prefix}btn-focus-box-shadow);\n    }\n  }\n\n  .btn-check:focus-visible + & {\n    border-color: var(--#{$prefix}btn-hover-border-color);\n    outline: 0;\n    // Avoid using mixin so we can pass custom focus shadow properly\n    @if $enable-shadows {\n      box-shadow: var(--#{$prefix}btn-box-shadow), var(--#{$prefix}btn-focus-box-shadow);\n    } @else {\n      box-shadow: var(--#{$prefix}btn-focus-box-shadow);\n    }\n  }\n\n  .btn-check:checked + &,\n  :not(.btn-check) + &:active,\n  &:first-child:active,\n  &.active,\n  &.show {\n    color: var(--#{$prefix}btn-active-color);\n    background-color: var(--#{$prefix}btn-active-bg);\n    // Remove CSS gradients if they're enabled\n    background-image: if($enable-gradients, none, null);\n    border-color: var(--#{$prefix}btn-active-border-color);\n    @include box-shadow(var(--#{$prefix}btn-active-shadow));\n\n    &:focus-visible {\n      // Avoid using mixin so we can pass custom focus shadow properly\n      @if $enable-shadows {\n        box-shadow: var(--#{$prefix}btn-active-shadow), var(--#{$prefix}btn-focus-box-shadow);\n      } @else {\n        box-shadow: var(--#{$prefix}btn-focus-box-shadow);\n      }\n    }\n  }\n\n  .btn-check:checked:focus-visible + & {\n    // Avoid using mixin so we can pass custom focus shadow properly\n    @if $enable-shadows {\n      box-shadow: var(--#{$prefix}btn-active-shadow), var(--#{$prefix}btn-focus-box-shadow);\n    } @else {\n      box-shadow: var(--#{$prefix}btn-focus-box-shadow);\n    }\n  }\n\n  &:disabled,\n  &.disabled,\n  fieldset:disabled & {\n    color: var(--#{$prefix}btn-disabled-color);\n    pointer-events: none;\n    background-color: var(--#{$prefix}btn-disabled-bg);\n    background-image: if($enable-gradients, none, null);\n    border-color: var(--#{$prefix}btn-disabled-border-color);\n    opacity: var(--#{$prefix}btn-disabled-opacity);\n    @include box-shadow(none);\n  }\n}\n\n\n//\n// Alternate buttons\n//\n\n// scss-docs-start btn-variant-loops\n@each $color, $value in $theme-colors {\n  .btn-#{$color} {\n    @if $color == \"light\" {\n      @include button-variant(\n        $value,\n        $value,\n        $hover-background: shade-color($value, $btn-hover-bg-shade-amount),\n        $hover-border: shade-color($value, $btn-hover-border-shade-amount),\n        $active-background: shade-color($value, $btn-active-bg-shade-amount),\n        $active-border: shade-color($value, $btn-active-border-shade-amount)\n      );\n    } @else if $color == \"dark\" {\n      @include button-variant(\n        $value,\n        $value,\n        $hover-background: tint-color($value, $btn-hover-bg-tint-amount),\n        $hover-border: tint-color($value, $btn-hover-border-tint-amount),\n        $active-background: tint-color($value, $btn-active-bg-tint-amount),\n        $active-border: tint-color($value, $btn-active-border-tint-amount)\n      );\n    } @else {\n      @include button-variant($value, $value);\n    }\n  }\n}\n\n@each $color, $value in $theme-colors {\n  .btn-outline-#{$color} {\n    @include button-outline-variant($value);\n  }\n}\n// scss-docs-end btn-variant-loops\n\n\n//\n// Link buttons\n//\n\n// Make a button look and behave like a link\n.btn-link {\n  --#{$prefix}btn-font-weight: #{$font-weight-normal};\n  --#{$prefix}btn-color: #{$btn-link-color};\n  --#{$prefix}btn-bg: transparent;\n  --#{$prefix}btn-border-color: transparent;\n  --#{$prefix}btn-hover-color: #{$btn-link-hover-color};\n  --#{$prefix}btn-hover-border-color: transparent;\n  --#{$prefix}btn-active-color: #{$btn-link-hover-color};\n  --#{$prefix}btn-active-border-color: transparent;\n  --#{$prefix}btn-disabled-color: #{$btn-link-disabled-color};\n  --#{$prefix}btn-disabled-border-color: transparent;\n  --#{$prefix}btn-box-shadow: 0 0 0 #000; // Can't use `none` as keyword negates all values when used with multiple shadows\n  --#{$prefix}btn-focus-shadow-rgb: #{$btn-link-focus-shadow-rgb};\n\n  text-decoration: $link-decoration;\n  @if $enable-gradients {\n    background-image: none;\n  }\n\n  &:hover,\n  &:focus-visible {\n    text-decoration: $link-hover-decoration;\n  }\n\n  &:focus-visible {\n    color: var(--#{$prefix}btn-color);\n  }\n\n  &:hover {\n    color: var(--#{$prefix}btn-hover-color);\n  }\n\n  // No need for an active state here\n}\n\n\n//\n// Button Sizes\n//\n\n.btn-lg {\n  @include button-size($btn-padding-y-lg, $btn-padding-x-lg, $btn-font-size-lg, $btn-border-radius-lg);\n}\n\n.btn-sm {\n  @include button-size($btn-padding-y-sm, $btn-padding-x-sm, $btn-font-size-sm, $btn-border-radius-sm);\n}\n","// Button variants\n//\n// Easily pump out default styles, as well as :hover, :focus, :active,\n// and disabled options for all buttons\n\n// scss-docs-start btn-variant-mixin\n@mixin button-variant(\n  $background,\n  $border,\n  $color: color-contrast($background),\n  $hover-background: if($color == $color-contrast-light, shade-color($background, $btn-hover-bg-shade-amount), tint-color($background, $btn-hover-bg-tint-amount)),\n  $hover-border: if($color == $color-contrast-light, shade-color($border, $btn-hover-border-shade-amount), tint-color($border, $btn-hover-border-tint-amount)),\n  $hover-color: color-contrast($hover-background),\n  $active-background: if($color == $color-contrast-light, shade-color($background, $btn-active-bg-shade-amount), tint-color($background, $btn-active-bg-tint-amount)),\n  $active-border: if($color == $color-contrast-light, shade-color($border, $btn-active-border-shade-amount), tint-color($border, $btn-active-border-tint-amount)),\n  $active-color: color-contrast($active-background),\n  $disabled-background: $background,\n  $disabled-border: $border,\n  $disabled-color: color-contrast($disabled-background)\n) {\n  --#{$prefix}btn-color: #{$color};\n  --#{$prefix}btn-bg: #{$background};\n  --#{$prefix}btn-border-color: #{$border};\n  --#{$prefix}btn-hover-color: #{$hover-color};\n  --#{$prefix}btn-hover-bg: #{$hover-background};\n  --#{$prefix}btn-hover-border-color: #{$hover-border};\n  --#{$prefix}btn-focus-shadow-rgb: #{to-rgb(mix($color, $border, 15%))};\n  --#{$prefix}btn-active-color: #{$active-color};\n  --#{$prefix}btn-active-bg: #{$active-background};\n  --#{$prefix}btn-active-border-color: #{$active-border};\n  --#{$prefix}btn-active-shadow: #{$btn-active-box-shadow};\n  --#{$prefix}btn-disabled-color: #{$disabled-color};\n  --#{$prefix}btn-disabled-bg: #{$disabled-background};\n  --#{$prefix}btn-disabled-border-color: #{$disabled-border};\n}\n// scss-docs-end btn-variant-mixin\n\n// scss-docs-start btn-outline-variant-mixin\n@mixin button-outline-variant(\n  $color,\n  $color-hover: color-contrast($color),\n  $active-background: $color,\n  $active-border: $color,\n  $active-color: color-contrast($active-background)\n) {\n  --#{$prefix}btn-color: #{$color};\n  --#{$prefix}btn-border-color: #{$color};\n  --#{$prefix}btn-hover-color: #{$color-hover};\n  --#{$prefix}btn-hover-bg: #{$active-background};\n  --#{$prefix}btn-hover-border-color: #{$active-border};\n  --#{$prefix}btn-focus-shadow-rgb: #{to-rgb($color)};\n  --#{$prefix}btn-active-color: #{$active-color};\n  --#{$prefix}btn-active-bg: #{$active-background};\n  --#{$prefix}btn-active-border-color: #{$active-border};\n  --#{$prefix}btn-active-shadow: #{$btn-active-box-shadow};\n  --#{$prefix}btn-disabled-color: #{$color};\n  --#{$prefix}btn-disabled-bg: transparent;\n  --#{$prefix}btn-disabled-border-color: #{$color};\n  --#{$prefix}gradient: none;\n}\n// scss-docs-end btn-outline-variant-mixin\n\n// scss-docs-start btn-size-mixin\n@mixin button-size($padding-y, $padding-x, $font-size, $border-radius) {\n  --#{$prefix}btn-padding-y: #{$padding-y};\n  --#{$prefix}btn-padding-x: #{$padding-x};\n  @include rfs($font-size, --#{$prefix}btn-font-size);\n  --#{$prefix}btn-border-radius: #{$border-radius};\n}\n// scss-docs-end btn-size-mixin\n",".fade {\n  @include transition($transition-fade);\n\n  &:not(.show) {\n    opacity: 0;\n  }\n}\n\n// scss-docs-start collapse-classes\n.collapse {\n  &:not(.show) {\n    display: none;\n  }\n}\n\n.collapsing {\n  height: 0;\n  overflow: hidden;\n  @include transition($transition-collapse);\n\n  &.collapse-horizontal {\n    width: 0;\n    height: auto;\n    @include transition($transition-collapse-width);\n  }\n}\n// scss-docs-end collapse-classes\n","// The dropdown wrapper (`<div>`)\n.dropup,\n.dropend,\n.dropdown,\n.dropstart,\n.dropup-center,\n.dropdown-center {\n  position: relative;\n}\n\n.dropdown-toggle {\n  white-space: nowrap;\n\n  // Generate the caret automatically\n  @include caret();\n}\n\n// The dropdown menu\n.dropdown-menu {\n  // scss-docs-start dropdown-css-vars\n  --#{$prefix}dropdown-zindex: #{$zindex-dropdown};\n  --#{$prefix}dropdown-min-width: #{$dropdown-min-width};\n  --#{$prefix}dropdown-padding-x: #{$dropdown-padding-x};\n  --#{$prefix}dropdown-padding-y: #{$dropdown-padding-y};\n  --#{$prefix}dropdown-spacer: #{$dropdown-spacer};\n  @include rfs($dropdown-font-size, --#{$prefix}dropdown-font-size);\n  --#{$prefix}dropdown-color: #{$dropdown-color};\n  --#{$prefix}dropdown-bg: #{$dropdown-bg};\n  --#{$prefix}dropdown-border-color: #{$dropdown-border-color};\n  --#{$prefix}dropdown-border-radius: #{$dropdown-border-radius};\n  --#{$prefix}dropdown-border-width: #{$dropdown-border-width};\n  --#{$prefix}dropdown-inner-border-radius: #{$dropdown-inner-border-radius};\n  --#{$prefix}dropdown-divider-bg: #{$dropdown-divider-bg};\n  --#{$prefix}dropdown-divider-margin-y: #{$dropdown-divider-margin-y};\n  --#{$prefix}dropdown-box-shadow: #{$dropdown-box-shadow};\n  --#{$prefix}dropdown-link-color: #{$dropdown-link-color};\n  --#{$prefix}dropdown-link-hover-color: #{$dropdown-link-hover-color};\n  --#{$prefix}dropdown-link-hover-bg: #{$dropdown-link-hover-bg};\n  --#{$prefix}dropdown-link-active-color: #{$dropdown-link-active-color};\n  --#{$prefix}dropdown-link-active-bg: #{$dropdown-link-active-bg};\n  --#{$prefix}dropdown-link-disabled-color: #{$dropdown-link-disabled-color};\n  --#{$prefix}dropdown-item-padding-x: #{$dropdown-item-padding-x};\n  --#{$prefix}dropdown-item-padding-y: #{$dropdown-item-padding-y};\n  --#{$prefix}dropdown-header-color: #{$dropdown-header-color};\n  --#{$prefix}dropdown-header-padding-x: #{$dropdown-header-padding-x};\n  --#{$prefix}dropdown-header-padding-y: #{$dropdown-header-padding-y};\n  // scss-docs-end dropdown-css-vars\n\n  position: absolute;\n  z-index: var(--#{$prefix}dropdown-zindex);\n  display: none; // none by default, but block on \"open\" of the menu\n  min-width: var(--#{$prefix}dropdown-min-width);\n  padding: var(--#{$prefix}dropdown-padding-y) var(--#{$prefix}dropdown-padding-x);\n  margin: 0; // Override default margin of ul\n  @include font-size(var(--#{$prefix}dropdown-font-size));\n  color: var(--#{$prefix}dropdown-color);\n  text-align: left; // Ensures proper alignment if parent has it changed (e.g., modal footer)\n  list-style: none;\n  background-color: var(--#{$prefix}dropdown-bg);\n  background-clip: padding-box;\n  border: var(--#{$prefix}dropdown-border-width) solid var(--#{$prefix}dropdown-border-color);\n  @include border-radius(var(--#{$prefix}dropdown-border-radius));\n  @include box-shadow(var(--#{$prefix}dropdown-box-shadow));\n\n  &[data-bs-popper] {\n    top: 100%;\n    left: 0;\n    margin-top: var(--#{$prefix}dropdown-spacer);\n  }\n\n  @if $dropdown-padding-y == 0 {\n    > .dropdown-item:first-child,\n    > li:first-child .dropdown-item {\n      @include border-top-radius(var(--#{$prefix}dropdown-inner-border-radius));\n    }\n    > .dropdown-item:last-child,\n    > li:last-child .dropdown-item {\n      @include border-bottom-radius(var(--#{$prefix}dropdown-inner-border-radius));\n    }\n\n  }\n}\n\n// scss-docs-start responsive-breakpoints\n// We deliberately hardcode the `bs-` prefix because we check\n// this custom property in JS to determine Popper's positioning\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  @include media-breakpoint-up($breakpoint) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    .dropdown-menu#{$infix}-start {\n      --bs-position: start;\n\n      &[data-bs-popper] {\n        right: auto;\n        left: 0;\n      }\n    }\n\n    .dropdown-menu#{$infix}-end {\n      --bs-position: end;\n\n      &[data-bs-popper] {\n        right: 0;\n        left: auto;\n      }\n    }\n  }\n}\n// scss-docs-end responsive-breakpoints\n\n// Allow for dropdowns to go bottom up (aka, dropup-menu)\n// Just add .dropup after the standard .dropdown class and you're set.\n.dropup {\n  .dropdown-menu[data-bs-popper] {\n    top: auto;\n    bottom: 100%;\n    margin-top: 0;\n    margin-bottom: var(--#{$prefix}dropdown-spacer);\n  }\n\n  .dropdown-toggle {\n    @include caret(up);\n  }\n}\n\n.dropend {\n  .dropdown-menu[data-bs-popper] {\n    top: 0;\n    right: auto;\n    left: 100%;\n    margin-top: 0;\n    margin-left: var(--#{$prefix}dropdown-spacer);\n  }\n\n  .dropdown-toggle {\n    @include caret(end);\n    &::after {\n      vertical-align: 0;\n    }\n  }\n}\n\n.dropstart {\n  .dropdown-menu[data-bs-popper] {\n    top: 0;\n    right: 100%;\n    left: auto;\n    margin-top: 0;\n    margin-right: var(--#{$prefix}dropdown-spacer);\n  }\n\n  .dropdown-toggle {\n    @include caret(start);\n    &::before {\n      vertical-align: 0;\n    }\n  }\n}\n\n\n// Dividers (basically an `<hr>`) within the dropdown\n.dropdown-divider {\n  height: 0;\n  margin: var(--#{$prefix}dropdown-divider-margin-y) 0;\n  overflow: hidden;\n  border-top: 1px solid var(--#{$prefix}dropdown-divider-bg);\n  opacity: 1; // Revisit in v6 to de-dupe styles that conflict with <hr> element\n}\n\n// Links, buttons, and more within the dropdown menu\n//\n// `<button>`-specific styles are denoted with `// For <button>s`\n.dropdown-item {\n  display: block;\n  width: 100%; // For `<button>`s\n  padding: var(--#{$prefix}dropdown-item-padding-y) var(--#{$prefix}dropdown-item-padding-x);\n  clear: both;\n  font-weight: $font-weight-normal;\n  color: var(--#{$prefix}dropdown-link-color);\n  text-align: inherit; // For `<button>`s\n  text-decoration: if($link-decoration == none, null, none);\n  white-space: nowrap; // prevent links from randomly breaking onto new lines\n  background-color: transparent; // For `<button>`s\n  border: 0; // For `<button>`s\n  @include border-radius(var(--#{$prefix}dropdown-item-border-radius, 0));\n\n  &:hover,\n  &:focus {\n    color: var(--#{$prefix}dropdown-link-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n    @include gradient-bg(var(--#{$prefix}dropdown-link-hover-bg));\n  }\n\n  &.active,\n  &:active {\n    color: var(--#{$prefix}dropdown-link-active-color);\n    text-decoration: none;\n    @include gradient-bg(var(--#{$prefix}dropdown-link-active-bg));\n  }\n\n  &.disabled,\n  &:disabled {\n    color: var(--#{$prefix}dropdown-link-disabled-color);\n    pointer-events: none;\n    background-color: transparent;\n    // Remove CSS gradients if they're enabled\n    background-image: if($enable-gradients, none, null);\n  }\n}\n\n.dropdown-menu.show {\n  display: block;\n}\n\n// Dropdown section headers\n.dropdown-header {\n  display: block;\n  padding: var(--#{$prefix}dropdown-header-padding-y) var(--#{$prefix}dropdown-header-padding-x);\n  margin-bottom: 0; // for use with heading elements\n  @include font-size($font-size-sm);\n  color: var(--#{$prefix}dropdown-header-color);\n  white-space: nowrap; // as with > li > a\n}\n\n// Dropdown text\n.dropdown-item-text {\n  display: block;\n  padding: var(--#{$prefix}dropdown-item-padding-y) var(--#{$prefix}dropdown-item-padding-x);\n  color: var(--#{$prefix}dropdown-link-color);\n}\n\n// Dark dropdowns\n.dropdown-menu-dark {\n  // scss-docs-start dropdown-dark-css-vars\n  --#{$prefix}dropdown-color: #{$dropdown-dark-color};\n  --#{$prefix}dropdown-bg: #{$dropdown-dark-bg};\n  --#{$prefix}dropdown-border-color: #{$dropdown-dark-border-color};\n  --#{$prefix}dropdown-box-shadow: #{$dropdown-dark-box-shadow};\n  --#{$prefix}dropdown-link-color: #{$dropdown-dark-link-color};\n  --#{$prefix}dropdown-link-hover-color: #{$dropdown-dark-link-hover-color};\n  --#{$prefix}dropdown-divider-bg: #{$dropdown-dark-divider-bg};\n  --#{$prefix}dropdown-link-hover-bg: #{$dropdown-dark-link-hover-bg};\n  --#{$prefix}dropdown-link-active-color: #{$dropdown-dark-link-active-color};\n  --#{$prefix}dropdown-link-active-bg: #{$dropdown-dark-link-active-bg};\n  --#{$prefix}dropdown-link-disabled-color: #{$dropdown-dark-link-disabled-color};\n  --#{$prefix}dropdown-header-color: #{$dropdown-dark-header-color};\n  // scss-docs-end dropdown-dark-css-vars\n}\n","// scss-docs-start caret-mixins\n@mixin caret-down($width: $caret-width) {\n  border-top: $width solid;\n  border-right: $width solid transparent;\n  border-bottom: 0;\n  border-left: $width solid transparent;\n}\n\n@mixin caret-up($width: $caret-width) {\n  border-top: 0;\n  border-right: $width solid transparent;\n  border-bottom: $width solid;\n  border-left: $width solid transparent;\n}\n\n@mixin caret-end($width: $caret-width) {\n  border-top: $width solid transparent;\n  border-right: 0;\n  border-bottom: $width solid transparent;\n  border-left: $width solid;\n}\n\n@mixin caret-start($width: $caret-width) {\n  border-top: $width solid transparent;\n  border-right: $width solid;\n  border-bottom: $width solid transparent;\n}\n\n@mixin caret(\n  $direction: down,\n  $width: $caret-width,\n  $spacing: $caret-spacing,\n  $vertical-align: $caret-vertical-align\n) {\n  @if $enable-caret {\n    &::after {\n      display: inline-block;\n      margin-left: $spacing;\n      vertical-align: $vertical-align;\n      content: \"\";\n      @if $direction == down {\n        @include caret-down($width);\n      } @else if $direction == up {\n        @include caret-up($width);\n      } @else if $direction == end {\n        @include caret-end($width);\n      }\n    }\n\n    @if $direction == start {\n      &::after {\n        display: none;\n      }\n\n      &::before {\n        display: inline-block;\n        margin-right: $spacing;\n        vertical-align: $vertical-align;\n        content: \"\";\n        @include caret-start($width);\n      }\n    }\n\n    &:empty::after {\n      margin-left: 0;\n    }\n  }\n}\n// scss-docs-end caret-mixins\n","// Make the div behave like a button\n.btn-group,\n.btn-group-vertical {\n  position: relative;\n  display: inline-flex;\n  vertical-align: middle; // match .btn alignment given font-size hack above\n\n  > .btn {\n    position: relative;\n    flex: 1 1 auto;\n  }\n\n  // Bring the hover, focused, and \"active\" buttons to the front to overlay\n  // the borders properly\n  > .btn-check:checked + .btn,\n  > .btn-check:focus + .btn,\n  > .btn:hover,\n  > .btn:focus,\n  > .btn:active,\n  > .btn.active {\n    z-index: 1;\n  }\n}\n\n// Optional: Group multiple button groups together for a toolbar\n.btn-toolbar {\n  display: flex;\n  flex-wrap: wrap;\n  justify-content: flex-start;\n\n  .input-group {\n    width: auto;\n  }\n}\n\n.btn-group {\n  @include border-radius($btn-border-radius);\n\n  // Prevent double borders when buttons are next to each other\n  > :not(.btn-check:first-child) + .btn,\n  > .btn-group:not(:first-child) {\n    margin-left: calc(#{$btn-border-width} * -1); // stylelint-disable-line function-disallowed-list\n  }\n\n  // Reset rounded corners\n  > .btn:not(:last-child):not(.dropdown-toggle),\n  > .btn.dropdown-toggle-split:first-child,\n  > .btn-group:not(:last-child) > .btn {\n    @include border-end-radius(0);\n  }\n\n  // The left radius should be 0 if the button is:\n  // - the \"third or more\" child\n  // - the second child and the previous element isn't `.btn-check` (making it the first child visually)\n  // - part of a btn-group which isn't the first child\n  > .btn:nth-child(n + 3),\n  > :not(.btn-check) + .btn,\n  > .btn-group:not(:first-child) > .btn {\n    @include border-start-radius(0);\n  }\n}\n\n// Sizing\n//\n// Remix the default button sizing classes into new ones for easier manipulation.\n\n.btn-group-sm > .btn { @extend .btn-sm; }\n.btn-group-lg > .btn { @extend .btn-lg; }\n\n\n//\n// Split button dropdowns\n//\n\n.dropdown-toggle-split {\n  padding-right: $btn-padding-x * .75;\n  padding-left: $btn-padding-x * .75;\n\n  &::after,\n  .dropup &::after,\n  .dropend &::after {\n    margin-left: 0;\n  }\n\n  .dropstart &::before {\n    margin-right: 0;\n  }\n}\n\n.btn-sm + .dropdown-toggle-split {\n  padding-right: $btn-padding-x-sm * .75;\n  padding-left: $btn-padding-x-sm * .75;\n}\n\n.btn-lg + .dropdown-toggle-split {\n  padding-right: $btn-padding-x-lg * .75;\n  padding-left: $btn-padding-x-lg * .75;\n}\n\n\n// The clickable button for toggling the menu\n// Set the same inset shadow as the :active state\n.btn-group.show .dropdown-toggle {\n  @include box-shadow($btn-active-box-shadow);\n\n  // Show no shadow for `.btn-link` since it has no other button styles.\n  &.btn-link {\n    @include box-shadow(none);\n  }\n}\n\n\n//\n// Vertical button groups\n//\n\n.btn-group-vertical {\n  flex-direction: column;\n  align-items: flex-start;\n  justify-content: center;\n\n  > .btn,\n  > .btn-group {\n    width: 100%;\n  }\n\n  > .btn:not(:first-child),\n  > .btn-group:not(:first-child) {\n    margin-top: calc(#{$btn-border-width} * -1); // stylelint-disable-line function-disallowed-list\n  }\n\n  // Reset rounded corners\n  > .btn:not(:last-child):not(.dropdown-toggle),\n  > .btn-group:not(:last-child) > .btn {\n    @include border-bottom-radius(0);\n  }\n\n  > .btn ~ .btn,\n  > .btn-group:not(:first-child) > .btn {\n    @include border-top-radius(0);\n  }\n}\n","// Base class\n//\n// Kickstart any navigation component with a set of style resets. Works with\n// `<nav>`s, `<ul>`s or `<ol>`s.\n\n.nav {\n  // scss-docs-start nav-css-vars\n  --#{$prefix}nav-link-padding-x: #{$nav-link-padding-x};\n  --#{$prefix}nav-link-padding-y: #{$nav-link-padding-y};\n  @include rfs($nav-link-font-size, --#{$prefix}nav-link-font-size);\n  --#{$prefix}nav-link-font-weight: #{$nav-link-font-weight};\n  --#{$prefix}nav-link-color: #{$nav-link-color};\n  --#{$prefix}nav-link-hover-color: #{$nav-link-hover-color};\n  --#{$prefix}nav-link-disabled-color: #{$nav-link-disabled-color};\n  // scss-docs-end nav-css-vars\n\n  display: flex;\n  flex-wrap: wrap;\n  padding-left: 0;\n  margin-bottom: 0;\n  list-style: none;\n}\n\n.nav-link {\n  display: block;\n  padding: var(--#{$prefix}nav-link-padding-y) var(--#{$prefix}nav-link-padding-x);\n  @include font-size(var(--#{$prefix}nav-link-font-size));\n  font-weight: var(--#{$prefix}nav-link-font-weight);\n  color: var(--#{$prefix}nav-link-color);\n  text-decoration: if($link-decoration == none, null, none);\n  background: none;\n  border: 0;\n  @include transition($nav-link-transition);\n\n  &:hover,\n  &:focus {\n    color: var(--#{$prefix}nav-link-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n  }\n\n  &:focus-visible {\n    outline: 0;\n    box-shadow: $nav-link-focus-box-shadow;\n  }\n\n  // Disabled state lightens text\n  &.disabled,\n  &:disabled {\n    color: var(--#{$prefix}nav-link-disabled-color);\n    pointer-events: none;\n    cursor: default;\n  }\n}\n\n//\n// Tabs\n//\n\n.nav-tabs {\n  // scss-docs-start nav-tabs-css-vars\n  --#{$prefix}nav-tabs-border-width: #{$nav-tabs-border-width};\n  --#{$prefix}nav-tabs-border-color: #{$nav-tabs-border-color};\n  --#{$prefix}nav-tabs-border-radius: #{$nav-tabs-border-radius};\n  --#{$prefix}nav-tabs-link-hover-border-color: #{$nav-tabs-link-hover-border-color};\n  --#{$prefix}nav-tabs-link-active-color: #{$nav-tabs-link-active-color};\n  --#{$prefix}nav-tabs-link-active-bg: #{$nav-tabs-link-active-bg};\n  --#{$prefix}nav-tabs-link-active-border-color: #{$nav-tabs-link-active-border-color};\n  // scss-docs-end nav-tabs-css-vars\n\n  border-bottom: var(--#{$prefix}nav-tabs-border-width) solid var(--#{$prefix}nav-tabs-border-color);\n\n  .nav-link {\n    margin-bottom: calc(-1 * var(--#{$prefix}nav-tabs-border-width)); // stylelint-disable-line function-disallowed-list\n    border: var(--#{$prefix}nav-tabs-border-width) solid transparent;\n    @include border-top-radius(var(--#{$prefix}nav-tabs-border-radius));\n\n    &:hover,\n    &:focus {\n      // Prevents active .nav-link tab overlapping focus outline of previous/next .nav-link\n      isolation: isolate;\n      border-color: var(--#{$prefix}nav-tabs-link-hover-border-color);\n    }\n  }\n\n  .nav-link.active,\n  .nav-item.show .nav-link {\n    color: var(--#{$prefix}nav-tabs-link-active-color);\n    background-color: var(--#{$prefix}nav-tabs-link-active-bg);\n    border-color: var(--#{$prefix}nav-tabs-link-active-border-color);\n  }\n\n  .dropdown-menu {\n    // Make dropdown border overlap tab border\n    margin-top: calc(-1 * var(--#{$prefix}nav-tabs-border-width)); // stylelint-disable-line function-disallowed-list\n    // Remove the top rounded corners here since there is a hard edge above the menu\n    @include border-top-radius(0);\n  }\n}\n\n\n//\n// Pills\n//\n\n.nav-pills {\n  // scss-docs-start nav-pills-css-vars\n  --#{$prefix}nav-pills-border-radius: #{$nav-pills-border-radius};\n  --#{$prefix}nav-pills-link-active-color: #{$nav-pills-link-active-color};\n  --#{$prefix}nav-pills-link-active-bg: #{$nav-pills-link-active-bg};\n  // scss-docs-end nav-pills-css-vars\n\n  .nav-link {\n    @include border-radius(var(--#{$prefix}nav-pills-border-radius));\n  }\n\n  .nav-link.active,\n  .show > .nav-link {\n    color: var(--#{$prefix}nav-pills-link-active-color);\n    @include gradient-bg(var(--#{$prefix}nav-pills-link-active-bg));\n  }\n}\n\n\n//\n// Underline\n//\n\n.nav-underline {\n  // scss-docs-start nav-underline-css-vars\n  --#{$prefix}nav-underline-gap: #{$nav-underline-gap};\n  --#{$prefix}nav-underline-border-width: #{$nav-underline-border-width};\n  --#{$prefix}nav-underline-link-active-color: #{$nav-underline-link-active-color};\n  // scss-docs-end nav-underline-css-vars\n\n  gap: var(--#{$prefix}nav-underline-gap);\n\n  .nav-link {\n    padding-right: 0;\n    padding-left: 0;\n    border-bottom: var(--#{$prefix}nav-underline-border-width) solid transparent;\n\n    &:hover,\n    &:focus {\n      border-bottom-color: currentcolor;\n    }\n  }\n\n  .nav-link.active,\n  .show > .nav-link {\n    font-weight: $font-weight-bold;\n    color: var(--#{$prefix}nav-underline-link-active-color);\n    border-bottom-color: currentcolor;\n  }\n}\n\n\n//\n// Justified variants\n//\n\n.nav-fill {\n  > .nav-link,\n  .nav-item {\n    flex: 1 1 auto;\n    text-align: center;\n  }\n}\n\n.nav-justified {\n  > .nav-link,\n  .nav-item {\n    flex-basis: 0;\n    flex-grow: 1;\n    text-align: center;\n  }\n}\n\n.nav-fill,\n.nav-justified {\n  .nav-item .nav-link {\n    width: 100%; // Make sure button will grow\n  }\n}\n\n\n// Tabbable tabs\n//\n// Hide tabbable panes to start, show them when `.active`\n\n.tab-content {\n  > .tab-pane {\n    display: none;\n  }\n  > .active {\n    display: block;\n  }\n}\n","// Navbar\n//\n// Provide a static navbar from which we expand to create full-width, fixed, and\n// other navbar variations.\n\n.navbar {\n  // scss-docs-start navbar-css-vars\n  --#{$prefix}navbar-padding-x: #{if($navbar-padding-x == null, 0, $navbar-padding-x)};\n  --#{$prefix}navbar-padding-y: #{$navbar-padding-y};\n  --#{$prefix}navbar-color: #{$navbar-light-color};\n  --#{$prefix}navbar-hover-color: #{$navbar-light-hover-color};\n  --#{$prefix}navbar-disabled-color: #{$navbar-light-disabled-color};\n  --#{$prefix}navbar-active-color: #{$navbar-light-active-color};\n  --#{$prefix}navbar-brand-padding-y: #{$navbar-brand-padding-y};\n  --#{$prefix}navbar-brand-margin-end: #{$navbar-brand-margin-end};\n  --#{$prefix}navbar-brand-font-size: #{$navbar-brand-font-size};\n  --#{$prefix}navbar-brand-color: #{$navbar-light-brand-color};\n  --#{$prefix}navbar-brand-hover-color: #{$navbar-light-brand-hover-color};\n  --#{$prefix}navbar-nav-link-padding-x: #{$navbar-nav-link-padding-x};\n  --#{$prefix}navbar-toggler-padding-y: #{$navbar-toggler-padding-y};\n  --#{$prefix}navbar-toggler-padding-x: #{$navbar-toggler-padding-x};\n  --#{$prefix}navbar-toggler-font-size: #{$navbar-toggler-font-size};\n  --#{$prefix}navbar-toggler-icon-bg: #{escape-svg($navbar-light-toggler-icon-bg)};\n  --#{$prefix}navbar-toggler-border-color: #{$navbar-light-toggler-border-color};\n  --#{$prefix}navbar-toggler-border-radius: #{$navbar-toggler-border-radius};\n  --#{$prefix}navbar-toggler-focus-width: #{$navbar-toggler-focus-width};\n  --#{$prefix}navbar-toggler-transition: #{$navbar-toggler-transition};\n  // scss-docs-end navbar-css-vars\n\n  position: relative;\n  display: flex;\n  flex-wrap: wrap; // allow us to do the line break for collapsing content\n  align-items: center;\n  justify-content: space-between; // space out brand from logo\n  padding: var(--#{$prefix}navbar-padding-y) var(--#{$prefix}navbar-padding-x);\n  @include gradient-bg();\n\n  // Because flex properties aren't inherited, we need to redeclare these first\n  // few properties so that content nested within behave properly.\n  // The `flex-wrap` property is inherited to simplify the expanded navbars\n  %container-flex-properties {\n    display: flex;\n    flex-wrap: inherit;\n    align-items: center;\n    justify-content: space-between;\n  }\n\n  > .container,\n  > .container-fluid {\n    @extend %container-flex-properties;\n  }\n\n  @each $breakpoint, $container-max-width in $container-max-widths {\n    > .container#{breakpoint-infix($breakpoint, $container-max-widths)} {\n      @extend %container-flex-properties;\n    }\n  }\n}\n\n\n// Navbar brand\n//\n// Used for brand, project, or site names.\n\n.navbar-brand {\n  padding-top: var(--#{$prefix}navbar-brand-padding-y);\n  padding-bottom: var(--#{$prefix}navbar-brand-padding-y);\n  margin-right: var(--#{$prefix}navbar-brand-margin-end);\n  @include font-size(var(--#{$prefix}navbar-brand-font-size));\n  color: var(--#{$prefix}navbar-brand-color);\n  text-decoration: if($link-decoration == none, null, none);\n  white-space: nowrap;\n\n  &:hover,\n  &:focus {\n    color: var(--#{$prefix}navbar-brand-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n  }\n}\n\n\n// Navbar nav\n//\n// Custom navbar navigation (doesn't require `.nav`, but does make use of `.nav-link`).\n\n.navbar-nav {\n  // scss-docs-start navbar-nav-css-vars\n  --#{$prefix}nav-link-padding-x: 0;\n  --#{$prefix}nav-link-padding-y: #{$nav-link-padding-y};\n  @include rfs($nav-link-font-size, --#{$prefix}nav-link-font-size);\n  --#{$prefix}nav-link-font-weight: #{$nav-link-font-weight};\n  --#{$prefix}nav-link-color: var(--#{$prefix}navbar-color);\n  --#{$prefix}nav-link-hover-color: var(--#{$prefix}navbar-hover-color);\n  --#{$prefix}nav-link-disabled-color: var(--#{$prefix}navbar-disabled-color);\n  // scss-docs-end navbar-nav-css-vars\n\n  display: flex;\n  flex-direction: column; // cannot use `inherit` to get the `.navbar`s value\n  padding-left: 0;\n  margin-bottom: 0;\n  list-style: none;\n\n  .nav-link {\n    &.active,\n    &.show {\n      color: var(--#{$prefix}navbar-active-color);\n    }\n  }\n\n  .dropdown-menu {\n    position: static;\n  }\n}\n\n\n// Navbar text\n//\n//\n\n.navbar-text {\n  padding-top: $nav-link-padding-y;\n  padding-bottom: $nav-link-padding-y;\n  color: var(--#{$prefix}navbar-color);\n\n  a,\n  a:hover,\n  a:focus  {\n    color: var(--#{$prefix}navbar-active-color);\n  }\n}\n\n\n// Responsive navbar\n//\n// Custom styles for responsive collapsing and toggling of navbar contents.\n// Powered by the collapse Bootstrap JavaScript plugin.\n\n// When collapsed, prevent the toggleable navbar contents from appearing in\n// the default flexbox row orientation. Requires the use of `flex-wrap: wrap`\n// on the `.navbar` parent.\n.navbar-collapse {\n  flex-basis: 100%;\n  flex-grow: 1;\n  // For always expanded or extra full navbars, ensure content aligns itself\n  // properly vertically. Can be easily overridden with flex utilities.\n  align-items: center;\n}\n\n// Button for toggling the navbar when in its collapsed state\n.navbar-toggler {\n  padding: var(--#{$prefix}navbar-toggler-padding-y) var(--#{$prefix}navbar-toggler-padding-x);\n  @include font-size(var(--#{$prefix}navbar-toggler-font-size));\n  line-height: 1;\n  color: var(--#{$prefix}navbar-color);\n  background-color: transparent; // remove default button style\n  border: var(--#{$prefix}border-width) solid var(--#{$prefix}navbar-toggler-border-color); // remove default button style\n  @include border-radius(var(--#{$prefix}navbar-toggler-border-radius));\n  @include transition(var(--#{$prefix}navbar-toggler-transition));\n\n  &:hover {\n    text-decoration: none;\n  }\n\n  &:focus {\n    text-decoration: none;\n    outline: 0;\n    box-shadow: 0 0 0 var(--#{$prefix}navbar-toggler-focus-width);\n  }\n}\n\n// Keep as a separate element so folks can easily override it with another icon\n// or image file as needed.\n.navbar-toggler-icon {\n  display: inline-block;\n  width: 1.5em;\n  height: 1.5em;\n  vertical-align: middle;\n  background-image: var(--#{$prefix}navbar-toggler-icon-bg);\n  background-repeat: no-repeat;\n  background-position: center;\n  background-size: 100%;\n}\n\n.navbar-nav-scroll {\n  max-height: var(--#{$prefix}scroll-height, 75vh);\n  overflow-y: auto;\n}\n\n// scss-docs-start navbar-expand-loop\n// Generate series of `.navbar-expand-*` responsive classes for configuring\n// where your navbar collapses.\n.navbar-expand {\n  @each $breakpoint in map-keys($grid-breakpoints) {\n    $next: breakpoint-next($breakpoint, $grid-breakpoints);\n    $infix: breakpoint-infix($next, $grid-breakpoints);\n\n    // stylelint-disable-next-line scss/selector-no-union-class-name\n    &#{$infix} {\n      @include media-breakpoint-up($next) {\n        flex-wrap: nowrap;\n        justify-content: flex-start;\n\n        .navbar-nav {\n          flex-direction: row;\n\n          .dropdown-menu {\n            position: absolute;\n          }\n\n          .nav-link {\n            padding-right: var(--#{$prefix}navbar-nav-link-padding-x);\n            padding-left: var(--#{$prefix}navbar-nav-link-padding-x);\n          }\n        }\n\n        .navbar-nav-scroll {\n          overflow: visible;\n        }\n\n        .navbar-collapse {\n          display: flex !important; // stylelint-disable-line declaration-no-important\n          flex-basis: auto;\n        }\n\n        .navbar-toggler {\n          display: none;\n        }\n\n        .offcanvas {\n          // stylelint-disable declaration-no-important\n          position: static;\n          z-index: auto;\n          flex-grow: 1;\n          width: auto !important;\n          height: auto !important;\n          visibility: visible !important;\n          background-color: transparent !important;\n          border: 0 !important;\n          transform: none !important;\n          @include box-shadow(none);\n          @include transition(none);\n          // stylelint-enable declaration-no-important\n\n          .offcanvas-header {\n            display: none;\n          }\n\n          .offcanvas-body {\n            display: flex;\n            flex-grow: 0;\n            padding: 0;\n            overflow-y: visible;\n          }\n        }\n      }\n    }\n  }\n}\n// scss-docs-end navbar-expand-loop\n\n// Navbar themes\n//\n// Styles for switching between navbars with light or dark background.\n\n.navbar-light {\n  @include deprecate(\"`.navbar-light`\", \"v5.2.0\", \"v6.0.0\", true);\n}\n\n.navbar-dark,\n.navbar[data-bs-theme=\"dark\"] {\n  // scss-docs-start navbar-dark-css-vars\n  --#{$prefix}navbar-color: #{$navbar-dark-color};\n  --#{$prefix}navbar-hover-color: #{$navbar-dark-hover-color};\n  --#{$prefix}navbar-disabled-color: #{$navbar-dark-disabled-color};\n  --#{$prefix}navbar-active-color: #{$navbar-dark-active-color};\n  --#{$prefix}navbar-brand-color: #{$navbar-dark-brand-color};\n  --#{$prefix}navbar-brand-hover-color: #{$navbar-dark-brand-hover-color};\n  --#{$prefix}navbar-toggler-border-color: #{$navbar-dark-toggler-border-color};\n  --#{$prefix}navbar-toggler-icon-bg: #{escape-svg($navbar-dark-toggler-icon-bg)};\n  // scss-docs-end navbar-dark-css-vars\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .navbar-toggler-icon {\n      --#{$prefix}navbar-toggler-icon-bg: #{escape-svg($navbar-dark-toggler-icon-bg)};\n    }\n  }\n}\n","//\n// Base styles\n//\n\n.card {\n  // scss-docs-start card-css-vars\n  --#{$prefix}card-spacer-y: #{$card-spacer-y};\n  --#{$prefix}card-spacer-x: #{$card-spacer-x};\n  --#{$prefix}card-title-spacer-y: #{$card-title-spacer-y};\n  --#{$prefix}card-title-color: #{$card-title-color};\n  --#{$prefix}card-subtitle-color: #{$card-subtitle-color};\n  --#{$prefix}card-border-width: #{$card-border-width};\n  --#{$prefix}card-border-color: #{$card-border-color};\n  --#{$prefix}card-border-radius: #{$card-border-radius};\n  --#{$prefix}card-box-shadow: #{$card-box-shadow};\n  --#{$prefix}card-inner-border-radius: #{$card-inner-border-radius};\n  --#{$prefix}card-cap-padding-y: #{$card-cap-padding-y};\n  --#{$prefix}card-cap-padding-x: #{$card-cap-padding-x};\n  --#{$prefix}card-cap-bg: #{$card-cap-bg};\n  --#{$prefix}card-cap-color: #{$card-cap-color};\n  --#{$prefix}card-height: #{$card-height};\n  --#{$prefix}card-color: #{$card-color};\n  --#{$prefix}card-bg: #{$card-bg};\n  --#{$prefix}card-img-overlay-padding: #{$card-img-overlay-padding};\n  --#{$prefix}card-group-margin: #{$card-group-margin};\n  // scss-docs-end card-css-vars\n\n  position: relative;\n  display: flex;\n  flex-direction: column;\n  min-width: 0; // See https://github.com/twbs/bootstrap/pull/22740#issuecomment-305868106\n  height: var(--#{$prefix}card-height);\n  color: var(--#{$prefix}body-color);\n  word-wrap: break-word;\n  background-color: var(--#{$prefix}card-bg);\n  background-clip: border-box;\n  border: var(--#{$prefix}card-border-width) solid var(--#{$prefix}card-border-color);\n  @include border-radius(var(--#{$prefix}card-border-radius));\n  @include box-shadow(var(--#{$prefix}card-box-shadow));\n\n  > hr {\n    margin-right: 0;\n    margin-left: 0;\n  }\n\n  > .list-group {\n    border-top: inherit;\n    border-bottom: inherit;\n\n    &:first-child {\n      border-top-width: 0;\n      @include border-top-radius(var(--#{$prefix}card-inner-border-radius));\n    }\n\n    &:last-child  {\n      border-bottom-width: 0;\n      @include border-bottom-radius(var(--#{$prefix}card-inner-border-radius));\n    }\n  }\n\n  // Due to specificity of the above selector (`.card > .list-group`), we must\n  // use a child selector here to prevent double borders.\n  > .card-header + .list-group,\n  > .list-group + .card-footer {\n    border-top: 0;\n  }\n}\n\n.card-body {\n  // Enable `flex-grow: 1` for decks and groups so that card blocks take up\n  // as much space as possible, ensuring footers are aligned to the bottom.\n  flex: 1 1 auto;\n  padding: var(--#{$prefix}card-spacer-y) var(--#{$prefix}card-spacer-x);\n  color: var(--#{$prefix}card-color);\n}\n\n.card-title {\n  margin-bottom: var(--#{$prefix}card-title-spacer-y);\n  color: var(--#{$prefix}card-title-color);\n}\n\n.card-subtitle {\n  margin-top: calc(-.5 * var(--#{$prefix}card-title-spacer-y)); // stylelint-disable-line function-disallowed-list\n  margin-bottom: 0;\n  color: var(--#{$prefix}card-subtitle-color);\n}\n\n.card-text:last-child {\n  margin-bottom: 0;\n}\n\n.card-link {\n  &:hover {\n    text-decoration: if($link-hover-decoration == underline, none, null);\n  }\n\n  + .card-link {\n    margin-left: var(--#{$prefix}card-spacer-x);\n  }\n}\n\n//\n// Optional textual caps\n//\n\n.card-header {\n  padding: var(--#{$prefix}card-cap-padding-y) var(--#{$prefix}card-cap-padding-x);\n  margin-bottom: 0; // Removes the default margin-bottom of <hN>\n  color: var(--#{$prefix}card-cap-color);\n  background-color: var(--#{$prefix}card-cap-bg);\n  border-bottom: var(--#{$prefix}card-border-width) solid var(--#{$prefix}card-border-color);\n\n  &:first-child {\n    @include border-radius(var(--#{$prefix}card-inner-border-radius) var(--#{$prefix}card-inner-border-radius) 0 0);\n  }\n}\n\n.card-footer {\n  padding: var(--#{$prefix}card-cap-padding-y) var(--#{$prefix}card-cap-padding-x);\n  color: var(--#{$prefix}card-cap-color);\n  background-color: var(--#{$prefix}card-cap-bg);\n  border-top: var(--#{$prefix}card-border-width) solid var(--#{$prefix}card-border-color);\n\n  &:last-child {\n    @include border-radius(0 0 var(--#{$prefix}card-inner-border-radius) var(--#{$prefix}card-inner-border-radius));\n  }\n}\n\n\n//\n// Header navs\n//\n\n.card-header-tabs {\n  margin-right: calc(-.5 * var(--#{$prefix}card-cap-padding-x)); // stylelint-disable-line function-disallowed-list\n  margin-bottom: calc(-1 * var(--#{$prefix}card-cap-padding-y)); // stylelint-disable-line function-disallowed-list\n  margin-left: calc(-.5 * var(--#{$prefix}card-cap-padding-x)); // stylelint-disable-line function-disallowed-list\n  border-bottom: 0;\n\n  .nav-link.active {\n    background-color: var(--#{$prefix}card-bg);\n    border-bottom-color: var(--#{$prefix}card-bg);\n  }\n}\n\n.card-header-pills {\n  margin-right: calc(-.5 * var(--#{$prefix}card-cap-padding-x)); // stylelint-disable-line function-disallowed-list\n  margin-left: calc(-.5 * var(--#{$prefix}card-cap-padding-x)); // stylelint-disable-line function-disallowed-list\n}\n\n// Card image\n.card-img-overlay {\n  position: absolute;\n  top: 0;\n  right: 0;\n  bottom: 0;\n  left: 0;\n  padding: var(--#{$prefix}card-img-overlay-padding);\n  @include border-radius(var(--#{$prefix}card-inner-border-radius));\n}\n\n.card-img,\n.card-img-top,\n.card-img-bottom {\n  width: 100%; // Required because we use flexbox and this inherently applies align-self: stretch\n}\n\n.card-img,\n.card-img-top {\n  @include border-top-radius(var(--#{$prefix}card-inner-border-radius));\n}\n\n.card-img,\n.card-img-bottom {\n  @include border-bottom-radius(var(--#{$prefix}card-inner-border-radius));\n}\n\n\n//\n// Card groups\n//\n\n.card-group {\n  // The child selector allows nested `.card` within `.card-group`\n  // to display properly.\n  > .card {\n    margin-bottom: var(--#{$prefix}card-group-margin);\n  }\n\n  @include media-breakpoint-up(sm) {\n    display: flex;\n    flex-flow: row wrap;\n    // The child selector allows nested `.card` within `.card-group`\n    // to display properly.\n    > .card {\n      // Flexbugs #4: https://github.com/philipwalton/flexbugs#flexbug-4\n      flex: 1 0 0%;\n      margin-bottom: 0;\n\n      + .card {\n        margin-left: 0;\n        border-left: 0;\n      }\n\n      // Handle rounded corners\n      @if $enable-rounded {\n        &:not(:last-child) {\n          @include border-end-radius(0);\n\n          .card-img-top,\n          .card-header {\n            // stylelint-disable-next-line property-disallowed-list\n            border-top-right-radius: 0;\n          }\n          .card-img-bottom,\n          .card-footer {\n            // stylelint-disable-next-line property-disallowed-list\n            border-bottom-right-radius: 0;\n          }\n        }\n\n        &:not(:first-child) {\n          @include border-start-radius(0);\n\n          .card-img-top,\n          .card-header {\n            // stylelint-disable-next-line property-disallowed-list\n            border-top-left-radius: 0;\n          }\n          .card-img-bottom,\n          .card-footer {\n            // stylelint-disable-next-line property-disallowed-list\n            border-bottom-left-radius: 0;\n          }\n        }\n      }\n    }\n  }\n}\n","//\n// Base styles\n//\n\n.accordion {\n  // scss-docs-start accordion-css-vars\n  --#{$prefix}accordion-color: #{$accordion-color};\n  --#{$prefix}accordion-bg: #{$accordion-bg};\n  --#{$prefix}accordion-transition: #{$accordion-transition};\n  --#{$prefix}accordion-border-color: #{$accordion-border-color};\n  --#{$prefix}accordion-border-width: #{$accordion-border-width};\n  --#{$prefix}accordion-border-radius: #{$accordion-border-radius};\n  --#{$prefix}accordion-inner-border-radius: #{$accordion-inner-border-radius};\n  --#{$prefix}accordion-btn-padding-x: #{$accordion-button-padding-x};\n  --#{$prefix}accordion-btn-padding-y: #{$accordion-button-padding-y};\n  --#{$prefix}accordion-btn-color: #{$accordion-button-color};\n  --#{$prefix}accordion-btn-bg: #{$accordion-button-bg};\n  --#{$prefix}accordion-btn-icon: #{escape-svg($accordion-button-icon)};\n  --#{$prefix}accordion-btn-icon-width: #{$accordion-icon-width};\n  --#{$prefix}accordion-btn-icon-transform: #{$accordion-icon-transform};\n  --#{$prefix}accordion-btn-icon-transition: #{$accordion-icon-transition};\n  --#{$prefix}accordion-btn-active-icon: #{escape-svg($accordion-button-active-icon)};\n  --#{$prefix}accordion-btn-focus-box-shadow: #{$accordion-button-focus-box-shadow};\n  --#{$prefix}accordion-body-padding-x: #{$accordion-body-padding-x};\n  --#{$prefix}accordion-body-padding-y: #{$accordion-body-padding-y};\n  --#{$prefix}accordion-active-color: #{$accordion-button-active-color};\n  --#{$prefix}accordion-active-bg: #{$accordion-button-active-bg};\n  // scss-docs-end accordion-css-vars\n}\n\n.accordion-button {\n  position: relative;\n  display: flex;\n  align-items: center;\n  width: 100%;\n  padding: var(--#{$prefix}accordion-btn-padding-y) var(--#{$prefix}accordion-btn-padding-x);\n  @include font-size($font-size-base);\n  color: var(--#{$prefix}accordion-btn-color);\n  text-align: left; // Reset button style\n  background-color: var(--#{$prefix}accordion-btn-bg);\n  border: 0;\n  @include border-radius(0);\n  overflow-anchor: none;\n  @include transition(var(--#{$prefix}accordion-transition));\n\n  &:not(.collapsed) {\n    color: var(--#{$prefix}accordion-active-color);\n    background-color: var(--#{$prefix}accordion-active-bg);\n    box-shadow: inset 0 calc(-1 * var(--#{$prefix}accordion-border-width)) 0 var(--#{$prefix}accordion-border-color); // stylelint-disable-line function-disallowed-list\n\n    &::after {\n      background-image: var(--#{$prefix}accordion-btn-active-icon);\n      transform: var(--#{$prefix}accordion-btn-icon-transform);\n    }\n  }\n\n  // Accordion icon\n  &::after {\n    flex-shrink: 0;\n    width: var(--#{$prefix}accordion-btn-icon-width);\n    height: var(--#{$prefix}accordion-btn-icon-width);\n    margin-left: auto;\n    content: \"\";\n    background-image: var(--#{$prefix}accordion-btn-icon);\n    background-repeat: no-repeat;\n    background-size: var(--#{$prefix}accordion-btn-icon-width);\n    @include transition(var(--#{$prefix}accordion-btn-icon-transition));\n  }\n\n  &:hover {\n    z-index: 2;\n  }\n\n  &:focus {\n    z-index: 3;\n    outline: 0;\n    box-shadow: var(--#{$prefix}accordion-btn-focus-box-shadow);\n  }\n}\n\n.accordion-header {\n  margin-bottom: 0;\n}\n\n.accordion-item {\n  color: var(--#{$prefix}accordion-color);\n  background-color: var(--#{$prefix}accordion-bg);\n  border: var(--#{$prefix}accordion-border-width) solid var(--#{$prefix}accordion-border-color);\n\n  &:first-of-type {\n    @include border-top-radius(var(--#{$prefix}accordion-border-radius));\n\n    > .accordion-header .accordion-button {\n      @include border-top-radius(var(--#{$prefix}accordion-inner-border-radius));\n    }\n  }\n\n  &:not(:first-of-type) {\n    border-top: 0;\n  }\n\n  // Only set a border-radius on the last item if the accordion is collapsed\n  &:last-of-type {\n    @include border-bottom-radius(var(--#{$prefix}accordion-border-radius));\n\n    > .accordion-header .accordion-button {\n      &.collapsed {\n        @include border-bottom-radius(var(--#{$prefix}accordion-inner-border-radius));\n      }\n    }\n\n    > .accordion-collapse {\n      @include border-bottom-radius(var(--#{$prefix}accordion-border-radius));\n    }\n  }\n}\n\n.accordion-body {\n  padding: var(--#{$prefix}accordion-body-padding-y) var(--#{$prefix}accordion-body-padding-x);\n}\n\n\n// Flush accordion items\n//\n// Remove borders and border-radius to keep accordion items edge-to-edge.\n\n.accordion-flush {\n  > .accordion-item {\n    border-right: 0;\n    border-left: 0;\n    @include border-radius(0);\n\n    &:first-child { border-top: 0; }\n    &:last-child { border-bottom: 0; }\n\n    // stylelint-disable selector-max-class\n    > .accordion-header .accordion-button {\n      &,\n      &.collapsed {\n        @include border-radius(0);\n      }\n    }\n    // stylelint-enable selector-max-class\n\n    > .accordion-collapse {\n      @include border-radius(0);\n    }\n  }\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .accordion-button::after {\n      --#{$prefix}accordion-btn-icon: #{escape-svg($accordion-button-icon-dark)};\n      --#{$prefix}accordion-btn-active-icon: #{escape-svg($accordion-button-active-icon-dark)};\n    }\n  }\n}\n",".breadcrumb {\n  // scss-docs-start breadcrumb-css-vars\n  --#{$prefix}breadcrumb-padding-x: #{$breadcrumb-padding-x};\n  --#{$prefix}breadcrumb-padding-y: #{$breadcrumb-padding-y};\n  --#{$prefix}breadcrumb-margin-bottom: #{$breadcrumb-margin-bottom};\n  @include rfs($breadcrumb-font-size, --#{$prefix}breadcrumb-font-size);\n  --#{$prefix}breadcrumb-bg: #{$breadcrumb-bg};\n  --#{$prefix}breadcrumb-border-radius: #{$breadcrumb-border-radius};\n  --#{$prefix}breadcrumb-divider-color: #{$breadcrumb-divider-color};\n  --#{$prefix}breadcrumb-item-padding-x: #{$breadcrumb-item-padding-x};\n  --#{$prefix}breadcrumb-item-active-color: #{$breadcrumb-active-color};\n  // scss-docs-end breadcrumb-css-vars\n\n  display: flex;\n  flex-wrap: wrap;\n  padding: var(--#{$prefix}breadcrumb-padding-y) var(--#{$prefix}breadcrumb-padding-x);\n  margin-bottom: var(--#{$prefix}breadcrumb-margin-bottom);\n  @include font-size(var(--#{$prefix}breadcrumb-font-size));\n  list-style: none;\n  background-color: var(--#{$prefix}breadcrumb-bg);\n  @include border-radius(var(--#{$prefix}breadcrumb-border-radius));\n}\n\n.breadcrumb-item {\n  // The separator between breadcrumbs (by default, a forward-slash: \"/\")\n  + .breadcrumb-item {\n    padding-left: var(--#{$prefix}breadcrumb-item-padding-x);\n\n    &::before {\n      float: left; // Suppress inline spacings and underlining of the separator\n      padding-right: var(--#{$prefix}breadcrumb-item-padding-x);\n      color: var(--#{$prefix}breadcrumb-divider-color);\n      content: var(--#{$prefix}breadcrumb-divider, escape-svg($breadcrumb-divider)) #{\"/* rtl:\"} var(--#{$prefix}breadcrumb-divider, escape-svg($breadcrumb-divider-flipped)) #{\"*/\"};\n    }\n  }\n\n  &.active {\n    color: var(--#{$prefix}breadcrumb-item-active-color);\n  }\n}\n",".pagination {\n  // scss-docs-start pagination-css-vars\n  --#{$prefix}pagination-padding-x: #{$pagination-padding-x};\n  --#{$prefix}pagination-padding-y: #{$pagination-padding-y};\n  @include rfs($pagination-font-size, --#{$prefix}pagination-font-size);\n  --#{$prefix}pagination-color: #{$pagination-color};\n  --#{$prefix}pagination-bg: #{$pagination-bg};\n  --#{$prefix}pagination-border-width: #{$pagination-border-width};\n  --#{$prefix}pagination-border-color: #{$pagination-border-color};\n  --#{$prefix}pagination-border-radius: #{$pagination-border-radius};\n  --#{$prefix}pagination-hover-color: #{$pagination-hover-color};\n  --#{$prefix}pagination-hover-bg: #{$pagination-hover-bg};\n  --#{$prefix}pagination-hover-border-color: #{$pagination-hover-border-color};\n  --#{$prefix}pagination-focus-color: #{$pagination-focus-color};\n  --#{$prefix}pagination-focus-bg: #{$pagination-focus-bg};\n  --#{$prefix}pagination-focus-box-shadow: #{$pagination-focus-box-shadow};\n  --#{$prefix}pagination-active-color: #{$pagination-active-color};\n  --#{$prefix}pagination-active-bg: #{$pagination-active-bg};\n  --#{$prefix}pagination-active-border-color: #{$pagination-active-border-color};\n  --#{$prefix}pagination-disabled-color: #{$pagination-disabled-color};\n  --#{$prefix}pagination-disabled-bg: #{$pagination-disabled-bg};\n  --#{$prefix}pagination-disabled-border-color: #{$pagination-disabled-border-color};\n  // scss-docs-end pagination-css-vars\n\n  display: flex;\n  @include list-unstyled();\n}\n\n.page-link {\n  position: relative;\n  display: block;\n  padding: var(--#{$prefix}pagination-padding-y) var(--#{$prefix}pagination-padding-x);\n  @include font-size(var(--#{$prefix}pagination-font-size));\n  color: var(--#{$prefix}pagination-color);\n  text-decoration: if($link-decoration == none, null, none);\n  background-color: var(--#{$prefix}pagination-bg);\n  border: var(--#{$prefix}pagination-border-width) solid var(--#{$prefix}pagination-border-color);\n  @include transition($pagination-transition);\n\n  &:hover {\n    z-index: 2;\n    color: var(--#{$prefix}pagination-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n    background-color: var(--#{$prefix}pagination-hover-bg);\n    border-color: var(--#{$prefix}pagination-hover-border-color);\n  }\n\n  &:focus {\n    z-index: 3;\n    color: var(--#{$prefix}pagination-focus-color);\n    background-color: var(--#{$prefix}pagination-focus-bg);\n    outline: $pagination-focus-outline;\n    box-shadow: var(--#{$prefix}pagination-focus-box-shadow);\n  }\n\n  &.active,\n  .active > & {\n    z-index: 3;\n    color: var(--#{$prefix}pagination-active-color);\n    @include gradient-bg(var(--#{$prefix}pagination-active-bg));\n    border-color: var(--#{$prefix}pagination-active-border-color);\n  }\n\n  &.disabled,\n  .disabled > & {\n    color: var(--#{$prefix}pagination-disabled-color);\n    pointer-events: none;\n    background-color: var(--#{$prefix}pagination-disabled-bg);\n    border-color: var(--#{$prefix}pagination-disabled-border-color);\n  }\n}\n\n.page-item {\n  &:not(:first-child) .page-link {\n    margin-left: $pagination-margin-start;\n  }\n\n  @if $pagination-margin-start == calc(#{$pagination-border-width} * -1) {\n    &:first-child {\n      .page-link {\n        @include border-start-radius(var(--#{$prefix}pagination-border-radius));\n      }\n    }\n\n    &:last-child {\n      .page-link {\n        @include border-end-radius(var(--#{$prefix}pagination-border-radius));\n      }\n    }\n  } @else {\n    // Add border-radius to all pageLinks in case they have left margin\n    .page-link {\n      @include border-radius(var(--#{$prefix}pagination-border-radius));\n    }\n  }\n}\n\n\n//\n// Sizing\n//\n\n.pagination-lg {\n  @include pagination-size($pagination-padding-y-lg, $pagination-padding-x-lg, $font-size-lg, $pagination-border-radius-lg);\n}\n\n.pagination-sm {\n  @include pagination-size($pagination-padding-y-sm, $pagination-padding-x-sm, $font-size-sm, $pagination-border-radius-sm);\n}\n","// Pagination\n\n// scss-docs-start pagination-mixin\n@mixin pagination-size($padding-y, $padding-x, $font-size, $border-radius) {\n  --#{$prefix}pagination-padding-x: #{$padding-x};\n  --#{$prefix}pagination-padding-y: #{$padding-y};\n  @include rfs($font-size, --#{$prefix}pagination-font-size);\n  --#{$prefix}pagination-border-radius: #{$border-radius};\n}\n// scss-docs-end pagination-mixin\n","// Base class\n//\n// Requires one of the contextual, color modifier classes for `color` and\n// `background-color`.\n\n.badge {\n  // scss-docs-start badge-css-vars\n  --#{$prefix}badge-padding-x: #{$badge-padding-x};\n  --#{$prefix}badge-padding-y: #{$badge-padding-y};\n  @include rfs($badge-font-size, --#{$prefix}badge-font-size);\n  --#{$prefix}badge-font-weight: #{$badge-font-weight};\n  --#{$prefix}badge-color: #{$badge-color};\n  --#{$prefix}badge-border-radius: #{$badge-border-radius};\n  // scss-docs-end badge-css-vars\n\n  display: inline-block;\n  padding: var(--#{$prefix}badge-padding-y) var(--#{$prefix}badge-padding-x);\n  @include font-size(var(--#{$prefix}badge-font-size));\n  font-weight: var(--#{$prefix}badge-font-weight);\n  line-height: 1;\n  color: var(--#{$prefix}badge-color);\n  text-align: center;\n  white-space: nowrap;\n  vertical-align: baseline;\n  @include border-radius(var(--#{$prefix}badge-border-radius));\n  @include gradient-bg();\n\n  // Empty badges collapse automatically\n  &:empty {\n    display: none;\n  }\n}\n\n// Quick fix for badges in buttons\n.btn .badge {\n  position: relative;\n  top: -1px;\n}\n","//\n// Base styles\n//\n\n.alert {\n  // scss-docs-start alert-css-vars\n  --#{$prefix}alert-bg: transparent;\n  --#{$prefix}alert-padding-x: #{$alert-padding-x};\n  --#{$prefix}alert-padding-y: #{$alert-padding-y};\n  --#{$prefix}alert-margin-bottom: #{$alert-margin-bottom};\n  --#{$prefix}alert-color: inherit;\n  --#{$prefix}alert-border-color: transparent;\n  --#{$prefix}alert-border: #{$alert-border-width} solid var(--#{$prefix}alert-border-color);\n  --#{$prefix}alert-border-radius: #{$alert-border-radius};\n  --#{$prefix}alert-link-color: inherit;\n  // scss-docs-end alert-css-vars\n\n  position: relative;\n  padding: var(--#{$prefix}alert-padding-y) var(--#{$prefix}alert-padding-x);\n  margin-bottom: var(--#{$prefix}alert-margin-bottom);\n  color: var(--#{$prefix}alert-color);\n  background-color: var(--#{$prefix}alert-bg);\n  border: var(--#{$prefix}alert-border);\n  @include border-radius(var(--#{$prefix}alert-border-radius));\n}\n\n// Headings for larger alerts\n.alert-heading {\n  // Specified to prevent conflicts of changing $headings-color\n  color: inherit;\n}\n\n// Provide class for links that match alerts\n.alert-link {\n  font-weight: $alert-link-font-weight;\n  color: var(--#{$prefix}alert-link-color);\n}\n\n\n// Dismissible alerts\n//\n// Expand the right padding and account for the close button's positioning.\n\n.alert-dismissible {\n  padding-right: $alert-dismissible-padding-r;\n\n  // Adjust close link position\n  .btn-close {\n    position: absolute;\n    top: 0;\n    right: 0;\n    z-index: $stretched-link-z-index + 1;\n    padding: $alert-padding-y * 1.25 $alert-padding-x;\n  }\n}\n\n\n// scss-docs-start alert-modifiers\n// Generate contextual modifier classes for colorizing the alert\n@each $state in map-keys($theme-colors) {\n  .alert-#{$state} {\n    --#{$prefix}alert-color: var(--#{$prefix}#{$state}-text-emphasis);\n    --#{$prefix}alert-bg: var(--#{$prefix}#{$state}-bg-subtle);\n    --#{$prefix}alert-border-color: var(--#{$prefix}#{$state}-border-subtle);\n    --#{$prefix}alert-link-color: var(--#{$prefix}#{$state}-text-emphasis);\n  }\n}\n// scss-docs-end alert-modifiers\n","// Disable animation if transitions are disabled\n\n// scss-docs-start progress-keyframes\n@if $enable-transitions {\n  @keyframes progress-bar-stripes {\n    0% { background-position-x: $progress-height; }\n  }\n}\n// scss-docs-end progress-keyframes\n\n.progress,\n.progress-stacked {\n  // scss-docs-start progress-css-vars\n  --#{$prefix}progress-height: #{$progress-height};\n  @include rfs($progress-font-size, --#{$prefix}progress-font-size);\n  --#{$prefix}progress-bg: #{$progress-bg};\n  --#{$prefix}progress-border-radius: #{$progress-border-radius};\n  --#{$prefix}progress-box-shadow: #{$progress-box-shadow};\n  --#{$prefix}progress-bar-color: #{$progress-bar-color};\n  --#{$prefix}progress-bar-bg: #{$progress-bar-bg};\n  --#{$prefix}progress-bar-transition: #{$progress-bar-transition};\n  // scss-docs-end progress-css-vars\n\n  display: flex;\n  height: var(--#{$prefix}progress-height);\n  overflow: hidden; // force rounded corners by cropping it\n  @include font-size(var(--#{$prefix}progress-font-size));\n  background-color: var(--#{$prefix}progress-bg);\n  @include border-radius(var(--#{$prefix}progress-border-radius));\n  @include box-shadow(var(--#{$prefix}progress-box-shadow));\n}\n\n.progress-bar {\n  display: flex;\n  flex-direction: column;\n  justify-content: center;\n  overflow: hidden;\n  color: var(--#{$prefix}progress-bar-color);\n  text-align: center;\n  white-space: nowrap;\n  background-color: var(--#{$prefix}progress-bar-bg);\n  @include transition(var(--#{$prefix}progress-bar-transition));\n}\n\n.progress-bar-striped {\n  @include gradient-striped();\n  background-size: var(--#{$prefix}progress-height) var(--#{$prefix}progress-height);\n}\n\n.progress-stacked > .progress {\n  overflow: visible;\n}\n\n.progress-stacked > .progress > .progress-bar {\n  width: 100%;\n}\n\n@if $enable-transitions {\n  .progress-bar-animated {\n    animation: $progress-bar-animation-timing progress-bar-stripes;\n\n    @if $enable-reduced-motion {\n      @media (prefers-reduced-motion: reduce) {\n        animation: none;\n      }\n    }\n  }\n}\n","// Base class\n//\n// Easily usable on <ul>, <ol>, or <div>.\n\n.list-group {\n  // scss-docs-start list-group-css-vars\n  --#{$prefix}list-group-color: #{$list-group-color};\n  --#{$prefix}list-group-bg: #{$list-group-bg};\n  --#{$prefix}list-group-border-color: #{$list-group-border-color};\n  --#{$prefix}list-group-border-width: #{$list-group-border-width};\n  --#{$prefix}list-group-border-radius: #{$list-group-border-radius};\n  --#{$prefix}list-group-item-padding-x: #{$list-group-item-padding-x};\n  --#{$prefix}list-group-item-padding-y: #{$list-group-item-padding-y};\n  --#{$prefix}list-group-action-color: #{$list-group-action-color};\n  --#{$prefix}list-group-action-hover-color: #{$list-group-action-hover-color};\n  --#{$prefix}list-group-action-hover-bg: #{$list-group-hover-bg};\n  --#{$prefix}list-group-action-active-color: #{$list-group-action-active-color};\n  --#{$prefix}list-group-action-active-bg: #{$list-group-action-active-bg};\n  --#{$prefix}list-group-disabled-color: #{$list-group-disabled-color};\n  --#{$prefix}list-group-disabled-bg: #{$list-group-disabled-bg};\n  --#{$prefix}list-group-active-color: #{$list-group-active-color};\n  --#{$prefix}list-group-active-bg: #{$list-group-active-bg};\n  --#{$prefix}list-group-active-border-color: #{$list-group-active-border-color};\n  // scss-docs-end list-group-css-vars\n\n  display: flex;\n  flex-direction: column;\n\n  // No need to set list-style: none; since .list-group-item is block level\n  padding-left: 0; // reset padding because ul and ol\n  margin-bottom: 0;\n  @include border-radius(var(--#{$prefix}list-group-border-radius));\n}\n\n.list-group-numbered {\n  list-style-type: none;\n  counter-reset: section;\n\n  > .list-group-item::before {\n    // Increments only this instance of the section counter\n    content: counters(section, \".\") \". \";\n    counter-increment: section;\n  }\n}\n\n// Interactive list items\n//\n// Use anchor or button elements instead of `li`s or `div`s to create interactive\n// list items. Includes an extra `.active` modifier class for selected items.\n\n.list-group-item-action {\n  width: 100%; // For `<button>`s (anchors become 100% by default though)\n  color: var(--#{$prefix}list-group-action-color);\n  text-align: inherit; // For `<button>`s (anchors inherit)\n\n  // Hover state\n  &:hover,\n  &:focus {\n    z-index: 1; // Place hover/focus items above their siblings for proper border styling\n    color: var(--#{$prefix}list-group-action-hover-color);\n    text-decoration: none;\n    background-color: var(--#{$prefix}list-group-action-hover-bg);\n  }\n\n  &:active {\n    color: var(--#{$prefix}list-group-action-active-color);\n    background-color: var(--#{$prefix}list-group-action-active-bg);\n  }\n}\n\n// Individual list items\n//\n// Use on `li`s or `div`s within the `.list-group` parent.\n\n.list-group-item {\n  position: relative;\n  display: block;\n  padding: var(--#{$prefix}list-group-item-padding-y) var(--#{$prefix}list-group-item-padding-x);\n  color: var(--#{$prefix}list-group-color);\n  text-decoration: if($link-decoration == none, null, none);\n  background-color: var(--#{$prefix}list-group-bg);\n  border: var(--#{$prefix}list-group-border-width) solid var(--#{$prefix}list-group-border-color);\n\n  &:first-child {\n    @include border-top-radius(inherit);\n  }\n\n  &:last-child {\n    @include border-bottom-radius(inherit);\n  }\n\n  &.disabled,\n  &:disabled {\n    color: var(--#{$prefix}list-group-disabled-color);\n    pointer-events: none;\n    background-color: var(--#{$prefix}list-group-disabled-bg);\n  }\n\n  // Include both here for `<a>`s and `<button>`s\n  &.active {\n    z-index: 2; // Place active items above their siblings for proper border styling\n    color: var(--#{$prefix}list-group-active-color);\n    background-color: var(--#{$prefix}list-group-active-bg);\n    border-color: var(--#{$prefix}list-group-active-border-color);\n  }\n\n  // stylelint-disable-next-line scss/selector-no-redundant-nesting-selector\n  & + .list-group-item {\n    border-top-width: 0;\n\n    &.active {\n      margin-top: calc(-1 * var(--#{$prefix}list-group-border-width)); // stylelint-disable-line function-disallowed-list\n      border-top-width: var(--#{$prefix}list-group-border-width);\n    }\n  }\n}\n\n// Horizontal\n//\n// Change the layout of list group items from vertical (default) to horizontal.\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  @include media-breakpoint-up($breakpoint) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    .list-group-horizontal#{$infix} {\n      flex-direction: row;\n\n      > .list-group-item {\n        &:first-child:not(:last-child) {\n          @include border-bottom-start-radius(var(--#{$prefix}list-group-border-radius));\n          @include border-top-end-radius(0);\n        }\n\n        &:last-child:not(:first-child) {\n          @include border-top-end-radius(var(--#{$prefix}list-group-border-radius));\n          @include border-bottom-start-radius(0);\n        }\n\n        &.active {\n          margin-top: 0;\n        }\n\n        + .list-group-item {\n          border-top-width: var(--#{$prefix}list-group-border-width);\n          border-left-width: 0;\n\n          &.active {\n            margin-left: calc(-1 * var(--#{$prefix}list-group-border-width)); // stylelint-disable-line function-disallowed-list\n            border-left-width: var(--#{$prefix}list-group-border-width);\n          }\n        }\n      }\n    }\n  }\n}\n\n\n// Flush list items\n//\n// Remove borders and border-radius to keep list group items edge-to-edge. Most\n// useful within other components (e.g., cards).\n\n.list-group-flush {\n  @include border-radius(0);\n\n  > .list-group-item {\n    border-width: 0 0 var(--#{$prefix}list-group-border-width);\n\n    &:last-child {\n      border-bottom-width: 0;\n    }\n  }\n}\n\n\n// scss-docs-start list-group-modifiers\n// List group contextual variants\n//\n// Add modifier classes to change text and background color on individual items.\n// Organizationally, this must come after the `:hover` states.\n\n@each $state in map-keys($theme-colors) {\n  .list-group-item-#{$state} {\n    --#{$prefix}list-group-color: var(--#{$prefix}#{$state}-text-emphasis);\n    --#{$prefix}list-group-bg: var(--#{$prefix}#{$state}-bg-subtle);\n    --#{$prefix}list-group-border-color: var(--#{$prefix}#{$state}-border-subtle);\n    --#{$prefix}list-group-action-hover-color: var(--#{$prefix}emphasis-color);\n    --#{$prefix}list-group-action-hover-bg: var(--#{$prefix}#{$state}-border-subtle);\n    --#{$prefix}list-group-action-active-color: var(--#{$prefix}emphasis-color);\n    --#{$prefix}list-group-action-active-bg: var(--#{$prefix}#{$state}-border-subtle);\n    --#{$prefix}list-group-active-color: var(--#{$prefix}#{$state}-bg-subtle);\n    --#{$prefix}list-group-active-bg: var(--#{$prefix}#{$state}-text-emphasis);\n    --#{$prefix}list-group-active-border-color: var(--#{$prefix}#{$state}-text-emphasis);\n  }\n}\n// scss-docs-end list-group-modifiers\n","// Transparent background and border properties included for button version.\n// iOS requires the button element instead of an anchor tag.\n// If you want the anchor version, it requires `href=\"#\"`.\n// See https://developer.mozilla.org/en-US/docs/Web/Events/click#Safari_Mobile\n\n.btn-close {\n  // scss-docs-start close-css-vars\n  --#{$prefix}btn-close-color: #{$btn-close-color};\n  --#{$prefix}btn-close-bg: #{ escape-svg($btn-close-bg) };\n  --#{$prefix}btn-close-opacity: #{$btn-close-opacity};\n  --#{$prefix}btn-close-hover-opacity: #{$btn-close-hover-opacity};\n  --#{$prefix}btn-close-focus-shadow: #{$btn-close-focus-shadow};\n  --#{$prefix}btn-close-focus-opacity: #{$btn-close-focus-opacity};\n  --#{$prefix}btn-close-disabled-opacity: #{$btn-close-disabled-opacity};\n  --#{$prefix}btn-close-white-filter: #{$btn-close-white-filter};\n  // scss-docs-end close-css-vars\n\n  box-sizing: content-box;\n  width: $btn-close-width;\n  height: $btn-close-height;\n  padding: $btn-close-padding-y $btn-close-padding-x;\n  color: var(--#{$prefix}btn-close-color);\n  background: transparent var(--#{$prefix}btn-close-bg) center / $btn-close-width auto no-repeat; // include transparent for button elements\n  border: 0; // for button elements\n  @include border-radius();\n  opacity: var(--#{$prefix}btn-close-opacity);\n\n  // Override <a>'s hover style\n  &:hover {\n    color: var(--#{$prefix}btn-close-color);\n    text-decoration: none;\n    opacity: var(--#{$prefix}btn-close-hover-opacity);\n  }\n\n  &:focus {\n    outline: 0;\n    box-shadow: var(--#{$prefix}btn-close-focus-shadow);\n    opacity: var(--#{$prefix}btn-close-focus-opacity);\n  }\n\n  &:disabled,\n  &.disabled {\n    pointer-events: none;\n    user-select: none;\n    opacity: var(--#{$prefix}btn-close-disabled-opacity);\n  }\n}\n\n@mixin btn-close-white() {\n  filter: var(--#{$prefix}btn-close-white-filter);\n}\n\n.btn-close-white {\n  @include btn-close-white();\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .btn-close {\n      @include btn-close-white();\n    }\n  }\n}\n",".toast {\n  // scss-docs-start toast-css-vars\n  --#{$prefix}toast-zindex: #{$zindex-toast};\n  --#{$prefix}toast-padding-x: #{$toast-padding-x};\n  --#{$prefix}toast-padding-y: #{$toast-padding-y};\n  --#{$prefix}toast-spacing: #{$toast-spacing};\n  --#{$prefix}toast-max-width: #{$toast-max-width};\n  @include rfs($toast-font-size, --#{$prefix}toast-font-size);\n  --#{$prefix}toast-color: #{$toast-color};\n  --#{$prefix}toast-bg: #{$toast-background-color};\n  --#{$prefix}toast-border-width: #{$toast-border-width};\n  --#{$prefix}toast-border-color: #{$toast-border-color};\n  --#{$prefix}toast-border-radius: #{$toast-border-radius};\n  --#{$prefix}toast-box-shadow: #{$toast-box-shadow};\n  --#{$prefix}toast-header-color: #{$toast-header-color};\n  --#{$prefix}toast-header-bg: #{$toast-header-background-color};\n  --#{$prefix}toast-header-border-color: #{$toast-header-border-color};\n  // scss-docs-end toast-css-vars\n\n  width: var(--#{$prefix}toast-max-width);\n  max-width: 100%;\n  @include font-size(var(--#{$prefix}toast-font-size));\n  color: var(--#{$prefix}toast-color);\n  pointer-events: auto;\n  background-color: var(--#{$prefix}toast-bg);\n  background-clip: padding-box;\n  border: var(--#{$prefix}toast-border-width) solid var(--#{$prefix}toast-border-color);\n  box-shadow: var(--#{$prefix}toast-box-shadow);\n  @include border-radius(var(--#{$prefix}toast-border-radius));\n\n  &.showing {\n    opacity: 0;\n  }\n\n  &:not(.show) {\n    display: none;\n  }\n}\n\n.toast-container {\n  --#{$prefix}toast-zindex: #{$zindex-toast};\n\n  position: absolute;\n  z-index: var(--#{$prefix}toast-zindex);\n  width: max-content;\n  max-width: 100%;\n  pointer-events: none;\n\n  > :not(:last-child) {\n    margin-bottom: var(--#{$prefix}toast-spacing);\n  }\n}\n\n.toast-header {\n  display: flex;\n  align-items: center;\n  padding: var(--#{$prefix}toast-padding-y) var(--#{$prefix}toast-padding-x);\n  color: var(--#{$prefix}toast-header-color);\n  background-color: var(--#{$prefix}toast-header-bg);\n  background-clip: padding-box;\n  border-bottom: var(--#{$prefix}toast-border-width) solid var(--#{$prefix}toast-header-border-color);\n  @include border-top-radius(calc(var(--#{$prefix}toast-border-radius) - var(--#{$prefix}toast-border-width)));\n\n  .btn-close {\n    margin-right: calc(-.5 * var(--#{$prefix}toast-padding-x)); // stylelint-disable-line function-disallowed-list\n    margin-left: var(--#{$prefix}toast-padding-x);\n  }\n}\n\n.toast-body {\n  padding: var(--#{$prefix}toast-padding-x);\n  word-wrap: break-word;\n}\n","// stylelint-disable function-disallowed-list\n\n// .modal-open      - body class for killing the scroll\n// .modal           - container to scroll within\n// .modal-dialog    - positioning shell for the actual modal\n// .modal-content   - actual modal w/ bg and corners and stuff\n\n\n// Container that the modal scrolls within\n.modal {\n  // scss-docs-start modal-css-vars\n  --#{$prefix}modal-zindex: #{$zindex-modal};\n  --#{$prefix}modal-width: #{$modal-md};\n  --#{$prefix}modal-padding: #{$modal-inner-padding};\n  --#{$prefix}modal-margin: #{$modal-dialog-margin};\n  --#{$prefix}modal-color: #{$modal-content-color};\n  --#{$prefix}modal-bg: #{$modal-content-bg};\n  --#{$prefix}modal-border-color: #{$modal-content-border-color};\n  --#{$prefix}modal-border-width: #{$modal-content-border-width};\n  --#{$prefix}modal-border-radius: #{$modal-content-border-radius};\n  --#{$prefix}modal-box-shadow: #{$modal-content-box-shadow-xs};\n  --#{$prefix}modal-inner-border-radius: #{$modal-content-inner-border-radius};\n  --#{$prefix}modal-header-padding-x: #{$modal-header-padding-x};\n  --#{$prefix}modal-header-padding-y: #{$modal-header-padding-y};\n  --#{$prefix}modal-header-padding: #{$modal-header-padding}; // Todo in v6: Split this padding into x and y\n  --#{$prefix}modal-header-border-color: #{$modal-header-border-color};\n  --#{$prefix}modal-header-border-width: #{$modal-header-border-width};\n  --#{$prefix}modal-title-line-height: #{$modal-title-line-height};\n  --#{$prefix}modal-footer-gap: #{$modal-footer-margin-between};\n  --#{$prefix}modal-footer-bg: #{$modal-footer-bg};\n  --#{$prefix}modal-footer-border-color: #{$modal-footer-border-color};\n  --#{$prefix}modal-footer-border-width: #{$modal-footer-border-width};\n  // scss-docs-end modal-css-vars\n\n  position: fixed;\n  top: 0;\n  left: 0;\n  z-index: var(--#{$prefix}modal-zindex);\n  display: none;\n  width: 100%;\n  height: 100%;\n  overflow-x: hidden;\n  overflow-y: auto;\n  // Prevent Chrome on Windows from adding a focus outline. For details, see\n  // https://github.com/twbs/bootstrap/pull/10951.\n  outline: 0;\n  // We deliberately don't use `-webkit-overflow-scrolling: touch;` due to a\n  // gnarly iOS Safari bug: https://bugs.webkit.org/show_bug.cgi?id=158342\n  // See also https://github.com/twbs/bootstrap/issues/17695\n}\n\n// Shell div to position the modal with bottom padding\n.modal-dialog {\n  position: relative;\n  width: auto;\n  margin: var(--#{$prefix}modal-margin);\n  // allow clicks to pass through for custom click handling to close modal\n  pointer-events: none;\n\n  // When fading in the modal, animate it to slide down\n  .modal.fade & {\n    @include transition($modal-transition);\n    transform: $modal-fade-transform;\n  }\n  .modal.show & {\n    transform: $modal-show-transform;\n  }\n\n  // When trying to close, animate focus to scale\n  .modal.modal-static & {\n    transform: $modal-scale-transform;\n  }\n}\n\n.modal-dialog-scrollable {\n  height: calc(100% - var(--#{$prefix}modal-margin) * 2);\n\n  .modal-content {\n    max-height: 100%;\n    overflow: hidden;\n  }\n\n  .modal-body {\n    overflow-y: auto;\n  }\n}\n\n.modal-dialog-centered {\n  display: flex;\n  align-items: center;\n  min-height: calc(100% - var(--#{$prefix}modal-margin) * 2);\n}\n\n// Actual modal\n.modal-content {\n  position: relative;\n  display: flex;\n  flex-direction: column;\n  width: 100%; // Ensure `.modal-content` extends the full width of the parent `.modal-dialog`\n  // counteract the pointer-events: none; in the .modal-dialog\n  color: var(--#{$prefix}modal-color);\n  pointer-events: auto;\n  background-color: var(--#{$prefix}modal-bg);\n  background-clip: padding-box;\n  border: var(--#{$prefix}modal-border-width) solid var(--#{$prefix}modal-border-color);\n  @include border-radius(var(--#{$prefix}modal-border-radius));\n  @include box-shadow(var(--#{$prefix}modal-box-shadow));\n  // Remove focus outline from opened modal\n  outline: 0;\n}\n\n// Modal background\n.modal-backdrop {\n  // scss-docs-start modal-backdrop-css-vars\n  --#{$prefix}backdrop-zindex: #{$zindex-modal-backdrop};\n  --#{$prefix}backdrop-bg: #{$modal-backdrop-bg};\n  --#{$prefix}backdrop-opacity: #{$modal-backdrop-opacity};\n  // scss-docs-end modal-backdrop-css-vars\n\n  @include overlay-backdrop(var(--#{$prefix}backdrop-zindex), var(--#{$prefix}backdrop-bg), var(--#{$prefix}backdrop-opacity));\n}\n\n// Modal header\n// Top section of the modal w/ title and dismiss\n.modal-header {\n  display: flex;\n  flex-shrink: 0;\n  align-items: center;\n  padding: var(--#{$prefix}modal-header-padding);\n  border-bottom: var(--#{$prefix}modal-header-border-width) solid var(--#{$prefix}modal-header-border-color);\n  @include border-top-radius(var(--#{$prefix}modal-inner-border-radius));\n\n  .btn-close {\n    padding: calc(var(--#{$prefix}modal-header-padding-y) * .5) calc(var(--#{$prefix}modal-header-padding-x) * .5);\n    margin: calc(-.5 * var(--#{$prefix}modal-header-padding-y)) calc(-.5 * var(--#{$prefix}modal-header-padding-x)) calc(-.5 * var(--#{$prefix}modal-header-padding-y)) auto;\n  }\n}\n\n// Title text within header\n.modal-title {\n  margin-bottom: 0;\n  line-height: var(--#{$prefix}modal-title-line-height);\n}\n\n// Modal body\n// Where all modal content resides (sibling of .modal-header and .modal-footer)\n.modal-body {\n  position: relative;\n  // Enable `flex-grow: 1` so that the body take up as much space as possible\n  // when there should be a fixed height on `.modal-dialog`.\n  flex: 1 1 auto;\n  padding: var(--#{$prefix}modal-padding);\n}\n\n// Footer (for actions)\n.modal-footer {\n  display: flex;\n  flex-shrink: 0;\n  flex-wrap: wrap;\n  align-items: center; // vertically center\n  justify-content: flex-end; // Right align buttons with flex property because text-align doesn't work on flex items\n  padding: calc(var(--#{$prefix}modal-padding) - var(--#{$prefix}modal-footer-gap) * .5);\n  background-color: var(--#{$prefix}modal-footer-bg);\n  border-top: var(--#{$prefix}modal-footer-border-width) solid var(--#{$prefix}modal-footer-border-color);\n  @include border-bottom-radius(var(--#{$prefix}modal-inner-border-radius));\n\n  // Place margin between footer elements\n  // This solution is far from ideal because of the universal selector usage,\n  // but is needed to fix https://github.com/twbs/bootstrap/issues/24800\n  > * {\n    margin: calc(var(--#{$prefix}modal-footer-gap) * .5); // Todo in v6: replace with gap on parent class\n  }\n}\n\n// Scale up the modal\n@include media-breakpoint-up(sm) {\n  .modal {\n    --#{$prefix}modal-margin: #{$modal-dialog-margin-y-sm-up};\n    --#{$prefix}modal-box-shadow: #{$modal-content-box-shadow-sm-up};\n  }\n\n  // Automatically set modal's width for larger viewports\n  .modal-dialog {\n    max-width: var(--#{$prefix}modal-width);\n    margin-right: auto;\n    margin-left: auto;\n  }\n\n  .modal-sm {\n    --#{$prefix}modal-width: #{$modal-sm};\n  }\n}\n\n@include media-breakpoint-up(lg) {\n  .modal-lg,\n  .modal-xl {\n    --#{$prefix}modal-width: #{$modal-lg};\n  }\n}\n\n@include media-breakpoint-up(xl) {\n  .modal-xl {\n    --#{$prefix}modal-width: #{$modal-xl};\n  }\n}\n\n// scss-docs-start modal-fullscreen-loop\n@each $breakpoint in map-keys($grid-breakpoints) {\n  $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n  $postfix: if($infix != \"\", $infix + \"-down\", \"\");\n\n  @include media-breakpoint-down($breakpoint) {\n    .modal-fullscreen#{$postfix} {\n      width: 100vw;\n      max-width: none;\n      height: 100%;\n      margin: 0;\n\n      .modal-content {\n        height: 100%;\n        border: 0;\n        @include border-radius(0);\n      }\n\n      .modal-header,\n      .modal-footer {\n        @include border-radius(0);\n      }\n\n      .modal-body {\n        overflow-y: auto;\n      }\n    }\n  }\n}\n// scss-docs-end modal-fullscreen-loop\n","// Shared between modals and offcanvases\n@mixin overlay-backdrop($zindex, $backdrop-bg, $backdrop-opacity) {\n  position: fixed;\n  top: 0;\n  left: 0;\n  z-index: $zindex;\n  width: 100vw;\n  height: 100vh;\n  background-color: $backdrop-bg;\n\n  // Fade for backdrop\n  &.fade { opacity: 0; }\n  &.show { opacity: $backdrop-opacity; }\n}\n","// Base class\n.tooltip {\n  // scss-docs-start tooltip-css-vars\n  --#{$prefix}tooltip-zindex: #{$zindex-tooltip};\n  --#{$prefix}tooltip-max-width: #{$tooltip-max-width};\n  --#{$prefix}tooltip-padding-x: #{$tooltip-padding-x};\n  --#{$prefix}tooltip-padding-y: #{$tooltip-padding-y};\n  --#{$prefix}tooltip-margin: #{$tooltip-margin};\n  @include rfs($tooltip-font-size, --#{$prefix}tooltip-font-size);\n  --#{$prefix}tooltip-color: #{$tooltip-color};\n  --#{$prefix}tooltip-bg: #{$tooltip-bg};\n  --#{$prefix}tooltip-border-radius: #{$tooltip-border-radius};\n  --#{$prefix}tooltip-opacity: #{$tooltip-opacity};\n  --#{$prefix}tooltip-arrow-width: #{$tooltip-arrow-width};\n  --#{$prefix}tooltip-arrow-height: #{$tooltip-arrow-height};\n  // scss-docs-end tooltip-css-vars\n\n  z-index: var(--#{$prefix}tooltip-zindex);\n  display: block;\n  margin: var(--#{$prefix}tooltip-margin);\n  @include deprecate(\"`$tooltip-margin`\", \"v5\", \"v5.x\", true);\n  // Our parent element can be arbitrary since tooltips are by default inserted as a sibling of their target element.\n  // So reset our font and text properties to avoid inheriting weird values.\n  @include reset-text();\n  @include font-size(var(--#{$prefix}tooltip-font-size));\n  // Allow breaking very long words so they don't overflow the tooltip's bounds\n  word-wrap: break-word;\n  opacity: 0;\n\n  &.show { opacity: var(--#{$prefix}tooltip-opacity); }\n\n  .tooltip-arrow {\n    display: block;\n    width: var(--#{$prefix}tooltip-arrow-width);\n    height: var(--#{$prefix}tooltip-arrow-height);\n\n    &::before {\n      position: absolute;\n      content: \"\";\n      border-color: transparent;\n      border-style: solid;\n    }\n  }\n}\n\n.bs-tooltip-top .tooltip-arrow {\n  bottom: calc(-1 * var(--#{$prefix}tooltip-arrow-height)); // stylelint-disable-line function-disallowed-list\n\n  &::before {\n    top: -1px;\n    border-width: var(--#{$prefix}tooltip-arrow-height) calc(var(--#{$prefix}tooltip-arrow-width) * .5) 0; // stylelint-disable-line function-disallowed-list\n    border-top-color: var(--#{$prefix}tooltip-bg);\n  }\n}\n\n/* rtl:begin:ignore */\n.bs-tooltip-end .tooltip-arrow {\n  left: calc(-1 * var(--#{$prefix}tooltip-arrow-height)); // stylelint-disable-line function-disallowed-list\n  width: var(--#{$prefix}tooltip-arrow-height);\n  height: var(--#{$prefix}tooltip-arrow-width);\n\n  &::before {\n    right: -1px;\n    border-width: calc(var(--#{$prefix}tooltip-arrow-width) * .5) var(--#{$prefix}tooltip-arrow-height) calc(var(--#{$prefix}tooltip-arrow-width) * .5) 0; // stylelint-disable-line function-disallowed-list\n    border-right-color: var(--#{$prefix}tooltip-bg);\n  }\n}\n\n/* rtl:end:ignore */\n\n.bs-tooltip-bottom .tooltip-arrow {\n  top: calc(-1 * var(--#{$prefix}tooltip-arrow-height)); // stylelint-disable-line function-disallowed-list\n\n  &::before {\n    bottom: -1px;\n    border-width: 0 calc(var(--#{$prefix}tooltip-arrow-width) * .5) var(--#{$prefix}tooltip-arrow-height); // stylelint-disable-line function-disallowed-list\n    border-bottom-color: var(--#{$prefix}tooltip-bg);\n  }\n}\n\n/* rtl:begin:ignore */\n.bs-tooltip-start .tooltip-arrow {\n  right: calc(-1 * var(--#{$prefix}tooltip-arrow-height)); // stylelint-disable-line function-disallowed-list\n  width: var(--#{$prefix}tooltip-arrow-height);\n  height: var(--#{$prefix}tooltip-arrow-width);\n\n  &::before {\n    left: -1px;\n    border-width: calc(var(--#{$prefix}tooltip-arrow-width) * .5) 0 calc(var(--#{$prefix}tooltip-arrow-width) * .5) var(--#{$prefix}tooltip-arrow-height); // stylelint-disable-line function-disallowed-list\n    border-left-color: var(--#{$prefix}tooltip-bg);\n  }\n}\n\n/* rtl:end:ignore */\n\n.bs-tooltip-auto {\n  &[data-popper-placement^=\"top\"] {\n    @extend .bs-tooltip-top;\n  }\n  &[data-popper-placement^=\"right\"] {\n    @extend .bs-tooltip-end;\n  }\n  &[data-popper-placement^=\"bottom\"] {\n    @extend .bs-tooltip-bottom;\n  }\n  &[data-popper-placement^=\"left\"] {\n    @extend .bs-tooltip-start;\n  }\n}\n\n// Wrapper for the tooltip content\n.tooltip-inner {\n  max-width: var(--#{$prefix}tooltip-max-width);\n  padding: var(--#{$prefix}tooltip-padding-y) var(--#{$prefix}tooltip-padding-x);\n  color: var(--#{$prefix}tooltip-color);\n  text-align: center;\n  background-color: var(--#{$prefix}tooltip-bg);\n  @include border-radius(var(--#{$prefix}tooltip-border-radius));\n}\n","@mixin reset-text {\n  font-family: $font-family-base;\n  // We deliberately do NOT reset font-size or overflow-wrap / word-wrap.\n  font-style: normal;\n  font-weight: $font-weight-normal;\n  line-height: $line-height-base;\n  text-align: left; // Fallback for where `start` is not supported\n  text-align: start;\n  text-decoration: none;\n  text-shadow: none;\n  text-transform: none;\n  letter-spacing: normal;\n  word-break: normal;\n  white-space: normal;\n  word-spacing: normal;\n  line-break: auto;\n}\n",".popover {\n  // scss-docs-start popover-css-vars\n  --#{$prefix}popover-zindex: #{$zindex-popover};\n  --#{$prefix}popover-max-width: #{$popover-max-width};\n  @include rfs($popover-font-size, --#{$prefix}popover-font-size);\n  --#{$prefix}popover-bg: #{$popover-bg};\n  --#{$prefix}popover-border-width: #{$popover-border-width};\n  --#{$prefix}popover-border-color: #{$popover-border-color};\n  --#{$prefix}popover-border-radius: #{$popover-border-radius};\n  --#{$prefix}popover-inner-border-radius: #{$popover-inner-border-radius};\n  --#{$prefix}popover-box-shadow: #{$popover-box-shadow};\n  --#{$prefix}popover-header-padding-x: #{$popover-header-padding-x};\n  --#{$prefix}popover-header-padding-y: #{$popover-header-padding-y};\n  @include rfs($popover-header-font-size, --#{$prefix}popover-header-font-size);\n  --#{$prefix}popover-header-color: #{$popover-header-color};\n  --#{$prefix}popover-header-bg: #{$popover-header-bg};\n  --#{$prefix}popover-body-padding-x: #{$popover-body-padding-x};\n  --#{$prefix}popover-body-padding-y: #{$popover-body-padding-y};\n  --#{$prefix}popover-body-color: #{$popover-body-color};\n  --#{$prefix}popover-arrow-width: #{$popover-arrow-width};\n  --#{$prefix}popover-arrow-height: #{$popover-arrow-height};\n  --#{$prefix}popover-arrow-border: var(--#{$prefix}popover-border-color);\n  // scss-docs-end popover-css-vars\n\n  z-index: var(--#{$prefix}popover-zindex);\n  display: block;\n  max-width: var(--#{$prefix}popover-max-width);\n  // Our parent element can be arbitrary since tooltips are by default inserted as a sibling of their target element.\n  // So reset our font and text properties to avoid inheriting weird values.\n  @include reset-text();\n  @include font-size(var(--#{$prefix}popover-font-size));\n  // Allow breaking very long words so they don't overflow the popover's bounds\n  word-wrap: break-word;\n  background-color: var(--#{$prefix}popover-bg);\n  background-clip: padding-box;\n  border: var(--#{$prefix}popover-border-width) solid var(--#{$prefix}popover-border-color);\n  @include border-radius(var(--#{$prefix}popover-border-radius));\n  @include box-shadow(var(--#{$prefix}popover-box-shadow));\n\n  .popover-arrow {\n    display: block;\n    width: var(--#{$prefix}popover-arrow-width);\n    height: var(--#{$prefix}popover-arrow-height);\n\n    &::before,\n    &::after {\n      position: absolute;\n      display: block;\n      content: \"\";\n      border-color: transparent;\n      border-style: solid;\n      border-width: 0;\n    }\n  }\n}\n\n.bs-popover-top {\n  > .popover-arrow {\n    bottom: calc(-1 * (var(--#{$prefix}popover-arrow-height)) - var(--#{$prefix}popover-border-width)); // stylelint-disable-line function-disallowed-list\n\n    &::before,\n    &::after {\n      border-width: var(--#{$prefix}popover-arrow-height) calc(var(--#{$prefix}popover-arrow-width) * .5) 0; // stylelint-disable-line function-disallowed-list\n    }\n\n    &::before {\n      bottom: 0;\n      border-top-color: var(--#{$prefix}popover-arrow-border);\n    }\n\n    &::after {\n      bottom: var(--#{$prefix}popover-border-width);\n      border-top-color: var(--#{$prefix}popover-bg);\n    }\n  }\n}\n\n/* rtl:begin:ignore */\n.bs-popover-end {\n  > .popover-arrow {\n    left: calc(-1 * (var(--#{$prefix}popover-arrow-height)) - var(--#{$prefix}popover-border-width)); // stylelint-disable-line function-disallowed-list\n    width: var(--#{$prefix}popover-arrow-height);\n    height: var(--#{$prefix}popover-arrow-width);\n\n    &::before,\n    &::after {\n      border-width: calc(var(--#{$prefix}popover-arrow-width) * .5) var(--#{$prefix}popover-arrow-height) calc(var(--#{$prefix}popover-arrow-width) * .5) 0; // stylelint-disable-line function-disallowed-list\n    }\n\n    &::before {\n      left: 0;\n      border-right-color: var(--#{$prefix}popover-arrow-border);\n    }\n\n    &::after {\n      left: var(--#{$prefix}popover-border-width);\n      border-right-color: var(--#{$prefix}popover-bg);\n    }\n  }\n}\n\n/* rtl:end:ignore */\n\n.bs-popover-bottom {\n  > .popover-arrow {\n    top: calc(-1 * (var(--#{$prefix}popover-arrow-height)) - var(--#{$prefix}popover-border-width)); // stylelint-disable-line function-disallowed-list\n\n    &::before,\n    &::after {\n      border-width: 0 calc(var(--#{$prefix}popover-arrow-width) * .5) var(--#{$prefix}popover-arrow-height); // stylelint-disable-line function-disallowed-list\n    }\n\n    &::before {\n      top: 0;\n      border-bottom-color: var(--#{$prefix}popover-arrow-border);\n    }\n\n    &::after {\n      top: var(--#{$prefix}popover-border-width);\n      border-bottom-color: var(--#{$prefix}popover-bg);\n    }\n  }\n\n  // This will remove the popover-header's border just below the arrow\n  .popover-header::before {\n    position: absolute;\n    top: 0;\n    left: 50%;\n    display: block;\n    width: var(--#{$prefix}popover-arrow-width);\n    margin-left: calc(-.5 * var(--#{$prefix}popover-arrow-width)); // stylelint-disable-line function-disallowed-list\n    content: \"\";\n    border-bottom: var(--#{$prefix}popover-border-width) solid var(--#{$prefix}popover-header-bg);\n  }\n}\n\n/* rtl:begin:ignore */\n.bs-popover-start {\n  > .popover-arrow {\n    right: calc(-1 * (var(--#{$prefix}popover-arrow-height)) - var(--#{$prefix}popover-border-width)); // stylelint-disable-line function-disallowed-list\n    width: var(--#{$prefix}popover-arrow-height);\n    height: var(--#{$prefix}popover-arrow-width);\n\n    &::before,\n    &::after {\n      border-width: calc(var(--#{$prefix}popover-arrow-width) * .5) 0 calc(var(--#{$prefix}popover-arrow-width) * .5) var(--#{$prefix}popover-arrow-height); // stylelint-disable-line function-disallowed-list\n    }\n\n    &::before {\n      right: 0;\n      border-left-color: var(--#{$prefix}popover-arrow-border);\n    }\n\n    &::after {\n      right: var(--#{$prefix}popover-border-width);\n      border-left-color: var(--#{$prefix}popover-bg);\n    }\n  }\n}\n\n/* rtl:end:ignore */\n\n.bs-popover-auto {\n  &[data-popper-placement^=\"top\"] {\n    @extend .bs-popover-top;\n  }\n  &[data-popper-placement^=\"right\"] {\n    @extend .bs-popover-end;\n  }\n  &[data-popper-placement^=\"bottom\"] {\n    @extend .bs-popover-bottom;\n  }\n  &[data-popper-placement^=\"left\"] {\n    @extend .bs-popover-start;\n  }\n}\n\n// Offset the popover to account for the popover arrow\n.popover-header {\n  padding: var(--#{$prefix}popover-header-padding-y) var(--#{$prefix}popover-header-padding-x);\n  margin-bottom: 0; // Reset the default from Reboot\n  @include font-size(var(--#{$prefix}popover-header-font-size));\n  color: var(--#{$prefix}popover-header-color);\n  background-color: var(--#{$prefix}popover-header-bg);\n  border-bottom: var(--#{$prefix}popover-border-width) solid var(--#{$prefix}popover-border-color);\n  @include border-top-radius(var(--#{$prefix}popover-inner-border-radius));\n\n  &:empty {\n    display: none;\n  }\n}\n\n.popover-body {\n  padding: var(--#{$prefix}popover-body-padding-y) var(--#{$prefix}popover-body-padding-x);\n  color: var(--#{$prefix}popover-body-color);\n}\n","// Notes on the classes:\n//\n// 1. .carousel.pointer-event should ideally be pan-y (to allow for users to scroll vertically)\n//    even when their scroll action started on a carousel, but for compatibility (with Firefox)\n//    we're preventing all actions instead\n// 2. The .carousel-item-start and .carousel-item-end is used to indicate where\n//    the active slide is heading.\n// 3. .active.carousel-item is the current slide.\n// 4. .active.carousel-item-start and .active.carousel-item-end is the current\n//    slide in its in-transition state. Only one of these occurs at a time.\n// 5. .carousel-item-next.carousel-item-start and .carousel-item-prev.carousel-item-end\n//    is the upcoming slide in transition.\n\n.carousel {\n  position: relative;\n}\n\n.carousel.pointer-event {\n  touch-action: pan-y;\n}\n\n.carousel-inner {\n  position: relative;\n  width: 100%;\n  overflow: hidden;\n  @include clearfix();\n}\n\n.carousel-item {\n  position: relative;\n  display: none;\n  float: left;\n  width: 100%;\n  margin-right: -100%;\n  backface-visibility: hidden;\n  @include transition($carousel-transition);\n}\n\n.carousel-item.active,\n.carousel-item-next,\n.carousel-item-prev {\n  display: block;\n}\n\n.carousel-item-next:not(.carousel-item-start),\n.active.carousel-item-end {\n  transform: translateX(100%);\n}\n\n.carousel-item-prev:not(.carousel-item-end),\n.active.carousel-item-start {\n  transform: translateX(-100%);\n}\n\n\n//\n// Alternate transitions\n//\n\n.carousel-fade {\n  .carousel-item {\n    opacity: 0;\n    transition-property: opacity;\n    transform: none;\n  }\n\n  .carousel-item.active,\n  .carousel-item-next.carousel-item-start,\n  .carousel-item-prev.carousel-item-end {\n    z-index: 1;\n    opacity: 1;\n  }\n\n  .active.carousel-item-start,\n  .active.carousel-item-end {\n    z-index: 0;\n    opacity: 0;\n    @include transition(opacity 0s $carousel-transition-duration);\n  }\n}\n\n\n//\n// Left/right controls for nav\n//\n\n.carousel-control-prev,\n.carousel-control-next {\n  position: absolute;\n  top: 0;\n  bottom: 0;\n  z-index: 1;\n  // Use flex for alignment (1-3)\n  display: flex; // 1. allow flex styles\n  align-items: center; // 2. vertically center contents\n  justify-content: center; // 3. horizontally center contents\n  width: $carousel-control-width;\n  padding: 0;\n  color: $carousel-control-color;\n  text-align: center;\n  background: none;\n  border: 0;\n  opacity: $carousel-control-opacity;\n  @include transition($carousel-control-transition);\n\n  // Hover/focus state\n  &:hover,\n  &:focus {\n    color: $carousel-control-color;\n    text-decoration: none;\n    outline: 0;\n    opacity: $carousel-control-hover-opacity;\n  }\n}\n.carousel-control-prev {\n  left: 0;\n  background-image: if($enable-gradients, linear-gradient(90deg, rgba($black, .25), rgba($black, .001)), null);\n}\n.carousel-control-next {\n  right: 0;\n  background-image: if($enable-gradients, linear-gradient(270deg, rgba($black, .25), rgba($black, .001)), null);\n}\n\n// Icons for within\n.carousel-control-prev-icon,\n.carousel-control-next-icon {\n  display: inline-block;\n  width: $carousel-control-icon-width;\n  height: $carousel-control-icon-width;\n  background-repeat: no-repeat;\n  background-position: 50%;\n  background-size: 100% 100%;\n}\n\n.carousel-control-prev-icon {\n  background-image: escape-svg($carousel-control-prev-icon-bg) #{\"/*rtl:\" + escape-svg($carousel-control-next-icon-bg) + \"*/\"};\n}\n.carousel-control-next-icon {\n  background-image: escape-svg($carousel-control-next-icon-bg) #{\"/*rtl:\" + escape-svg($carousel-control-prev-icon-bg) + \"*/\"};\n}\n\n// Optional indicator pips/controls\n//\n// Add a container (such as a list) with the following class and add an item (ideally a focusable control,\n// like a button) with data-bs-target for each slide your carousel holds.\n\n.carousel-indicators {\n  position: absolute;\n  right: 0;\n  bottom: 0;\n  left: 0;\n  z-index: 2;\n  display: flex;\n  justify-content: center;\n  padding: 0;\n  // Use the .carousel-control's width as margin so we don't overlay those\n  margin-right: $carousel-control-width;\n  margin-bottom: 1rem;\n  margin-left: $carousel-control-width;\n\n  [data-bs-target] {\n    box-sizing: content-box;\n    flex: 0 1 auto;\n    width: $carousel-indicator-width;\n    height: $carousel-indicator-height;\n    padding: 0;\n    margin-right: $carousel-indicator-spacer;\n    margin-left: $carousel-indicator-spacer;\n    text-indent: -999px;\n    cursor: pointer;\n    background-color: $carousel-indicator-active-bg;\n    background-clip: padding-box;\n    border: 0;\n    // Use transparent borders to increase the hit area by 10px on top and bottom.\n    border-top: $carousel-indicator-hit-area-height solid transparent;\n    border-bottom: $carousel-indicator-hit-area-height solid transparent;\n    opacity: $carousel-indicator-opacity;\n    @include transition($carousel-indicator-transition);\n  }\n\n  .active {\n    opacity: $carousel-indicator-active-opacity;\n  }\n}\n\n\n// Optional captions\n//\n//\n\n.carousel-caption {\n  position: absolute;\n  right: (100% - $carousel-caption-width) * .5;\n  bottom: $carousel-caption-spacer;\n  left: (100% - $carousel-caption-width) * .5;\n  padding-top: $carousel-caption-padding-y;\n  padding-bottom: $carousel-caption-padding-y;\n  color: $carousel-caption-color;\n  text-align: center;\n}\n\n// Dark mode carousel\n\n@mixin carousel-dark() {\n  .carousel-control-prev-icon,\n  .carousel-control-next-icon {\n    filter: $carousel-dark-control-icon-filter;\n  }\n\n  .carousel-indicators [data-bs-target] {\n    background-color: $carousel-dark-indicator-active-bg;\n  }\n\n  .carousel-caption {\n    color: $carousel-dark-caption-color;\n  }\n}\n\n.carousel-dark {\n  @include carousel-dark();\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    @if $color-mode-type == \"media-query\" {\n      .carousel {\n        @include carousel-dark();\n      }\n    } @else {\n      .carousel,\n      &.carousel {\n        @include carousel-dark();\n      }\n    }\n  }\n}\n","// scss-docs-start clearfix\n@mixin clearfix() {\n  &::after {\n    display: block;\n    clear: both;\n    content: \"\";\n  }\n}\n// scss-docs-end clearfix\n","//\n// Rotating border\n//\n\n.spinner-grow,\n.spinner-border {\n  display: inline-block;\n  width: var(--#{$prefix}spinner-width);\n  height: var(--#{$prefix}spinner-height);\n  vertical-align: var(--#{$prefix}spinner-vertical-align);\n  // stylelint-disable-next-line property-disallowed-list\n  border-radius: 50%;\n  animation: var(--#{$prefix}spinner-animation-speed) linear infinite var(--#{$prefix}spinner-animation-name);\n}\n\n// scss-docs-start spinner-border-keyframes\n@keyframes spinner-border {\n  to { transform: rotate(360deg) #{\"/* rtl:ignore */\"}; }\n}\n// scss-docs-end spinner-border-keyframes\n\n.spinner-border {\n  // scss-docs-start spinner-border-css-vars\n  --#{$prefix}spinner-width: #{$spinner-width};\n  --#{$prefix}spinner-height: #{$spinner-height};\n  --#{$prefix}spinner-vertical-align: #{$spinner-vertical-align};\n  --#{$prefix}spinner-border-width: #{$spinner-border-width};\n  --#{$prefix}spinner-animation-speed: #{$spinner-animation-speed};\n  --#{$prefix}spinner-animation-name: spinner-border;\n  // scss-docs-end spinner-border-css-vars\n\n  border: var(--#{$prefix}spinner-border-width) solid currentcolor;\n  border-right-color: transparent;\n}\n\n.spinner-border-sm {\n  // scss-docs-start spinner-border-sm-css-vars\n  --#{$prefix}spinner-width: #{$spinner-width-sm};\n  --#{$prefix}spinner-height: #{$spinner-height-sm};\n  --#{$prefix}spinner-border-width: #{$spinner-border-width-sm};\n  // scss-docs-end spinner-border-sm-css-vars\n}\n\n//\n// Growing circle\n//\n\n// scss-docs-start spinner-grow-keyframes\n@keyframes spinner-grow {\n  0% {\n    transform: scale(0);\n  }\n  50% {\n    opacity: 1;\n    transform: none;\n  }\n}\n// scss-docs-end spinner-grow-keyframes\n\n.spinner-grow {\n  // scss-docs-start spinner-grow-css-vars\n  --#{$prefix}spinner-width: #{$spinner-width};\n  --#{$prefix}spinner-height: #{$spinner-height};\n  --#{$prefix}spinner-vertical-align: #{$spinner-vertical-align};\n  --#{$prefix}spinner-animation-speed: #{$spinner-animation-speed};\n  --#{$prefix}spinner-animation-name: spinner-grow;\n  // scss-docs-end spinner-grow-css-vars\n\n  background-color: currentcolor;\n  opacity: 0;\n}\n\n.spinner-grow-sm {\n  --#{$prefix}spinner-width: #{$spinner-width-sm};\n  --#{$prefix}spinner-height: #{$spinner-height-sm};\n}\n\n@if $enable-reduced-motion {\n  @media (prefers-reduced-motion: reduce) {\n    .spinner-border,\n    .spinner-grow {\n      --#{$prefix}spinner-animation-speed: #{$spinner-animation-speed * 2};\n    }\n  }\n}\n","// stylelint-disable function-disallowed-list\n\n%offcanvas-css-vars {\n  // scss-docs-start offcanvas-css-vars\n  --#{$prefix}offcanvas-zindex: #{$zindex-offcanvas};\n  --#{$prefix}offcanvas-width: #{$offcanvas-horizontal-width};\n  --#{$prefix}offcanvas-height: #{$offcanvas-vertical-height};\n  --#{$prefix}offcanvas-padding-x: #{$offcanvas-padding-x};\n  --#{$prefix}offcanvas-padding-y: #{$offcanvas-padding-y};\n  --#{$prefix}offcanvas-color: #{$offcanvas-color};\n  --#{$prefix}offcanvas-bg: #{$offcanvas-bg-color};\n  --#{$prefix}offcanvas-border-width: #{$offcanvas-border-width};\n  --#{$prefix}offcanvas-border-color: #{$offcanvas-border-color};\n  --#{$prefix}offcanvas-box-shadow: #{$offcanvas-box-shadow};\n  --#{$prefix}offcanvas-transition: #{transform $offcanvas-transition-duration ease-in-out};\n  --#{$prefix}offcanvas-title-line-height: #{$offcanvas-title-line-height};\n  // scss-docs-end offcanvas-css-vars\n}\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  $next: breakpoint-next($breakpoint, $grid-breakpoints);\n  $infix: breakpoint-infix($next, $grid-breakpoints);\n\n  .offcanvas#{$infix} {\n    @extend %offcanvas-css-vars;\n  }\n}\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  $next: breakpoint-next($breakpoint, $grid-breakpoints);\n  $infix: breakpoint-infix($next, $grid-breakpoints);\n\n  .offcanvas#{$infix} {\n    @include media-breakpoint-down($next) {\n      position: fixed;\n      bottom: 0;\n      z-index: var(--#{$prefix}offcanvas-zindex);\n      display: flex;\n      flex-direction: column;\n      max-width: 100%;\n      color: var(--#{$prefix}offcanvas-color);\n      visibility: hidden;\n      background-color: var(--#{$prefix}offcanvas-bg);\n      background-clip: padding-box;\n      outline: 0;\n      @include box-shadow(var(--#{$prefix}offcanvas-box-shadow));\n      @include transition(var(--#{$prefix}offcanvas-transition));\n\n      &.offcanvas-start {\n        top: 0;\n        left: 0;\n        width: var(--#{$prefix}offcanvas-width);\n        border-right: var(--#{$prefix}offcanvas-border-width) solid var(--#{$prefix}offcanvas-border-color);\n        transform: translateX(-100%);\n      }\n\n      &.offcanvas-end {\n        top: 0;\n        right: 0;\n        width: var(--#{$prefix}offcanvas-width);\n        border-left: var(--#{$prefix}offcanvas-border-width) solid var(--#{$prefix}offcanvas-border-color);\n        transform: translateX(100%);\n      }\n\n      &.offcanvas-top {\n        top: 0;\n        right: 0;\n        left: 0;\n        height: var(--#{$prefix}offcanvas-height);\n        max-height: 100%;\n        border-bottom: var(--#{$prefix}offcanvas-border-width) solid var(--#{$prefix}offcanvas-border-color);\n        transform: translateY(-100%);\n      }\n\n      &.offcanvas-bottom {\n        right: 0;\n        left: 0;\n        height: var(--#{$prefix}offcanvas-height);\n        max-height: 100%;\n        border-top: var(--#{$prefix}offcanvas-border-width) solid var(--#{$prefix}offcanvas-border-color);\n        transform: translateY(100%);\n      }\n\n      &.showing,\n      &.show:not(.hiding) {\n        transform: none;\n      }\n\n      &.showing,\n      &.hiding,\n      &.show {\n        visibility: visible;\n      }\n    }\n\n    @if not ($infix == \"\") {\n      @include media-breakpoint-up($next) {\n        --#{$prefix}offcanvas-height: auto;\n        --#{$prefix}offcanvas-border-width: 0;\n        background-color: transparent !important; // stylelint-disable-line declaration-no-important\n\n        .offcanvas-header {\n          display: none;\n        }\n\n        .offcanvas-body {\n          display: flex;\n          flex-grow: 0;\n          padding: 0;\n          overflow-y: visible;\n          // Reset `background-color` in case `.bg-*` classes are used in offcanvas\n          background-color: transparent !important; // stylelint-disable-line declaration-no-important\n        }\n      }\n    }\n  }\n}\n\n.offcanvas-backdrop {\n  @include overlay-backdrop($zindex-offcanvas-backdrop, $offcanvas-backdrop-bg, $offcanvas-backdrop-opacity);\n}\n\n.offcanvas-header {\n  display: flex;\n  align-items: center;\n  padding: var(--#{$prefix}offcanvas-padding-y) var(--#{$prefix}offcanvas-padding-x);\n\n  .btn-close {\n    padding: calc(var(--#{$prefix}offcanvas-padding-y) * .5) calc(var(--#{$prefix}offcanvas-padding-x) * .5);\n    margin: calc(-.5 * var(--#{$prefix}offcanvas-padding-y)) calc(-.5 * var(--#{$prefix}offcanvas-padding-x)) calc(-.5 * var(--#{$prefix}offcanvas-padding-y)) auto;\n  }\n}\n\n.offcanvas-title {\n  margin-bottom: 0;\n  line-height: var(--#{$prefix}offcanvas-title-line-height);\n}\n\n.offcanvas-body {\n  flex-grow: 1;\n  padding: var(--#{$prefix}offcanvas-padding-y) var(--#{$prefix}offcanvas-padding-x);\n  overflow-y: auto;\n}\n",".placeholder {\n  display: inline-block;\n  min-height: 1em;\n  vertical-align: middle;\n  cursor: wait;\n  background-color: currentcolor;\n  opacity: $placeholder-opacity-max;\n\n  &.btn::before {\n    display: inline-block;\n    content: \"\";\n  }\n}\n\n// Sizing\n.placeholder-xs {\n  min-height: .6em;\n}\n\n.placeholder-sm {\n  min-height: .8em;\n}\n\n.placeholder-lg {\n  min-height: 1.2em;\n}\n\n// Animation\n.placeholder-glow {\n  .placeholder {\n    animation: placeholder-glow 2s ease-in-out infinite;\n  }\n}\n\n@keyframes placeholder-glow {\n  50% {\n    opacity: $placeholder-opacity-min;\n  }\n}\n\n.placeholder-wave {\n  mask-image: linear-gradient(130deg, $black 55%, rgba(0, 0, 0, (1 - $placeholder-opacity-min)) 75%, $black 95%);\n  mask-size: 200% 100%;\n  animation: placeholder-wave 2s linear infinite;\n}\n\n@keyframes placeholder-wave {\n  100% {\n    mask-position: -200% 0%;\n  }\n}\n","// All-caps `RGBA()` function used because of this Sass bug: https://github.com/sass/node-sass/issues/2251\n@each $color, $value in $theme-colors {\n  .text-bg-#{$color} {\n    color: color-contrast($value) if($enable-important-utilities, !important, null);\n    background-color: RGBA(var(--#{$prefix}#{$color}-rgb), var(--#{$prefix}bg-opacity, 1)) if($enable-important-utilities, !important, null);\n  }\n}\n","// All-caps `RGBA()` function used because of this Sass bug: https://github.com/sass/node-sass/issues/2251\n@each $color, $value in $theme-colors {\n  .link-#{$color} {\n    color: RGBA(var(--#{$prefix}#{$color}-rgb), var(--#{$prefix}link-opacity, 1)) if($enable-important-utilities, !important, null);\n    text-decoration-color: RGBA(var(--#{$prefix}#{$color}-rgb), var(--#{$prefix}link-underline-opacity, 1)) if($enable-important-utilities, !important, null);\n\n    @if $link-shade-percentage != 0 {\n      &:hover,\n      &:focus {\n        $hover-color: if(color-contrast($value) == $color-contrast-light, shade-color($value, $link-shade-percentage), tint-color($value, $link-shade-percentage));\n        color: RGBA(#{to-rgb($hover-color)}, var(--#{$prefix}link-opacity, 1)) if($enable-important-utilities, !important, null);\n        text-decoration-color: RGBA(to-rgb($hover-color), var(--#{$prefix}link-underline-opacity, 1)) if($enable-important-utilities, !important, null);\n      }\n    }\n  }\n}\n\n// One-off special link helper as a bridge until v6\n.link-body-emphasis {\n  color: RGBA(var(--#{$prefix}emphasis-color-rgb), var(--#{$prefix}link-opacity, 1)) if($enable-important-utilities, !important, null);\n  text-decoration-color: RGBA(var(--#{$prefix}emphasis-color-rgb), var(--#{$prefix}link-underline-opacity, 1)) if($enable-important-utilities, !important, null);\n\n  @if $link-shade-percentage != 0 {\n    &:hover,\n    &:focus {\n      color: RGBA(var(--#{$prefix}emphasis-color-rgb), var(--#{$prefix}link-opacity, .75)) if($enable-important-utilities, !important, null);\n      text-decoration-color: RGBA(var(--#{$prefix}emphasis-color-rgb), var(--#{$prefix}link-underline-opacity, .75)) if($enable-important-utilities, !important, null);\n    }\n  }\n}\n",".focus-ring:focus {\n  outline: 0;\n  // By default, there is no `--bs-focus-ring-x`, `--bs-focus-ring-y`, or `--bs-focus-ring-blur`, but we provide CSS variables with fallbacks to initial `0` values\n  box-shadow: var(--#{$prefix}focus-ring-x, 0) var(--#{$prefix}focus-ring-y, 0) var(--#{$prefix}focus-ring-blur, 0) var(--#{$prefix}focus-ring-width) var(--#{$prefix}focus-ring-color);\n}\n",".icon-link {\n  display: inline-flex;\n  gap: $icon-link-gap;\n  align-items: center;\n  text-decoration-color: rgba(var(--#{$prefix}link-color-rgb), var(--#{$prefix}link-opacity, .5));\n  text-underline-offset: $icon-link-underline-offset;\n  backface-visibility: hidden;\n\n  > .bi {\n    flex-shrink: 0;\n    width: $icon-link-icon-size;\n    height: $icon-link-icon-size;\n    fill: currentcolor;\n    @include transition($icon-link-icon-transition);\n  }\n}\n\n.icon-link-hover {\n  &:hover,\n  &:focus-visible {\n    > .bi {\n      transform: var(--#{$prefix}icon-link-transform, $icon-link-icon-transform);\n    }\n  }\n}\n","// Credit: Nicolas Gallagher and SUIT CSS.\n\n.ratio {\n  position: relative;\n  width: 100%;\n\n  &::before {\n    display: block;\n    padding-top: var(--#{$prefix}aspect-ratio);\n    content: \"\";\n  }\n\n  > * {\n    position: absolute;\n    top: 0;\n    left: 0;\n    width: 100%;\n    height: 100%;\n  }\n}\n\n@each $key, $ratio in $aspect-ratios {\n  .ratio-#{$key} {\n    --#{$prefix}aspect-ratio: #{$ratio};\n  }\n}\n","// Shorthand\n\n.fixed-top {\n  position: fixed;\n  top: 0;\n  right: 0;\n  left: 0;\n  z-index: $zindex-fixed;\n}\n\n.fixed-bottom {\n  position: fixed;\n  right: 0;\n  bottom: 0;\n  left: 0;\n  z-index: $zindex-fixed;\n}\n\n// Responsive sticky top and bottom\n@each $breakpoint in map-keys($grid-breakpoints) {\n  @include media-breakpoint-up($breakpoint) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    .sticky#{$infix}-top {\n      position: sticky;\n      top: 0;\n      z-index: $zindex-sticky;\n    }\n\n    .sticky#{$infix}-bottom {\n      position: sticky;\n      bottom: 0;\n      z-index: $zindex-sticky;\n    }\n  }\n}\n","// scss-docs-start stacks\n.hstack {\n  display: flex;\n  flex-direction: row;\n  align-items: center;\n  align-self: stretch;\n}\n\n.vstack {\n  display: flex;\n  flex: 1 1 auto;\n  flex-direction: column;\n  align-self: stretch;\n}\n// scss-docs-end stacks\n","//\n// Visually hidden\n//\n\n.visually-hidden,\n.visually-hidden-focusable:not(:focus):not(:focus-within) {\n  @include visually-hidden();\n}\n","// stylelint-disable declaration-no-important\n\n// Hide content visually while keeping it accessible to assistive technologies\n//\n// See: https://www.a11yproject.com/posts/2013-01-11-how-to-hide-content/\n// See: https://kittygiraudel.com/2016/10/13/css-hide-and-seek/\n\n@mixin visually-hidden() {\n  width: 1px !important;\n  height: 1px !important;\n  padding: 0 !important;\n  margin: -1px !important; // Fix for https://github.com/twbs/bootstrap/issues/25686\n  overflow: hidden !important;\n  clip: rect(0, 0, 0, 0) !important;\n  white-space: nowrap !important;\n  border: 0 !important;\n\n  // Fix for positioned table caption that could become anonymous cells\n  &:not(caption) {\n    position: absolute !important;\n  }\n}\n\n// Use to only display content when it's focused, or one of its child elements is focused\n// (i.e. when focus is within the element/container that the class was applied to)\n//\n// Useful for \"Skip to main content\" links; see https://www.w3.org/TR/2013/NOTE-WCAG20-TECHS-20130905/G1\n\n@mixin visually-hidden-focusable() {\n  &:not(:focus):not(:focus-within) {\n    @include visually-hidden();\n  }\n}\n","//\n// Stretched link\n//\n\n.stretched-link {\n  &::#{$stretched-link-pseudo-element} {\n    position: absolute;\n    top: 0;\n    right: 0;\n    bottom: 0;\n    left: 0;\n    z-index: $stretched-link-z-index;\n    content: \"\";\n  }\n}\n","//\n// Text truncation\n//\n\n.text-truncate {\n  @include text-truncate();\n}\n","// Text truncate\n// Requires inline-block or block for proper styling\n\n@mixin text-truncate() {\n  overflow: hidden;\n  text-overflow: ellipsis;\n  white-space: nowrap;\n}\n",".vr {\n  display: inline-block;\n  align-self: stretch;\n  width: $vr-border-width;\n  min-height: 1em;\n  background-color: currentcolor;\n  opacity: $hr-opacity;\n}\n","// Utility generator\n// Used to generate utilities & print utilities\n@mixin generate-utility($utility, $infix: \"\", $is-rfs-media-query: false) {\n  $values: map-get($utility, values);\n\n  // If the values are a list or string, convert it into a map\n  @if type-of($values) == \"string\" or type-of(nth($values, 1)) != \"list\" {\n    $values: zip($values, $values);\n  }\n\n  @each $key, $value in $values {\n    $properties: map-get($utility, property);\n\n    // Multiple properties are possible, for example with vertical or horizontal margins or paddings\n    @if type-of($properties) == \"string\" {\n      $properties: append((), $properties);\n    }\n\n    // Use custom class if present\n    $property-class: if(map-has-key($utility, class), map-get($utility, class), nth($properties, 1));\n    $property-class: if($property-class == null, \"\", $property-class);\n\n    // Use custom CSS variable name if present, otherwise default to `class`\n    $css-variable-name: if(map-has-key($utility, css-variable-name), map-get($utility, css-variable-name), map-get($utility, class));\n\n    // State params to generate pseudo-classes\n    $state: if(map-has-key($utility, state), map-get($utility, state), ());\n\n    $infix: if($property-class == \"\" and str-slice($infix, 1, 1) == \"-\", str-slice($infix, 2), $infix);\n\n    // Don't prefix if value key is null (e.g. with shadow class)\n    $property-class-modifier: if($key, if($property-class == \"\" and $infix == \"\", \"\", \"-\") + $key, \"\");\n\n    @if map-get($utility, rfs) {\n      // Inside the media query\n      @if $is-rfs-media-query {\n        $val: rfs-value($value);\n\n        // Do not render anything if fluid and non fluid values are the same\n        $value: if($val == rfs-fluid-value($value), null, $val);\n      }\n      @else {\n        $value: rfs-fluid-value($value);\n      }\n    }\n\n    $is-css-var: map-get($utility, css-var);\n    $is-local-vars: map-get($utility, local-vars);\n    $is-rtl: map-get($utility, rtl);\n\n    @if $value != null {\n      @if $is-rtl == false {\n        /* rtl:begin:remove */\n      }\n\n      @if $is-css-var {\n        .#{$property-class + $infix + $property-class-modifier} {\n          --#{$prefix}#{$css-variable-name}: #{$value};\n        }\n\n        @each $pseudo in $state {\n          .#{$property-class + $infix + $property-class-modifier}-#{$pseudo}:#{$pseudo} {\n            --#{$prefix}#{$css-variable-name}: #{$value};\n          }\n        }\n      } @else {\n        .#{$property-class + $infix + $property-class-modifier} {\n          @each $property in $properties {\n            @if $is-local-vars {\n              @each $local-var, $variable in $is-local-vars {\n                --#{$prefix}#{$local-var}: #{$variable};\n              }\n            }\n            #{$property}: $value if($enable-important-utilities, !important, null);\n          }\n        }\n\n        @each $pseudo in $state {\n          .#{$property-class + $infix + $property-class-modifier}-#{$pseudo}:#{$pseudo} {\n            @each $property in $properties {\n              @if $is-local-vars {\n                @each $local-var, $variable in $is-local-vars {\n                  --#{$prefix}#{$local-var}: #{$variable};\n                }\n              }\n              #{$property}: $value if($enable-important-utilities, !important, null);\n            }\n          }\n        }\n      }\n\n      @if $is-rtl == false {\n        /* rtl:end:remove */\n      }\n    }\n  }\n}\n","// Loop over each breakpoint\n@each $breakpoint in map-keys($grid-breakpoints) {\n\n  // Generate media query if needed\n  @include media-breakpoint-up($breakpoint) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    // Loop over each utility property\n    @each $key, $utility in $utilities {\n      // The utility can be disabled with `false`, thus check if the utility is a map first\n      // Only proceed if responsive media queries are enabled or if it's the base media query\n      @if type-of($utility) == \"map\" and (map-get($utility, responsive) or $infix == \"\") {\n        @include generate-utility($utility, $infix);\n      }\n    }\n  }\n}\n\n// RFS rescaling\n@media (min-width: $rfs-mq-value) {\n  @each $breakpoint in map-keys($grid-breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    @if (map-get($grid-breakpoints, $breakpoint) < $rfs-breakpoint) {\n      // Loop over each utility property\n      @each $key, $utility in $utilities {\n        // The utility can be disabled with `false`, thus check if the utility is a map first\n        // Only proceed if responsive media queries are enabled or if it's the base media query\n        @if type-of($utility) == \"map\" and map-get($utility, rfs) and (map-get($utility, responsive) or $infix == \"\") {\n          @include generate-utility($utility, $infix, true);\n        }\n      }\n    }\n  }\n}\n\n\n// Print utilities\n@media print {\n  @each $key, $utility in $utilities {\n    // The utility can be disabled with `false`, thus check if the utility is a map first\n    // Then check if the utility needs print styles\n    @if type-of($utility) == \"map\" and map-get($utility, print) == true {\n      @include generate-utility($utility, \"-print\");\n    }\n  }\n}\n"],"names":[],"sourceRoot":""}
\ No newline at end of file
diff --git a/docs/html/_static/styles/pydata-sphinx-theme.css b/docs/html/_static/styles/pydata-sphinx-theme.css
new file mode 100644
index 0000000..3610bf3
--- /dev/null
+++ b/docs/html/_static/styles/pydata-sphinx-theme.css
@@ -0,0 +1,2 @@
+html{--pst-header-height:4rem;--pst-header-article-height:calc(var(--pst-header-height)*2/3);--pst-sidebar-secondary:17rem;--pst-font-size-base:1rem;--pst-font-size-h1:2.5rem;--pst-font-size-h2:2rem;--pst-font-size-h3:1.75rem;--pst-font-size-h4:1.5rem;--pst-font-size-h5:1.25rem;--pst-font-size-h6:1.1rem;--pst-font-size-milli:0.9rem;--pst-sidebar-font-size:0.9rem;--pst-sidebar-font-size-mobile:1.1rem;--pst-sidebar-header-font-size:1.2rem;--pst-sidebar-header-font-weight:600;--pst-admonition-font-weight-heading:600;--pst-font-weight-caption:300;--pst-font-weight-heading:400;--pst-font-family-base-system:-apple-system,"BlinkMacSystemFont","Segoe UI","Helvetica Neue","Arial",sans-serif,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol";--pst-font-family-monospace-system:"SFMono-Regular","Menlo","Consolas","Monaco","Liberation Mono","Lucida Console",monospace;--pst-font-family-base:var(--pst-font-family-base-system);--pst-font-family-heading:var(--pst-font-family-base-system);--pst-font-family-monospace:var(--pst-font-family-monospace-system);--pst-font-size-icon:1.5rem;--pst-icon-check-circle:"";--pst-icon-info-circle:"";--pst-icon-exclamation-triangle:"";--pst-icon-exclamation-circle:"";--pst-icon-times-circle:"";--pst-icon-lightbulb:"";--pst-icon-download:"";--pst-icon-angle-left:"";--pst-icon-angle-right:"";--pst-icon-external-link:"";--pst-icon-search-minus:"";--pst-icon-github:"";--pst-icon-gitlab:"";--pst-icon-share:"";--pst-icon-bell:"";--pst-icon-pencil:"";--pst-breadcrumb-divider:"";--pst-icon-admonition-default:var(--pst-icon-bell);--pst-icon-admonition-note:var(--pst-icon-info-circle);--pst-icon-admonition-attention:var(--pst-icon-exclamation-circle);--pst-icon-admonition-caution:var(--pst-icon-exclamation-triangle);--pst-icon-admonition-warning:var(--pst-icon-exclamation-triangle);--pst-icon-admonition-danger:var(--pst-icon-exclamation-triangle);--pst-icon-admonition-error:var(--pst-icon-times-circle);--pst-icon-admonition-hint:var(--pst-icon-lightbulb);--pst-icon-admonition-tip:var(--pst-icon-lightbulb);--pst-icon-admonition-important:var(--pst-icon-exclamation-circle);--pst-icon-admonition-seealso:var(--pst-icon-share);--pst-icon-admonition-todo:var(--pst-icon-pencil);--pst-icon-versionmodified-default:var(--pst-icon-exclamation-circle);--pst-icon-versionmodified-added:var(--pst-icon-exclamation-circle);--pst-icon-versionmodified-changed:var(--pst-icon-exclamation-circle);--pst-icon-versionmodified-deprecated:var(--pst-icon-exclamation-circle);font-size:var(--pst-font-size-base);scroll-padding-top:calc(var(--pst-header-height) + 1rem)}body{background-color:var(--pst-color-background);color:var(--pst-color-text-base);display:flex;flex-direction:column;font-family:var(--pst-font-family-base);font-weight:400;line-height:1.65;min-height:100vh}body::-webkit-scrollbar-track{background-color:var(--pst-color-background)}p{font-size:1em;margin-bottom:1.15rem}p.rubric{border-bottom:1px solid var(--pst-color-border)}p.centered{text-align:center}a{word-wrap:break-word;color:var(--pst-color-link);text-decoration:underline;text-decoration-thickness:max(1px,.0625rem);text-underline-offset:.1578em}a:hover{color:var(--pst-color-link-hover);text-decoration-skip-ink:none;text-decoration-thickness:max(3px,.1875rem,.12em)}a:active,a:visited{color:var(--pst-color-link)}a:visited:hover{color:var(--pst-color-link-hover)}a.headerlink{color:var(--pst-color-secondary);font-size:.8em;margin-left:.2em;opacity:.7;padding:0 4px;text-decoration:none;transition:all .2s ease-out;user-select:none}a.headerlink:hover{opacity:1}a.github:before,a.gitlab:before{color:var(--pst-color-text-muted);font:var(--fa-font-brands);margin-right:.25rem}a.github:before{content:var(--pst-icon-github)}a.gitlab:before{content:var(--pst-icon-gitlab)}h1,h2,h3,h4,h5,h6{font-family:var(--pst-font-family-heading);font-weight:var(--pst-font-weight-heading);line-height:1.15;margin:2.75rem 0 1.05rem}h1{font-size:var(--pst-font-size-h1);margin-top:0}h1,h2{color:var(--pst-heading-color)}h2{font-size:var(--pst-font-size-h2)}h3{font-size:var(--pst-font-size-h3)}h3,h4{color:var(--pst-heading-color)}h4{font-size:var(--pst-font-size-h4)}h5{font-size:var(--pst-font-size-h5)}h5,h6{color:var(--pst-color-text-base)}h6{font-size:var(--pst-font-size-h6)}.text_small,small{font-size:var(--pst-font-size-milli)}hr{border:0;border-top:1px solid var(--pst-color-border)}code,kbd,pre,samp{font-family:var(--pst-font-family-monospace)}kbd{background-color:var(--pst-color-on-background);color:var(--pst-color-text-muted)}kbd:not(.compound){border:1px solid var(--pst-color-border);box-shadow:1px 1px 1px var(--pst-color-shadow);margin:0 .1rem;padding:.1rem .4rem}code{color:var(--pst-color-inline-code)}pre{background-color:var(--pst-color-surface);border:1px solid var(--pst-color-border);border-radius:.25rem;color:var(--pst-color-text-base);line-height:1.2em;margin:1.5em 0;padding:1rem}pre .linenos{opacity:.8;padding-right:10px}#pst-back-to-top{background-color:var(--pst-color-secondary);border:none;color:var(--pst-color-secondary-text);display:none;left:50vw;position:fixed;top:90vh;transform:translate(-50%);z-index:1080}#pst-back-to-top 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table.dataframe tbody tr:hover{background-color:var(--pst-color-table-row-hover-bg)}html[data-theme=dark] .bd-content .nboutput .output_area.rendered_html:not(:has(table.dataframe)),html[data-theme=dark] .bd-content .nboutput .output_area .widget-subarea{color:var(--pst-color-on-background);background-color:var(--pst-color-text-base);border-radius:.25rem;padding:.5rem}html[data-theme=dark] .bd-content .nboutput .output_area.stderr{background-color:var(--pst-color-danger)}div.nblast.container{margin-bottom:1rem}div.cell_output .output{max-width:100%;overflow-x:auto}.bd-content div.cell_output table.dataframe{color:var(--pst-color-table);border:1px solid var(--pst-color-table-outer-border)}.bd-content div.cell_output table.dataframe th~th,.bd-content div.cell_output table.dataframe th~td,.bd-content div.cell_output table.dataframe td~th,.bd-content div.cell_output table.dataframe td~td{border-left:1px solid var(--pst-color-table-inner-border)}.bd-content div.cell_output table.dataframe thead tr{background-color:var(--pst-color-table-heading-bg);border-bottom:2px solid var(--pst-color-primary)}.bd-content div.cell_output table.dataframe tbody tr:nth-child(odd){background-color:var(--pst-color-table-row-zebra-low-bg)}.bd-content div.cell_output table.dataframe tbody tr:nth-child(even){background-color:var(--pst-color-table-row-zebra-high-bg)}.bd-content div.cell_output table.dataframe tbody tr:hover{background-color:var(--pst-color-table-row-hover-bg)}html[data-theme=dark] .bd-content div.cell_output img,html[data-theme=dark] .bd-content div.cell_output .text_html:not(:has(table.dataframe)),html[data-theme=dark] .bd-content div.cell_output .widget-subarea{color:var(--pst-color-on-background);background-color:var(--pst-color-text-base);border-radius:.25rem;padding:.5rem}.bd-content div.cell_input{display:flex;flex-direction:column;justify-content:stretch}.bd-content div.cell_input,.bd-content div.output{border-radius:.25rem}.bd-content div.output table{table-layout:auto}html[data-theme=dark] .bd-content img.leaflet-tile.leaflet-tile-loaded{border-radius:0;padding:0}.bd-search-container div#search-results>h2{font-size:var(--pst-font-size-icon);margin-top:1rem}.bd-search-container div#search-results p.search-summary{color:var(--pst-color-text-muted)}.bd-search-container ul.search{margin:0;list-style:none}.bd-search-container ul.search li{background-image:none;padding:1rem 0;margin:1rem 0;border-top:1px solid var(--pst-color-text-muted)}.bd-search-container ul.search li>a{font-size:1.2em}.bd-search-container ul.search li div.context,.bd-search-container ul.search li p.context{color:var(--pst-color-text-base);margin:.5em 0 0}.bd-search-container ul.search li div.context a::before,.bd-search-container ul.search li p.context a::before{content:\"#\";padding-right:.2em;color:var(--pst-color-text-muted)}","html {\n  /*****************************************************************************\n  * Overall Layout Variables\n  */\n\n  // Header height will impact the top offset for many sections\n  // Article header is 66% of Header\n  --pst-header-height: 4rem;\n  --pst-header-article-height: calc(var(--pst-header-height) * 2 / 3);\n  --pst-sidebar-secondary: 17rem;\n}\n\n/*******************************************************************************\n* Breakpoints that trigger UI changes\n*\n* Note that media-breakpoint-down begins at the next highest level!\n* So we should choose a media-breakpoint-down one *lower* than when we want to start\n* example: media-breakpoint-up(md) and media-breakpoint-down(sm) trigger at the same time\n* ref: https://github.com/twbs/bootstrap/issues/31214\n*/\n$breakpoint-sidebar-primary: lg; // When we collapse the primary sidebar\n$breakpoint-sidebar-secondary: xl; // When we collapse the secondary sidebar\n$breakpoint-page-width: 88rem; // taken from sphinx-basic-ng, which we are ultimately going to inherit\n\n/*******************************************************************************\n* Define the animation behaviour\n*/\n$animation-time: 200ms;\n\n/*******************************************************************************\n* UI shaping and padding\n*/\n$admonition-border-radius: 0.25rem;\n\n// In this theme, some focus rings have rounded corners while others do not.\n// This variable sets the border radius for the rounded focus rings.\n$focus-ring-radius: 0.125rem; // 2px at 100% zoom and 16px base font.\n\n$navbar-link-padding-y: 0.25rem;\n","html {\n  /*****************************************************************************\n  * Font features used in this theme\n  */\n\n  // base font size - applied at body/html level\n  --pst-font-size-base: 1rem;\n\n  // heading font sizes based on bootstrap sizing\n  --pst-font-size-h1: 2.5rem;\n  --pst-font-size-h2: 2rem;\n  --pst-font-size-h3: 1.75rem;\n  --pst-font-size-h4: 1.5rem;\n  --pst-font-size-h5: 1.25rem;\n  --pst-font-size-h6: 1.1rem;\n\n  // smaller than heading font sizes\n  --pst-font-size-milli: 0.9rem;\n\n  // Sidebar styles\n  --pst-sidebar-font-size: 0.9rem;\n  --pst-sidebar-font-size-mobile: 1.1rem;\n  --pst-sidebar-header-font-size: 1.2rem;\n  --pst-sidebar-header-font-weight: 600;\n\n  // Admonition styles\n  --pst-admonition-font-weight-heading: 600;\n\n  // Font weights\n  --pst-font-weight-caption: 300;\n  --pst-font-weight-heading: 400;\n\n  // Font family\n  // These are adapted from https://systemfontstack.com/ */\n  --pst-font-family-base-system: -apple-system, \"BlinkMacSystemFont\", \"Segoe UI\",\n    \"Helvetica Neue\", \"Arial\", sans-serif, \"Apple Color Emoji\", \"Segoe UI Emoji\",\n    \"Segoe UI Symbol\";\n  --pst-font-family-monospace-system: \"SFMono-Regular\", \"Menlo\", \"Consolas\",\n    \"Monaco\", \"Liberation Mono\", \"Lucida Console\", monospace;\n  --pst-font-family-base: var(--pst-font-family-base-system);\n  --pst-font-family-heading: var(--pst-font-family-base-system);\n  --pst-font-family-monospace: var(--pst-font-family-monospace-system);\n}\n\n$line-height-body: 1.65;\n","html {\n  /*****************************************************************************\n  * Icon\n  */\n\n  // Font size across all icons\n  --pst-font-size-icon: 1.5rem;\n\n  // Font Awesome default icons\n  --pst-icon-check-circle: \"\\f058\"; // fa-solid fa-circle-check\n  --pst-icon-info-circle: \"\\f05a\"; // fa-solid fa-circle-info\n  --pst-icon-exclamation-triangle: \"\\f071\"; // fa-solid fa-triangle-exclamation\n  --pst-icon-exclamation-circle: \"\\f06a\"; // fa-solid fa-circle-exclamation\n  --pst-icon-times-circle: \"\\f057\"; // fa-solid fa-circle-xmark\n  --pst-icon-lightbulb: \"\\f0eb\"; // fa-solid fa-lightbulb\n  --pst-icon-download: \"\\f019\"; // fa-solid fa-download\n  --pst-icon-angle-left: \"\\f104\"; // fa-solid fa-angle-left\n  --pst-icon-angle-right: \"\\f105\"; // fa-solid fa-angle-right\n  --pst-icon-external-link: \"\\f35d\"; // fa-solid fa-up-right-from-square\n  --pst-icon-search-minus: \"\\f010\"; // fa-solid fa-magnifying-glass-minus\n  --pst-icon-github: \"\\f09b\"; // fa-brands fa-github\n  --pst-icon-gitlab: \"\\f296\"; // fa-brands fa-gitlab\n  --pst-icon-share: \"\\f064\"; // fa-solid fa-share\n  --pst-icon-bell: \"\\f0f3\"; // fa-solid fa-bell\n  --pst-icon-pencil: \"\\f303\"; // fa-solid fa-pencil\n\n  // Bootstrap icons\n  --pst-breadcrumb-divider: \"\\f105\";\n}\n","html {\n  /*****************************************************************************\n  * Admonitions\n  **/\n\n  --pst-icon-admonition-default: var(--pst-icon-bell);\n  --pst-icon-admonition-note: var(--pst-icon-info-circle);\n  --pst-icon-admonition-attention: var(--pst-icon-exclamation-circle);\n  --pst-icon-admonition-caution: var(--pst-icon-exclamation-triangle);\n  --pst-icon-admonition-warning: var(--pst-icon-exclamation-triangle);\n  --pst-icon-admonition-danger: var(--pst-icon-exclamation-triangle);\n  --pst-icon-admonition-error: var(--pst-icon-times-circle);\n  --pst-icon-admonition-hint: var(--pst-icon-lightbulb);\n  --pst-icon-admonition-tip: var(--pst-icon-lightbulb);\n  --pst-icon-admonition-important: var(--pst-icon-exclamation-circle);\n  --pst-icon-admonition-seealso: var(--pst-icon-share);\n  --pst-icon-admonition-todo: var(--pst-icon-pencil);\n}\n","html {\n  /*****************************************************************************\n  * versionmodified\n  **/\n\n  --pst-icon-versionmodified-default: var(--pst-icon-exclamation-circle);\n  --pst-icon-versionmodified-added: var(--pst-icon-exclamation-circle);\n  --pst-icon-versionmodified-changed: var(--pst-icon-exclamation-circle);\n  --pst-icon-versionmodified-deprecated: var(--pst-icon-exclamation-circle);\n}\n","html {\n  font-size: var(--pst-font-size-base);\n  scroll-padding-top: calc(var(--pst-header-height) + 1rem);\n}\n\nbody {\n  background-color: var(--pst-color-background);\n  font-family: var(--pst-font-family-base);\n  font-weight: 400;\n  line-height: $line-height-body;\n  color: var(--pst-color-text-base);\n  min-height: 100vh;\n  display: flex;\n  flex-direction: column;\n\n  // hack to avoid the black background on some browser including Safari\n  &::-webkit-scrollbar-track {\n    background-color: var(--pst-color-background);\n  }\n}\n\np {\n  margin-bottom: 1.15rem;\n  font-size: 1em;\n\n  /* section header in docstring pages */\n  &.rubric {\n    border-bottom: 1px solid var(--pst-color-border);\n  }\n\n  &.centered {\n    text-align: center;\n  }\n}\n\na {\n  @include link-style-default;\n\n  // The # anchor that appears on hover over headings\n  &.headerlink {\n    color: var(--pst-color-secondary);\n    opacity: 0.7;\n    font-size: 0.8em;\n    padding: 0 4px;\n    margin-left: 0.2em;\n    text-decoration: none;\n    transition: all 0.2s ease-out;\n    user-select: none;\n\n    &:hover {\n      opacity: 1;\n    }\n  }\n\n  // set up a icon next to the shorten links from github and gitlab\n  &.github,\n  &.gitlab {\n    &::before {\n      color: var(--pst-color-text-muted);\n      font: var(--fa-font-brands);\n      margin-right: 0.25rem;\n    }\n  }\n\n  &.github::before {\n    content: var(--pst-icon-github);\n  }\n\n  &.gitlab::before {\n    content: var(--pst-icon-gitlab);\n  }\n}\n\n%heading-style {\n  margin: 2.75rem 0 1.05rem;\n  font-family: var(--pst-font-family-heading);\n  font-weight: var(--pst-font-weight-heading);\n  line-height: 1.15;\n}\n\nh1 {\n  @extend %heading-style;\n\n  margin-top: 0;\n  font-size: var(--pst-font-size-h1);\n  color: var(--pst-heading-color);\n}\n\nh2 {\n  @extend %heading-style;\n\n  font-size: var(--pst-font-size-h2);\n  color: var(--pst-heading-color);\n}\n\nh3 {\n  @extend %heading-style;\n\n  font-size: var(--pst-font-size-h3);\n  color: var(--pst-heading-color);\n}\n\nh4 {\n  @extend %heading-style;\n\n  font-size: var(--pst-font-size-h4);\n  color: var(--pst-heading-color);\n}\n\nh5 {\n  @extend %heading-style;\n\n  font-size: var(--pst-font-size-h5);\n  color: var(--pst-color-text-base);\n}\n\nh6 {\n  @extend %heading-style;\n\n  font-size: var(--pst-font-size-h6);\n  color: var(--pst-color-text-base);\n}\n\nsmall,\n.text_small {\n  font-size: var(--pst-font-size-milli);\n}\n\nhr {\n  border: 0;\n  border-top: 1px solid var(--pst-color-border);\n}\n\npre,\ncode,\nkbd,\nsamp {\n  font-family: var(--pst-font-family-monospace);\n}\n\nkbd {\n  // use theme negative\n  background-color: var(--pst-color-on-background);\n  color: var(--pst-color-text-muted);\n\n  // Compound keyboard elements will have nested kbd in them, so this prevents double lines\n  &:not(.compound) {\n    border: 1px solid var(--pst-color-border);\n    margin: 0 0.1rem;\n    padding: 0.1rem 0.4rem;\n    box-shadow: 1px 1px 1px var(--pst-color-shadow);\n  }\n}\n\ncode {\n  color: var(--pst-color-inline-code);\n}\n\npre {\n  margin: 1.5em 0;\n  padding: 1rem;\n  background-color: var(--pst-color-surface);\n  color: var(--pst-color-text-base);\n  line-height: 1.2em;\n  border: 1px solid var(--pst-color-border);\n  border-radius: $admonition-border-radius;\n\n  .linenos {\n    // minimum opacity to make the line numbers WCAG AA conformant\n    opacity: 0.8;\n    padding-right: 10px;\n  }\n}\n\n// the back to top btn\n#pst-back-to-top {\n  z-index: $zindex-tooltip;\n  position: fixed;\n  display: none;\n  top: 90vh;\n  left: 50vw;\n  transform: translate(-50%);\n  color: var(--pst-color-secondary-text);\n  background-color: var(--pst-color-secondary);\n  border: none;\n\n  .fa-arrow-up {\n    // Using margin instead of a space character prevents the space between the\n    // icon and the text from being underlined when the button is hovered.\n    margin-inline-end: 0.5em;\n  }\n\n  &:hover {\n    @include link-style-hover;\n\n    text-decoration-thickness: 1px;\n    background-color: var(--pst-violet-600);\n    color: var(--pst-color-secondary-text);\n  }\n\n  &:focus-visible {\n    box-shadow: none;\n    outline: $focus-ring-outline;\n    outline-color: var(--pst-color-secondary);\n    outline-offset: $focus-ring-width;\n  }\n}\n\n// Focus ring\n// ----------\n// Note: The Bootstrap stylesheet provides the focus ring (customized by this\n// theme via Sass variables in _bootstrap.scss) in some cases. This rule covers\n// all other cases.\n:focus-visible {\n  outline: $focus-ring-outline;\n  box-shadow: none; // override Bootstrap\n}\n","// Consistent styling for links\n// ============================\n\n@use \"sass:string\";\n\n// Define some useful variables for links styling consistency\n\n// The default thickness of the underline for links will be either:\n//  - 1px\n//  - 0.0625rem if it's thicker than 1px because the user has changed the text\n//    size in their browser\n$link-underline-thickness: string.unquote(\"max(1px, .0625rem)\") !default;\n\n// Offset of link underlines from text baseline\n// The default is 3px expressed as ems, as calculated against the default body\n// font size (on desktop).\n$link-underline-offset: 0.1578em !default;\n\n// Thickness of link underlines in hover state\n// The default for each link will be the thickest of the following:\n//  - 3px\n//  - 0.1875rem, if it's thicker than 3px because the user has changed the text\n//    size in their browser\n//  - 0.12em (relative to the link's text size)\n$link-hover-decoration-thickness: string.unquote(\n  \"max(3px, .1875rem, .12em)\"\n) !default;\n\n// Ensures links have an underline decoration by default - needed to meet\n// WCAG SC 1.4.1\n@mixin link-decoration {\n  text-decoration: underline;\n\n  @if $link-underline-thickness {\n    text-decoration-thickness: $link-underline-thickness;\n  }\n\n  @if $link-underline-offset {\n    text-underline-offset: $link-underline-offset;\n  }\n}\n\n// Ensures links have an underline decoration on hover - distinct from the\n// default behaviour\n@mixin link-decoration-hover {\n  @if $link-hover-decoration-thickness {\n    text-decoration-thickness: $link-hover-decoration-thickness;\n\n    // Disable ink skipping on underlines on hover.\n    text-decoration-skip-ink: none;\n  }\n}\n\n// Simple hover style - can be used alone or in conjunction with other mixins\n// Add the text underline and change in thickness on hover.\n// Intended for use with the `:hover` pseudo-class.\n@mixin link-style-hover {\n  @include link-decoration;\n  @include link-decoration-hover;\n\n  color: var(--pst-color-link-hover);\n}\n\n// Default link styles\n// -------------------\n// Defines: default unvisited, visited, hover, and active.\n// TODO: @trallard to improve focus styles in subsequent PR\n@mixin link-style-default {\n  // So that really long links don't spill out of their container\n  word-wrap: break-word;\n  color: var(--pst-color-link);\n\n  @include link-decoration;\n\n  &:hover {\n    color: var(--pst-color-link-hover);\n\n    @include link-decoration-hover;\n  }\n\n  // TODO: @trallard to add active styles in subsequent PR\n  &:active {\n    color: var(--pst-color-link);\n  }\n\n  // Visited should still be hoverable\n  &:visited {\n    color: var(--pst-color-link);\n\n    &:hover {\n      color: var(--pst-color-link-hover);\n    }\n  }\n}\n\n// Text link styles\n// ----------------\n// Makes links use the muted text colour and removes the underline.\n// Use this mixin for navigation bar links.\n@mixin link-style-text {\n  color: var(--pst-color-text-muted);\n  text-decoration: none;\n\n  &:hover {\n    color: var(--pst-color-link-hover);\n\n    @include link-decoration;\n    @include link-decoration-hover;\n  }\n}\n\n// Sidebar and TOC links\n// ---------------------\n// Makes links use the muted text colour and removes the underline.\n// Use this mixin for navigation the primary sidebar and table of contents.\n// Active and hover should work together rather than one overriding the other.\n@mixin link-sidebar {\n  color: var(--pst-color-text-muted);\n  text-decoration: none;\n\n  &:hover {\n    text-decoration: underline;\n    background-color: transparent;\n    color: var(--pst-color-link-hover);\n\n    @include link-decoration-hover;\n  }\n\n  // TODO: @trallard to update active styles in subsequent PR\n  &:active {\n    color: var(--pst-color-link-hover);\n  }\n\n  &:focus-visible {\n    box-shadow: $focus-ring-box-shadow;\n    outline: none;\n    z-index: 10; // keep focus ring on top (prevent the link-sidebar-current notch from lying on top of the ring)\n  }\n}\n\n// Sidebar current page link styles\n// --------------------------------\n// Adds a vertical line on the left hand side of the link to indicate that\n// it's the current page. Note this is distinct from an active state.\n// Used on the primary sidebar and the TOC.\n// We want the side box shadow to have the same thickness as the hover underline\n@mixin link-sidebar-current {\n  font-weight: 600;\n  color: var(--pst-color-primary);\n\n  @if $link-hover-decoration-thickness {\n    $notch-shadow: inset\n      $link-hover-decoration-thickness\n      0\n      0\n      var(--pst-color-primary);\n\n    box-shadow: $notch-shadow;\n\n    &:focus-visible {\n      box-shadow: $notch-shadow, $focus-ring-box-shadow;\n      outline: none;\n    }\n  }\n}\n\n// Heaver navbar text and icon links\n// ---------------------------------\n// (includes light/dark mode button)\n\n// This mixin makes it possible to show hover/underline and focus/ring styles at\n// the same time. The trick is to use:\n//    - a pseudo-element with bottom border for the hover underline\n//    - a CSS outline for the focus ring.\n\n// Normally we use box-shadow for underline and outline for focus ring. But we\n// cannot apply box-shadow and outline together on the same element because the\n// border-radius value that we use to round the outline will also round the\n// box-shadow used for the underline. We also cannot use text-underline because\n// it does not work on non-text links, nor do we want to use it on text links\n// that we want to treat as blocks, such as the header nav links because the\n// underline will wrap across two lines if the link text also wraps across two\n// lines.\n@mixin link-style-block {\n  color: var(--pst-color-text-muted);\n\n  // Set position relative so that the child ::before pseudo-element's absolute\n  // position is relative to this element.\n  position: relative;\n\n  // Set up pseudo-element used for hover underline styles\n  &::before {\n    content: \"\";\n    display: block;\n    position: absolute;\n    inset: 0;\n    background-color: transparent;\n\n    @if $link-hover-decoration-thickness {\n      bottom: calc(-1 * $link-hover-decoration-thickness);\n      margin: $link-hover-decoration-thickness 0;\n    }\n  }\n\n  &:hover {\n    color: var(--pst-color-secondary);\n    text-decoration: none; // override the link-style-hover mixin\n    &::before {\n      @if $link-hover-decoration-thickness {\n        border-bottom: $link-hover-decoration-thickness\n          solid\n          var(--pst-color-secondary);\n      }\n    }\n  }\n\n  &:focus-visible {\n    box-shadow: none; // override Bootstrap\n    outline: 3px solid var(--pst-color-accent);\n    outline-offset: 3px;\n  }\n}\n","// Variables\n//\n// Variables should follow the `$component-state-property-size` formula for\n// consistent naming. Ex: $nav-link-disabled-color and $modal-content-box-shadow-xs.\n\n// Color system\n\n// scss-docs-start gray-color-variables\n$white:    #fff !default;\n$gray-100: #f8f9fa !default;\n$gray-200: #e9ecef !default;\n$gray-300: #dee2e6 !default;\n$gray-400: #ced4da !default;\n$gray-500: #adb5bd !default;\n$gray-600: #6c757d !default;\n$gray-700: #495057 !default;\n$gray-800: #343a40 !default;\n$gray-900: #212529 !default;\n$black:    #000 !default;\n// scss-docs-end gray-color-variables\n\n// fusv-disable\n// scss-docs-start gray-colors-map\n$grays: (\n  \"100\": $gray-100,\n  \"200\": $gray-200,\n  \"300\": $gray-300,\n  \"400\": $gray-400,\n  \"500\": $gray-500,\n  \"600\": $gray-600,\n  \"700\": $gray-700,\n  \"800\": $gray-800,\n  \"900\": $gray-900\n) !default;\n// scss-docs-end gray-colors-map\n// fusv-enable\n\n// scss-docs-start color-variables\n$blue:    #0d6efd !default;\n$indigo:  #6610f2 !default;\n$purple:  #6f42c1 !default;\n$pink:    #d63384 !default;\n$red:     #dc3545 !default;\n$orange:  #fd7e14 !default;\n$yellow:  #ffc107 !default;\n$green:   #198754 !default;\n$teal:    #20c997 !default;\n$cyan:    #0dcaf0 !default;\n// scss-docs-end color-variables\n\n// scss-docs-start colors-map\n$colors: (\n  \"blue\":       $blue,\n  \"indigo\":     $indigo,\n  \"purple\":     $purple,\n  \"pink\":       $pink,\n  \"red\":        $red,\n  \"orange\":     $orange,\n  \"yellow\":     $yellow,\n  \"green\":      $green,\n  \"teal\":       $teal,\n  \"cyan\":       $cyan,\n  \"black\":      $black,\n  \"white\":      $white,\n  \"gray\":       $gray-600,\n  \"gray-dark\":  $gray-800\n) !default;\n// scss-docs-end colors-map\n\n// The contrast ratio to reach against white, to determine if color changes from \"light\" to \"dark\". Acceptable values for WCAG 2.0 are 3, 4.5 and 7.\n// See https://www.w3.org/TR/WCAG20/#visual-audio-contrast-contrast\n$min-contrast-ratio:   4.5 !default;\n\n// Customize the light and dark text colors for use in our color contrast function.\n$color-contrast-dark:      $black !default;\n$color-contrast-light:     $white !default;\n\n// fusv-disable\n$blue-100: tint-color($blue, 80%) !default;\n$blue-200: tint-color($blue, 60%) !default;\n$blue-300: tint-color($blue, 40%) !default;\n$blue-400: tint-color($blue, 20%) !default;\n$blue-500: $blue !default;\n$blue-600: shade-color($blue, 20%) !default;\n$blue-700: shade-color($blue, 40%) !default;\n$blue-800: shade-color($blue, 60%) !default;\n$blue-900: shade-color($blue, 80%) !default;\n\n$indigo-100: tint-color($indigo, 80%) !default;\n$indigo-200: tint-color($indigo, 60%) !default;\n$indigo-300: tint-color($indigo, 40%) !default;\n$indigo-400: tint-color($indigo, 20%) !default;\n$indigo-500: $indigo !default;\n$indigo-600: shade-color($indigo, 20%) !default;\n$indigo-700: shade-color($indigo, 40%) !default;\n$indigo-800: shade-color($indigo, 60%) !default;\n$indigo-900: shade-color($indigo, 80%) !default;\n\n$purple-100: tint-color($purple, 80%) !default;\n$purple-200: tint-color($purple, 60%) !default;\n$purple-300: tint-color($purple, 40%) !default;\n$purple-400: tint-color($purple, 20%) !default;\n$purple-500: $purple !default;\n$purple-600: shade-color($purple, 20%) !default;\n$purple-700: shade-color($purple, 40%) !default;\n$purple-800: shade-color($purple, 60%) !default;\n$purple-900: shade-color($purple, 80%) !default;\n\n$pink-100: tint-color($pink, 80%) !default;\n$pink-200: tint-color($pink, 60%) !default;\n$pink-300: tint-color($pink, 40%) !default;\n$pink-400: tint-color($pink, 20%) !default;\n$pink-500: $pink !default;\n$pink-600: shade-color($pink, 20%) !default;\n$pink-700: shade-color($pink, 40%) !default;\n$pink-800: shade-color($pink, 60%) !default;\n$pink-900: shade-color($pink, 80%) !default;\n\n$red-100: tint-color($red, 80%) !default;\n$red-200: tint-color($red, 60%) !default;\n$red-300: tint-color($red, 40%) !default;\n$red-400: tint-color($red, 20%) !default;\n$red-500: $red !default;\n$red-600: shade-color($red, 20%) !default;\n$red-700: shade-color($red, 40%) !default;\n$red-800: shade-color($red, 60%) !default;\n$red-900: shade-color($red, 80%) !default;\n\n$orange-100: tint-color($orange, 80%) !default;\n$orange-200: tint-color($orange, 60%) !default;\n$orange-300: tint-color($orange, 40%) !default;\n$orange-400: tint-color($orange, 20%) !default;\n$orange-500: $orange !default;\n$orange-600: shade-color($orange, 20%) !default;\n$orange-700: shade-color($orange, 40%) !default;\n$orange-800: shade-color($orange, 60%) !default;\n$orange-900: shade-color($orange, 80%) !default;\n\n$yellow-100: tint-color($yellow, 80%) !default;\n$yellow-200: tint-color($yellow, 60%) !default;\n$yellow-300: tint-color($yellow, 40%) !default;\n$yellow-400: tint-color($yellow, 20%) !default;\n$yellow-500: $yellow !default;\n$yellow-600: shade-color($yellow, 20%) !default;\n$yellow-700: shade-color($yellow, 40%) !default;\n$yellow-800: shade-color($yellow, 60%) !default;\n$yellow-900: shade-color($yellow, 80%) !default;\n\n$green-100: tint-color($green, 80%) !default;\n$green-200: tint-color($green, 60%) !default;\n$green-300: tint-color($green, 40%) !default;\n$green-400: tint-color($green, 20%) !default;\n$green-500: $green !default;\n$green-600: shade-color($green, 20%) !default;\n$green-700: shade-color($green, 40%) !default;\n$green-800: shade-color($green, 60%) !default;\n$green-900: shade-color($green, 80%) !default;\n\n$teal-100: tint-color($teal, 80%) !default;\n$teal-200: tint-color($teal, 60%) !default;\n$teal-300: tint-color($teal, 40%) !default;\n$teal-400: tint-color($teal, 20%) !default;\n$teal-500: $teal !default;\n$teal-600: shade-color($teal, 20%) !default;\n$teal-700: shade-color($teal, 40%) !default;\n$teal-800: shade-color($teal, 60%) !default;\n$teal-900: shade-color($teal, 80%) !default;\n\n$cyan-100: tint-color($cyan, 80%) !default;\n$cyan-200: tint-color($cyan, 60%) !default;\n$cyan-300: tint-color($cyan, 40%) !default;\n$cyan-400: tint-color($cyan, 20%) !default;\n$cyan-500: $cyan !default;\n$cyan-600: shade-color($cyan, 20%) !default;\n$cyan-700: shade-color($cyan, 40%) !default;\n$cyan-800: shade-color($cyan, 60%) !default;\n$cyan-900: shade-color($cyan, 80%) !default;\n\n$blues: (\n  \"blue-100\": $blue-100,\n  \"blue-200\": $blue-200,\n  \"blue-300\": $blue-300,\n  \"blue-400\": $blue-400,\n  \"blue-500\": $blue-500,\n  \"blue-600\": $blue-600,\n  \"blue-700\": $blue-700,\n  \"blue-800\": $blue-800,\n  \"blue-900\": $blue-900\n) !default;\n\n$indigos: (\n  \"indigo-100\": $indigo-100,\n  \"indigo-200\": $indigo-200,\n  \"indigo-300\": $indigo-300,\n  \"indigo-400\": $indigo-400,\n  \"indigo-500\": $indigo-500,\n  \"indigo-600\": $indigo-600,\n  \"indigo-700\": $indigo-700,\n  \"indigo-800\": $indigo-800,\n  \"indigo-900\": $indigo-900\n) !default;\n\n$purples: (\n  \"purple-100\": $purple-100,\n  \"purple-200\": $purple-200,\n  \"purple-300\": $purple-300,\n  \"purple-400\": $purple-400,\n  \"purple-500\": $purple-500,\n  \"purple-600\": $purple-600,\n  \"purple-700\": $purple-700,\n  \"purple-800\": $purple-800,\n  \"purple-900\": $purple-900\n) !default;\n\n$pinks: (\n  \"pink-100\": $pink-100,\n  \"pink-200\": $pink-200,\n  \"pink-300\": $pink-300,\n  \"pink-400\": $pink-400,\n  \"pink-500\": $pink-500,\n  \"pink-600\": $pink-600,\n  \"pink-700\": $pink-700,\n  \"pink-800\": $pink-800,\n  \"pink-900\": $pink-900\n) !default;\n\n$reds: (\n  \"red-100\": $red-100,\n  \"red-200\": $red-200,\n  \"red-300\": $red-300,\n  \"red-400\": $red-400,\n  \"red-500\": $red-500,\n  \"red-600\": $red-600,\n  \"red-700\": $red-700,\n  \"red-800\": $red-800,\n  \"red-900\": $red-900\n) !default;\n\n$oranges: (\n  \"orange-100\": $orange-100,\n  \"orange-200\": $orange-200,\n  \"orange-300\": $orange-300,\n  \"orange-400\": $orange-400,\n  \"orange-500\": $orange-500,\n  \"orange-600\": $orange-600,\n  \"orange-700\": $orange-700,\n  \"orange-800\": $orange-800,\n  \"orange-900\": $orange-900\n) !default;\n\n$yellows: (\n  \"yellow-100\": $yellow-100,\n  \"yellow-200\": $yellow-200,\n  \"yellow-300\": $yellow-300,\n  \"yellow-400\": $yellow-400,\n  \"yellow-500\": $yellow-500,\n  \"yellow-600\": $yellow-600,\n  \"yellow-700\": $yellow-700,\n  \"yellow-800\": $yellow-800,\n  \"yellow-900\": $yellow-900\n) !default;\n\n$greens: (\n  \"green-100\": $green-100,\n  \"green-200\": $green-200,\n  \"green-300\": $green-300,\n  \"green-400\": $green-400,\n  \"green-500\": $green-500,\n  \"green-600\": $green-600,\n  \"green-700\": $green-700,\n  \"green-800\": $green-800,\n  \"green-900\": $green-900\n) !default;\n\n$teals: (\n  \"teal-100\": $teal-100,\n  \"teal-200\": $teal-200,\n  \"teal-300\": $teal-300,\n  \"teal-400\": $teal-400,\n  \"teal-500\": $teal-500,\n  \"teal-600\": $teal-600,\n  \"teal-700\": $teal-700,\n  \"teal-800\": $teal-800,\n  \"teal-900\": $teal-900\n) !default;\n\n$cyans: (\n  \"cyan-100\": $cyan-100,\n  \"cyan-200\": $cyan-200,\n  \"cyan-300\": $cyan-300,\n  \"cyan-400\": $cyan-400,\n  \"cyan-500\": $cyan-500,\n  \"cyan-600\": $cyan-600,\n  \"cyan-700\": $cyan-700,\n  \"cyan-800\": $cyan-800,\n  \"cyan-900\": $cyan-900\n) !default;\n// fusv-enable\n\n// scss-docs-start theme-color-variables\n$primary:       $blue !default;\n$secondary:     $gray-600 !default;\n$success:       $green !default;\n$info:          $cyan !default;\n$warning:       $yellow !default;\n$danger:        $red !default;\n$light:         $gray-100 !default;\n$dark:          $gray-900 !default;\n// scss-docs-end theme-color-variables\n\n// scss-docs-start theme-colors-map\n$theme-colors: (\n  \"primary\":    $primary,\n  \"secondary\":  $secondary,\n  \"success\":    $success,\n  \"info\":       $info,\n  \"warning\":    $warning,\n  \"danger\":     $danger,\n  \"light\":      $light,\n  \"dark\":       $dark\n) !default;\n// scss-docs-end theme-colors-map\n\n// scss-docs-start theme-text-variables\n$primary-text-emphasis:   shade-color($primary, 60%) !default;\n$secondary-text-emphasis: shade-color($secondary, 60%) !default;\n$success-text-emphasis:   shade-color($success, 60%) !default;\n$info-text-emphasis:      shade-color($info, 60%) !default;\n$warning-text-emphasis:   shade-color($warning, 60%) !default;\n$danger-text-emphasis:    shade-color($danger, 60%) !default;\n$light-text-emphasis:     $gray-700 !default;\n$dark-text-emphasis:      $gray-700 !default;\n// scss-docs-end theme-text-variables\n\n// scss-docs-start theme-bg-subtle-variables\n$primary-bg-subtle:       tint-color($primary, 80%) !default;\n$secondary-bg-subtle:     tint-color($secondary, 80%) !default;\n$success-bg-subtle:       tint-color($success, 80%) !default;\n$info-bg-subtle:          tint-color($info, 80%) !default;\n$warning-bg-subtle:       tint-color($warning, 80%) !default;\n$danger-bg-subtle:        tint-color($danger, 80%) !default;\n$light-bg-subtle:         mix($gray-100, $white) !default;\n$dark-bg-subtle:          $gray-400 !default;\n// scss-docs-end theme-bg-subtle-variables\n\n// scss-docs-start theme-border-subtle-variables\n$primary-border-subtle:   tint-color($primary, 60%) !default;\n$secondary-border-subtle: tint-color($secondary, 60%) !default;\n$success-border-subtle:   tint-color($success, 60%) !default;\n$info-border-subtle:      tint-color($info, 60%) !default;\n$warning-border-subtle:   tint-color($warning, 60%) !default;\n$danger-border-subtle:    tint-color($danger, 60%) !default;\n$light-border-subtle:     $gray-200 !default;\n$dark-border-subtle:      $gray-500 !default;\n// scss-docs-end theme-border-subtle-variables\n\n// Characters which are escaped by the escape-svg function\n$escaped-characters: (\n  (\"<\", \"%3c\"),\n  (\">\", \"%3e\"),\n  (\"#\", \"%23\"),\n  (\"(\", \"%28\"),\n  (\")\", \"%29\"),\n) !default;\n\n// Options\n//\n// Quickly modify global styling by enabling or disabling optional features.\n\n$enable-caret:                true !default;\n$enable-rounded:              true !default;\n$enable-shadows:              false !default;\n$enable-gradients:            false !default;\n$enable-transitions:          true !default;\n$enable-reduced-motion:       true !default;\n$enable-smooth-scroll:        true !default;\n$enable-grid-classes:         true !default;\n$enable-container-classes:    true !default;\n$enable-cssgrid:              false !default;\n$enable-button-pointers:      true !default;\n$enable-rfs:                  true !default;\n$enable-validation-icons:     true !default;\n$enable-negative-margins:     false !default;\n$enable-deprecation-messages: true !default;\n$enable-important-utilities:  true !default;\n\n$enable-dark-mode:            true !default;\n$color-mode-type:             data !default; // `data` or `media-query`\n\n// Prefix for :root CSS variables\n\n$variable-prefix:             bs- !default; // Deprecated in v5.2.0 for the shorter `$prefix`\n$prefix:                      $variable-prefix !default;\n\n// Gradient\n//\n// The gradient which is added to components if `$enable-gradients` is `true`\n// This gradient is also added to elements with `.bg-gradient`\n// scss-docs-start variable-gradient\n$gradient: linear-gradient(180deg, rgba($white, .15), rgba($white, 0)) !default;\n// scss-docs-end variable-gradient\n\n// Spacing\n//\n// Control the default styling of most Bootstrap elements by modifying these\n// variables. Mostly focused on spacing.\n// You can add more entries to the $spacers map, should you need more variation.\n\n// scss-docs-start spacer-variables-maps\n$spacer: 1rem !default;\n$spacers: (\n  0: 0,\n  1: $spacer * .25,\n  2: $spacer * .5,\n  3: $spacer,\n  4: $spacer * 1.5,\n  5: $spacer * 3,\n) !default;\n// scss-docs-end spacer-variables-maps\n\n// Position\n//\n// Define the edge positioning anchors of the position utilities.\n\n// scss-docs-start position-map\n$position-values: (\n  0: 0,\n  50: 50%,\n  100: 100%\n) !default;\n// scss-docs-end position-map\n\n// Body\n//\n// Settings for the `<body>` element.\n\n$body-text-align:           null !default;\n$body-color:                $gray-900 !default;\n$body-bg:                   $white !default;\n\n$body-secondary-color:      rgba($body-color, .75) !default;\n$body-secondary-bg:         $gray-200 !default;\n\n$body-tertiary-color:       rgba($body-color, .5) !default;\n$body-tertiary-bg:          $gray-100 !default;\n\n$body-emphasis-color:       $black !default;\n\n// Links\n//\n// Style anchor elements.\n\n$link-color:                              $primary !default;\n$link-decoration:                         underline !default;\n$link-shade-percentage:                   20% !default;\n$link-hover-color:                        shift-color($link-color, $link-shade-percentage) !default;\n$link-hover-decoration:                   null !default;\n\n$stretched-link-pseudo-element:           after !default;\n$stretched-link-z-index:                  1 !default;\n\n// Icon links\n// scss-docs-start icon-link-variables\n$icon-link-gap:               .375rem !default;\n$icon-link-underline-offset:  .25em !default;\n$icon-link-icon-size:         1em !default;\n$icon-link-icon-transition:   .2s ease-in-out transform !default;\n$icon-link-icon-transform:    translate3d(.25em, 0, 0) !default;\n// scss-docs-end icon-link-variables\n\n// Paragraphs\n//\n// Style p element.\n\n$paragraph-margin-bottom:   1rem !default;\n\n\n// Grid breakpoints\n//\n// Define the minimum dimensions at which your layout will change,\n// adapting to different screen sizes, for use in media queries.\n\n// scss-docs-start grid-breakpoints\n$grid-breakpoints: (\n  xs: 0,\n  sm: 576px,\n  md: 768px,\n  lg: 992px,\n  xl: 1200px,\n  xxl: 1400px\n) !default;\n// scss-docs-end grid-breakpoints\n\n@include _assert-ascending($grid-breakpoints, \"$grid-breakpoints\");\n@include _assert-starts-at-zero($grid-breakpoints, \"$grid-breakpoints\");\n\n\n// Grid containers\n//\n// Define the maximum width of `.container` for different screen sizes.\n\n// scss-docs-start container-max-widths\n$container-max-widths: (\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1140px,\n  xxl: 1320px\n) !default;\n// scss-docs-end container-max-widths\n\n@include _assert-ascending($container-max-widths, \"$container-max-widths\");\n\n\n// Grid columns\n//\n// Set the number of columns and specify the width of the gutters.\n\n$grid-columns:                12 !default;\n$grid-gutter-width:           1.5rem !default;\n$grid-row-columns:            6 !default;\n\n// Container padding\n\n$container-padding-x: $grid-gutter-width !default;\n\n\n// Components\n//\n// Define common padding and border radius sizes and more.\n\n// scss-docs-start border-variables\n$border-width:                1px !default;\n$border-widths: (\n  1: 1px,\n  2: 2px,\n  3: 3px,\n  4: 4px,\n  5: 5px\n) !default;\n$border-style:                solid !default;\n$border-color:                $gray-300 !default;\n$border-color-translucent:    rgba($black, .175) !default;\n// scss-docs-end border-variables\n\n// scss-docs-start border-radius-variables\n$border-radius:               .375rem !default;\n$border-radius-sm:            .25rem !default;\n$border-radius-lg:            .5rem !default;\n$border-radius-xl:            1rem !default;\n$border-radius-xxl:           2rem !default;\n$border-radius-pill:          50rem !default;\n// scss-docs-end border-radius-variables\n// fusv-disable\n$border-radius-2xl:           $border-radius-xxl !default; // Deprecated in v5.3.0\n// fusv-enable\n\n// scss-docs-start box-shadow-variables\n$box-shadow:                  0 .5rem 1rem rgba($black, .15) !default;\n$box-shadow-sm:               0 .125rem .25rem rgba($black, .075) !default;\n$box-shadow-lg:               0 1rem 3rem rgba($black, .175) !default;\n$box-shadow-inset:            inset 0 1px 2px rgba($black, .075) !default;\n// scss-docs-end box-shadow-variables\n\n$component-active-color:      $white !default;\n$component-active-bg:         $primary !default;\n\n// scss-docs-start focus-ring-variables\n$focus-ring-width:      .25rem !default;\n$focus-ring-opacity:    .25 !default;\n$focus-ring-color:      rgba($primary, $focus-ring-opacity) !default;\n$focus-ring-blur:       0 !default;\n$focus-ring-box-shadow: 0 0 $focus-ring-blur $focus-ring-width $focus-ring-color !default;\n// scss-docs-end focus-ring-variables\n\n// scss-docs-start caret-variables\n$caret-width:                 .3em !default;\n$caret-vertical-align:        $caret-width * .85 !default;\n$caret-spacing:               $caret-width * .85 !default;\n// scss-docs-end caret-variables\n\n$transition-base:             all .2s ease-in-out !default;\n$transition-fade:             opacity .15s linear !default;\n// scss-docs-start collapse-transition\n$transition-collapse:         height .35s ease !default;\n$transition-collapse-width:   width .35s ease !default;\n// scss-docs-end collapse-transition\n\n// stylelint-disable function-disallowed-list\n// scss-docs-start aspect-ratios\n$aspect-ratios: (\n  \"1x1\": 100%,\n  \"4x3\": calc(3 / 4 * 100%),\n  \"16x9\": calc(9 / 16 * 100%),\n  \"21x9\": calc(9 / 21 * 100%)\n) !default;\n// scss-docs-end aspect-ratios\n// stylelint-enable function-disallowed-list\n\n// Typography\n//\n// Font, line-height, and color for body text, headings, and more.\n\n// scss-docs-start font-variables\n// stylelint-disable value-keyword-case\n$font-family-sans-serif:      system-ui, -apple-system, \"Segoe UI\", Roboto, \"Helvetica Neue\", \"Noto Sans\", \"Liberation Sans\", Arial, sans-serif, \"Apple Color Emoji\", \"Segoe UI Emoji\", \"Segoe UI Symbol\", \"Noto Color Emoji\" !default;\n$font-family-monospace:       SFMono-Regular, Menlo, Monaco, Consolas, \"Liberation Mono\", \"Courier New\", monospace !default;\n// stylelint-enable value-keyword-case\n$font-family-base:            var(--#{$prefix}font-sans-serif) !default;\n$font-family-code:            var(--#{$prefix}font-monospace) !default;\n\n// $font-size-root affects the value of `rem`, which is used for as well font sizes, paddings, and margins\n// $font-size-base affects the font size of the body text\n$font-size-root:              null !default;\n$font-size-base:              1rem !default; // Assumes the browser default, typically `16px`\n$font-size-sm:                $font-size-base * .875 !default;\n$font-size-lg:                $font-size-base * 1.25 !default;\n\n$font-weight-lighter:         lighter !default;\n$font-weight-light:           300 !default;\n$font-weight-normal:          400 !default;\n$font-weight-medium:          500 !default;\n$font-weight-semibold:        600 !default;\n$font-weight-bold:            700 !default;\n$font-weight-bolder:          bolder !default;\n\n$font-weight-base:            $font-weight-normal !default;\n\n$line-height-base:            1.5 !default;\n$line-height-sm:              1.25 !default;\n$line-height-lg:              2 !default;\n\n$h1-font-size:                $font-size-base * 2.5 !default;\n$h2-font-size:                $font-size-base * 2 !default;\n$h3-font-size:                $font-size-base * 1.75 !default;\n$h4-font-size:                $font-size-base * 1.5 !default;\n$h5-font-size:                $font-size-base * 1.25 !default;\n$h6-font-size:                $font-size-base !default;\n// scss-docs-end font-variables\n\n// scss-docs-start font-sizes\n$font-sizes: (\n  1: $h1-font-size,\n  2: $h2-font-size,\n  3: $h3-font-size,\n  4: $h4-font-size,\n  5: $h5-font-size,\n  6: $h6-font-size\n) !default;\n// scss-docs-end font-sizes\n\n// scss-docs-start headings-variables\n$headings-margin-bottom:      $spacer * .5 !default;\n$headings-font-family:        null !default;\n$headings-font-style:         null !default;\n$headings-font-weight:        500 !default;\n$headings-line-height:        1.2 !default;\n$headings-color:              inherit !default;\n// scss-docs-end headings-variables\n\n// scss-docs-start display-headings\n$display-font-sizes: (\n  1: 5rem,\n  2: 4.5rem,\n  3: 4rem,\n  4: 3.5rem,\n  5: 3rem,\n  6: 2.5rem\n) !default;\n\n$display-font-family: null !default;\n$display-font-style:  null !default;\n$display-font-weight: 300 !default;\n$display-line-height: $headings-line-height !default;\n// scss-docs-end display-headings\n\n// scss-docs-start type-variables\n$lead-font-size:              $font-size-base * 1.25 !default;\n$lead-font-weight:            300 !default;\n\n$small-font-size:             .875em !default;\n\n$sub-sup-font-size:           .75em !default;\n\n// fusv-disable\n$text-muted:                  var(--#{$prefix}secondary-color) !default; // Deprecated in 5.3.0\n// fusv-enable\n\n$initialism-font-size:        $small-font-size !default;\n\n$blockquote-margin-y:         $spacer !default;\n$blockquote-font-size:        $font-size-base * 1.25 !default;\n$blockquote-footer-color:     $gray-600 !default;\n$blockquote-footer-font-size: $small-font-size !default;\n\n$hr-margin-y:                 $spacer !default;\n$hr-color:                    inherit !default;\n\n// fusv-disable\n$hr-bg-color:                 null !default; // Deprecated in v5.2.0\n$hr-height:                   null !default; // Deprecated in v5.2.0\n// fusv-enable\n\n$hr-border-color:             null !default; // Allows for inherited colors\n$hr-border-width:             var(--#{$prefix}border-width) !default;\n$hr-opacity:                  .25 !default;\n\n// scss-docs-start vr-variables\n$vr-border-width:             var(--#{$prefix}border-width) !default;\n// scss-docs-end vr-variables\n\n$legend-margin-bottom:        .5rem !default;\n$legend-font-size:            1.5rem !default;\n$legend-font-weight:          null !default;\n\n$dt-font-weight:              $font-weight-bold !default;\n\n$list-inline-padding:         .5rem !default;\n\n$mark-padding:                .1875em !default;\n$mark-color:                  $body-color !default;\n$mark-bg:                     $yellow-100 !default;\n// scss-docs-end type-variables\n\n\n// Tables\n//\n// Customizes the `.table` component with basic values, each used across all table variations.\n\n// scss-docs-start table-variables\n$table-cell-padding-y:        .5rem !default;\n$table-cell-padding-x:        .5rem !default;\n$table-cell-padding-y-sm:     .25rem !default;\n$table-cell-padding-x-sm:     .25rem !default;\n\n$table-cell-vertical-align:   top !default;\n\n$table-color:                 var(--#{$prefix}emphasis-color) !default;\n$table-bg:                    var(--#{$prefix}body-bg) !default;\n$table-accent-bg:             transparent !default;\n\n$table-th-font-weight:        null !default;\n\n$table-striped-color:         $table-color !default;\n$table-striped-bg-factor:     .05 !default;\n$table-striped-bg:            rgba(var(--#{$prefix}emphasis-color-rgb), $table-striped-bg-factor) !default;\n\n$table-active-color:          $table-color !default;\n$table-active-bg-factor:      .1 !default;\n$table-active-bg:             rgba(var(--#{$prefix}emphasis-color-rgb), $table-active-bg-factor) !default;\n\n$table-hover-color:           $table-color !default;\n$table-hover-bg-factor:       .075 !default;\n$table-hover-bg:              rgba(var(--#{$prefix}emphasis-color-rgb), $table-hover-bg-factor) !default;\n\n$table-border-factor:         .2 !default;\n$table-border-width:          var(--#{$prefix}border-width) !default;\n$table-border-color:          var(--#{$prefix}border-color) !default;\n\n$table-striped-order:         odd !default;\n$table-striped-columns-order: even !default;\n\n$table-group-separator-color: currentcolor !default;\n\n$table-caption-color:         var(--#{$prefix}secondary-color) !default;\n\n$table-bg-scale:              -80% !default;\n// scss-docs-end table-variables\n\n// scss-docs-start table-loop\n$table-variants: (\n  \"primary\":    shift-color($primary, $table-bg-scale),\n  \"secondary\":  shift-color($secondary, $table-bg-scale),\n  \"success\":    shift-color($success, $table-bg-scale),\n  \"info\":       shift-color($info, $table-bg-scale),\n  \"warning\":    shift-color($warning, $table-bg-scale),\n  \"danger\":     shift-color($danger, $table-bg-scale),\n  \"light\":      $light,\n  \"dark\":       $dark,\n) !default;\n// scss-docs-end table-loop\n\n\n// Buttons + Forms\n//\n// Shared variables that are reassigned to `$input-` and `$btn-` specific variables.\n\n// scss-docs-start input-btn-variables\n$input-btn-padding-y:         .375rem !default;\n$input-btn-padding-x:         .75rem !default;\n$input-btn-font-family:       null !default;\n$input-btn-font-size:         $font-size-base !default;\n$input-btn-line-height:       $line-height-base !default;\n\n$input-btn-focus-width:         $focus-ring-width !default;\n$input-btn-focus-color-opacity: $focus-ring-opacity !default;\n$input-btn-focus-color:         $focus-ring-color !default;\n$input-btn-focus-blur:          $focus-ring-blur !default;\n$input-btn-focus-box-shadow:    $focus-ring-box-shadow !default;\n\n$input-btn-padding-y-sm:      .25rem !default;\n$input-btn-padding-x-sm:      .5rem !default;\n$input-btn-font-size-sm:      $font-size-sm !default;\n\n$input-btn-padding-y-lg:      .5rem !default;\n$input-btn-padding-x-lg:      1rem !default;\n$input-btn-font-size-lg:      $font-size-lg !default;\n\n$input-btn-border-width:      var(--#{$prefix}border-width) !default;\n// scss-docs-end input-btn-variables\n\n\n// Buttons\n//\n// For each of Bootstrap's buttons, define text, background, and border color.\n\n// scss-docs-start btn-variables\n$btn-color:                   var(--#{$prefix}body-color) !default;\n$btn-padding-y:               $input-btn-padding-y !default;\n$btn-padding-x:               $input-btn-padding-x !default;\n$btn-font-family:             $input-btn-font-family !default;\n$btn-font-size:               $input-btn-font-size !default;\n$btn-line-height:             $input-btn-line-height !default;\n$btn-white-space:             null !default; // Set to `nowrap` to prevent text wrapping\n\n$btn-padding-y-sm:            $input-btn-padding-y-sm !default;\n$btn-padding-x-sm:            $input-btn-padding-x-sm !default;\n$btn-font-size-sm:            $input-btn-font-size-sm !default;\n\n$btn-padding-y-lg:            $input-btn-padding-y-lg !default;\n$btn-padding-x-lg:            $input-btn-padding-x-lg !default;\n$btn-font-size-lg:            $input-btn-font-size-lg !default;\n\n$btn-border-width:            $input-btn-border-width !default;\n\n$btn-font-weight:             $font-weight-normal !default;\n$btn-box-shadow:              inset 0 1px 0 rgba($white, .15), 0 1px 1px rgba($black, .075) !default;\n$btn-focus-width:             $input-btn-focus-width !default;\n$btn-focus-box-shadow:        $input-btn-focus-box-shadow !default;\n$btn-disabled-opacity:        .65 !default;\n$btn-active-box-shadow:       inset 0 3px 5px rgba($black, .125) !default;\n\n$btn-link-color:              var(--#{$prefix}link-color) !default;\n$btn-link-hover-color:        var(--#{$prefix}link-hover-color) !default;\n$btn-link-disabled-color:     $gray-600 !default;\n$btn-link-focus-shadow-rgb:   to-rgb(mix(color-contrast($link-color), $link-color, 15%)) !default;\n\n// Allows for customizing button radius independently from global border radius\n$btn-border-radius:           var(--#{$prefix}border-radius) !default;\n$btn-border-radius-sm:        var(--#{$prefix}border-radius-sm) !default;\n$btn-border-radius-lg:        var(--#{$prefix}border-radius-lg) !default;\n\n$btn-transition:              color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$btn-hover-bg-shade-amount:       15% !default;\n$btn-hover-bg-tint-amount:        15% !default;\n$btn-hover-border-shade-amount:   20% !default;\n$btn-hover-border-tint-amount:    10% !default;\n$btn-active-bg-shade-amount:      20% !default;\n$btn-active-bg-tint-amount:       20% !default;\n$btn-active-border-shade-amount:  25% !default;\n$btn-active-border-tint-amount:   10% !default;\n// scss-docs-end btn-variables\n\n\n// Forms\n\n// scss-docs-start form-text-variables\n$form-text-margin-top:                  .25rem !default;\n$form-text-font-size:                   $small-font-size !default;\n$form-text-font-style:                  null !default;\n$form-text-font-weight:                 null !default;\n$form-text-color:                       var(--#{$prefix}secondary-color) !default;\n// scss-docs-end form-text-variables\n\n// scss-docs-start form-label-variables\n$form-label-margin-bottom:              .5rem !default;\n$form-label-font-size:                  null !default;\n$form-label-font-style:                 null !default;\n$form-label-font-weight:                null !default;\n$form-label-color:                      null !default;\n// scss-docs-end form-label-variables\n\n// scss-docs-start form-input-variables\n$input-padding-y:                       $input-btn-padding-y !default;\n$input-padding-x:                       $input-btn-padding-x !default;\n$input-font-family:                     $input-btn-font-family !default;\n$input-font-size:                       $input-btn-font-size !default;\n$input-font-weight:                     $font-weight-base !default;\n$input-line-height:                     $input-btn-line-height !default;\n\n$input-padding-y-sm:                    $input-btn-padding-y-sm !default;\n$input-padding-x-sm:                    $input-btn-padding-x-sm !default;\n$input-font-size-sm:                    $input-btn-font-size-sm !default;\n\n$input-padding-y-lg:                    $input-btn-padding-y-lg !default;\n$input-padding-x-lg:                    $input-btn-padding-x-lg !default;\n$input-font-size-lg:                    $input-btn-font-size-lg !default;\n\n$input-bg:                              var(--#{$prefix}body-bg) !default;\n$input-disabled-color:                  null !default;\n$input-disabled-bg:                     var(--#{$prefix}secondary-bg) !default;\n$input-disabled-border-color:           null !default;\n\n$input-color:                           var(--#{$prefix}body-color) !default;\n$input-border-color:                    var(--#{$prefix}border-color) !default;\n$input-border-width:                    $input-btn-border-width !default;\n$input-box-shadow:                      var(--#{$prefix}box-shadow-inset) !default;\n\n$input-border-radius:                   var(--#{$prefix}border-radius) !default;\n$input-border-radius-sm:                var(--#{$prefix}border-radius-sm) !default;\n$input-border-radius-lg:                var(--#{$prefix}border-radius-lg) !default;\n\n$input-focus-bg:                        $input-bg !default;\n$input-focus-border-color:              tint-color($component-active-bg, 50%) !default;\n$input-focus-color:                     $input-color !default;\n$input-focus-width:                     $input-btn-focus-width !default;\n$input-focus-box-shadow:                $input-btn-focus-box-shadow !default;\n\n$input-placeholder-color:               var(--#{$prefix}secondary-color) !default;\n$input-plaintext-color:                 var(--#{$prefix}body-color) !default;\n\n$input-height-border:                   calc(#{$input-border-width} * 2) !default; // stylelint-disable-line function-disallowed-list\n\n$input-height-inner:                    add($input-line-height * 1em, $input-padding-y * 2) !default;\n$input-height-inner-half:               add($input-line-height * .5em, $input-padding-y) !default;\n$input-height-inner-quarter:            add($input-line-height * .25em, $input-padding-y * .5) !default;\n\n$input-height:                          add($input-line-height * 1em, add($input-padding-y * 2, $input-height-border, false)) !default;\n$input-height-sm:                       add($input-line-height * 1em, add($input-padding-y-sm * 2, $input-height-border, false)) !default;\n$input-height-lg:                       add($input-line-height * 1em, add($input-padding-y-lg * 2, $input-height-border, false)) !default;\n\n$input-transition:                      border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$form-color-width:                      3rem !default;\n// scss-docs-end form-input-variables\n\n// scss-docs-start form-check-variables\n$form-check-input-width:                  1em !default;\n$form-check-min-height:                   $font-size-base * $line-height-base !default;\n$form-check-padding-start:                $form-check-input-width + .5em !default;\n$form-check-margin-bottom:                .125rem !default;\n$form-check-label-color:                  null !default;\n$form-check-label-cursor:                 null !default;\n$form-check-transition:                   null !default;\n\n$form-check-input-active-filter:          brightness(90%) !default;\n\n$form-check-input-bg:                     $input-bg !default;\n$form-check-input-border:                 var(--#{$prefix}border-width) solid var(--#{$prefix}border-color) !default;\n$form-check-input-border-radius:          .25em !default;\n$form-check-radio-border-radius:          50% !default;\n$form-check-input-focus-border:           $input-focus-border-color !default;\n$form-check-input-focus-box-shadow:       $focus-ring-box-shadow !default;\n\n$form-check-input-checked-color:          $component-active-color !default;\n$form-check-input-checked-bg-color:       $component-active-bg !default;\n$form-check-input-checked-border-color:   $form-check-input-checked-bg-color !default;\n$form-check-input-checked-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 20 20'><path fill='none' stroke='#{$form-check-input-checked-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='3' d='m6 10 3 3 6-6'/></svg>\") !default;\n$form-check-radio-checked-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='2' fill='#{$form-check-input-checked-color}'/></svg>\") !default;\n\n$form-check-input-indeterminate-color:          $component-active-color !default;\n$form-check-input-indeterminate-bg-color:       $component-active-bg !default;\n$form-check-input-indeterminate-border-color:   $form-check-input-indeterminate-bg-color !default;\n$form-check-input-indeterminate-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 20 20'><path fill='none' stroke='#{$form-check-input-indeterminate-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='3' d='M6 10h8'/></svg>\") !default;\n\n$form-check-input-disabled-opacity:        .5 !default;\n$form-check-label-disabled-opacity:        $form-check-input-disabled-opacity !default;\n$form-check-btn-check-disabled-opacity:    $btn-disabled-opacity !default;\n\n$form-check-inline-margin-end:    1rem !default;\n// scss-docs-end form-check-variables\n\n// scss-docs-start form-switch-variables\n$form-switch-color:               rgba($black, .25) !default;\n$form-switch-width:               2em !default;\n$form-switch-padding-start:       $form-switch-width + .5em !default;\n$form-switch-bg-image:            url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-color}'/></svg>\") !default;\n$form-switch-border-radius:       $form-switch-width !default;\n$form-switch-transition:          background-position .15s ease-in-out !default;\n\n$form-switch-focus-color:         $input-focus-border-color !default;\n$form-switch-focus-bg-image:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-focus-color}'/></svg>\") !default;\n\n$form-switch-checked-color:       $component-active-color !default;\n$form-switch-checked-bg-image:    url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-checked-color}'/></svg>\") !default;\n$form-switch-checked-bg-position: right center !default;\n// scss-docs-end form-switch-variables\n\n// scss-docs-start input-group-variables\n$input-group-addon-padding-y:           $input-padding-y !default;\n$input-group-addon-padding-x:           $input-padding-x !default;\n$input-group-addon-font-weight:         $input-font-weight !default;\n$input-group-addon-color:               $input-color !default;\n$input-group-addon-bg:                  var(--#{$prefix}tertiary-bg) !default;\n$input-group-addon-border-color:        $input-border-color !default;\n// scss-docs-end input-group-variables\n\n// scss-docs-start form-select-variables\n$form-select-padding-y:             $input-padding-y !default;\n$form-select-padding-x:             $input-padding-x !default;\n$form-select-font-family:           $input-font-family !default;\n$form-select-font-size:             $input-font-size !default;\n$form-select-indicator-padding:     $form-select-padding-x * 3 !default; // Extra padding for background-image\n$form-select-font-weight:           $input-font-weight !default;\n$form-select-line-height:           $input-line-height !default;\n$form-select-color:                 $input-color !default;\n$form-select-bg:                    $input-bg !default;\n$form-select-disabled-color:        null !default;\n$form-select-disabled-bg:           $input-disabled-bg !default;\n$form-select-disabled-border-color: $input-disabled-border-color !default;\n$form-select-bg-position:           right $form-select-padding-x center !default;\n$form-select-bg-size:               16px 12px !default; // In pixels because image dimensions\n$form-select-indicator-color:       $gray-800 !default;\n$form-select-indicator:             url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16'><path fill='none' stroke='#{$form-select-indicator-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='2' d='m2 5 6 6 6-6'/></svg>\") !default;\n\n$form-select-feedback-icon-padding-end: $form-select-padding-x * 2.5 + $form-select-indicator-padding !default;\n$form-select-feedback-icon-position:    center right $form-select-indicator-padding !default;\n$form-select-feedback-icon-size:        $input-height-inner-half $input-height-inner-half !default;\n\n$form-select-border-width:        $input-border-width !default;\n$form-select-border-color:        $input-border-color !default;\n$form-select-border-radius:       $input-border-radius !default;\n$form-select-box-shadow:          var(--#{$prefix}box-shadow-inset) !default;\n\n$form-select-focus-border-color:  $input-focus-border-color !default;\n$form-select-focus-width:         $input-focus-width !default;\n$form-select-focus-box-shadow:    0 0 0 $form-select-focus-width $input-btn-focus-color !default;\n\n$form-select-padding-y-sm:        $input-padding-y-sm !default;\n$form-select-padding-x-sm:        $input-padding-x-sm !default;\n$form-select-font-size-sm:        $input-font-size-sm !default;\n$form-select-border-radius-sm:    $input-border-radius-sm !default;\n\n$form-select-padding-y-lg:        $input-padding-y-lg !default;\n$form-select-padding-x-lg:        $input-padding-x-lg !default;\n$form-select-font-size-lg:        $input-font-size-lg !default;\n$form-select-border-radius-lg:    $input-border-radius-lg !default;\n\n$form-select-transition:          $input-transition !default;\n// scss-docs-end form-select-variables\n\n// scss-docs-start form-range-variables\n$form-range-track-width:          100% !default;\n$form-range-track-height:         .5rem !default;\n$form-range-track-cursor:         pointer !default;\n$form-range-track-bg:             var(--#{$prefix}secondary-bg) !default;\n$form-range-track-border-radius:  1rem !default;\n$form-range-track-box-shadow:     var(--#{$prefix}box-shadow-inset) !default;\n\n$form-range-thumb-width:                   1rem !default;\n$form-range-thumb-height:                  $form-range-thumb-width !default;\n$form-range-thumb-bg:                      $component-active-bg !default;\n$form-range-thumb-border:                  0 !default;\n$form-range-thumb-border-radius:           1rem !default;\n$form-range-thumb-box-shadow:              0 .1rem .25rem rgba($black, .1) !default;\n$form-range-thumb-focus-box-shadow:        0 0 0 1px $body-bg, $input-focus-box-shadow !default;\n$form-range-thumb-focus-box-shadow-width:  $input-focus-width !default; // For focus box shadow issue in Edge\n$form-range-thumb-active-bg:               tint-color($component-active-bg, 70%) !default;\n$form-range-thumb-disabled-bg:             var(--#{$prefix}secondary-color) !default;\n$form-range-thumb-transition:              background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n// scss-docs-end form-range-variables\n\n// scss-docs-start form-file-variables\n$form-file-button-color:          $input-color !default;\n$form-file-button-bg:             var(--#{$prefix}tertiary-bg) !default;\n$form-file-button-hover-bg:       var(--#{$prefix}secondary-bg) !default;\n// scss-docs-end form-file-variables\n\n// scss-docs-start form-floating-variables\n$form-floating-height:                  add(3.5rem, $input-height-border) !default;\n$form-floating-line-height:             1.25 !default;\n$form-floating-padding-x:               $input-padding-x !default;\n$form-floating-padding-y:               1rem !default;\n$form-floating-input-padding-t:         1.625rem !default;\n$form-floating-input-padding-b:         .625rem !default;\n$form-floating-label-height:            1.5em !default;\n$form-floating-label-opacity:           .65 !default;\n$form-floating-label-transform:         scale(.85) translateY(-.5rem) translateX(.15rem) !default;\n$form-floating-label-disabled-color:    $gray-600 !default;\n$form-floating-transition:              opacity .1s ease-in-out, transform .1s ease-in-out !default;\n// scss-docs-end form-floating-variables\n\n// Form validation\n\n// scss-docs-start form-feedback-variables\n$form-feedback-margin-top:          $form-text-margin-top !default;\n$form-feedback-font-size:           $form-text-font-size !default;\n$form-feedback-font-style:          $form-text-font-style !default;\n$form-feedback-valid-color:         $success !default;\n$form-feedback-invalid-color:       $danger !default;\n\n$form-feedback-icon-valid-color:    $form-feedback-valid-color !default;\n$form-feedback-icon-valid:          url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 8 8'><path fill='#{$form-feedback-icon-valid-color}' d='M2.3 6.73.6 4.53c-.4-1.04.46-1.4 1.1-.8l1.1 1.4 3.4-3.8c.6-.63 1.6-.27 1.2.7l-4 4.6c-.43.5-.8.4-1.1.1z'/></svg>\") !default;\n$form-feedback-icon-invalid-color:  $form-feedback-invalid-color !default;\n$form-feedback-icon-invalid:        url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 12 12' width='12' height='12' fill='none' stroke='#{$form-feedback-icon-invalid-color}'><circle cx='6' cy='6' r='4.5'/><path stroke-linejoin='round' d='M5.8 3.6h.4L6 6.5z'/><circle cx='6' cy='8.2' r='.6' fill='#{$form-feedback-icon-invalid-color}' stroke='none'/></svg>\") !default;\n// scss-docs-end form-feedback-variables\n\n// scss-docs-start form-validation-colors\n$form-valid-color:                  $form-feedback-valid-color !default;\n$form-valid-border-color:           $form-feedback-valid-color !default;\n$form-invalid-color:                $form-feedback-invalid-color !default;\n$form-invalid-border-color:         $form-feedback-invalid-color !default;\n// scss-docs-end form-validation-colors\n\n// scss-docs-start form-validation-states\n$form-validation-states: (\n  \"valid\": (\n    \"color\": var(--#{$prefix}form-valid-color),\n    \"icon\": $form-feedback-icon-valid,\n    \"tooltip-color\": #fff,\n    \"tooltip-bg-color\": var(--#{$prefix}success),\n    \"focus-box-shadow\": 0 0 $input-btn-focus-blur $input-focus-width rgba(var(--#{$prefix}success-rgb), $input-btn-focus-color-opacity),\n    \"border-color\": var(--#{$prefix}form-valid-border-color),\n  ),\n  \"invalid\": (\n    \"color\": var(--#{$prefix}form-invalid-color),\n    \"icon\": $form-feedback-icon-invalid,\n    \"tooltip-color\": #fff,\n    \"tooltip-bg-color\": var(--#{$prefix}danger),\n    \"focus-box-shadow\": 0 0 $input-btn-focus-blur $input-focus-width rgba(var(--#{$prefix}danger-rgb), $input-btn-focus-color-opacity),\n    \"border-color\": var(--#{$prefix}form-invalid-border-color),\n  )\n) !default;\n// scss-docs-end form-validation-states\n\n// Z-index master list\n//\n// Warning: Avoid customizing these values. They're used for a bird's eye view\n// of components dependent on the z-axis and are designed to all work together.\n\n// scss-docs-start zindex-stack\n$zindex-dropdown:                   1000 !default;\n$zindex-sticky:                     1020 !default;\n$zindex-fixed:                      1030 !default;\n$zindex-offcanvas-backdrop:         1040 !default;\n$zindex-offcanvas:                  1045 !default;\n$zindex-modal-backdrop:             1050 !default;\n$zindex-modal:                      1055 !default;\n$zindex-popover:                    1070 !default;\n$zindex-tooltip:                    1080 !default;\n$zindex-toast:                      1090 !default;\n// scss-docs-end zindex-stack\n\n// scss-docs-start zindex-levels-map\n$zindex-levels: (\n  n1: -1,\n  0: 0,\n  1: 1,\n  2: 2,\n  3: 3\n) !default;\n// scss-docs-end zindex-levels-map\n\n\n// Navs\n\n// scss-docs-start nav-variables\n$nav-link-padding-y:                .5rem !default;\n$nav-link-padding-x:                1rem !default;\n$nav-link-font-size:                null !default;\n$nav-link-font-weight:              null !default;\n$nav-link-color:                    var(--#{$prefix}link-color) !default;\n$nav-link-hover-color:              var(--#{$prefix}link-hover-color) !default;\n$nav-link-transition:               color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out !default;\n$nav-link-disabled-color:           var(--#{$prefix}secondary-color) !default;\n$nav-link-focus-box-shadow:         $focus-ring-box-shadow !default;\n\n$nav-tabs-border-color:             var(--#{$prefix}border-color) !default;\n$nav-tabs-border-width:             var(--#{$prefix}border-width) !default;\n$nav-tabs-border-radius:            var(--#{$prefix}border-radius) !default;\n$nav-tabs-link-hover-border-color:  var(--#{$prefix}secondary-bg) var(--#{$prefix}secondary-bg) $nav-tabs-border-color !default;\n$nav-tabs-link-active-color:        var(--#{$prefix}emphasis-color) !default;\n$nav-tabs-link-active-bg:           var(--#{$prefix}body-bg) !default;\n$nav-tabs-link-active-border-color: var(--#{$prefix}border-color) var(--#{$prefix}border-color) $nav-tabs-link-active-bg !default;\n\n$nav-pills-border-radius:           var(--#{$prefix}border-radius) !default;\n$nav-pills-link-active-color:       $component-active-color !default;\n$nav-pills-link-active-bg:          $component-active-bg !default;\n\n$nav-underline-gap:                 1rem !default;\n$nav-underline-border-width:        .125rem !default;\n$nav-underline-link-active-color:   var(--#{$prefix}emphasis-color) !default;\n// scss-docs-end nav-variables\n\n\n// Navbar\n\n// scss-docs-start navbar-variables\n$navbar-padding-y:                  $spacer * .5 !default;\n$navbar-padding-x:                  null !default;\n\n$navbar-nav-link-padding-x:         .5rem !default;\n\n$navbar-brand-font-size:            $font-size-lg !default;\n// Compute the navbar-brand padding-y so the navbar-brand will have the same height as navbar-text and nav-link\n$nav-link-height:                   $font-size-base * $line-height-base + $nav-link-padding-y * 2 !default;\n$navbar-brand-height:               $navbar-brand-font-size * $line-height-base !default;\n$navbar-brand-padding-y:            ($nav-link-height - $navbar-brand-height) * .5 !default;\n$navbar-brand-margin-end:           1rem !default;\n\n$navbar-toggler-padding-y:          .25rem !default;\n$navbar-toggler-padding-x:          .75rem !default;\n$navbar-toggler-font-size:          $font-size-lg !default;\n$navbar-toggler-border-radius:      $btn-border-radius !default;\n$navbar-toggler-focus-width:        $btn-focus-width !default;\n$navbar-toggler-transition:         box-shadow .15s ease-in-out !default;\n\n$navbar-light-color:                rgba(var(--#{$prefix}emphasis-color-rgb), .65) !default;\n$navbar-light-hover-color:          rgba(var(--#{$prefix}emphasis-color-rgb), .8) !default;\n$navbar-light-active-color:         rgba(var(--#{$prefix}emphasis-color-rgb), 1) !default;\n$navbar-light-disabled-color:       rgba(var(--#{$prefix}emphasis-color-rgb), .3) !default;\n$navbar-light-icon-color:           rgba($body-color, .75) !default;\n$navbar-light-toggler-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'><path stroke='#{$navbar-light-icon-color}' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/></svg>\") !default;\n$navbar-light-toggler-border-color: rgba(var(--#{$prefix}emphasis-color-rgb), .15) !default;\n$navbar-light-brand-color:          $navbar-light-active-color !default;\n$navbar-light-brand-hover-color:    $navbar-light-active-color !default;\n// scss-docs-end navbar-variables\n\n// scss-docs-start navbar-dark-variables\n$navbar-dark-color:                 rgba($white, .55) !default;\n$navbar-dark-hover-color:           rgba($white, .75) !default;\n$navbar-dark-active-color:          $white !default;\n$navbar-dark-disabled-color:        rgba($white, .25) !default;\n$navbar-dark-icon-color:            $navbar-dark-color !default;\n$navbar-dark-toggler-icon-bg:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'><path stroke='#{$navbar-dark-icon-color}' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/></svg>\") !default;\n$navbar-dark-toggler-border-color:  rgba($white, .1) !default;\n$navbar-dark-brand-color:           $navbar-dark-active-color !default;\n$navbar-dark-brand-hover-color:     $navbar-dark-active-color !default;\n// scss-docs-end navbar-dark-variables\n\n\n// Dropdowns\n//\n// Dropdown menu container and contents.\n\n// scss-docs-start dropdown-variables\n$dropdown-min-width:                10rem !default;\n$dropdown-padding-x:                0 !default;\n$dropdown-padding-y:                .5rem !default;\n$dropdown-spacer:                   .125rem !default;\n$dropdown-font-size:                $font-size-base !default;\n$dropdown-color:                    var(--#{$prefix}body-color) !default;\n$dropdown-bg:                       var(--#{$prefix}body-bg) !default;\n$dropdown-border-color:             var(--#{$prefix}border-color-translucent) !default;\n$dropdown-border-radius:            var(--#{$prefix}border-radius) !default;\n$dropdown-border-width:             var(--#{$prefix}border-width) !default;\n$dropdown-inner-border-radius:      calc(#{$dropdown-border-radius} - #{$dropdown-border-width}) !default; // stylelint-disable-line function-disallowed-list\n$dropdown-divider-bg:               $dropdown-border-color !default;\n$dropdown-divider-margin-y:         $spacer * .5 !default;\n$dropdown-box-shadow:               var(--#{$prefix}box-shadow) !default;\n\n$dropdown-link-color:               var(--#{$prefix}body-color) !default;\n$dropdown-link-hover-color:         $dropdown-link-color !default;\n$dropdown-link-hover-bg:            var(--#{$prefix}tertiary-bg) !default;\n\n$dropdown-link-active-color:        $component-active-color !default;\n$dropdown-link-active-bg:           $component-active-bg !default;\n\n$dropdown-link-disabled-color:      var(--#{$prefix}tertiary-color) !default;\n\n$dropdown-item-padding-y:           $spacer * .25 !default;\n$dropdown-item-padding-x:           $spacer !default;\n\n$dropdown-header-color:             $gray-600 !default;\n$dropdown-header-padding-x:         $dropdown-item-padding-x !default;\n$dropdown-header-padding-y:         $dropdown-padding-y !default;\n// fusv-disable\n$dropdown-header-padding:           $dropdown-header-padding-y $dropdown-header-padding-x !default; // Deprecated in v5.2.0\n// fusv-enable\n// scss-docs-end dropdown-variables\n\n// scss-docs-start dropdown-dark-variables\n$dropdown-dark-color:               $gray-300 !default;\n$dropdown-dark-bg:                  $gray-800 !default;\n$dropdown-dark-border-color:        $dropdown-border-color !default;\n$dropdown-dark-divider-bg:          $dropdown-divider-bg !default;\n$dropdown-dark-box-shadow:          null !default;\n$dropdown-dark-link-color:          $dropdown-dark-color !default;\n$dropdown-dark-link-hover-color:    $white !default;\n$dropdown-dark-link-hover-bg:       rgba($white, .15) !default;\n$dropdown-dark-link-active-color:   $dropdown-link-active-color !default;\n$dropdown-dark-link-active-bg:      $dropdown-link-active-bg !default;\n$dropdown-dark-link-disabled-color: $gray-500 !default;\n$dropdown-dark-header-color:        $gray-500 !default;\n// scss-docs-end dropdown-dark-variables\n\n\n// Pagination\n\n// scss-docs-start pagination-variables\n$pagination-padding-y:              .375rem !default;\n$pagination-padding-x:              .75rem !default;\n$pagination-padding-y-sm:           .25rem !default;\n$pagination-padding-x-sm:           .5rem !default;\n$pagination-padding-y-lg:           .75rem !default;\n$pagination-padding-x-lg:           1.5rem !default;\n\n$pagination-font-size:              $font-size-base !default;\n\n$pagination-color:                  var(--#{$prefix}link-color) !default;\n$pagination-bg:                     var(--#{$prefix}body-bg) !default;\n$pagination-border-radius:          var(--#{$prefix}border-radius) !default;\n$pagination-border-width:           var(--#{$prefix}border-width) !default;\n$pagination-margin-start:           calc(#{$pagination-border-width} * -1) !default; // stylelint-disable-line function-disallowed-list\n$pagination-border-color:           var(--#{$prefix}border-color) !default;\n\n$pagination-focus-color:            var(--#{$prefix}link-hover-color) !default;\n$pagination-focus-bg:               var(--#{$prefix}secondary-bg) !default;\n$pagination-focus-box-shadow:       $focus-ring-box-shadow !default;\n$pagination-focus-outline:          0 !default;\n\n$pagination-hover-color:            var(--#{$prefix}link-hover-color) !default;\n$pagination-hover-bg:               var(--#{$prefix}tertiary-bg) !default;\n$pagination-hover-border-color:     var(--#{$prefix}border-color) !default; // Todo in v6: remove this?\n\n$pagination-active-color:           $component-active-color !default;\n$pagination-active-bg:              $component-active-bg !default;\n$pagination-active-border-color:    $component-active-bg !default;\n\n$pagination-disabled-color:         var(--#{$prefix}secondary-color) !default;\n$pagination-disabled-bg:            var(--#{$prefix}secondary-bg) !default;\n$pagination-disabled-border-color:  var(--#{$prefix}border-color) !default;\n\n$pagination-transition:              color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$pagination-border-radius-sm:       var(--#{$prefix}border-radius-sm) !default;\n$pagination-border-radius-lg:       var(--#{$prefix}border-radius-lg) !default;\n// scss-docs-end pagination-variables\n\n\n// Placeholders\n\n// scss-docs-start placeholders\n$placeholder-opacity-max:           .5 !default;\n$placeholder-opacity-min:           .2 !default;\n// scss-docs-end placeholders\n\n// Cards\n\n// scss-docs-start card-variables\n$card-spacer-y:                     $spacer !default;\n$card-spacer-x:                     $spacer !default;\n$card-title-spacer-y:               $spacer * .5 !default;\n$card-title-color:                  null !default;\n$card-subtitle-color:               null !default;\n$card-border-width:                 var(--#{$prefix}border-width) !default;\n$card-border-color:                 var(--#{$prefix}border-color-translucent) !default;\n$card-border-radius:                var(--#{$prefix}border-radius) !default;\n$card-box-shadow:                   null !default;\n$card-inner-border-radius:          subtract($card-border-radius, $card-border-width) !default;\n$card-cap-padding-y:                $card-spacer-y * .5 !default;\n$card-cap-padding-x:                $card-spacer-x !default;\n$card-cap-bg:                       rgba(var(--#{$prefix}body-color-rgb), .03) !default;\n$card-cap-color:                    null !default;\n$card-height:                       null !default;\n$card-color:                        null !default;\n$card-bg:                           var(--#{$prefix}body-bg) !default;\n$card-img-overlay-padding:          $spacer !default;\n$card-group-margin:                 $grid-gutter-width * .5 !default;\n// scss-docs-end card-variables\n\n// Accordion\n\n// scss-docs-start accordion-variables\n$accordion-padding-y:                     1rem !default;\n$accordion-padding-x:                     1.25rem !default;\n$accordion-color:                         var(--#{$prefix}body-color) !default;\n$accordion-bg:                            var(--#{$prefix}body-bg) !default;\n$accordion-border-width:                  var(--#{$prefix}border-width) !default;\n$accordion-border-color:                  var(--#{$prefix}border-color) !default;\n$accordion-border-radius:                 var(--#{$prefix}border-radius) !default;\n$accordion-inner-border-radius:           subtract($accordion-border-radius, $accordion-border-width) !default;\n\n$accordion-body-padding-y:                $accordion-padding-y !default;\n$accordion-body-padding-x:                $accordion-padding-x !default;\n\n$accordion-button-padding-y:              $accordion-padding-y !default;\n$accordion-button-padding-x:              $accordion-padding-x !default;\n$accordion-button-color:                  var(--#{$prefix}body-color) !default;\n$accordion-button-bg:                     var(--#{$prefix}accordion-bg) !default;\n$accordion-transition:                    $btn-transition, border-radius .15s ease !default;\n$accordion-button-active-bg:              var(--#{$prefix}primary-bg-subtle) !default;\n$accordion-button-active-color:           var(--#{$prefix}primary-text-emphasis) !default;\n\n// fusv-disable\n$accordion-button-focus-border-color:     $input-focus-border-color !default; // Deprecated in v5.3.3\n// fusv-enable\n$accordion-button-focus-box-shadow:       $btn-focus-box-shadow !default;\n\n$accordion-icon-width:                    1.25rem !default;\n$accordion-icon-color:                    $body-color !default;\n$accordion-icon-active-color:             $primary-text-emphasis !default;\n$accordion-icon-transition:               transform .2s ease-in-out !default;\n$accordion-icon-transform:                rotate(-180deg) !default;\n\n$accordion-button-icon:         url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='none' stroke='#{$accordion-icon-color}' stroke-linecap='round' stroke-linejoin='round'><path d='M2 5L8 11L14 5'/></svg>\") !default;\n$accordion-button-active-icon:  url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='none' stroke='#{$accordion-icon-active-color}' stroke-linecap='round' stroke-linejoin='round'><path d='M2 5L8 11L14 5'/></svg>\") !default;\n// scss-docs-end accordion-variables\n\n// Tooltips\n\n// scss-docs-start tooltip-variables\n$tooltip-font-size:                 $font-size-sm !default;\n$tooltip-max-width:                 200px !default;\n$tooltip-color:                     var(--#{$prefix}body-bg) !default;\n$tooltip-bg:                        var(--#{$prefix}emphasis-color) !default;\n$tooltip-border-radius:             var(--#{$prefix}border-radius) !default;\n$tooltip-opacity:                   .9 !default;\n$tooltip-padding-y:                 $spacer * .25 !default;\n$tooltip-padding-x:                 $spacer * .5 !default;\n$tooltip-margin:                    null !default; // TODO: remove this in v6\n\n$tooltip-arrow-width:               .8rem !default;\n$tooltip-arrow-height:              .4rem !default;\n// fusv-disable\n$tooltip-arrow-color:               null !default; // Deprecated in Bootstrap 5.2.0 for CSS variables\n// fusv-enable\n// scss-docs-end tooltip-variables\n\n// Form tooltips must come after regular tooltips\n// scss-docs-start tooltip-feedback-variables\n$form-feedback-tooltip-padding-y:     $tooltip-padding-y !default;\n$form-feedback-tooltip-padding-x:     $tooltip-padding-x !default;\n$form-feedback-tooltip-font-size:     $tooltip-font-size !default;\n$form-feedback-tooltip-line-height:   null !default;\n$form-feedback-tooltip-opacity:       $tooltip-opacity !default;\n$form-feedback-tooltip-border-radius: $tooltip-border-radius !default;\n// scss-docs-end tooltip-feedback-variables\n\n\n// Popovers\n\n// scss-docs-start popover-variables\n$popover-font-size:                 $font-size-sm !default;\n$popover-bg:                        var(--#{$prefix}body-bg) !default;\n$popover-max-width:                 276px !default;\n$popover-border-width:              var(--#{$prefix}border-width) !default;\n$popover-border-color:              var(--#{$prefix}border-color-translucent) !default;\n$popover-border-radius:             var(--#{$prefix}border-radius-lg) !default;\n$popover-inner-border-radius:       calc(#{$popover-border-radius} - #{$popover-border-width}) !default; // stylelint-disable-line function-disallowed-list\n$popover-box-shadow:                var(--#{$prefix}box-shadow) !default;\n\n$popover-header-font-size:          $font-size-base !default;\n$popover-header-bg:                 var(--#{$prefix}secondary-bg) !default;\n$popover-header-color:              $headings-color !default;\n$popover-header-padding-y:          .5rem !default;\n$popover-header-padding-x:          $spacer !default;\n\n$popover-body-color:                var(--#{$prefix}body-color) !default;\n$popover-body-padding-y:            $spacer !default;\n$popover-body-padding-x:            $spacer !default;\n\n$popover-arrow-width:               1rem !default;\n$popover-arrow-height:              .5rem !default;\n// scss-docs-end popover-variables\n\n// fusv-disable\n// Deprecated in Bootstrap 5.2.0 for CSS variables\n$popover-arrow-color:               $popover-bg !default;\n$popover-arrow-outer-color:         var(--#{$prefix}border-color-translucent) !default;\n// fusv-enable\n\n\n// Toasts\n\n// scss-docs-start toast-variables\n$toast-max-width:                   350px !default;\n$toast-padding-x:                   .75rem !default;\n$toast-padding-y:                   .5rem !default;\n$toast-font-size:                   .875rem !default;\n$toast-color:                       null !default;\n$toast-background-color:            rgba(var(--#{$prefix}body-bg-rgb), .85) !default;\n$toast-border-width:                var(--#{$prefix}border-width) !default;\n$toast-border-color:                var(--#{$prefix}border-color-translucent) !default;\n$toast-border-radius:               var(--#{$prefix}border-radius) !default;\n$toast-box-shadow:                  var(--#{$prefix}box-shadow) !default;\n$toast-spacing:                     $container-padding-x !default;\n\n$toast-header-color:                var(--#{$prefix}secondary-color) !default;\n$toast-header-background-color:     rgba(var(--#{$prefix}body-bg-rgb), .85) !default;\n$toast-header-border-color:         $toast-border-color !default;\n// scss-docs-end toast-variables\n\n\n// Badges\n\n// scss-docs-start badge-variables\n$badge-font-size:                   .75em !default;\n$badge-font-weight:                 $font-weight-bold !default;\n$badge-color:                       $white !default;\n$badge-padding-y:                   .35em !default;\n$badge-padding-x:                   .65em !default;\n$badge-border-radius:               var(--#{$prefix}border-radius) !default;\n// scss-docs-end badge-variables\n\n\n// Modals\n\n// scss-docs-start modal-variables\n$modal-inner-padding:               $spacer !default;\n\n$modal-footer-margin-between:       .5rem !default;\n\n$modal-dialog-margin:               .5rem !default;\n$modal-dialog-margin-y-sm-up:       1.75rem !default;\n\n$modal-title-line-height:           $line-height-base !default;\n\n$modal-content-color:               null !default;\n$modal-content-bg:                  var(--#{$prefix}body-bg) !default;\n$modal-content-border-color:        var(--#{$prefix}border-color-translucent) !default;\n$modal-content-border-width:        var(--#{$prefix}border-width) !default;\n$modal-content-border-radius:       var(--#{$prefix}border-radius-lg) !default;\n$modal-content-inner-border-radius: subtract($modal-content-border-radius, $modal-content-border-width) !default;\n$modal-content-box-shadow-xs:       var(--#{$prefix}box-shadow-sm) !default;\n$modal-content-box-shadow-sm-up:    var(--#{$prefix}box-shadow) !default;\n\n$modal-backdrop-bg:                 $black !default;\n$modal-backdrop-opacity:            .5 !default;\n\n$modal-header-border-color:         var(--#{$prefix}border-color) !default;\n$modal-header-border-width:         $modal-content-border-width !default;\n$modal-header-padding-y:            $modal-inner-padding !default;\n$modal-header-padding-x:            $modal-inner-padding !default;\n$modal-header-padding:              $modal-header-padding-y $modal-header-padding-x !default; // Keep this for backwards compatibility\n\n$modal-footer-bg:                   null !default;\n$modal-footer-border-color:         $modal-header-border-color !default;\n$modal-footer-border-width:         $modal-header-border-width !default;\n\n$modal-sm:                          300px !default;\n$modal-md:                          500px !default;\n$modal-lg:                          800px !default;\n$modal-xl:                          1140px !default;\n\n$modal-fade-transform:              translate(0, -50px) !default;\n$modal-show-transform:              none !default;\n$modal-transition:                  transform .3s ease-out !default;\n$modal-scale-transform:             scale(1.02) !default;\n// scss-docs-end modal-variables\n\n\n// Alerts\n//\n// Define alert colors, border radius, and padding.\n\n// scss-docs-start alert-variables\n$alert-padding-y:               $spacer !default;\n$alert-padding-x:               $spacer !default;\n$alert-margin-bottom:           1rem !default;\n$alert-border-radius:           var(--#{$prefix}border-radius) !default;\n$alert-link-font-weight:        $font-weight-bold !default;\n$alert-border-width:            var(--#{$prefix}border-width) !default;\n$alert-dismissible-padding-r:   $alert-padding-x * 3 !default; // 3x covers width of x plus default padding on either side\n// scss-docs-end alert-variables\n\n// fusv-disable\n$alert-bg-scale:                -80% !default; // Deprecated in v5.2.0, to be removed in v6\n$alert-border-scale:            -70% !default; // Deprecated in v5.2.0, to be removed in v6\n$alert-color-scale:             40% !default; // Deprecated in v5.2.0, to be removed in v6\n// fusv-enable\n\n// Progress bars\n\n// scss-docs-start progress-variables\n$progress-height:                   1rem !default;\n$progress-font-size:                $font-size-base * .75 !default;\n$progress-bg:                       var(--#{$prefix}secondary-bg) !default;\n$progress-border-radius:            var(--#{$prefix}border-radius) !default;\n$progress-box-shadow:               var(--#{$prefix}box-shadow-inset) !default;\n$progress-bar-color:                $white !default;\n$progress-bar-bg:                   $primary !default;\n$progress-bar-animation-timing:     1s linear infinite !default;\n$progress-bar-transition:           width .6s ease !default;\n// scss-docs-end progress-variables\n\n\n// List group\n\n// scss-docs-start list-group-variables\n$list-group-color:                  var(--#{$prefix}body-color) !default;\n$list-group-bg:                     var(--#{$prefix}body-bg) !default;\n$list-group-border-color:           var(--#{$prefix}border-color) !default;\n$list-group-border-width:           var(--#{$prefix}border-width) !default;\n$list-group-border-radius:          var(--#{$prefix}border-radius) !default;\n\n$list-group-item-padding-y:         $spacer * .5 !default;\n$list-group-item-padding-x:         $spacer !default;\n// fusv-disable\n$list-group-item-bg-scale:          -80% !default; // Deprecated in v5.3.0\n$list-group-item-color-scale:       40% !default; // Deprecated in v5.3.0\n// fusv-enable\n\n$list-group-hover-bg:               var(--#{$prefix}tertiary-bg) !default;\n$list-group-active-color:           $component-active-color !default;\n$list-group-active-bg:              $component-active-bg !default;\n$list-group-active-border-color:    $list-group-active-bg !default;\n\n$list-group-disabled-color:         var(--#{$prefix}secondary-color) !default;\n$list-group-disabled-bg:            $list-group-bg !default;\n\n$list-group-action-color:           var(--#{$prefix}secondary-color) !default;\n$list-group-action-hover-color:     var(--#{$prefix}emphasis-color) !default;\n\n$list-group-action-active-color:    var(--#{$prefix}body-color) !default;\n$list-group-action-active-bg:       var(--#{$prefix}secondary-bg) !default;\n// scss-docs-end list-group-variables\n\n\n// Image thumbnails\n\n// scss-docs-start thumbnail-variables\n$thumbnail-padding:                 .25rem !default;\n$thumbnail-bg:                      var(--#{$prefix}body-bg) !default;\n$thumbnail-border-width:            var(--#{$prefix}border-width) !default;\n$thumbnail-border-color:            var(--#{$prefix}border-color) !default;\n$thumbnail-border-radius:           var(--#{$prefix}border-radius) !default;\n$thumbnail-box-shadow:              var(--#{$prefix}box-shadow-sm) !default;\n// scss-docs-end thumbnail-variables\n\n\n// Figures\n\n// scss-docs-start figure-variables\n$figure-caption-font-size:          $small-font-size !default;\n$figure-caption-color:              var(--#{$prefix}secondary-color) !default;\n// scss-docs-end figure-variables\n\n\n// Breadcrumbs\n\n// scss-docs-start breadcrumb-variables\n$breadcrumb-font-size:              null !default;\n$breadcrumb-padding-y:              0 !default;\n$breadcrumb-padding-x:              0 !default;\n$breadcrumb-item-padding-x:         .5rem !default;\n$breadcrumb-margin-bottom:          1rem !default;\n$breadcrumb-bg:                     null !default;\n$breadcrumb-divider-color:          var(--#{$prefix}secondary-color) !default;\n$breadcrumb-active-color:           var(--#{$prefix}secondary-color) !default;\n$breadcrumb-divider:                quote(\"/\") !default;\n$breadcrumb-divider-flipped:        $breadcrumb-divider !default;\n$breadcrumb-border-radius:          null !default;\n// scss-docs-end breadcrumb-variables\n\n// Carousel\n\n// scss-docs-start carousel-variables\n$carousel-control-color:             $white !default;\n$carousel-control-width:             15% !default;\n$carousel-control-opacity:           .5 !default;\n$carousel-control-hover-opacity:     .9 !default;\n$carousel-control-transition:        opacity .15s ease !default;\n\n$carousel-indicator-width:           30px !default;\n$carousel-indicator-height:          3px !default;\n$carousel-indicator-hit-area-height: 10px !default;\n$carousel-indicator-spacer:          3px !default;\n$carousel-indicator-opacity:         .5 !default;\n$carousel-indicator-active-bg:       $white !default;\n$carousel-indicator-active-opacity:  1 !default;\n$carousel-indicator-transition:      opacity .6s ease !default;\n\n$carousel-caption-width:             70% !default;\n$carousel-caption-color:             $white !default;\n$carousel-caption-padding-y:         1.25rem !default;\n$carousel-caption-spacer:            1.25rem !default;\n\n$carousel-control-icon-width:        2rem !default;\n\n$carousel-control-prev-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$carousel-control-color}'><path d='M11.354 1.646a.5.5 0 0 1 0 .708L5.707 8l5.647 5.646a.5.5 0 0 1-.708.708l-6-6a.5.5 0 0 1 0-.708l6-6a.5.5 0 0 1 .708 0z'/></svg>\") !default;\n$carousel-control-next-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$carousel-control-color}'><path d='M4.646 1.646a.5.5 0 0 1 .708 0l6 6a.5.5 0 0 1 0 .708l-6 6a.5.5 0 0 1-.708-.708L10.293 8 4.646 2.354a.5.5 0 0 1 0-.708z'/></svg>\") !default;\n\n$carousel-transition-duration:       .6s !default;\n$carousel-transition:                transform $carousel-transition-duration ease-in-out !default; // Define transform transition first if using multiple transitions (e.g., `transform 2s ease, opacity .5s ease-out`)\n// scss-docs-end carousel-variables\n\n// scss-docs-start carousel-dark-variables\n$carousel-dark-indicator-active-bg:  $black !default;\n$carousel-dark-caption-color:        $black !default;\n$carousel-dark-control-icon-filter:  invert(1) grayscale(100) !default;\n// scss-docs-end carousel-dark-variables\n\n\n// Spinners\n\n// scss-docs-start spinner-variables\n$spinner-width:           2rem !default;\n$spinner-height:          $spinner-width !default;\n$spinner-vertical-align:  -.125em !default;\n$spinner-border-width:    .25em !default;\n$spinner-animation-speed: .75s !default;\n\n$spinner-width-sm:        1rem !default;\n$spinner-height-sm:       $spinner-width-sm !default;\n$spinner-border-width-sm: .2em !default;\n// scss-docs-end spinner-variables\n\n\n// Close\n\n// scss-docs-start close-variables\n$btn-close-width:            1em !default;\n$btn-close-height:           $btn-close-width !default;\n$btn-close-padding-x:        .25em !default;\n$btn-close-padding-y:        $btn-close-padding-x !default;\n$btn-close-color:            $black !default;\n$btn-close-bg:               url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$btn-close-color}'><path d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/></svg>\") !default;\n$btn-close-focus-shadow:     $focus-ring-box-shadow !default;\n$btn-close-opacity:          .5 !default;\n$btn-close-hover-opacity:    .75 !default;\n$btn-close-focus-opacity:    1 !default;\n$btn-close-disabled-opacity: .25 !default;\n$btn-close-white-filter:     invert(1) grayscale(100%) brightness(200%) !default;\n// scss-docs-end close-variables\n\n\n// Offcanvas\n\n// scss-docs-start offcanvas-variables\n$offcanvas-padding-y:               $modal-inner-padding !default;\n$offcanvas-padding-x:               $modal-inner-padding !default;\n$offcanvas-horizontal-width:        400px !default;\n$offcanvas-vertical-height:         30vh !default;\n$offcanvas-transition-duration:     .3s !default;\n$offcanvas-border-color:            $modal-content-border-color !default;\n$offcanvas-border-width:            $modal-content-border-width !default;\n$offcanvas-title-line-height:       $modal-title-line-height !default;\n$offcanvas-bg-color:                var(--#{$prefix}body-bg) !default;\n$offcanvas-color:                   var(--#{$prefix}body-color) !default;\n$offcanvas-box-shadow:              $modal-content-box-shadow-xs !default;\n$offcanvas-backdrop-bg:             $modal-backdrop-bg !default;\n$offcanvas-backdrop-opacity:        $modal-backdrop-opacity !default;\n// scss-docs-end offcanvas-variables\n\n// Code\n\n$code-font-size:                    $small-font-size !default;\n$code-color:                        $pink !default;\n\n$kbd-padding-y:                     .1875rem !default;\n$kbd-padding-x:                     .375rem !default;\n$kbd-font-size:                     $code-font-size !default;\n$kbd-color:                         var(--#{$prefix}body-bg) !default;\n$kbd-bg:                            var(--#{$prefix}body-color) !default;\n$nested-kbd-font-weight:            null !default; // Deprecated in v5.2.0, removing in v6\n\n$pre-color:                         null !default;\n\n@import \"variables-dark\"; // TODO: can be removed safely in v6, only here to avoid breaking changes in v5.3\n","// Override bootstrap variables\n$spacer: 1rem;\n$container-max-widths: (\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1400px,\n);\n$grid-breakpoints: (\n  xs: 0,\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1200px,\n);\n$dropdown-link-hover-bg: var(--pst-color-surface);\n\n// --pst-color-surface can also be assigned to the dark variant because it is\n// scoped to different values depending on light/dark theme\n$dropdown-dark-link-hover-bg: var(--pst-color-surface);\n$dropdown-link-active-bg: var(--pst-color-surface);\n$dropdown-dark-link-active-bg: var(--pst-color-surface);\n$focus-ring-width: 0.1875rem; // 3px at 100% zoom (0.1875 * 16px base font = 3px)\n$focus-ring-opacity: 1;\n$focus-ring-color: var(--pst-color-accent);\n$focus-ring-blur: 0;\n$focus-ring-box-shadow: 0 0 $focus-ring-blur $focus-ring-width $focus-ring-color;\n\n// outline creates the same style of focus ring, it just uses CSS outline instead of box shadow\n$focus-ring-outline: $focus-ring-color solid $focus-ring-width;\n$btn-focus-box-shadow: $focus-ring-box-shadow;\n","/***\n * Rules for the UX/UI of skip navigation link btn.\n * It's only visible to people\n * navigating with keyboard for accessibility purposes\n * ref: https://www.youtube.com/watch?v=VUR0I5mqq7I\n ***/\n\n.skip-link {\n  position: fixed;\n  top: 0;\n  left: 0;\n  right: 0;\n  text-align: center;\n  background-color: var(--pst-color-warning);\n  padding: 0.5rem;\n  z-index: $zindex-modal;\n  border-bottom: 1px solid var(--pst-color-border);\n\n  // This shows / hides the button\n  transform: translateY(-100%);\n  transition: transform 150ms ease-in-out;\n\n  &:focus-within {\n    transform: translateY(0%);\n  }\n\n  a {\n    // Ensure we are using a WCAG conformant colour\n    color: var(--pst-color-warning-text) !important;\n\n    &:focus-visible {\n      // use color with sufficient contrast\n      outline-color: $foundation-black;\n    }\n  }\n}\n","/*******************************************************************************\n* master color map. Only the colors that actually differ between light and dark\n* themes are specified separately.\n*\n* To see the full list of colors see https://www.figma.com/file/rUrrHGhUBBIAAjQ82x6pz9/PyData-Design-system---proposal-for-implementation-(2)?node-id=1234%3A765&t=ifcFT1JtnrSshGfi-1\n*/\n\n@use \"sass:map\";\n@use \"sass:meta\";\n@use \"sass:string\";\n\n/**\n* Function to get items from nested maps\n*/\n// @param {Map} $map - Map\n// @param {Arglist} $keys - Keys to fetch\n// @return {*}\n@function map-deep-get($map, $keys...) {\n  @each $key in $keys {\n    $map: map.get($map, $key);\n  }\n\n  @return $map;\n}\n\n/* Assign base colors for the PyData theme */\n$color-palette: (\n  // Primary color\n  \"teal\":\n    (\n      \"50\": #f4fbfc,\n      \"100\": #e9f6f8,\n      \"200\": #d0ecf1,\n      \"300\": #abdde6,\n      \"400\": #3fb1c5,\n      \"500\": #0a7d91,\n      \"600\": #085d6c,\n      \"700\": #064752,\n      \"800\": #042c33,\n      \"900\": #021b1f,\n    ),\n  // Secondary color\n  \"violet\":\n    (\n      \"50\": #f4eefb,\n      \"100\": #e0c7ff,\n      \"200\": #d5b4fd,\n      \"300\": #b780ff,\n      \"400\": #9c5ffd,\n      \"500\": #8045e5,\n      \"600\": #6432bd,\n      \"700\": #4b258f,\n      \"800\": #341a61,\n      \"900\": #1e0e39,\n    ),\n  // Neutrals\n  \"gray\":\n    (\n      \"50\": #f9f9fa,\n      \"100\": #f3f4f5,\n      \"200\": #e5e7ea,\n      \"300\": #d1d5da,\n      \"400\": #9ca4af,\n      \"500\": #677384,\n      \"600\": #48566b,\n      \"700\": #29313d,\n      \"800\": #222832,\n      \"900\": #14181e,\n    ),\n  // Accent color\n  \"pink\":\n    (\n      \"50\": #fcf8fd,\n      \"100\": #fcf0fa,\n      \"200\": #f8dff5,\n      \"300\": #f3c7ee,\n      \"400\": #e47fd7,\n      \"500\": #c132af,\n      \"600\": #912583,\n      \"700\": #6e1c64,\n      \"800\": #46123f,\n      \"900\": #2b0b27,\n    ),\n  \"foundation\": (\n    \"white\": #ffffff,\n    // gray-900\n    \"black\": #14181e,\n  )\n);\n\n:root {\n  // Add theme colours to the html root element\n  @each $group-color, $color in $color-palette {\n    @each $color-name, $definition in $color {\n      --pst-#{$group-color}-#{$color-name}: #{$definition};\n    }\n  }\n}\n\n// Static SCSS variables used thoroughout the theme\n// Minimum contrast ratio used for the theme.\n// Acceptable values for WCAG 2.0 are 3, 4.5 and 7.\n// See https://www.w3.org/TR/WCAG20/#visual-audio-contrast-contrast\n// 4.5 - is for text that is 14pt or less\n$min-contrast-ratio-4: 4.5;\n\n// 3 is for text that is 18pt or bold, or for non-text elements\n$min-contrast-ratio-3: 3;\n\n// Customize the light and dark text colors for use in our color contrast function.\n$foundation-black: #14181e;\n$foundation-white: #fff;\n\n// This is a custom - calculated  color between gray 100 and 200 - to reduce\n// the contrast ratio (avoid a jarring effect)\n$base-light-text: #ced6dd;\n\n// used in sphinx_design - gray 100\n$foundation-light-gray: #f3f4f5;\n\n// used in sphinx_design - gray 700\n$foundation-muted-gray: #29313d;\n\n// used in sphinx_design - gray 800\n$foundation-dark-gray: #222832;\n$pst-semantic-colors: (\n  \"primary\": (\n    \"light\": #{map-deep-get($color-palette, \"teal\", \"500\")},\n    \"bg-light\": #{map-deep-get($color-palette, \"teal\", \"200\")},\n    \"dark\": #{map-deep-get($color-palette, \"teal\", \"400\")},\n    \"bg-dark\": #{map-deep-get($color-palette, \"teal\", \"800\")},\n  ),\n  \"secondary\": (\n    \"light\": #{map-deep-get($color-palette, \"violet\", \"500\")},\n    \"bg-light\": #{map-deep-get($color-palette, \"violet\", \"100\")},\n    \"dark\": #{map-deep-get($color-palette, \"violet\", \"400\")},\n    \"bg-dark\": #{map-deep-get($color-palette, \"violet\", \"800\")},\n  ),\n  \"accent\": (\n    \"light\": #{map-deep-get($color-palette, \"pink\", \"500\")},\n    \"bg-light\": #{map-deep-get($color-palette, \"pink\", \"200\")},\n    \"dark\": #{map-deep-get($color-palette, \"pink\", \"400\")},\n    \"bg-dark\": #{map-deep-get($color-palette, \"pink\", \"800\")},\n  ),\n  \"info\": (\n    \"light\": #276be9,\n    \"bg-light\": #dce7fc,\n    \"dark\": #79a3f2,\n    \"bg-dark\": #06245d,\n  ),\n  \"warning\": (\n    \"light\": #f66a0a,\n    \"bg-light\": #f8e3d0,\n    \"dark\": #ff9245,\n    \"bg-dark\": #652a02,\n  ),\n  \"success\": (\n    \"light\": #00843f,\n    \"bg-light\": #d6ece1,\n    \"dark\": #5fb488,\n    \"bg-dark\": #002f17,\n  ),\n  // This is based on the warning color\n  \"attention\":\n    (\n      \"light\": var(--pst-color-warning),\n      \"bg-light\": var(--pst-color-warning-bg),\n      \"dark\": var(--pst-color-warning),\n      \"bg-dark\": var(--pst-color-warning-bg),\n    ),\n  \"danger\": (\n    \"light\": #d72d47,\n    \"bg-light\": #f9e1e4,\n    \"dark\": #e78894,\n    \"bg-dark\": #4e111b,\n  ),\n  \"text-base\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"800\")},\n    \"dark\": $base-light-text,\n  ),\n  \"text-muted\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"600\")},\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"400\")},\n  ),\n  \"heading-color\": (\n    \"light\": #{$foundation-white},\n    \"dark\": #{$foundation-black},\n  ),\n  \"shadow\": (\n    \"light\": rgba(0, 0, 0, 0.1),\n    \"dark\": rgba(0, 0, 0, 0.2),\n  ),\n  \"border\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"300\")},\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"600\")},\n  ),\n  \"border-muted\": (\n    \"light\": rgba(23, 23, 26, 0.2),\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"700\")},\n  ),\n  \"blockquote-notch\": (\n    // These colors have a contrast ratio > 3.0 against both the background and\n    // surface colors that the notch is sandwiched between\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"500\")},\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"400\")},\n  ),\n  \"inline-code\": (\n    \"light\": #{map-deep-get($color-palette, \"pink\", \"600\")},\n    \"dark\": #{map-deep-get($color-palette, \"pink\", \"300\")},\n  ),\n  \"inline-code-links\": (\n    // need to make sure there is enough contrast against the code bg\n    \"light\": #{map-deep-get($color-palette, \"teal\", \"600\")},\n    // keep primary color for dark mode\n    \"dark\": #{map-deep-get($color-palette, \"teal\", \"400\")},\n  ),\n  \"target\": (\n    \"light\": #f3cf95,\n    \"dark\": #675c04,\n  ),\n  \"table\": (\n    \"light\": #{map-deep-get($color-palette, \"foundation\", \"black\")},\n    \"dark\": #{map-deep-get($color-palette, \"foundation\", \"white\")},\n  ),\n  \"table-row-hover\": (\n    \"bg-light\": #{map-deep-get($color-palette, \"violet\", \"300\")},\n    \"bg-dark\": #{map-deep-get($color-palette, \"violet\", \"600\")},\n  ),\n  \"table-inner-border\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"200\")},\n    \"dark\": #364150,\n  ),\n  // DEPTH COLORS - you can see the elevation colours and shades\n  // in the Figma file https://www.figma.com/file/rUrrHGhUBBIAAjQ82x6pz9/PyData-Design-system---proposal-for-implementation-(2)?node-id=1492%3A922&t=sQeQZehkOzposYEg-1\n  // background: color of the canvas / the furthest back layer\n  \"background\":\n    (\n      \"light\": #{map-deep-get($color-palette, \"foundation\", \"white\")},\n      \"dark\": #{map-deep-get($color-palette, \"foundation\", \"black\")},\n    ),\n  // on-background: provides slight contrast against background\n  // (by use of shadows in light theme)\n  \"on-background\":\n    (\n      \"light\": #{map-deep-get($color-palette, \"foundation\", \"white\")},\n      \"dark\": #{map-deep-get($color-palette, \"gray\", \"800\")},\n    ),\n  \"surface\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"100\")},\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"700\")},\n  ),\n  // on_surface: object on top of surface object (without shadows)\n  \"on-surface\":\n    (\n      \"light\": #{map-deep-get($color-palette, \"gray\", \"800\")},\n      \"dark\": $foundation-light-gray,\n    ),\n);\n\n/*******************************************************************************\n* write the color rules for each theme (light/dark)\n*/\n\n/* NOTE:\n * Mixins enable us to reuse the same definitions for the different modes\n * https://sass-lang.com/documentation/at-rules/mixin\n * #{something} inserts a variable into a CSS selector or property name\n * https://sass-lang.com/documentation/interpolation\n */\n@mixin theme-colors($mode) {\n  // check if this color is defined differently for light/dark\n  @each $col-name, $definition in $pst-semantic-colors {\n    @if meta.type-of($definition) == map {\n      @each $key, $val in $definition {\n        @if string.index($key, $mode) {\n          // since now we define the bg colours in the semantic colours and not\n          // by changing opacity, we need to check if the key contains bg and the\n          // correct mode (light/dark)\n          @if string.index($key, \"bg\") {\n            --pst-color-#{$col-name}-bg: #{$val};\n          } @else {\n            --pst-color-#{$col-name}: #{$val};\n          }\n        }\n      }\n    } @else {\n      --pst-color-#{$col-name}: #{$definition};\n    }\n  }\n\n  // assign the \"duplicate\" colors (ones that just reference other variables)\n  & {\n    --pst-color-link: var(--pst-color-primary);\n    --pst-color-link-hover: var(--pst-color-secondary);\n    --pst-color-table-outer-border: var(--pst-color-surface);\n    --pst-color-table-heading-bg: var(--pst-color-surface);\n    --pst-color-table-row-zebra-high-bg: var(--pst-color-on-background);\n    --pst-color-table-row-zebra-low-bg: var(--pst-color-surface);\n  }\n\n  // adapt to light/dark-specific content\n  @if $mode == \"light\" {\n    .only-dark,\n    .only-dark ~ figcaption {\n      display: none !important;\n    }\n  } @else {\n    .only-light,\n    .only-light ~ figcaption {\n      display: none !important;\n    }\n\n    /* Adjust images in dark mode (unless they have class .only-dark or\n     * .dark-light, in which case assume they're already optimized for dark\n     * mode).\n     */\n    img:not(.only-dark, .dark-light) {\n      filter: brightness(0.8) contrast(1.2);\n    }\n\n    /* Give images a light background in dark mode in case they have\n    *  transparency and black text (unless they have class .only-dark or .dark-light, in\n    *  which case assume they're already optimized for dark mode).\n    */\n    .bd-content img:not(.only-dark, .dark-light) {\n      background-color: rgb(255 255 255);\n      border-radius: 0.25rem;\n    }\n\n    // MathJax SVG outputs should be filled to same color as text.\n    .MathJax_SVG * {\n      fill: var(--pst-color-text-base);\n    }\n  }\n}\n\n/* Defaults to light mode if data-theme is not set */\nhtml:not([data-theme]) {\n  @include theme-colors(\"light\");\n}\n\n/* NOTE: @each {...} is like a for-loop\n * https://sass-lang.com/documentation/at-rules/control/each\n */\n@each $mode in (light, dark) {\n  html[data-theme=\"#{$mode}\"] {\n    @include theme-colors($mode);\n\n    color-scheme: $mode;\n  }\n}\n\n// assign classes too, for runtime use of theme colors\n@each $col-name, $definition in $pst-semantic-colors {\n  .pst-color-#{$col-name} {\n    color: var(--pst-color-#{$col-name});\n  }\n}\n",".bd-container {\n  flex-grow: 1;\n  display: flex;\n  justify-content: center;\n\n  .bd-container__inner {\n    display: flex;\n  }\n}\n\n.bd-page-width {\n  width: 100%;\n\n  @include media-breakpoint-up(lg) {\n    max-width: $breakpoint-page-width;\n  }\n}\n","// Breakpoint viewport sizes and media queries.\n//\n// Breakpoints are defined as a map of (name: minimum width), order from small to large:\n//\n//    (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px)\n//\n// The map defined in the `$grid-breakpoints` global variable is used as the `$breakpoints` argument by default.\n\n// Name of the next breakpoint, or null for the last breakpoint.\n//\n//    >> breakpoint-next(sm)\n//    md\n//    >> breakpoint-next(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    md\n//    >> breakpoint-next(sm, $breakpoint-names: (xs sm md lg xl xxl))\n//    md\n@function breakpoint-next($name, $breakpoints: $grid-breakpoints, $breakpoint-names: map-keys($breakpoints)) {\n  $n: index($breakpoint-names, $name);\n  @if not $n {\n    @error \"breakpoint `#{$name}` not found in `#{$breakpoints}`\";\n  }\n  @return if($n < length($breakpoint-names), nth($breakpoint-names, $n + 1), null);\n}\n\n// Minimum breakpoint width. Null for the smallest (first) breakpoint.\n//\n//    >> breakpoint-min(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    576px\n@function breakpoint-min($name, $breakpoints: $grid-breakpoints) {\n  $min: map-get($breakpoints, $name);\n  @return if($min != 0, $min, null);\n}\n\n// Maximum breakpoint width.\n// The maximum value is reduced by 0.02px to work around the limitations of\n// `min-` and `max-` prefixes and viewports with fractional widths.\n// See https://www.w3.org/TR/mediaqueries-4/#mq-min-max\n// Uses 0.02px rather than 0.01px to work around a current rounding bug in Safari.\n// See https://bugs.webkit.org/show_bug.cgi?id=178261\n//\n//    >> breakpoint-max(md, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    767.98px\n@function breakpoint-max($name, $breakpoints: $grid-breakpoints) {\n  $max: map-get($breakpoints, $name);\n  @return if($max and $max > 0, $max - .02, null);\n}\n\n// Returns a blank string if smallest breakpoint, otherwise returns the name with a dash in front.\n// Useful for making responsive utilities.\n//\n//    >> breakpoint-infix(xs, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    \"\"  (Returns a blank string)\n//    >> breakpoint-infix(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    \"-sm\"\n@function breakpoint-infix($name, $breakpoints: $grid-breakpoints) {\n  @return if(breakpoint-min($name, $breakpoints) == null, \"\", \"-#{$name}\");\n}\n\n// Media of at least the minimum breakpoint width. No query for the smallest breakpoint.\n// Makes the @content apply to the given breakpoint and wider.\n@mixin media-breakpoint-up($name, $breakpoints: $grid-breakpoints) {\n  $min: breakpoint-min($name, $breakpoints);\n  @if $min {\n    @media (min-width: $min) {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Media of at most the maximum breakpoint width. No query for the largest breakpoint.\n// Makes the @content apply to the given breakpoint and narrower.\n@mixin media-breakpoint-down($name, $breakpoints: $grid-breakpoints) {\n  $max: breakpoint-max($name, $breakpoints);\n  @if $max {\n    @media (max-width: $max) {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Media that spans multiple breakpoint widths.\n// Makes the @content apply between the min and max breakpoints\n@mixin media-breakpoint-between($lower, $upper, $breakpoints: $grid-breakpoints) {\n  $min: breakpoint-min($lower, $breakpoints);\n  $max: breakpoint-max($upper, $breakpoints);\n\n  @if $min != null and $max != null {\n    @media (min-width: $min) and (max-width: $max) {\n      @content;\n    }\n  } @else if $max == null {\n    @include media-breakpoint-up($lower, $breakpoints) {\n      @content;\n    }\n  } @else if $min == null {\n    @include media-breakpoint-down($upper, $breakpoints) {\n      @content;\n    }\n  }\n}\n\n// Media between the breakpoint's minimum and maximum widths.\n// No minimum for the smallest breakpoint, and no maximum for the largest one.\n// Makes the @content apply only to the given breakpoint, not viewports any wider or narrower.\n@mixin media-breakpoint-only($name, $breakpoints: $grid-breakpoints) {\n  $min:  breakpoint-min($name, $breakpoints);\n  $next: breakpoint-next($name, $breakpoints);\n  $max:  breakpoint-max($next, $breakpoints);\n\n  @if $min != null and $max != null {\n    @media (min-width: $min) and (max-width: $max) {\n      @content;\n    }\n  } @else if $max == null {\n    @include media-breakpoint-up($name, $breakpoints) {\n      @content;\n    }\n  } @else if $min == null {\n    @include media-breakpoint-down($next, $breakpoints) {\n      @content;\n    }\n  }\n}\n",".pst-async-banner-revealer {\n  // Setting height to 0 and overflow to hidden allows us to add up the heights\n  // of this element's children before revealing them.\n  height: 0;\n  overflow: hidden;\n\n  // Height to be set by JavaScript, which should trigger the following\n  // transition rule (unless the user has set their system to reduce motion).\n  transition: height 300ms ease-in-out;\n\n  @media (prefers-reduced-motion) {\n    transition: none;\n  }\n}\n\n#bd-header-version-warning,\n.bd-header-announcement {\n  min-height: 3rem;\n  width: 100%;\n  display: flex;\n  position: relative;\n  align-items: center;\n  justify-content: center;\n  text-align: center;\n  padding: 0.5rem 12.5%; // Horizontal padding so the width is 75%\n  // One breakpoint less than $breakpoint-sidebar-primary. See variables/_layout.scss for more info.\n  @include media-breakpoint-down(lg) {\n    // Announcements can take a bit more width on mobile\n    padding: 0.5rem 2%;\n  }\n\n  p {\n    font-weight: bold;\n    margin: 0;\n  }\n\n  // Ensure there is enough contrast against the background\n  a {\n    color: var(--pst-color-inline-code-links);\n  }\n\n  // The \"Switch to stable version\" link (styled like a button)\n  .pst-button-link-to-stable-version {\n    @include box-shadow;\n\n    $background-color: var(--pst-color-danger);\n    $hover-background-color: var(--pst-color-danger-highlight);\n    $color: var(--pst-color-danger-text);\n\n    background-color: $background-color;\n    border-color: $background-color;\n    color: $color;\n    border-radius: 0.25rem;\n\n    &:hover {\n      background-color: $hover-background-color;\n      border-color: $hover-background-color;\n      color: $color;\n    }\n\n    &:focus-visible {\n      outline: $focus-ring-width solid $background-color;\n      outline-offset: $focus-ring-width;\n    }\n  }\n}\n\n// Bg color is now defined in the theme color palette - using our secondary color\n.bd-header-announcement {\n  background-color: var(--pst-color-secondary-bg);\n}\n\n#bd-header-version-warning {\n  background-color: var(--pst-color-danger-bg);\n}\n","/*********************************************\n* SASS Mixins\n*********************************************/\n\n/**\n * A consistent box shadow style we apply across elements.\n */\n@mixin box-shadow() {\n  box-shadow:\n    0 0.2rem 0.5rem var(--pst-color-shadow),\n    0 0 0.0625rem var(--pst-color-shadow) !important;\n}\n\n/**\n   * Set background of some cell outputs to white-ish to make sure colors work\n   * This is because many libraries make output that only looks good on white\n   */\n@mixin cell-output-background {\n  color: var(--pst-color-on-background);\n  background-color: var(--pst-color-text-base);\n  border-radius: 0.25rem;\n  padding: 0.5rem;\n}\n\n@mixin table-colors {\n  color: var(--pst-color-table);\n  border: 1px solid var(--pst-color-table-outer-border);\n\n  th,\n  td {\n    ~ th,\n    ~ td {\n      border-left: 1px solid var(--pst-color-table-inner-border);\n    }\n  }\n\n  thead {\n    tr {\n      background-color: var(--pst-color-table-heading-bg);\n      border-bottom: 2px solid var(--pst-color-primary);\n    }\n  }\n\n  tbody {\n    tr {\n      &:nth-child(odd) {\n        background-color: var(--pst-color-table-row-zebra-low-bg);\n      }\n\n      &:nth-child(even) {\n        background-color: var(--pst-color-table-row-zebra-high-bg);\n      }\n\n      &:hover {\n        background-color: var(--pst-color-table-row-hover-bg);\n      }\n    }\n  }\n}\n\n// Minimum mouse hit area\n// ----------------------\n// Ensures that the element has a minimum hit area that conforms to\n// accessibility guidelines. For WCAG AA, we need 24px x 24px, see:\n// https://www.w3.org/WAI/WCAG22/Understanding/target-size-minimum.html\n@mixin min-hit-area() {\n  box-sizing: border-box;\n  min-width: 24px;\n  min-height: 24px;\n}\n","/**\n * Main content area\n */\n.bd-main {\n  flex-grow: 1;\n  flex-direction: column;\n  display: flex;\n  min-width: 0;\n\n  .bd-content {\n    display: flex;\n    justify-content: center;\n    height: 100%;\n\n    .bd-article-container {\n      justify-content: start;\n      display: flex;\n      flex-direction: column;\n\n      // Max-width is slightly more than the W3 since our docs often have images.\n      // We shoot for about 100 characters per line instead of 80.\n      // ref: https://www.w3.org/WAI/tutorials/page-structure/styling/#line-length\n      width: 100%;\n      max-width: 60em;\n      overflow-x: auto; // Prevent wide content from pushing off the secondary sidebar\n      padding: 1rem;\n\n      .bd-article {\n        // Give a bit more verticle spacing on wide screens\n        @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n          padding-top: 1.5rem;\n          padding-left: 2rem;\n        }\n      }\n    }\n  }\n}\n",".bd-footer {\n  width: 100%;\n  border-top: 1px solid var(--pst-color-border);\n\n  .bd-footer__inner {\n    display: flex;\n    flex-grow: 1;\n    padding: 1rem;\n    margin: auto;\n  }\n\n  .footer-items__start,\n  .footer-items__center,\n  .footer-items__end {\n    display: flex;\n    flex-direction: column;\n    gap: 0.5rem;\n    justify-content: center;\n    flex-grow: 1;\n  }\n\n  .footer-items__center {\n    text-align: center;\n  }\n\n  .footer-items__end {\n    text-align: end;\n  }\n\n  // So that paragraphs don't take up extra room\n  .footer-item p {\n    margin-bottom: 0;\n  }\n}\n",".bd-footer-article {\n  margin-top: auto;\n\n  .footer-article-items {\n    display: flex;\n    flex-direction: column;\n  }\n}\n",".bd-footer-content {\n  .footer-content-items {\n    display: flex;\n    flex-direction: column;\n    margin-top: auto;\n  }\n}\n","/**\n * Header at the top of the page\n * It includes the announcement bar and the navigation bar.\n */\n\n// Styling for the Icon links can be found in components/_icon-links.scss\n\n// If we want the shadow to only point downward in the future, set\n// box-shadow to: 0 0.125rem 0.25rem -0.125rem rgba(0, 0, 0, 0.11);\n.bd-header {\n  position: sticky;\n  top: 0;\n  z-index: $zindex-fixed;\n\n  // Overrides bootstrap\n  background-color: var(--pst-color-on-background) !important;\n  box-shadow: 0 0.125rem 0.25rem 0 var(--pst-color-shadow);\n  width: 100%;\n  padding: 0;\n  max-width: 100vw;\n  justify-content: center;\n\n  .bd-header__inner {\n    display: flex;\n    align-items: center;\n    height: fit-content;\n    padding-left: 1rem;\n    padding-right: 1rem;\n  }\n\n  :focus-visible {\n    border-radius: $focus-ring-radius;\n  }\n\n  // These items will define the height of the header\n  .navbar-item {\n    height: var(--pst-header-height);\n    max-height: var(--pst-header-height);\n    display: flex;\n    align-items: center;\n  }\n\n  // Hide the navbar header items on mobile because they're in the sidebar\n  .navbar-header-items {\n    display: none;\n    flex-shrink: 1;\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      display: inherit;\n      flex-grow: 1;\n      padding: 0 0 0 0.5rem;\n    }\n  }\n\n  .navbar-header-items__end,\n  .navbar-header-items__center,\n  .navbar-header-items__start {\n    display: flex;\n    align-items: center;\n    flex-flow: wrap;\n\n    // In case we wrap our items to multiple rows on small screens\n    row-gap: 0;\n  }\n\n  .navbar-header-items__end,\n  .navbar-header-items__center {\n    column-gap: 1rem;\n  }\n\n  // A little smaller because this is displayed by default on mobile\n  .navbar-header-items__start {\n    flex-shrink: 0;\n    margin-right: auto;\n    gap: 0.5rem;\n  }\n\n  .navbar-header-items__end {\n    // End navbar items should snap to the right\n    justify-content: end;\n  }\n\n  // Contains the navigation links within the navbar\n  ul.navbar-nav {\n    display: flex;\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      // Align on wide screens so the dropdown button is centered properly\n      align-items: baseline;\n    }\n\n    > li.nav-item {\n      margin-inline: 2px; // breathing room so hover and focus styles do not overlap\n\n      > .nav-link {\n        @include link-style-block;\n\n        padding-inline: 6px;\n      }\n\n      &.current {\n        > .nav-link {\n          color: var(--pst-color-primary);\n\n          // Underline the current navbar item\n          &::before {\n            border-bottom: 3px solid var(--pst-color-primary);\n          }\n        }\n      }\n\n      &.dropdown {\n        margin-inline: 4px;\n\n        button {\n          padding-inline: 8px;\n        }\n\n        > .dropdown-toggle {\n          border-radius: $focus-ring-radius; // make border radius the same for both hover ring and focus ring\n          color: var(--pst-color-text-muted);\n\n          &:focus-visible {\n            box-shadow: $focus-ring-box-shadow;\n          }\n\n          &:hover {\n            text-decoration: none;\n            box-shadow: 0 0 0 $focus-ring-width var(--pst-color-link-hover); // purple focus ring\n            // Brighten the text on hover (muted -> base)\n            color: var(--pst-color-text-base);\n          }\n        }\n      }\n    }\n\n    li a.nav-link.dropdown-item {\n      @include link-style-text;\n    }\n\n    // Dropdowns for the extra links\n    .dropdown {\n      button {\n        display: unset;\n        border: none;\n\n        &:hover {\n          @include link-style-hover;\n        }\n      }\n\n      .dropdown-menu {\n        z-index: $zindex-popover;\n        border: 1px solid var(--pst-color-border);\n        box-shadow: 0 0 0.3rem 0.1rem var(--pst-color-shadow);\n        background-color: var(--pst-color-on-background);\n        padding: 0.5rem 0;\n        margin: 0.5rem 0;\n        min-width: 20rem;\n\n        .dropdown-item {\n          // Give the items in the dropdown some breathing room but let the hit\n          // and hover area of the items extend to the edges of the menu\n          padding: 0.25rem 1.5rem;\n\n          // Override Bootstrap\n          &:focus:not(:hover, :active) {\n            background-color: inherit;\n          }\n\n          &:focus-visible {\n            z-index: 10; // keep focus ring on top (prevent the hover background of the next dropdown item from covering the ring)\n          }\n        }\n\n        // Hide the menu unless show has been clicked\n        &:not(.show) {\n          display: none;\n        }\n      }\n    }\n  }\n\n  // **************************************************************\n  // Showing and hiding the sidebar toggle buttons and header items\n  // **************************************************************\n\n  // Toggle buttons\n  button.sidebar-toggle {\n    font-size: var(--pst-font-size-icon);\n    color: var(--pst-color-muted);\n    margin-bottom: 0;\n    background-color: inherit;\n    padding: 0.5rem;\n  }\n\n  button.primary-toggle {\n    margin-right: 1rem;\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      display: none;\n    }\n  }\n\n  button.secondary-toggle {\n    margin-left: 1rem;\n\n    @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n      display: none;\n    }\n  }\n}\n\n// inline the element in the navbar as long as they fit and use display block when collapsing\n@include media-breakpoint-up($breakpoint-sidebar-primary) {\n  .navbar-center-items .navbar-item {\n    display: inline-block;\n  }\n}\n\n.nav-link {\n  &:hover {\n    @include link-style-hover;\n  }\n\n  // Override Bootstrap\n  transition: none;\n\n  &.nav-external::after {\n    font: var(--fa-font-solid);\n    content: var(--pst-icon-external-link);\n    font-size: 0.75em;\n    margin-left: 0.3em;\n  }\n}\n\n.bd-navbar-elements li.nav-item i {\n  font-size: 0.7rem;\n  padding-left: 2px;\n  vertical-align: middle;\n}\n\n// THe elements next to the hamburger menu only show on narrow screens\n.navbar-persistent--mobile {\n  margin-left: auto;\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    display: none;\n  }\n}\n\n// The navbar-persistent content should only show on wide screens\n.navbar-persistent--container {\n  display: none;\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    display: flex;\n  }\n}\n",".header-article__inner {\n  display: flex;\n  padding: 0 0.5rem;\n\n  // The items define the height so that it disappears if there are no items\n  .header-article-item {\n    min-height: var(--pst-header-article-height);\n    height: var(--pst-header-article-height);\n  }\n\n  .header-article-items__start,\n  .header-article-items__end {\n    display: flex;\n    align-items: start;\n    gap: 0.5rem;\n  }\n\n  .header-article-items__end {\n    margin-left: auto;\n  }\n}\n","/**\n * The primary sidebar on the left.\n * e.g., between-pages navigation.\n */\n\n$sidebar-padding-right: 1rem;\n\n.bd-sidebar-primary {\n  display: flex;\n  flex-direction: column;\n  gap: 1rem;\n  max-height: calc(100vh - var(--pst-header-height));\n  position: sticky;\n  top: var(--pst-header-height);\n\n  @include make-col(3);\n\n  // Borders padding and whitespace\n  padding: 2rem $sidebar-padding-right 1rem 1rem;\n  border-right: 1px solid var(--pst-color-border);\n  background-color: var(--pst-color-background);\n  overflow-y: auto;\n  font-size: var(--pst-sidebar-font-size-mobile);\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    font-size: var(--pst-sidebar-font-size);\n  }\n\n  :focus-visible {\n    border-radius: $focus-ring-radius;\n  }\n\n  // override bootstrap when navlink are displayed in the sidebar\n  .nav-link {\n    font-size: var(--pst-sidebar-font-size-mobile);\n  }\n\n  &.no-sidebar {\n    border-right: 0;\n  }\n\n  &.hide-on-wide {\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      display: none;\n    }\n  }\n\n  // Headers shouldn't be colored in the sidebars and some extensions add headers\n  h1,\n  h2,\n  h3,\n  h4 {\n    color: var(--pst-color-text-base);\n  }\n\n  .sidebar-primary-items__start,\n  .sidebar-primary-items__end {\n    .sidebar-primary-item {\n      padding: 0.5rem 0;\n    }\n  }\n\n  // Hide the sidebar header items on widescreen since they are visible in the header\n  .sidebar-header-items {\n    display: flex;\n    flex-direction: column;\n\n    .sidebar-header-items__title {\n      font-weight: var(--pst-sidebar-header-font-weight);\n      font-size: var(--pst-sidebar-header-font-size);\n      color: var(--pst-color-text-base);\n      margin-bottom: 0.5rem;\n    }\n\n    // The dropdown toggle for extra links just shows them all instead.\n    .nav-item.dropdown {\n      // On mobile, the dropdown behaves like any other link, no hiding\n      button {\n        display: none;\n      }\n\n      .dropdown-menu {\n        display: flex;\n        flex-direction: column;\n        padding: 0;\n        margin: 0;\n        border: none;\n        background-color: inherit;\n        font-size: inherit;\n\n        .dropdown-item {\n          &:hover,\n          &:focus {\n            // In the mobile sidebar, the dropdown menu is inlined with the\n            // other links, which do not have background-color changes on hover\n            // and focus\n            background-color: unset;\n          }\n        }\n      }\n    }\n\n    .bd-navbar-elements {\n      .nav-link {\n        &:focus-visible {\n          box-shadow: none; // Override Bootstrap\n          outline: $focus-ring-outline;\n          outline-offset: $focus-ring-width;\n        }\n      }\n    }\n\n    // Center header items get displayed vertically, end items are displayed horizontally\n    .sidebar-header-items__center {\n      display: flex;\n      flex-direction: column;\n    }\n\n    // Positioning end items\n    .sidebar-header-items__end {\n      display: flex;\n      align-items: center;\n      gap: 1rem;\n    }\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      display: none;\n    }\n  }\n\n  .sidebar-primary-items__start {\n    // Add a border on mobile to separate it from the header sidebar area\n    border-top: 1px solid var(--pst-color-border);\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      border-top: none;\n    }\n  }\n\n  .sidebar-primary-items__end {\n    margin-top: auto;\n    margin-bottom: 1em;\n  }\n\n  .list-caption {\n    list-style: none;\n    padding-left: 0;\n\n    // Level 0 TOC heading is put inside the <summary> tag\n    // so let the <summary> tag take up more space\n    li.toctree-l0.has-children {\n      > details {\n        > summary {\n          position: relative;\n          height: auto;\n          width: auto;\n          display: flex;\n          justify-content: space-between;\n          align-items: baseline;\n\n          .toctree-toggle {\n            // Prevent toggle icon from getting squished by summary being a\n            // flexbox\n            flex: 0 0 auto;\n\n            // Make the level 0 chevron icon slightly bigger than descendant\n            // levels\n            .fa-chevron-down {\n              font-size: 1rem;\n            }\n          }\n        }\n      }\n    }\n  }\n\n  li.has-children {\n    $toctree-toggle-width: 30px;\n\n    position: relative;\n\n    > .reference,\n    .caption {\n      margin-right: calc(\n        $toctree-toggle-width + $focus-ring-width\n      ); // keep clear of the toggle icon\n\n      padding-top: 0.25rem; // align caption text with toggle chevron\n    }\n\n    > details {\n      > summary {\n        // Remove browser default toggle icon\n        list-style: none;\n\n        &::-webkit-details-marker {\n          display: none;\n        }\n\n        // The summary element is natively focusable, but delegate the focus state to the toggle icon\n        &:focus-visible {\n          outline: none;\n\n          > .toctree-toggle {\n            outline: $focus-ring-outline;\n            outline-offset: -$focus-ring-width; // Prevent right side of focus ring from disappearing underneath the sidebar's right edge\n          }\n        }\n\n        // Container for expand/collapse chevron icon\n        .toctree-toggle {\n          cursor: pointer;\n\n          // Position it so that it's aligned with the top right corner of the\n          // last positioned element, in this case the li.has-children\n          position: absolute;\n          top: 0;\n          right: 0;\n\n          // Give it dimensions\n          width: $toctree-toggle-width;\n          height: $toctree-toggle-width; // make it square\n\n          // Vertically and horizontally center the icon within the container\n          display: inline-flex;\n          justify-content: center;\n          align-items: center;\n\n          .fa-chevron-down {\n            font-size: 0.75rem;\n          }\n        }\n      }\n\n      // The section is open/expanded, rotate the toggle icon (chevron) so it\n      // points up instead of down\n      &[open] {\n        > summary {\n          .fa-chevron-down {\n            transform: rotate(180deg);\n          }\n        }\n      }\n    }\n  }\n}\n\n/* Between-page links and captions */\nnav.bd-links {\n  margin-right: -$sidebar-padding-right; // align toctree toggle chevrons with right edge of sidebar and allow text to flow closer to the right edge\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    display: block;\n  }\n\n  ul {\n    display: block;\n    list-style: none;\n\n    // Reduce padding of nested `ul` items a bit\n    ul {\n      padding: 0 0 0 1rem;\n    }\n  }\n\n  li > a {\n    display: block;\n    padding: 0.25rem 0.65rem;\n\n    @include link-sidebar;\n\n    box-shadow: none;\n    margin-right: $focus-ring-width; // prevent the right side focus ring from disappearing under the sidebar right edge\n\n    &.reference.external {\n      &::after {\n        font: var(--fa-font-solid);\n        content: var(--pst-icon-external-link);\n        font-size: 0.75em;\n        margin-left: 0.3em;\n      }\n    }\n  }\n\n  .current > a {\n    @include link-sidebar-current;\n\n    background-color: transparent;\n  }\n\n  // Title\n  p.bd-links__title {\n    font-size: var(--pst-sidebar-header-font-size);\n    font-weight: var(--pst-sidebar-header-font-weight);\n    margin-bottom: 0.5rem;\n  }\n\n  // Toctree captions\n  p.caption {\n    font-weight: var(--pst-sidebar-header-font-weight);\n    position: relative;\n    margin-top: 1.25rem;\n    margin-bottom: 0.5rem;\n    color: var(--pst-color-text-base);\n\n    &:first-child {\n      margin-top: 0;\n    }\n\n    font-size: var(--pst-sidebar-font-size-mobile);\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      font-size: var(--pst-sidebar-font-size);\n    }\n  }\n}\n","// Grid system\n//\n// Generate semantic grid columns with these mixins.\n\n@mixin make-row($gutter: $grid-gutter-width) {\n  --#{$prefix}gutter-x: #{$gutter};\n  --#{$prefix}gutter-y: 0;\n  display: flex;\n  flex-wrap: wrap;\n  // TODO: Revisit calc order after https://github.com/react-bootstrap/react-bootstrap/issues/6039 is fixed\n  margin-top: calc(-1 * var(--#{$prefix}gutter-y)); // stylelint-disable-line function-disallowed-list\n  margin-right: calc(-.5 * var(--#{$prefix}gutter-x)); // stylelint-disable-line function-disallowed-list\n  margin-left: calc(-.5 * var(--#{$prefix}gutter-x)); // stylelint-disable-line function-disallowed-list\n}\n\n@mixin make-col-ready() {\n  // Add box sizing if only the grid is loaded\n  box-sizing: if(variable-exists(include-column-box-sizing) and $include-column-box-sizing, border-box, null);\n  // Prevent columns from becoming too narrow when at smaller grid tiers by\n  // always setting `width: 100%;`. This works because we set the width\n  // later on to override this initial width.\n  flex-shrink: 0;\n  width: 100%;\n  max-width: 100%; // Prevent `.col-auto`, `.col` (& responsive variants) from breaking out the grid\n  padding-right: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  padding-left: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  margin-top: var(--#{$prefix}gutter-y);\n}\n\n@mixin make-col($size: false, $columns: $grid-columns) {\n  @if $size {\n    flex: 0 0 auto;\n    width: percentage(divide($size, $columns));\n\n  } @else {\n    flex: 1 1 0;\n    max-width: 100%;\n  }\n}\n\n@mixin make-col-auto() {\n  flex: 0 0 auto;\n  width: auto;\n}\n\n@mixin make-col-offset($size, $columns: $grid-columns) {\n  $num: divide($size, $columns);\n  margin-left: if($num == 0, 0, percentage($num));\n}\n\n// Row columns\n//\n// Specify on a parent element(e.g., .row) to force immediate children into NN\n// number of columns. Supports wrapping to new lines, but does not do a Masonry\n// style grid.\n@mixin row-cols($count) {\n  > * {\n    flex: 0 0 auto;\n    width: percentage(divide(1, $count));\n  }\n}\n\n// Framework grid generation\n//\n// Used only by Bootstrap to generate the correct number of grid classes given\n// any value of `$grid-columns`.\n\n@mixin make-grid-columns($columns: $grid-columns, $gutter: $grid-gutter-width, $breakpoints: $grid-breakpoints) {\n  @each $breakpoint in map-keys($breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $breakpoints);\n\n    @include media-breakpoint-up($breakpoint, $breakpoints) {\n      // Provide basic `.col-{bp}` classes for equal-width flexbox columns\n      .col#{$infix} {\n        flex: 1 0 0%; // Flexbugs #4: https://github.com/philipwalton/flexbugs#flexbug-4\n      }\n\n      .row-cols#{$infix}-auto > * {\n        @include make-col-auto();\n      }\n\n      @if $grid-row-columns > 0 {\n        @for $i from 1 through $grid-row-columns {\n          .row-cols#{$infix}-#{$i} {\n            @include row-cols($i);\n          }\n        }\n      }\n\n      .col#{$infix}-auto {\n        @include make-col-auto();\n      }\n\n      @if $columns > 0 {\n        @for $i from 1 through $columns {\n          .col#{$infix}-#{$i} {\n            @include make-col($i, $columns);\n          }\n        }\n\n        // `$columns - 1` because offsetting by the width of an entire row isn't possible\n        @for $i from 0 through ($columns - 1) {\n          @if not ($infix == \"\" and $i == 0) { // Avoid emitting useless .offset-0\n            .offset#{$infix}-#{$i} {\n              @include make-col-offset($i, $columns);\n            }\n          }\n        }\n      }\n\n      // Gutters\n      //\n      // Make use of `.g-*`, `.gx-*` or `.gy-*` utilities to change spacing between the columns.\n      @each $key, $value in $gutters {\n        .g#{$infix}-#{$key},\n        .gx#{$infix}-#{$key} {\n          --#{$prefix}gutter-x: #{$value};\n        }\n\n        .g#{$infix}-#{$key},\n        .gy#{$infix}-#{$key} {\n          --#{$prefix}gutter-y: #{$value};\n        }\n      }\n    }\n  }\n}\n\n@mixin make-cssgrid($columns: $grid-columns, $breakpoints: $grid-breakpoints) {\n  @each $breakpoint in map-keys($breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $breakpoints);\n\n    @include media-breakpoint-up($breakpoint, $breakpoints) {\n      @if $columns > 0 {\n        @for $i from 1 through $columns {\n          .g-col#{$infix}-#{$i} {\n            grid-column: auto / span $i;\n          }\n        }\n\n        // Start with `1` because `0` is an invalid value.\n        // Ends with `$columns - 1` because offsetting by the width of an entire row isn't possible.\n        @for $i from 1 through ($columns - 1) {\n          .g-start#{$infix}-#{$i} {\n            grid-column-start: $i;\n          }\n        }\n      }\n    }\n  }\n}\n","/**\n * Secondary sidebar on the right.\n * e.g., in-page table of contents.\n */\n.bd-sidebar-secondary {\n  display: flex;\n  order: 2;\n  flex-shrink: 0;\n  flex-direction: column;\n  position: sticky;\n  top: var(--pst-header-height);\n  max-height: calc(100vh - var(--pst-header-height));\n  padding: 2rem 1rem 1rem;\n  width: var(--pst-sidebar-secondary);\n  font-size: var(--pst-sidebar-font-size-mobile);\n\n  @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n    font-size: var(--pst-sidebar-font-size);\n  }\n\n  // Color and border\n  background-color: var(--pst-color-background);\n  overflow-y: auto;\n}\n\n.sidebar-secondary-item {\n  padding: 0.5rem;\n\n  @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n    border-left: 1px solid var(--pst-color-border);\n    padding-left: 1rem;\n  }\n\n  i {\n    padding-right: 0.5rem;\n  }\n}\n","/*******************************************************************************\n* Rules for the UX/UI of sidebar sliding drawers on mobile\n* Note that this sheet controls styles across many parts of the theme\n* It is aggregated into this one sheet instead of being split across\n* components in order to keep it easier to debug in one place.\n* It is broken up into major sections below.\n*/\n\n/*******************************************************************************\n* Buttons and overlays\n*/\ninput.sidebar-toggle {\n  display: none;\n}\n\n// Background overlays\nlabel.overlay {\n  background-color: black;\n  opacity: 0.5;\n  height: 0;\n  width: 0;\n  position: fixed;\n  top: 0;\n  left: 0;\n  transition: opacity $animation-time ease-out;\n  z-index: $zindex-modal-backdrop;\n}\n\ninput {\n  // Show the correct overlay when its input is checked\n  &#pst-primary-sidebar-checkbox:checked + label.overlay.overlay-primary,\n  &#pst-secondary-sidebar-checkbox:checked + label.overlay.overlay-secondary {\n    height: 100vh;\n    width: 100vw;\n  }\n\n  // Primary sidebar slides in from the left\n  &#pst-primary-sidebar-checkbox:checked ~ .bd-container .bd-sidebar-primary {\n    visibility: visible;\n    margin-left: 0;\n  }\n\n  // Secondary sidebar slides in from the right\n  &#pst-secondary-sidebar-checkbox:checked\n    ~ .bd-container\n    .bd-sidebar-secondary {\n    visibility: visible;\n    margin-right: 0;\n  }\n}\n\n/*******************************************************************************\n* Sidebar drawer behavior\n*/\n\n/**\n * Behavior for sliding drawer elements that will be toggled with an input\n *\n * NOTE: We use this mixin to define the toggle behavior on narrow screens,\n * And the wide-screen behavior of the sections is defined in their own section\n * .scss files.\n */\n@mixin sliding-drawer($side: \"left\") {\n  position: fixed;\n  top: 0;\n  z-index: $zindex-modal;\n  height: 100vh;\n  max-height: 100vh;\n  width: 75%;\n  flex-grow: 0.75;\n  max-width: 350px;\n  transition:\n    visibility $animation-time ease-out,\n    margin $animation-time ease-out;\n  visibility: hidden;\n\n  @if $side == \"right\" {\n    margin-right: -75%;\n    right: 0;\n  } @else {\n    margin-left: -75%;\n    left: 0;\n  }\n}\n\n// Primary sidebar hides/shows at earlier widths\n@include media-breakpoint-up($breakpoint-sidebar-primary) {\n  .sidebar-toggle.primary-toggle {\n    display: none;\n  }\n\n  input#pst-primary-sidebar-checkbox {\n    &:checked + label.overlay.overlay-primary {\n      height: 0;\n      width: 0;\n    }\n  }\n\n  .bd-sidebar-primary {\n    margin-left: 0;\n    visibility: visible;\n  }\n}\n\n.bd-sidebar-primary {\n  @include media-breakpoint-down($breakpoint-sidebar-primary) {\n    @include sliding-drawer(\"left\");\n  }\n}\n\n.bd-sidebar-secondary {\n  @include media-breakpoint-down($breakpoint-sidebar-secondary) {\n    @include sliding-drawer(\"right\");\n  }\n}\n","/**\n * Breadcrumbs for parent pages meant for the article header\n */\nul.bd-breadcrumbs {\n  list-style: none;\n  padding-left: 0;\n  display: flex;\n  flex-wrap: wrap;\n\n  // Font size slightly smaller to avoid cluttering up space too much\n  font-size: 0.8rem;\n\n  li.breadcrumb-item {\n    display: flex;\n    align-items: center;\n\n    // Style should look like heavier in-page links\n    // keeping visited in the default link colour\n    font-weight: bold;\n\n    a {\n      @include link-style-text;\n    }\n\n    // Items that aren't the home have a caret to the left\n    &:not(.breadcrumb-home)::before {\n      font: var(--fa-font-solid);\n      font-size: 0.8rem;\n      content: var(--pst-breadcrumb-divider);\n      color: var(--pst-color-text-muted);\n      padding: 0 0.5rem;\n    }\n  }\n}\n","/**\n * Icon links in the navbar\n */\n\n.pst-navbar-icon {\n  // Extra specificity needed for overrides\n  html & {\n    @include min-hit-area;\n    @include link-style-block;\n\n    display: flex;\n    align-items: center;\n    justify-content: center;\n\n    // Bootstrap overrides\n    border-radius: 0;\n    border: none;\n    font-size: 1rem;\n    line-height: $line-height-body; // Override Boostrap, which defines a separate line-height for buttons\n    padding: $navbar-link-padding-y 0; // Horizontal white space in nav bar between items is controlled via column gap rule on the container.\n\n    // Make the navbar icon links have the same size as the navbar text links\n    height: calc(2 * $navbar-link-padding-y + $line-height-body * 1rem);\n  }\n}\n\nul.navbar-icon-links {\n  display: flex;\n  flex-flow: row wrap;\n  column-gap: 1rem;\n  justify-content: space-evenly;\n  align-items: center;\n  padding-left: 0;\n  margin-bottom: 0;\n  list-style: none;\n\n  // Icons styling\n  i {\n    &.fa-brands,\n    &.fa-regular,\n    &.fa-solid {\n      vertical-align: middle;\n      font-style: normal;\n      font-size: var(--pst-font-size-icon);\n    }\n\n    /* Social media buttons hard-code the brand color */\n    &.fa-square-twitter::before {\n      color: #55acee;\n    }\n\n    &.fa-square-gitlab::before {\n      color: #548;\n    }\n\n    &.fa-bitbucket::before {\n      color: #0052cc;\n    }\n  }\n\n  // Force images to be icon-sized\n  img.icon-link-image {\n    height: 1.5em;\n    border-radius: 0.2rem;\n  }\n\n  .fa-pydata {\n    stroke: var(--pst-color-background);\n    stroke-linejoin: round;\n    stroke-width: 0.35;\n  }\n}\n","/**\n * Logo in the navbar\n */\n\n.navbar-brand {\n  position: relative;\n  height: var(--pst-header-height);\n  max-height: var(--pst-header-height);\n  padding: 0.5rem 0;\n  width: auto;\n  margin: 0;\n  display: flex;\n\n  // Ensure that the logo stays the same length while other content shrinks\n  flex-shrink: 0;\n  align-items: center;\n  gap: 0.5rem;\n\n  // If there's no logo image, we use a p element w/ the site title\n  p {\n    color: var(--pst-color-text-base);\n    margin-bottom: 0;\n  }\n\n  // If there's a logo, it'll be in an img block\n  img {\n    max-width: 100%;\n    height: 100%;\n    width: auto;\n  }\n\n  &:hover,\n  &:visited:hover {\n    @include link-style-hover;\n\n    color: var(--pst-color-text-base);\n  }\n}\n","/**\n * Navigation links in the navbar and icon links\n */\nul.navbar-nav {\n  // Reduce padding of nested `ul` items a bit\n  ul {\n    padding: 0 0 0 1rem;\n  }\n\n  // Navbar links - do not have an underline by default\n  li {\n    display: flex;\n    flex-direction: column;\n\n    a {\n      display: flex;\n      align-items: center;\n      height: 100%;\n      padding-top: $navbar-link-padding-y;\n      padding-bottom: $navbar-link-padding-y;\n\n      @include link-style-text;\n    }\n  }\n}\n","/**\n * The list of in-page TOC links\n */\n.page-toc {\n  .section-nav {\n    padding-left: 0;\n    border-bottom: none;\n\n    ul {\n      padding-left: 1rem;\n    }\n  }\n\n  // override bootstrap settings\n  .nav-link {\n    font-size: var(--pst-sidebar-font-size-mobile);\n\n    @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n      font-size: var(--pst-sidebar-font-size);\n    }\n  }\n\n  .onthispage {\n    color: var(--pst-color-text-base);\n    font-weight: var(--pst-sidebar-header-font-weight);\n    margin-bottom: 0.5rem;\n  }\n}\n","/**\n* Previous / Next navigation buttons\n**/\n.prev-next-area {\n  width: 100%;\n\n  p {\n    color: var(--pst-color-text-muted);\n    margin: 0 0.3em;\n    line-height: 1.3em;\n  }\n\n  i {\n    font-size: 1.2em;\n  }\n\n  a {\n    // So that buttons align with icons\n    display: flex;\n    align-items: center;\n    border: none;\n    padding: 10px;\n    max-width: 45%;\n    overflow-x: hidden;\n    color: var(--pst-color-text-muted);\n    text-decoration: none;\n\n    p.prev-next-title {\n      @include link-style-default;\n\n      font-weight: var(--pst-admonition-font-weight-heading);\n      font-size: 1.1em;\n    }\n\n    &:hover,\n    &:visited:hover {\n      p.prev-next-title {\n        @include link-style-hover;\n      }\n    }\n\n    .prev-next-info {\n      flex-direction: column;\n      margin: 0 0.5em;\n\n      .prev-next-subtitle {\n        text-transform: capitalize;\n      }\n    }\n\n    &.left-prev {\n      float: left;\n    }\n\n    &.right-next {\n      float: right;\n\n      div.prev-next-info {\n        text-align: right;\n      }\n    }\n  }\n}\n","/**\n * Search field\n **/\n.bd-search {\n  position: relative;\n  padding-left: 0.5rem;\n  gap: 0.5rem;\n  background-color: var(--pst-color-background);\n  border-radius: $admonition-border-radius;\n  border: 1px solid var(--pst-color-border);\n  color: var(--pst-color-text-base);\n\n  // Background should always be same color regardless of active or not\n  &:active {\n    background-color: var(--pst-color-background);\n    color: var(--pst-color-text-muted);\n  }\n\n  .icon {\n    position: absolute;\n    color: var(--pst-color-border);\n    left: 25px;\n  }\n\n  .fa-solid.fa-magnifying-glass {\n    position: absolute;\n    left: calc((2.5rem - 0.7em) / 2);\n    color: var(--pst-color-text-muted);\n  }\n\n  input {\n    // Inner-text of the search bar\n    &::placeholder {\n      color: var(--pst-color-text-muted);\n    }\n\n    // Remove the little \"x\" that pops up when you start typing\n    &::-webkit-search-cancel-button,\n    &::-webkit-search-decoration {\n      appearance: none;\n    }\n  }\n\n  // Shows off the keyboard shortcuts for the button\n  .search-button__kbd-shortcut {\n    display: flex;\n    position: absolute;\n    right: 0.5rem;\n    color: var(--pst-color-border);\n  }\n}\n\n.form-control {\n  background-color: var(--pst-color-background);\n  color: var(--pst-color-text-base);\n\n  &:focus,\n  &:focus-visible {\n    border: none;\n    background-color: var(--pst-color-background);\n    color: var(--pst-color-text-muted);\n  }\n}\n\n/**\n * Search button - located in the navbar\n */\n\n// Search link icon should be a bit bigger since it is separate from icon links\n.search-button i {\n  font-size: 1.3rem;\n}\n\n// __search-container will only show up when we use the search pop-up bar\n.search-button__search-container,\n.search-button__overlay {\n  display: none;\n}\n\n.search-button__wrapper.show {\n  .search-button__search-container {\n    display: flex;\n\n    // Center in middle of screen just underneath header\n    position: fixed;\n    z-index: $zindex-modal;\n    top: 30%;\n    left: 50%;\n    transform: translate(-50%, -50%);\n    right: 1rem;\n    margin-top: 0.5rem;\n    width: 90%;\n    max-width: 800px;\n  }\n\n  .search-button__overlay {\n    display: flex;\n    position: fixed;\n    z-index: $zindex-modal-backdrop;\n    background-color: black;\n    opacity: 0.5;\n    width: 100%;\n    height: 100%;\n    top: 0;\n    left: 0;\n  }\n\n  form.bd-search {\n    flex-grow: 1;\n    padding-top: 0;\n    padding-bottom: 0;\n  }\n\n  // Font and input text a bit bigger\n  svg,\n  input {\n    font-size: var(--pst-font-size-icon);\n  }\n}\n\n/**\n * The search button component that looks like a field.\n * Lives at components/search-button-field.html\n */\n.search-button-field {\n  $search-button-border-radius: 1.5em;\n\n  display: inline-flex;\n  align-items: center;\n  border: var(--pst-color-border) solid 1px;\n  border-radius: $search-button-border-radius;\n  color: var(--pst-color-text-muted);\n  padding: 0.5em;\n  background-color: var(--pst-color-surface);\n\n  &:hover {\n    box-shadow: 0 0 0 $focus-ring-width var(--pst-color-link-hover);\n  }\n\n  &:focus-visible {\n    border-radius: $search-button-border-radius;\n  }\n\n  // The keyboard shotcut text\n  .search-button__default-text {\n    font-size: var(--bs-nav-link-font-size);\n    font-weight: var(--bs-nav-link-font-weight);\n    margin-right: 0.5em;\n    margin-left: 0.5em;\n  }\n\n  .kbd-shortcut__modifier {\n    font-size: 0.75em;\n  }\n\n  // Ensures that all the text lines up in the middle\n  > * {\n    align-items: center;\n  }\n\n  // Only the icon should be visible on narrow screens\n  > :not(svg) {\n    display: none;\n\n    @include media-breakpoint-up(lg) {\n      display: flex;\n    }\n  }\n}\n","/**\n * The 'Hide Search Matches' button.\n * This only shows up when a person lands on a page after clicking a search result.\n * Clicking it removes the highlighting of the search term from the page.\n * We want it to behave like a button.\n */\ndiv#searchbox {\n  // Leave `#searchbox` rules empty so that it doesn't show at all when it is empty\n  p.highlight-link {\n    margin: 1rem 0;\n    width: fit-content;\n\n    // A bit more margin on wide screens to mimic article behavior\n    @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n      margin-left: 2rem;\n    }\n\n    // Put outer shadow on this one so that we can darken the link w/ an inner shadow\n    @include box-shadow;\n\n    // Style the button to look like a Sphinx Design button\n    a {\n      border-radius: 0.25rem;\n      font-size: 1.25rem;\n      padding: 0.75rem;\n      background-color: var(--pst-color-primary);\n      color: var(--pst-color-primary-text);\n      text-decoration: none;\n\n      // The box shadow is inset so that it darkens the button on hover\n      transition: box-shadow 0.25s ease-out;\n\n      &:hover {\n        box-shadow: inset 0 0 50px 50px rgb(0 0 0 / 25%);\n      }\n\n      &::before {\n        content: var(--pst-icon-search-minus);\n        color: unset;\n        font: var(--fa-font-solid);\n        margin-right: 0.5rem;\n      }\n    }\n  }\n}\n","/**\n * Light/dark theme switcher\n */\n\n.theme-switch-button {\n  .theme-switch {\n    display: none;\n\n    &:active {\n      text-decoration: none;\n      color: var(--pst-color-link-hover);\n    }\n\n    .fa-lg {\n      aspect-ratio: 1 / 1;\n    }\n  }\n}\n\n@each $mode in auto, light, dark {\n  html[data-mode=\"#{$mode}\"]\n    .theme-switch-button\n    .theme-switch[data-mode=\"#{$mode}\"] {\n    display: inline; // inline needed for span height to be calculated using inherited font size and line height\n  }\n}\n","button.btn.version-switcher__button {\n  border-color: var(--pst-color-border);\n  color: var(--pst-color-text-base);\n\n  // Add a margin on narrow screens to avoid feeling cramped\n  margin-bottom: 1em;\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    margin-bottom: unset;\n  }\n\n  &:hover {\n    box-shadow: 0 0 0 $focus-ring-width var(--pst-color-secondary);\n    border-color: transparent;\n  }\n\n  &:active {\n    color: var(--pst-color-text-base);\n    border-color: var(--pst-color-border);\n  }\n\n  &:focus-visible {\n    border-color: transparent;\n  }\n}\n\n.version-switcher__menu {\n  border-color: var(--pst-color-border);\n  border-radius: var(--bs-dropdown-border-radius);\n\n  a.list-group-item {\n    background-color: var(--pst-color-on-background);\n    color: var(--pst-color-text-base);\n    padding: 0.75rem 1.25rem;\n\n    &:not(:last-child) {\n      border-bottom: 1px solid var(--pst-color-border);\n    }\n\n    &:hover {\n      @include link-style-hover;\n\n      background-color: var(--pst-color-surface);\n    }\n\n    &.active {\n      @include link-sidebar-current;\n\n      position: relative;\n      z-index: 1;\n\n      span::before {\n        content: \"\";\n        width: 100%;\n        height: 100%;\n        position: absolute;\n        z-index: -1;\n        left: 0;\n        top: 0;\n      }\n    }\n\n    &:focus-visible {\n      z-index: 10; // keep focus ring on top (prevent the hover background of the next dropdown item from covering the ring)\n    }\n  }\n}\n\n// Font behavior on mobile\nbutton.version-switcher__button,\n.version-switcher__menu {\n  font-size: 1.1em; // A bit smaller than other menu font\n  z-index: $zindex-modal; // higher than the sidebars\n\n  // Make sure it meets WCAG target size requirement no matter the version\n  // string displayed in the button\n  @include min-hit-area;\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    font-size: unset;\n  }\n}\n","/* Collapsing of the TOC sidebar while scrolling */\n\n/* Nav: hide second level (shown on .active) */\n\nnav.page-toc {\n  // A little extra space before the buttons\n  margin-bottom: 1rem;\n}\n\n.bd-toc .nav {\n  .nav {\n    display: none;\n\n    // So we can manually specify a level as visible in the config\n    &.visible {\n      display: block;\n    }\n  }\n\n  > .active > ul {\n    display: block;\n  }\n}\n\n// Each entry of the in-page TOC\n.toc-entry {\n  display: block;\n\n  a > code {\n    color: var(--pst-color-text-muted);\n  }\n\n  a.nav-link {\n    display: block;\n    padding: 0.125rem 0;\n\n    // Padding w/ negative margin so the top TOC item highlight overlaps w/ the TOC border\n    padding-left: 1rem;\n    margin-left: -1rem;\n\n    @include link-sidebar;\n\n    &.active {\n      @include link-sidebar-current;\n\n      background-color: transparent;\n\n      &:hover {\n        color: var(--pst-color-link-hover);\n      }\n    }\n\n    &:focus-visible {\n      border-radius: $focus-ring-radius;\n    }\n  }\n}\n","div.versionadded,\ndiv.versionchanged,\ndiv.deprecated {\n  vertical-align: middle;\n  margin: 1.5625em auto;\n  padding: 0 0.6rem;\n  overflow: hidden;\n\n  /* break-inside has replaced page-break-inside and is widely usable since 2019 */\n  page-break-inside: avoid;\n  break-inside: avoid;\n  border-left: 0.2rem solid;\n  border-color: var(--pst-color-info);\n  border-radius: $admonition-border-radius;\n  background-color: var(--pst-color-on-background);\n\n  @include box-shadow;\n\n  position: relative;\n\n  > p {\n    margin-bottom: 0.6rem;\n    margin-top: 0.6rem;\n  }\n}\n\ndiv.versionadded {\n  border-color: var(--pst-color-success);\n  background-color: var(--pst-color-success-bg);\n}\n\ndiv.versionchanged {\n  border-color: var(--pst-color-warning);\n  background-color: var(--pst-color-warning-bg);\n}\n\ndiv.deprecated {\n  border-color: var(--pst-color-danger);\n  background-color: var(--pst-color-danger-bg);\n}\n\nspan.versionmodified {\n  font-weight: 600;\n\n  &::before {\n    margin-right: 0.6rem;\n    color: var(--pst-color-info);\n    font: var(--fa-font-solid);\n    content: var(--pst-icon-versionmodified-default);\n  }\n}\n\nspan.versionmodified.added {\n  &::before {\n    color: var(--pst-color-success);\n    content: var(--pst-icon-versionmodified-added);\n  }\n}\n\nspan.versionmodified.changed {\n  &::before {\n    color: var(--pst-color-warning);\n    content: var(--pst-icon-versionmodified-changed);\n  }\n}\n\nspan.versionmodified.deprecated {\n  &::before {\n    color: var(--pst-color-danger);\n    content: var(--pst-icon-versionmodified-deprecated);\n  }\n}\n",".sidebar-indices-items {\n  display: flex;\n  flex-direction: column;\n  border-top: 1px solid var(--pst-color-border);\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    border-top: none;\n  }\n\n  .sidebar-indices-items__title {\n    font-weight: var(--pst-sidebar-header-font-weight);\n    font-size: var(--pst-sidebar-header-font-size);\n    color: var(--pst-color-text-base);\n    margin-bottom: 0.5rem;\n  }\n\n  ul.indices-link {\n    margin-right: -1rem;\n    list-style: none;\n    padding: 0;\n\n    li > a {\n      display: block;\n      padding: 0.25rem 0;\n      color: var(--pst-color-text-muted);\n\n      &:hover {\n        color: var(--pst-color-primary);\n        text-decoration: none;\n        background-color: transparent;\n      }\n    }\n  }\n}\n",".bd-sidebar-primary div#rtd-footer-container {\n  position: sticky;\n  bottom: -1rem;\n  margin: -1rem; // ignore sidebar padding\n\n  .rst-versions.rst-badge {\n    position: unset;\n    font-size: 0.9em;\n    font-family: var(--pst-font-family-base);\n    max-width: unset;\n\n    .rst-current-version {\n      display: flex;\n      align-items: center;\n      gap: 0.2rem;\n      height: 2.5rem;\n      transition: background-color 0.2s ease-out;\n      background-color: var(--pst-color-background);\n      color: var(--pst-color-success);\n      border-top: 1px solid var(--pst-color-border);\n    }\n\n    .fa.fa-book {\n      color: var(--pst-color-text-muted);\n      margin-right: auto;\n\n      &::after {\n        color: var(--pst-color-text-base);\n        content: \"Read The Docs\";\n        font-family: var(--pst-font-family-base);\n        font-weight: var(--pst-admonition-font-weight-heading);\n      }\n    }\n\n    .fa.fa-caret-down {\n      color: var(--pst-color-text-muted);\n    }\n  }\n\n  .rst-versions.rst-badge.shift-up {\n    .rst-current-version {\n      border-bottom: 1px solid var(--pst-color-border);\n    }\n  }\n\n  .rst-other-versions {\n    background-color: var(--pst-color-surface);\n    color: var(--pst-color-text-base);\n\n    dl {\n      dd a {\n        color: var(--pst-color-text-muted);\n      }\n    }\n\n    hr {\n      background-color: var(--pst-color-border);\n    }\n\n    small a {\n      color: var(--pst-color-link);\n    }\n\n    input {\n      padding-left: 0.5rem;\n      border: 1px solid var(--pst-color-border);\n      background-color: var(--pst-color-surface);\n    }\n  }\n}\n","/**\n * Admonitions and blocks of styled content.\n * Admonitions CSS originally inspired by https://squidfunk.github.io/mkdocs-material/getting-started/\n */\n$admonition-border-radius: 0.25rem;\n$admonition-left-border-width: 0.2rem;\n\ndiv.admonition,\n.admonition {\n  margin: 1.5625em auto;\n  padding: 0 0.6rem 0.8rem;\n  overflow: hidden;\n\n  /* break-inside has replaced page-break-inside and is widely usable since 2019 */\n  page-break-inside: avoid;\n  break-inside: avoid;\n  border-left: $admonition-left-border-width solid;\n  border-color: var(--pst-color-info);\n  border-radius: $admonition-border-radius;\n  background-color: var(--pst-color-on-background);\n\n  @include box-shadow;\n\n  // Last item should have no spacing since we'll control that w/ padding\n  *:last-child {\n    margin-bottom: 0;\n  }\n\n  // Items after the title should be indented\n  p.admonition-title ~ * {\n    margin-left: 1.4rem;\n    margin-right: 1.4rem;\n  }\n\n  // Lists need to have left margin so they don't spill into it\n  > ol,\n  > ul {\n    margin-left: 1em;\n  }\n\n  // Defaults for all admonitions\n  > .admonition-title {\n    margin: 0 -0.6rem;\n    padding: 0.4rem 0.6rem 0.4rem 2rem;\n    font-weight: var(--pst-admonition-font-weight-heading);\n    position: relative;\n\n    @include legacy-backdrop-placeholder;\n\n    background-color: var(--pst-color-info-bg);\n\n    // now that we use solid colors we want the title on top\n    z-index: 1;\n\n    &::after {\n      position: absolute;\n      left: 0.5rem;\n      width: 1rem;\n      height: 1rem;\n      color: var(--pst-color-info);\n      font: var(--fa-font-solid);\n      line-height: inherit;\n      content: var(--pst-icon-admonition-default);\n      opacity: 1;\n    }\n\n    // Next element after title needs some extra upper-space\n    + * {\n      margin-top: 0.4em;\n    }\n  }\n\n  &.attention {\n    border-color: var(--pst-color-attention);\n\n    > .admonition-title {\n      background-color: var(--pst-color-attention-bg);\n\n      &::after {\n        color: var(--pst-color-attention);\n        content: var(--pst-icon-admonition-attention);\n      }\n    }\n  }\n\n  &.caution {\n    border-color: var(--pst-color-warning);\n\n    > .admonition-title {\n      background-color: var(--pst-color-warning-bg);\n\n      &::after {\n        color: var(--pst-color-warning);\n        content: var(--pst-icon-admonition-caution);\n      }\n    }\n  }\n\n  &.warning {\n    border-color: var(--pst-color-warning);\n\n    > .admonition-title {\n      background-color: var(--pst-color-warning-bg);\n\n      &::after {\n        color: var(--pst-color-warning);\n        content: var(--pst-icon-admonition-warning);\n      }\n    }\n  }\n\n  &.danger {\n    border-color: var(--pst-color-danger);\n\n    > .admonition-title {\n      background-color: var(--pst-color-danger-bg);\n\n      &::after {\n        color: var(--pst-color-danger);\n        content: var(--pst-icon-admonition-danger);\n      }\n    }\n  }\n\n  &.error {\n    border-color: var(--pst-color-danger);\n\n    > .admonition-title {\n      background-color: var(--pst-color-danger-bg);\n\n      &::after {\n        color: var(--pst-color-danger);\n        content: var(--pst-icon-admonition-error);\n      }\n    }\n  }\n\n  &.hint {\n    border-color: var(--pst-color-success);\n\n    > .admonition-title {\n      background-color: var(--pst-color-success-bg);\n\n      &::after {\n        color: var(--pst-color-success);\n        content: var(--pst-icon-admonition-hint);\n      }\n    }\n  }\n\n  &.tip {\n    border-color: var(--pst-color-success);\n\n    > .admonition-title {\n      background-color: var(--pst-color-success-bg);\n\n      &::after {\n        color: var(--pst-color-success);\n        content: var(--pst-icon-admonition-tip);\n      }\n    }\n  }\n\n  &.important {\n    border-color: var(--pst-color-attention);\n\n    > .admonition-title {\n      background-color: var(--pst-color-attention-bg);\n\n      &::after {\n        color: var(--pst-color-attention);\n        content: var(--pst-icon-admonition-important);\n      }\n    }\n  }\n\n  &.note {\n    border-color: var(--pst-color-info);\n\n    > .admonition-title {\n      background-color: var(--pst-color-info-bg);\n\n      &::after {\n        color: var(--pst-color-info);\n        content: var(--pst-icon-admonition-note);\n      }\n    }\n  }\n\n  &.seealso {\n    border-color: var(--pst-color-success);\n\n    > .admonition-title {\n      background-color: var(--pst-color-success-bg);\n\n      &::after {\n        color: var(--pst-color-success);\n        content: var(--pst-icon-admonition-seealso);\n      }\n    }\n  }\n\n  &.admonition-todo {\n    border-color: var(--pst-color-secondary);\n\n    > .admonition-title {\n      background-color: var(--pst-color-secondary-bg);\n\n      &::after {\n        color: var(--pst-color-secondary);\n        content: var(--pst-icon-admonition-todo);\n      }\n    }\n  }\n\n  /**\n   * Special-case for a `sidebar` class that makes the admonition float to\n   * the right like the {sidebar} directive.\n   */\n  &.sidebar {\n    max-width: 40%;\n    float: right;\n    clear: both;\n    margin-left: 0.5rem;\n    margin-top: 0;\n\n    // Undo the .sidebar directive border\n    border-width: 0 0 0 $admonition-left-border-width;\n\n    // TODO: these semantic-color-names border-color rules might no longer be\n    //       needed when we drop support for Sphinx 4 / docutils 0.17\n    &.attention,\n    &.important {\n      border-color: var(--pst-color-attention);\n    }\n\n    &.caution,\n    &.warning {\n      border-color: var(--pst-color-warning);\n    }\n\n    &.danger,\n    &.error {\n      border-color: var(--pst-color-danger);\n    }\n\n    &.hint,\n    &.tip,\n    &.seealso {\n      border-color: var(--pst-color-success);\n    }\n\n    &.note,\n    &.todo {\n      border-color: var(--pst-color-info);\n    }\n\n    // No inner margin since we have less horizontal space w/ the sidebar\n    p.admonition-title ~ * {\n      margin-left: 0;\n      margin-right: 0;\n    }\n  }\n}\n\n/**************************************************************\n * Similar content blocks that are not technically admonitions.\n */\n\n/**\n * Topics and the {contents} directive\n */\n// Docutils <= 0.17\ndiv.topic,\ndiv.topic.contents,\n// Docutils >= 0.18\nnav.contents,\naside.topic {\n  display: flex;\n  flex-direction: column;\n  background-color: var(--pst-color-surface);\n  border-color: var(--pst-color-border);\n  border-radius: $admonition-border-radius;\n  padding: 1rem 1.25rem;\n\n  @include box-shadow;\n\n  .topic-title {\n    margin: 0 0 0.5rem;\n  }\n\n  // Over-ride text color to ensure enough contrast\n  p {\n    color: var(--pst-color-on-surface) !important;\n  }\n\n  // Over-ride large default padding\n  ul.simple {\n    padding-left: 1rem;\n\n    ul {\n      // So that sub-lists will have a bit less padding\n      padding-left: 2em;\n    }\n  }\n}\n\n/**\n * Sidebar directive\n */\naside.sidebar {\n  border: 1px solid var(--pst-color-border);\n  background-color: var(--pst-color-surface);\n  border-radius: $admonition-border-radius;\n\n  // to match the admonition-styled sidebars:\n  margin-left: 0.5rem;\n  padding: 0;\n\n  > *:last-child {\n    padding-bottom: 1rem;\n  }\n\n  p.sidebar-title {\n    position: relative;\n    margin-bottom: 0;\n    padding-top: 0.5rem;\n    padding-bottom: 0.5rem;\n    border-bottom: 1px solid var(--pst-color-border);\n    font-family: var(--pst-font-family-heading);\n    font-weight: var(--pst-admonition-font-weight-heading);\n  }\n\n  // Add margin to the first non-`.sidebar-title` item\n  > *:not(.sidebar-title):first-child,\n  > p.sidebar-title + * {\n    margin-top: 1rem;\n  }\n\n  > * {\n    padding-left: 1rem;\n    padding-right: 1rem;\n  }\n}\n\n/**\n * Rubrics look kind of like section headers\n */\np.rubric {\n  display: flex;\n  flex-direction: column;\n}\n\n/**\n * Seealso is kind of like a vertically-collapsed admonition\n */\n.seealso dd {\n  margin-top: 0;\n  margin-bottom: 0;\n}\n","/**\n * Miscellaneous color functions and mixins\n**/\n\n@use \"sass:list\";\n@use \"sass:map\";\n@use \"sass:meta\";\n@use \"sass:math\";\n@use \"sass:string\";\n\n// We must add ::before pseudo-element to some theme components (such as admonitions)\n// because users were instructed to customize the background color this way.\n@mixin legacy-backdrop-placeholder {\n  &::before {\n    content: \"\";\n    width: 100%;\n    height: 100%;\n    position: absolute;\n    left: 0;\n    top: 0;\n    z-index: -1;\n\n    // So that hovering over the text within background is still possible.\n    // Otherwise the background overlays the text and you cannot click or select easily.\n    // ref: https://developer.mozilla.org/en-US/docs/Web/CSS/pointer-events\n    pointer-events: none;\n  }\n}\n\n/**\n* Function to get items from nested maps\n*/\n// @param {Map} $map - Map\n// @param {Arglist} $keys - Keys to fetc\n// @return {*}\n@function map-deep-get($map, $keys...) {\n  @each $key in $keys {\n    $map: map.get($map, $key);\n  }\n\n  @return $map;\n}\n\n/**\n  * Function to check if the color needs converting to a \"color\" type\n  * if it is a string we cannot use other color manipulation functions\n  * It is used to create the sphinx-design colours as these are often interpolated\n  */\n// @param {String/Color} $color - Color definition from map\n// @return {Color} - Color type (in hex)\n@function check-color($color) {\n  @if meta.type-of($color) == string {\n    $color: from-hex($color);\n  }\n\n  @return $color;\n}\n\n/**\n  * Function to convert the string representation of a color to a color type (hex)\n  */\n// @param {String} $string - String representation of a color\n@function from-hex($string) {\n  $string-lower: string.to-lower-case($string);\n  $r: \"\";\n  $g: \"\";\n  $b: \"\";\n  $hex: \"0\" \"1\" \"2\" \"3\" \"4\" \"5\" \"6\" \"7\" \"8\" \"9\" \"a\" \"b\" \"c\" \"d\" \"e\" \"f\";\n  $length: string.length($string);\n  $max: if($length == 4, 1, 2);\n\n  // Check for length accuracy\n  @if $length != 4 and $length != 7 {\n    @return $string;\n  }\n\n  // Loop from the second character (omitting #)\n  @for $i from 2 through $length {\n    $c: string.slice($string-lower, $i, $i);\n\n    // If wrong character, return\n    @if not list.index($hex, $c) {\n      @return $string;\n    }\n\n    @if string.length($r) < $max {\n      $r: $r + $c;\n    } @else if string.length($g) < $max {\n      $g: $g + $c;\n    } @else if string.length($b) < $max {\n      $b: $b + $c;\n    }\n  }\n\n  @if $length == 4 {\n    $r: $r + $r;\n    $g: $g + $g;\n    $b: $b + $b;\n  }\n\n  @return rgb(hex-to-dec($r), hex-to-dec($g), hex-to-dec($b));\n}\n\n@function hex-to-dec($string) {\n  $hex: \"0\" \"1\" \"2\" \"3\" \"4\" \"5\" \"6\" \"7\" \"8\" \"9\" \"a\" \"b\" \"c\" \"d\" \"e\" \"f\";\n  $string: string.to-lower-case($string);\n  $length: string.length($string);\n  $dec: 0;\n\n  @for $i from 1 through $length {\n    $factor: 1 + (15 * ($length - $i));\n    $index: list.index($hex, string.slice($string, $i, $i));\n    $dec: $dec + $factor * ($index - 1);\n  }\n\n  @return $dec;\n}\n","// Style API docs from sphinx' autodoc / autosummary\n\n/*******************************************************************************\n* Styling for field lists\n*/\n\n/* grey highlighting of 'parameter' and 'returns' field */\ntable.field-list {\n  border-collapse: separate;\n  border-spacing: 10px;\n  margin-left: 1px;\n\n  th.field-name {\n    padding: 1px 8px 1px 5px;\n    white-space: nowrap;\n    background-color: var(--pst-color-surface);\n  }\n\n  /* italic font for parameter types */\n  td.field-body {\n    p {\n      font-style: italic;\n\n      > strong {\n        font-style: normal;\n      }\n    }\n\n    /* reduced space around parameter description */\n    blockquote {\n      border-left: none;\n      margin: 0 0 0.3em;\n      padding-left: 30px;\n    }\n  }\n}\n\n/*******************************************************************************\n* Styling for autosummary tables\n*/\n\n.table.autosummary {\n  // The first column (with the signature) should not wrap\n  td:first-child {\n    white-space: nowrap;\n  }\n}\n\n/* overriding basic.css to use our own monospace font */\n.sig {\n  font-family: var(--pst-font-family-monospace);\n}\n\n/* C++ specific styling - overriding the basic.css to avoid custom colors */\n\n.sig-inline.c-texpr,\n.sig-inline.cpp-texpr {\n  font-family: unset;\n}\n\n.sig.c .k,\n.sig.c .kt,\n.sig.cpp .k,\n.sig.cpp .kt {\n  color: var(--pst-color-text-base);\n}\n\n.sig.c .m,\n.sig.cpp .m {\n  color: var(--pst-color-text-base);\n}\n\n.sig.c .s,\n.sig.c .sc,\n.sig.cpp .s,\n.sig.cpp .sc {\n  color: var(--pst-color-text-base);\n}\n\n// addition\n\n// .sig.c .sig-name .n,\n// .sig.cpp .sig-name .n {\n//   color: var(--pst-color-inline-code);\n// }\n\n.sig-name {\n  color: var(--pst-color-inline-code);\n}\n\n.sig-param .o,\n.sig-param .default_value {\n  color: var(--pst-color-text-muted);\n  font-weight: normal;\n}\n\n// change target color for dark theme\ndt:target,\nspan.highlighted {\n  background-color: var(--pst-color-target);\n}\n\n.viewcode-back {\n  font-family: var(--pst-font-family-base);\n}\n\n.viewcode-block:target {\n  border-top: 1px solid var(--pst-color-border);\n  border-bottom: 1px solid var(--pst-color-border);\n  position: relative;\n  background-color: var(--pst-color-target);\n}\n\ndl > dt > a:has(.viewcode-link) {\n  // Sphinx applies a `float:right` rule to the .viewcode-line span, which\n  // exposes a browser glitch in the focus ring. It seems the browser creates\n  // two separate boxes, an empty box where the anchor element gets laid out and\n  // then another box around the anchor's contents that have been floated right.\n  // Firefox draws the focus ring around the empty anchor element box. Chrome\n  // draws two focus rings: one around the empty anchor and one around the\n  // floated-right element. To fix the glitch, we apply the float rule on the\n  // parent rather than the child.\n  float: right;\n\n  .viewcode-link {\n    float: none;\n  }\n}\n\n/*******************************************************************************\n* Styling for autosummary titles like \"parameters\" and \"returns\"\n*/\n\n// the API selector\n// from https://github.com/pradyunsg/furo/blob/main/src/furo/assets/styles/content/_api.sass#L6)\ndl[class]:not(.option-list, .field-list, .footnote, .glossary, .simple) {\n  // increase margin bottom after the dl elements\n  margin-bottom: 3rem;\n\n  dd {\n    margin-left: 2rem;\n\n    // Fix until this will be solved to Sphinx https://github.com/sphinx-doc/sphinx/issues/10815\n    & > dl.simple > dt {\n      display: flex;\n    }\n  }\n\n  dl.field-list {\n    display: grid;\n    grid-template-columns: unset;\n  }\n\n  dt.field-odd,\n  dt.field-even {\n    margin-top: 0.2rem;\n    margin-bottom: 0.1rem;\n    background-color: var(--pst-color-surface);\n  }\n}\n","/**\n * Code block styling\n * Note that we inherit a lot of styling from Bootstrap so not many rules here.\n */\n\n// General code block behavior\n// Unset bootstrap behavior\ndiv[class*=\"highlight-\"],\ndiv.highlight,\ndiv.literal-block-wrapper {\n  display: flex;\n  flex-direction: column;\n  width: unset;\n  border-radius: $admonition-border-radius;\n  break-inside: avoid;\n}\n\n// Code blocks with captions\n// There's a wrapper when the code block has a title\ndiv.literal-block-wrapper {\n  border: 1px solid var(--pst-color-border);\n  border-radius: $admonition-border-radius;\n\n  // This is where the title goes\n  div.code-block-caption {\n    margin: 0;\n    border-bottom: 1px solid var(--pst-color-border);\n    padding: 0.5rem;\n    font-size: 1rem;\n    font-weight: var(--pst-font-weight-caption);\n\n    a.headerlink {\n      font-size: inherit;\n    }\n  }\n\n  // Remove the upper border radius since we want it to connect with the title\n  // Remove the box shadow so the wrapper gets the shadow instead\n  div[class*=\"highlight-\"] {\n    margin: 0;\n    border-radius: 0;\n\n    pre {\n      border: none;\n      box-shadow: none;\n    }\n  }\n}\n\n/**\n * In-line code\n */\ncode.literal {\n  padding: 0.1rem 0.25rem;\n  background-color: var(--pst-color-surface);\n  border: 1px solid var(--pst-color-border);\n  border-radius: 0.25rem;\n}\n\na > code {\n  color: var(--pst-color-inline-code-links);\n}\n\n// Fix for Sphinx's \"highlight\" directive - this is an issue with our accessible pygments theme\n// and the colour we are using for the background of the code blocks.\nhtml[data-theme=\"light\"] .highlight .nf {\n  color: #0078a1 !important;\n}\n\n// Minimum opacity needed for linenos to be WCAG AA conformant\nspan.linenos {\n  opacity: 0.8 !important;\n}\n","figure > a,\nfigure > a > img,\nfigure > img,\nfigure > video {\n  display: block;\n  margin-left: auto;\n  margin-right: auto;\n}\n\nfigure {\n  a.headerlink {\n    // So that header link doesn't push caption to be off-center.\n    position: absolute;\n    font-size: inherit;\n  }\n\n  // Default headerlink hover doesn't trigger on figures\n  &:hover a.headerlink {\n    visibility: visible;\n  }\n\n  figcaption {\n    font-family: var(--pst-font-family-heading);\n    font-weight: var(--pst-font-weight-caption);\n    color: var(--pst-color-text-muted);\n    margin-left: auto;\n    margin-right: auto;\n    margin-top: 0.3rem;\n    text-align: center;\n\n    & > p:last-child {\n      // Don't add extra margin to already existing figure bottom margin\n      margin-bottom: 0;\n    }\n\n    p {\n      text-align: start;\n      display: inline-block;\n    }\n\n    table.table {\n      width: fit-content;\n      margin-left: auto;\n      margin-right: auto;\n    }\n  }\n}\n","// For consistency, add bracket around footnotes/citations which are\n// cited more than once. E.g. [Newton](1,2) instead of Newton(1,2)\ndt.label > span.brackets:not(:only-child)::before {\n  content: \"[\";\n}\n\ndt.label > span.brackets:not(:only-child)::after {\n  content: \"]\";\n}\n\n// Make footnote as a superscript\na.footnote-reference {\n  vertical-align: super;\n  font-size: small;\n}\n\n// Docutils 0.18 uses an `aside.footnote` container with different internal structure\naside.footnote {\n  margin-bottom: 0.5rem;\n\n  &:last-child {\n    margin-bottom: 1rem;\n  }\n\n  span.label,\n  span.backrefs {\n    font-weight: bold;\n  }\n\n  &:target {\n    background-color: var(--pst-color-target);\n  }\n}\n","/**\n * Hacky fixes that don't fit cleanly into other sections\n */\n\n// Ensure user highlighting/selecting behaves properly\n// From https://stackoverflow.com/a/34372191\ntable.highlighttable td.linenos,\nspan.linenos,\ndiv.doctest > div.highlight span.gp {\n  /* gp: Generic.Prompt */\n  user-select: none;\n}\n","// Override bootstrap by restoring the basic theme default.\ndd {\n  margin-top: 3px;\n  margin-bottom: 10px;\n  margin-left: 30px;\n}\n\nol,\nul {\n  padding-inline-start: 2rem;\n\n  li > p:first-child {\n    margin-bottom: 0.25rem;\n    margin-top: 0.25rem;\n  }\n}\n","// GitHub blockquote style\nblockquote {\n  padding: 1em;\n  color: var(--pst-color-text-muted);\n  border-left: 0.25em solid var(--pst-color-blockquote-notch);\n  border-radius: $admonition-border-radius;\n  position: relative;\n\n  p {\n    color: var(--pst-color-text-base);\n  }\n\n  // remove padding from included line-block to avoid duplication\n  .line-block {\n    margin: 0;\n  }\n\n  // remove margin bottom for the last p\n  p:last-child {\n    margin-bottom: 0;\n  }\n\n  @include legacy-backdrop-placeholder;\n\n  background-color: var(--pst-color-surface);\n\n  // Ensure there is enough contrast against the background\n  a {\n    color: var(--pst-color-inline-code-links);\n  }\n\n  // hack to make the text in the blockquote selectable\n  &::before {\n    z-index: -1;\n  }\n}\n","/**\n * Span-level styling within content\n */\n\nspan.guilabel {\n  border: 1px solid var(--pst-color-info);\n  font-size: 80%;\n  font-weight: 700;\n  border-radius: 4px;\n  padding: 2.4px 6px;\n  margin: auto 2px;\n  position: relative;\n\n  @include legacy-backdrop-placeholder;\n\n  background-color: var(--pst-color-info-bg);\n}\n\na.reference.download::before {\n  content: var(--pst-icon-download);\n  font: var(--fa-font-solid);\n  font-size: 0.8em;\n  padding: 0 0.25em;\n  color: var(--pst-color-text-muted);\n}\n","/**\n * Tables\n */\n\ntable {\n  // default to table-center\n  margin-left: auto;\n  margin-right: auto;\n\n  &.table-right {\n    margin-right: 0;\n  }\n\n  &.table-left {\n    margin-left: 0;\n  }\n}\n\n// customize table caption from bootstrap\n// to display them on top and centered\ntable caption {\n  text-align: center;\n  caption-side: top;\n  color: var(--pst-color-text-muted);\n}\n\n// MyST Markdown tables use these classes to control alignment\nth,\ntd {\n  &.text-left {\n    text-align: left;\n  }\n\n  &.text-right {\n    text-align: right;\n  }\n\n  &.text-center {\n    text-align: center;\n  }\n}\n\n// override bootstrap table colors\n.table {\n  @include table-colors;\n\n  --bs-table-bg: transparent; // background\n  --bs-table-color: var(\n    --pst-color-text-base\n  ); // ensure text and bullets are visible\n}\n\n.pst-scrollable-table-container {\n  // Put a horizontal scrollbar just below tables that are too wide to fit\n  // within the main column\n  overflow-x: auto;\n}\n","/**\n * Style the toctree component in pages (avoid modifying the navbars)\n */\n.toctree-wrapper {\n  p.caption {\n    font-size: 1.5em;\n    margin-bottom: 0;\n  }\n\n  & > ul {\n    padding-left: 0;\n  }\n\n  li[class^=\"toctree-l\"] {\n    list-style: none;\n    margin-bottom: 0.2em;\n\n    & > a {\n      list-style: none;\n      font-size: 1.1em;\n    }\n\n    & > ul {\n      list-style: none;\n      padding-inline-start: 1.5em;\n    }\n  }\n\n  // slightly bigger font for l1\n  .toctree-l1 > a {\n    font-size: 1.3em;\n  }\n}\n\ndiv.topic.contents,  // Docutils <= 0.17\nnav.contents  // Docutils >= 0.18\n{\n  // Style similarly to toctree\n  ul.simple {\n    list-style: none;\n    padding-left: 0;\n  }\n}\n","/**\n * Mathematics via MathJax.\n *\n * This is designed for MathJax v3\n * ref: https://www.sphinx-doc.org/en/master/usage/extensions/math.html#module-sphinx.ext.mathjax\n */\n\n// Applies to all math elements\nspan.math,\ndiv.math {\n  align-items: center;\n  display: flex;\n  max-width: 100%;\n\n  // This will be over-ridden for the y-direction and divs\n  overflow: hidden;\n}\n\n// Inline-only\nspan.math {\n  display: inline-flex;\n}\n\n// Block-level only\ndiv.math {\n  gap: 0.5em;\n\n  // So that the eqno shows up after the equation\n  flex-direction: row-reverse;\n\n  // The equation number / link\n  span.eqno a.headerlink {\n    position: relative;\n    font-size: 1em;\n  }\n\n  // The math container\n  mjx-container {\n    flex-grow: 1;\n    padding-bottom: 0.2rem;\n    overflow: auto;\n\n    // Set height to 0 so that it does not cause scrollbars to appear\n    // ref: https://github.com/mathjax/MathJax/issues/2521\n    mjx-assistive-mml {\n      height: 0;\n    }\n  }\n}\n","/**\n * ABlog\n * ref: https://ablog.readthedocs.io/\n */\n\n/**\n * Sidebar template components\n */\n.ablog-sidebar-item {\n  h2,\n  h3 {\n    font-size: var(--pst-sidebar-header-font-size);\n\n    // Remove unnecessary vertical whitespace\n    margin-top: 0.5rem;\n\n    // The headers are all links, but this makes them hard to parse\n    // So we change the colors to make them look like headers\n    a {\n      color: var(--pst-color-text-base);\n    }\n  }\n\n  ul {\n    // No bullet points for the primary sidebar items\n    list-style: none;\n    padding-left: 0;\n\n    // Otherwise a scrollbar randomly shows up\n    overflow-y: hidden;\n\n    // List of recent post items\n    display: flex;\n    flex-direction: column;\n    gap: 0.5em;\n    margin-bottom: 0;\n\n    // The ablog cloud should move horizontally\n    &.ablog-cloud {\n      flex-flow: row wrap;\n      gap: 0.5rem;\n\n      // Vertical-align tag clouds\n      li {\n        // Center the tag cloud items\n        display: flex;\n        align-items: center;\n      }\n    }\n  }\n}\n\n/**\n * Previous / next buttons at the bottom\n */\n.ablog__prev-next {\n  font-size: 1.2em;\n  display: flex;\n  padding: 1rem 0;\n\n  // The bottom previous / next arrows\n  > span {\n    // To ensure that the whole thing fits on one line even if there are long titles\n    display: flex;\n    max-width: 45%;\n\n    // Links within each span have the collection of icon + text\n    a {\n      display: flex;\n      align-items: center;\n      margin-left: auto;\n      gap: 1rem;\n      line-height: 1.5rem;\n\n      i::before {\n        color: var(--pst-color-text-base);\n      }\n    }\n  }\n\n  // The first span is for the previous page and aligns to the left\n  span.ablog__prev {\n    i.fa-arrow-circle-left::before {\n      content: var(--pst-icon-angle-left);\n    }\n  }\n\n  // The second span is just an empty space so we remove it because we're\n  // positioning with flex\n  span.ablog__spacer {\n    display: none;\n  }\n\n  // The third span is aligned to the right\n  span.ablog__next {\n    margin-left: auto;\n    text-align: right;\n\n    i.fa-arrow-circle-right::before {\n      content: var(--pst-icon-angle-right);\n    }\n  }\n}\n\n/**\n * {postlist} directive and posts page\n */\n.ablog__collection,\n.postlist {\n  padding-left: 0;\n\n  .ablog-post {\n    list-style: none;\n\n    // Post metadata tags (author, links ,etc) should be a bit smaller\n    .ablog-archive {\n      display: flex;\n      flex-flow: row wrap;\n      gap: 1rem;\n      list-style: none;\n      font-size: 0.75rem;\n      padding-left: 0;\n    }\n\n    // Title line should be a bit bigger and bold to stand out\n    .ablog-post-title {\n      margin-top: 0;\n      font-size: 1.25rem;\n\n      a {\n        font-weight: bold;\n      }\n    }\n\n    // Read more button should be a bit bigger\n    .ablog-post-expand {\n      margin-bottom: 0.5rem;\n    }\n  }\n}\n","/**\n * Special cases for Bootstrap functionality\n */\n\n// Bootstrap adds margin to their general container class. However, sphinx/docutils\n// can also generate output with the container class, but in those cases we should\n// not add the margin from bootstrap. Same for max-width.\n.docutils.container {\n  padding-left: unset;\n  padding-right: unset;\n  margin-left: unset;\n  margin-right: unset;\n  max-width: unset;\n  width: unset;\n}\n\n.btn {\n  --bs-btn-focus-box-shadow: #{$btn-focus-box-shadow};\n}\n","/**\n * Sphinx Copybutton\n * ref: https://sphinx-copybutton.readthedocs.io/\n */\n\ndiv.highlight button.copybtn {\n  // Nicer spacing\n  display: flex;\n  align-items: center;\n  justify-content: center;\n\n  // Don't over-ride the success color\n  &:not(.success) {\n    color: var(--pst-color-muted);\n  }\n\n  border: none;\n  background-color: var(--pst-color-surface);\n\n  &:hover {\n    &:not(.success) {\n      color: var(--pst-color-text);\n      background-color: var(--pst-color-shadow);\n    }\n  }\n\n  // Tooltip styling\n  &.o-tooltip--left::after {\n    color: var(--pst-color-text);\n    background-color: var(--pst-color-surface);\n  }\n\n  &:focus {\n    // For keyboard users, make the copy button visible when focussed.\n    opacity: 1;\n  }\n\n  &:focus-visible {\n    outline: $focus-ring-outline;\n  }\n}\n\ndiv.highlight:has(button.copybtn) {\n  // Make sure the code block has enough height for the copy button.\n  // Sphinx-copybutton sets 0.3em top offset plus 1.7em height:\n  // https://github.com/executablebooks/sphinx-copybutton/blob/master/sphinx_copybutton/_static/copybutton.css\n  min-height: 2em;\n}\n","// adapt ethical ad to the theme\n#ethical-ad-placement {\n  .ethical-sidebar a,\n  .ethical-sidebar a:visited,\n  .ethical-sidebar a:hover,\n  .ethical-sidebar a:active,\n  .ethical-footer a,\n  .ethical-footer a:visited,\n  .ethical-footer a:hover,\n  .ethical-footer a:active {\n    color: var(--pst-color-text-base);\n  }\n\n  .ethical-sidebar,\n  .ethical-footer {\n    background-color: var(--pst-color-background);\n    border: 1px solid var(--pst-color-border);\n    border-radius: 5px;\n    color: var(--pst-color-text-base);\n    font-size: 14px;\n    line-height: 20px;\n  }\n}\n","/**\n * Styles for various Sphinx execution libraries.\n * For now, where these define output sections, we simply revert their background\n * to be a \"light theme\" background. This ensures that inputs/outputs behave similarly,\n * because the CSS is often controlled by each package.\n * In the future, we might add dark theme support for specific packages.\n */\n\n/******************************************************************************\n * Jupyter Sphinx\n */\n\n.bd-content div.jupyter_container {\n  // We don't want borders around the whole container, just around code cells\n  border: none;\n  background-color: unset;\n  box-shadow: none;\n\n  // Code cells should have the same style as our other code objects\n  div.output,\n  div.highlight {\n    border-radius: 0.25rem;\n  }\n\n  div.highlight {\n    background-color: var(--pst-color-surface);\n  }\n\n  // Ensure the style is the same as our code cells. Jupyter Sphinx makes it tiny.\n  .cell_input,\n  .cell_output {\n    border-radius: 0.25rem;\n\n    pre {\n      padding: 1rem;\n    }\n  }\n}\n","/* Styles for graphviz generated output from Sphinx */\n\n/* Style the inheritance diagram such that it has a dark mode */\nhtml[data-theme=\"dark\"] div.graphviz > object.inheritance {\n  filter: brightness(0.8) invert(0.82) contrast(1.2);\n  color-scheme: normal;\n}\n","/**\n * Special-cases for packages in the PyData ecosystem\n */\n\n// xarray output display in bootstrap\n.xr-wrap[hidden] {\n  display: block !important;\n}\n\n// ipywidgets\n.jp-OutputArea-output.lm-Widget {\n  // override overflow:hidden rule from Lumino (.lm-Widget) to allow scrolling\n  overflow: auto;\n}\n","/*******************************************************************************\n * Special-cases for the sphinx-design library, mainly to make it compatible\n * with the dark/light themes of pydata-sphinx-theme.\n *\n * NOTE: sphinx-design uses !important quite liberally, so here we must do the\n * same for our overrides to have any effect.\n */\n@use \"../variables/color\" as pst-color;\n@use \"sass:color\";\n@use \"sass:map\";\n@use \"sass:meta\";\n@use \"sass:string\";\n\n/*******************************************************************************\n  * Color and variables\n  *\n  * This is a list of the semantic color names from sphinx-design (we only\n  * need to override variables that sphinx-design has actually defined).\n  * https://github.com/executablebooks/sphinx-design/blob/9226a12a/style/_colors.scss#L31-L43\n  */\n$sd-semantic-color-names: (\n  \"primary\",\n  \"secondary\",\n  \"success\",\n  \"info\",\n  \"warning\",\n  \"danger\",\n  \"light\",\n  \"muted\",\n  \"dark\",\n  \"black\",\n  \"white\"\n);\n\n/**\n  * Here we create some extra --pst-color-* variables and use\n  * them to override the value of the corresponding sphinx-design variables.\n  * This is easier than re-writing the sphinx-design rules. Even easier would be\n  * directly assigning our values to the --sd-color-* variables, but then our\n  * downstream users couldn't override *our* colors and have it affect buttons\n  * and badges.\n  *\n  * First, define the extra keys needed to cover the full range of semantic\n  * color names used in sphinx-design, then merge them with the names we\n  * already define for our own needs.\n  * see https://sphinx-design.readthedocs.io/en/latest/css_variables.html\n  */\n$extra-semantic-colors: (\n  \"white\": $foundation-white,\n  \"light\": (\n    light: $foundation-light-gray,\n    bg-light: color.scale($foundation-light-gray, $lightness: 30%),\n    dark: $foundation-light-gray,\n    bg-dark: color.scale($foundation-light-gray, $lightness: -30%),\n  ),\n  \"muted\": (\n    light: $foundation-muted-gray,\n    bg-light: color.scale($foundation-muted-gray, $lightness: 30%),\n    dark: $foundation-light-gray,\n    bg-dark: color.scale($foundation-muted-gray, $lightness: -30%),\n  ),\n  \"dark\": $foundation-dark-gray,\n  \"black\": $foundation-black,\n);\n$all-colors: map.merge($pst-semantic-colors, $extra-semantic-colors);\n\n@mixin create-sd-colors($value, $name) {\n  // define the pst variables, so that downstream user overrides will work\n  --pst-color-#{$name}: #{$value};\n\n  // we are now using a11y-combination to calculate the text color - this is based\n  // on the WCAG color contrast guidelines\n  --pst-color-#{$name}-text: #{a11y-combination($value)};\n\n  // TODO: highlight seems to be used for buttons @trallard to fix on a11y follow-up work\n  --pst-color-#{$name}-highlight: #{color.adjust($value, $lightness: -15%)};\n\n  // override the sphinx-design variables\n  --sd-color-#{$name}: var(--pst-color-#{$name});\n  --sd-color-#{$name}-text: var(--pst-color-#{$name}-text);\n\n  // TODO: highlight seems to be used for buttons @trallard to fix on a11y follow-up work\n  --sd-color-#{$name}-highlight: var(--pst-color-#{$name}-highlight);\n}\n\n// Now we override the --sd-color-* variables.\n@each $mode in (light, dark) {\n  html[data-theme=\"#{$mode}\"] {\n    // check if this color is defined differently for light/dark\n    @each $name in $sd-semantic-color-names {\n      $definition: map.get($all-colors, $name);\n\n      @if meta.type-of($definition) == map {\n        @each $key, $value in $definition {\n          @if string.index($key, $mode) {\n            // since now we define the bg colours in the semantic colours and not\n            // by changing opacity, we need to check if the key contains bg and the\n            // correct mode (light/dark)\n            @if string.index($key, \"bg\") {\n              --sd-color-#{$name}-bg: #{$value};\n\n              // create local variable\n              $value: check-color($value);\n\n              --sd-color-#{$name}-bg-text: #{a11y-combination($value)};\n            } @else {\n              $value: check-color($value);\n\n              @include create-sd-colors($value, $name);\n            }\n          }\n        }\n      } @else {\n        $value: map.get($all-colors, $name);\n\n        @include create-sd-colors($value, $name);\n      }\n    }\n  }\n}\n\n// Make sure the color border variables are set using our variables\n@each $mode in (light, dark) {\n  html[data-theme=\"#{$mode}\"] {\n    --sd-color-card-border: var(--pst-color-border);\n  }\n}\n\n/*******************************************************************************\n  * shadows\n  */\nhtml[data-theme=\"light\"] {\n  .sd-shadow-xs,\n  .sd-shadow-sm,\n  .sd-shadow-md,\n  .sd-shadow-lg {\n    @include box-shadow;\n  }\n}\n\n/*******************************************************************************\n  * cards\n  */\n\n.bd-content .sd-card {\n  border: 1px solid var(--pst-color-border);\n\n  // TODO - --pst-color-panel-background is not defined... where is this coming from?\n  .sd-card-header {\n    background-color: var(--pst-color-panel-background);\n    border-bottom: 1px solid var(--pst-color-border);\n  }\n\n  .sd-card-footer {\n    background-color: var(--pst-color-panel-background);\n    border-top: 1px solid var(--pst-color-border);\n  }\n\n  .sd-card-body {\n    background-color: var(--pst-color-panel-background);\n  }\n\n  // Focus ring for link-cards\n  .sd-stretched-link:focus-visible {\n    // Don't put the focus ring on the <a> element (it has zero height in Sphinx Design cards)\n    outline: none;\n\n    // Put the focus ring on the <a> element's ::after pseudo-element\n    &::after {\n      outline: $focus-ring-outline;\n      border-radius: 0.25rem; // copied from Sphinx Design CSS for .sd-card\n    }\n  }\n\n  &.sd-card-hover:hover {\n    border-color: var(--pst-color-link-hover);\n  }\n}\n\n/*******************************************************************************\n  * tabs\n  */\n\n.bd-content .sd-tab-set {\n  > input {\n    // Active tab label\n    &:checked + label {\n      border-style: solid solid none;\n      border-color: var(--pst-color-primary) var(--pst-color-primary)\n        transparent; // top LR bottom\n\n      border-width: 0.125rem 0.125rem 0;\n      border-radius: 0.125rem 0.125rem 0 0;\n      background-color: var(--pst-color-on-background);\n      transform: translateY(0.125rem);\n      color: var(--pst-color-primary);\n    }\n\n    &:focus-visible + label {\n      border: 0.125rem solid var(--pst-color-accent);\n      border-radius: 0.125rem;\n      background-color: var(--pst-color-accent-bg);\n      color: var(--pst-color-on-surface);\n    }\n\n    // Hover label\n    &:not(:checked, :focus-visible) + label:hover {\n      border-color: transparent;\n      color: var(--pst-color-secondary);\n      text-decoration-line: underline;\n      text-decoration-thickness: $link-hover-decoration-thickness;\n    }\n  }\n\n  // Tab label\n  > label {\n    color: var(--pst-color-on-surface);\n    border: 0.125rem solid transparent;\n    border-radius: 0.125rem 0.125rem 0 0;\n    background-color: var(--pst-color-surface);\n    padding: 0 0.75em;\n    margin-inline-end: 0.25rem;\n    line-height: 1.95;\n\n    html[data-theme=\"dark\"] & {\n      background-color: var(--pst-color-on-background);\n    }\n  }\n\n  // panel\n  .sd-tab-content {\n    border: 0.125rem solid var(--pst-color-primary);\n    border-radius: 0.1875rem;\n    box-shadow: unset;\n    padding: 0.625rem;\n    background-color: var(--pst-color-on-background);\n  }\n}\n\n/*******************************************************************************\n* Dropdowns\n*/\n\ndetails.sd-dropdown {\n  // Remove all borders to over-ride SD behavior, and we'll add our own later\n  border: 0 !important;\n\n  // Drop shadow should behave same as admonitions\n  @include box-shadow;\n\n  // Header is where the \"clickable\" box goes\n  summary.sd-card-header {\n    border: 0 !important;\n    display: flex;\n    align-items: center;\n    position: relative; // So background color works\n    font-weight: 600;\n    padding-top: 0.5rem;\n    padding-bottom: 0.5rem;\n\n    // Set a variable that we can re-use for colors later\n    // We must set this in the current and content sibling container\n    // so that it is defined in both places\n    --pst-sd-dropdown-color: var(--pst-gray-500);\n    --pst-sd-dropdown-bg-color: var(--pst-color-surface);\n\n    & + div.sd-summary-content {\n      border: 0;\n\n      --pst-sd-dropdown-color: var(--sd-color-card-border);\n    }\n\n    @each $name in $sd-semantic-color-names {\n      &.sd-bg-#{$name} {\n        --pst-sd-dropdown-color: var(--sd-color-#{$name});\n        --pst-sd-dropdown-bg-color: var(--sd-color-#{$name}-bg);\n\n        // Otherwise it won't be defined in the sibling element\n        & + div.sd-summary-content {\n          --pst-sd-dropdown-color: var(--sd-color-#{$name});\n          --pst-sd-dropdown-bg-color: var(--sd-color-#{$name}-bg);\n        }\n      }\n      &.sd-bg-text-#{$name} {\n        // Use the WCAG conformant text color\n        color: var(--sd-color-#{$name}-bg-text) !important;\n      }\n    }\n\n    @include legacy-backdrop-placeholder;\n\n    background-color: var(--pst-sd-dropdown-bg-color) !important;\n\n    // Add a left border with the same structure as our admonitions\n    border-left: 0.2rem solid var(--pst-sd-dropdown-color) !important;\n\n    // stylelint-disable-next-line no-duplicate-selectors\n    & + div.sd-summary-content {\n      border-left: 0.2rem solid var(--pst-sd-dropdown-color) !important;\n      border-bottom-left-radius: calc(0.25rem - 1px);\n      background-color: var(--pst-color-on-background);\n    }\n\n    span.sd-summary-icon {\n      display: inline-flex;\n      align-items: center;\n      color: var(--pst-sd-dropdown-color) !important;\n\n      svg {\n        opacity: 1;\n      }\n    }\n\n    // Positioning of the caret\n    .sd-summary-up,\n    .sd-summary-down {\n      top: 0.7rem;\n    }\n\n    // Focus ring\n    &:focus-visible {\n      outline: $focus-ring-outline;\n      outline-offset: -$focus-ring-width;\n    }\n  }\n}\n\n/*******************************************************************************\n* Buttons (which in Sphinx Design are actually links that look like buttons)\n* ref: https://sphinx-design.readthedocs.io/en/pydata-theme/badges_buttons.html#buttons\n*/\nhtml {\n  .sd-btn {\n    min-width: 2.25rem;\n    padding: 0.3125rem 0.75rem 0.4375rem; // 5px 12px 7px\n\n    &:hover {\n      @include link-style-hover; // override Sphinx Design\n\n      text-decoration-thickness: 1px;\n    }\n  }\n\n  @each $name in $sd-semantic-color-names {\n    .sd-btn-#{$name},\n    .sd-btn-outline-#{$name} {\n      &:focus-visible {\n        // Override Sphinx Design's use of -highlight colors. The -highlight\n        // colors are 15% darker, so this would create the effect of darkening\n        // the button when focused but we just want the button to have a focus\n        // ring of the same (non-highlight) color.\n        background-color: var(--sd-color-#{$name}) !important;\n        border-color: var(--sd-color-#{$name}) !important;\n        outline: var(--sd-color-#{$name}) solid $focus-ring-width;\n        outline-offset: $focus-ring-width;\n      }\n    }\n  }\n}\n","/**\n * Sphinx togglebutton\n */\n\n.bd-content {\n  @mixin chevron-down {\n    .toggle-chevron-right {\n      transform: rotate(90deg); // point chevron down\n      transition: none; // match non-animated behavior of other chevrons on site\n    }\n  }\n\n  @mixin chevron-up {\n    .toggle-chevron-right {\n      transform: rotate(-90deg); // point chevron up\n      transition: none; // match non-animated behavior of other chevrons on site\n    }\n  }\n\n  // Admonition toggles\n  .admonition {\n    button.toggle-button {\n      color: inherit;\n\n      // When disclosure widget is closed\n      &.toggle-button-hidden {\n        @include chevron-down;\n      }\n\n      // When open\n      @include chevron-up;\n    }\n\n    // Focus ring\n    // ----------\n    // Sphinx-togglebutton makes the entire admonition header clickable, but\n    // only the button within the header is focusable. We want the entire\n    // clickable area to be surrounded with a focus ring, so that's why we use\n    // the :focus-within selector, rather than a :focus-visible selector on the\n    // button.\n    &:focus-within {\n      overflow: visible;\n\n      // The complicated focus ring styles here are a consequence of the markup\n      // and border styles for this particular admonition class. (For the other\n      // type of admonition on this site, the focus ring style is achieved with\n      // simple `outline` and `outline-offset` rules on the admonition's\n      // header.) The problem is that Sphinx-togglebutton puts the admonition's\n      // left border on the outermost container (rather than separately setting\n      // the left border on the container's children). This makes it complicated\n      // to get the focus ring to simultaneously cover the left border in the\n      // header and align perfectly on the right with the body.\n      .admonition-title:focus-within::before {\n        content: \"\";\n        transform: translateX(\n          -$admonition-left-border-width\n        ); // align left edges of admonition and ring\n\n        width: calc(100% + $admonition-left-border-width); // align right edges\n        height: 100%;\n        border: $focus-ring-outline;\n        border-radius: $focus-ring-width;\n      }\n\n      // When expanded, sharpen the bottom left and right corners of the focus ring\n      &:not(.toggle-hidden) .admonition-title:focus-within::before {\n        border-bottom-left-radius: 0;\n        border-bottom-right-radius: 0;\n      }\n    }\n  }\n\n  // Details buttons\n  details.toggle-details {\n    // Over-ride border color to re-use our primary color\n    summary {\n      border-left: 3px solid var(--pst-color-primary);\n\n      @include chevron-down;\n    }\n\n    // When expanded, sharpen the bottom left and right corners of the focus ring\n    &[open] {\n      @include chevron-up;\n\n      :focus-visible {\n        border-bottom-left-radius: 0;\n        border-bottom-right-radius: 0;\n      }\n    }\n  }\n}\n","/**\n * Styles for various Sphinx execution libraries to display pre-executed notebooks.\n * For now, where these define output sections, we simply revert their background\n * to be a \"light theme\" background. This ensures that inputs/outputs behave similarly,\n * because the CSS is often controlled by each package.\n * In the future, we might add dark theme support for specific packages.\n */\n\n/*******************************************************************************\n * nbsphinx\n */\nhtml div.rendered_html,\n// NBsphinx ipywidgets output selector\nhtml .jp-RenderedHTMLCommon {\n  // Add some margin around the element box for the focus ring. Otherwise the\n  // focus ring gets clipped because the containing elements have `overflow:\n  // hidden` applied to them (via the `.lm-Widget` selector)\n  margin: $focus-ring-width;\n\n  table {\n    table-layout: auto;\n  }\n}\n\n.bd-content .nboutput {\n  .output_area {\n    &.rendered_html,\n    .jp-RenderedHTMLCommon {\n      // pandas\n      table.dataframe {\n        @include table-colors;\n      }\n    }\n\n    // Dark theme special-cases\n    html[data-theme=\"dark\"] & {\n      &.rendered_html:not(:has(table.dataframe)),\n      // ipywidgets\n      .widget-subarea {\n        @include cell-output-background;\n      }\n\n      &.stderr {\n        background-color: var(--pst-color-danger);\n      }\n    }\n  }\n}\n\n// Add extra padding to the final item in an nbsphinx container\ndiv.nblast.container {\n  margin-bottom: 1rem;\n}\n\n/*******************************************************************************\n * myst NB\n */\n\ndiv.cell_output .output {\n  max-width: 100%;\n  overflow-x: auto;\n}\n\n.bd-content div.cell_output {\n  // pandas\n  table.dataframe {\n    @include table-colors;\n  }\n\n  html[data-theme=\"dark\"] & {\n    img,\n    .text_html:not(:has(table.dataframe)),\n    // ipywidgets\n    .widget-subarea {\n      @include cell-output-background;\n    }\n  }\n}\n\n// Prevent tables from scrunching together\n.bd-content {\n  div.cell_input {\n    display: flex;\n    flex-direction: column;\n    justify-content: stretch;\n  }\n\n  div.cell_input,\n  div.output {\n    border-radius: $admonition-border-radius;\n  }\n\n  div.output {\n    table {\n      table-layout: auto;\n    }\n  }\n}\n","/**\n * style for the various mapping libs based on leaflet (folium, geemap, ipyleaflet)\n * mainly ensure the good display of the maps in both themes and avoid the customization\n * of the tiles\n */\n\n/**\n * avoid border override from pydata-sphinx-theme\n * minimal selctor to get the priority\n */\nhtml[data-theme=\"dark\"] .bd-content img.leaflet-tile.leaflet-tile-loaded {\n  border-radius: 0;\n  padding: 0;\n}\n","/**\n * /search.html page special-cases\n */\n\n.bd-search-container {\n  /*******************************************\n  * Search results\n  */\n  // Whitespace\n  div#search-results {\n    > h2 {\n      font-size: var(--pst-font-size-icon);\n      margin-top: 1rem;\n    }\n\n    p.search-summary {\n      color: var(--pst-color-text-muted);\n    }\n  }\n\n  ul.search {\n    margin: 0;\n    list-style: none;\n\n    li {\n      background-image: none;\n      padding: 1rem 0;\n      margin: 1rem 0;\n      border-top: 1px solid var(--pst-color-text-muted);\n\n      // First link is the page title, it should be a bit bigger\n      > a {\n        font-size: 1.2em;\n      }\n\n      div.context,\n      p.context {\n        color: var(--pst-color-text-base);\n        margin: 0.5em 0 0;\n\n        // Add a # before page section titles to make it clear they are sections\n        a::before {\n          content: \"#\";\n          padding-right: 0.2em;\n          color: var(--pst-color-text-muted);\n        }\n      }\n    }\n  }\n}\n"],"names":[],"sourceRoot":""}
\ No newline at end of file
diff --git a/docs/html/_static/styles/sphinx-book-theme.css b/docs/html/_static/styles/sphinx-book-theme.css
new file mode 100644
index 0000000..5197540
--- /dev/null
+++ b/docs/html/_static/styles/sphinx-book-theme.css
@@ -0,0 +1,9 @@
+/*! sphinx-book-theme CSS
+ * BSD 3-Clause License
+ * Copyright (c) 2020, EBP
+ * All rights reserved.
+ *
+ * This follows the 7-1 pattern described here:
+ * https://sass-guidelin.es/#architecture
+ */html[data-theme=dark],html[data-theme=light]{--pst-font-size-base:none;--pst-color-secondary:#e89217}html[data-theme=light]{--sbt-color-announcement:#616161;--pst-color-primary:#176de8}html[data-theme=dark]{--pst-color-primary:#528fe4;--sbt-color-announcement:#616161;--pst-color-background:#121212}html{scroll-padding-top:4rem}.sbt-scroll-pixel-helper{height:0;left:0;position:absolute;top:0;width:0}.d-n,.onlyprint{display:none}@media print{.onlyprint{display:block!important}.noprint{display:none!important}}.bd-article-container h1,.bd-article-container h2,.bd-article-container h3,.bd-article-container h4,.bd-article-container h5,.bd-article-container p.caption{color:var(--pst-color-muted)}.bd-article-container h1,.bd-article-container h2{font-weight:500}a.brackets:before{color:inherit;font-family:inherit;margin-right:0}table{position:relative}@media print{.bd-main .bd-content{height:auto;margin-left:2rem}.bd-main .bd-content #jb-print-docs-body{margin-left:0}.bd-main .bd-content #jb-print-docs-body h1{font-size:3em;margin-bottom:0;text-align:center}.bd-main .bd-content .bd-article{padding-top:0}.bd-main .bd-content .bd-article h1:first-of-type{display:none}.bd-main .bd-content .container{min-width:0!important}.bd-main .bd-content h1{margin-bottom:1em;margin-top:1em}.bd-main .bd-content h1,.bd-main .bd-content h2,.bd-main .bd-content h3,.bd-main .bd-content h4{break-after:avoid;color:#000}.bd-main .bd-content table{break-inside:avoid}.bd-main .bd-content pre{word-wrap:break-word}.bd-main .bd-content a.headerlink{display:none}.bd-main .bd-content aside.margin,.bd-main .bd-content aside.sidebar,.bd-main .bd-content blockquote.epigraph{border:none}.bd-main .bd-content .footer{margin-top:1em}.bd-main .bd-content #jb-print-toc{margin-bottom:1.5rem;margin-left:0}.bd-main .bd-content #jb-print-toc .section-nav{border-left:0!important;list-style-type:disc!important;margin-left:3em!important}.bd-main .bd-content #jb-print-toc .section-nav 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var(--pst-color-shadow),\n    0 0 0.0625rem var(--pst-color-shadow) !important;\n}\n",".bd-header {\n  // Turn off sticky positioning so that it scrolls\n  position: inherit;\n\n  // Turn these off because we'll show it in the article header\n  // The primary sidebar toggle\n  button.sidebar-toggle {\n    display: none;\n  }\n}\n","/*********************************************\n* Left Nav Bar *\n*********************************************/\n\n.bd-sidebar-primary {\n  // So that it sticks to the top of the page instead of the header height\n  top: 0;\n  max-height: 100vh;\n\n  // Tighter padding so the spacing is more equal\n  padding: 1rem;\n\n  // Draw open/close animation\n  transition:\n    margin-left $animation-time ease 0s,\n    opacity $animation-time ease 0s,\n    visibility $animation-time ease 0s;\n\n  @media (max-width: $breakpoint-md) {\n    // So that tooltips don't overlap\n    z-index: $zindex-tooltip + 1;\n  }\n\n  // Remove the top border since there's often no header items above\n  .sidebar-primary-items__start {\n    border-top: none;\n  }\n\n  // On wide screens we have a smaller sidebar width than PST\n  @media (min-width: $breakpoint-lg) {\n    flex-basis: $sidebar-primary-width-widescreen;\n  }\n}\n\n// Default visibility for left sidebar is the reverse of what it is on mobile.\n// It is shown by default, and hidden with a click.\n// Mobile behavior is defined in the pydata sphinx theme\n@media (min-width: $breakpoint-lg) {\n  input#pst-primary-sidebar-checkbox:checked\n    ~ .bd-container\n    .bd-sidebar-primary {\n    margin-left: -$sidebar-primary-width-widescreen;\n    visibility: hidden;\n    opacity: 0;\n  }\n}\n","/*********************************************\n* Table of Contents (right nav bar)          *\n*********************************************/\n\n.bd-sidebar-secondary {\n  // So that it sticks to the top of the page instead of the header height\n  top: 0;\n\n  @media (max-width: $breakpoint-xl) {\n    // So that tooltips don't overlap\n    z-index: $zindex-tooltip + 1;\n  }\n\n  // So that we can center the TOC button with the article header buttons\n  .sidebar-secondary-items {\n    padding: 0;\n    display: flex;\n    gap: 0.5rem;\n\n    // Remove the manual padding since we'll use `gap`\n    .sidebar-secondary-item {\n      padding-top: 0;\n      padding-bottom: 0;\n    }\n  }\n\n  // The 'on this page' title div should take up the full header\n  .onthispage {\n    height: $header-article-height;\n    min-height: $header-article-height;\n    display: flex;\n    gap: 0.5rem;\n    align-items: center;\n    margin: 0;\n    color: var(--pst-color-muted);\n  }\n\n  // Wide screen behavior\n  @media (min-width: $breakpoint-xl) {\n    background: var(--pst-color-background);\n    height: fit-content;\n    transition: max-height 0.4s ease;\n\n    // To make sure it shows above the page content\n    z-index: $zindex-middle;\n\n    // Remove padding so that it's flush on the top and left\n    padding: 0;\n\n    // This is the container div for the secondary sidebar nav\n    .toc-item {\n      // Border is a bit less prominent\n      border-left-color: var(--pst-color-surface);\n      // Flush with the top so that we can fix the height of the toc header div\n      padding-top: 0;\n\n      // The table of contents <ul> after the title div\n      nav.page-toc {\n        // Remove extra margin the PST adds for other sidebar items\n        margin-bottom: 0;\n\n        // Items after \"on this page' should fade in and out\n        transition: opacity 0.4s ease;\n      }\n    }\n\n    // Rules for when sidebar is hidden\n    // the `not:hover` means that hovering over the sidebar will show it again.\n    &.hide:not(:hover) {\n      // Cap the height, and remove the scrollbar\n      max-height: $header-article-height;\n      overflow-y: hidden;\n\n      // Shadow matches what we have in the article header\n      .scrolled & {\n        box-shadow: 0 6px 6px -6px rgba(0, 0, 0, 0.3);\n      }\n\n      // Show a down carot\n      .onthispage:after {\n        opacity: 1;\n        content: \"\\f107\";\n        font-family: \"Font Awesome 5 Free\";\n        font-weight: 900;\n        padding-left: 0.5em;\n        transition: opacity 0.3s ease;\n      }\n\n      // The page-toc list is hidden\n      nav.page-toc {\n        opacity: 0;\n      }\n    }\n  }\n}\n","/*********************************************\n* Footer - content *\n*********************************************/\nfooter {\n  font-size: var(--sbt-font-size-small-1);\n  &.bd-footer-content {\n    display: flex;\n    flex-wrap: wrap;\n    padding: 15px;\n    border-top: 1px solid #ccc;\n    font-size: $sbt-font-size-small-1;\n    .bd-footer-content__inner {\n      padding-left: 0px;\n      p {\n        margin-bottom: 0px;\n      }\n    }\n  }\n}\n",".bd-footer-article {\n  padding: 0rem 1rem;\n  @media (max-width: $breakpoint-md) {\n    // More room for text on mobile\n    padding: 0rem 0.5rem;\n  }\n}\n","#pst-back-to-top {\n  // We should only display the back-to-top button when the TOC is not available.\n  // AKA, on narrow-ish screens.\n  // This over-rides a PST default\n  @media (min-width: $breakpoint-xl) {\n    display: none !important;\n  }\n\n  // Reduce size to be slightly less intrusive on the page\n  font-size: 0.8rem;\n}\n","/**\n * Icon links design for the primary sidebar, where it defaults in this theme.\n */\n.bd-sidebar-primary {\n  .navbar-icon-links {\n    column-gap: 0.5rem;\n\n    .nav-link {\n      // There are few kinds of elements that can be icon links and each is different\n      i,\n      span {\n        font-size: 1.2rem;\n      }\n\n      // Images usually fill more vertical space so we make them a bit smaller\n      img {\n        font-size: 0.8rem;\n      }\n    }\n  }\n}\n",".navbar-brand {\n  height: unset;\n  max-height: unset;\n\n  // Auto-center the image and title and make items vertical\n  flex-direction: column;\n  justify-content: center;\n  gap: 0.25rem;\n\n  // To prevent site text from having underlines\n  &:hover {\n    text-decoration: none;\n  }\n  .logo__title {\n    font-size: 1.25rem;\n    white-space: normal;\n    text-align: center;\n  }\n  // Prevent images from stretching vertically\n  .logo__image {\n    height: unset;\n  }\n}\n","// Clean up the search page so that it has less unnecessary text\n.bd-search-container {\n  margin: 2em;\n\n  #search-results {\n    h2:first-child {\n      display: none;\n    }\n  }\n}\n\n// Sidebar search button field\n.search-button-field {\n  width: 100%;\n  font-size: 0.9rem;\n  // Hide by default and we'll add a rule to show it on wide sreen below\n  display: none;\n\n  .search-button__kbd-shortcut {\n    margin-left: auto;\n  }\n}\n\n// Rules to switch off visibility of the field button and the header button\n@media (min-width: $breakpoint-lg) {\n  .search-button {\n    display: none !important;\n  }\n\n  .search-button-field {\n    display: flex;\n  }\n}\n\n// The \"clear search results\" button\n// Make it line up with our content better\ndiv#searchbox {\n  padding-right: 2rem;\n  padding-left: 2rem;\n  @media (max-width: $breakpoint-md) {\n    padding-right: 1rem;\n    padding-left: 1rem;\n  }\n\n  p.highlight-link {\n    // Remove the extra margin that is added\n    @media (min-width: $breakpoint-md) {\n      margin-left: 0;\n      a {\n        font-size: 1rem;\n      }\n    }\n  }\n}\n","// Images\nimg {\n  max-width: 100%;\n\n  &.align-center {\n    margin-left: auto;\n    margin-right: auto;\n    display: block;\n  }\n\n  &.align-left {\n    clear: left;\n    float: left;\n    margin-right: 1em;\n  }\n\n  &.align-right {\n    clear: right;\n    float: right;\n    margin-left: 1em;\n  }\n}\n\n// Figures\ndiv.figure {\n  width: 100%;\n  margin-bottom: 1em;\n  text-align: center;\n\n  &.align-left {\n    text-align: left;\n    p.caption {\n      margin-left: 0;\n    }\n  }\n  &.align-right {\n    text-align: right;\n    p.caption {\n      margin-right: 0;\n    }\n  }\n\n  p.caption {\n    margin: 0.5em 10%;\n  }\n\n  &.margin,\n  &.margin-caption {\n    p.caption {\n      margin: 0.5em 0;\n    }\n  }\n\n  &.margin-caption p.caption {\n    text-align: left;\n  }\n\n  span.caption-number {\n    font-weight: bold;\n  }\n\n  span {\n    font-size: 0.9rem;\n  }\n}\n","/**\n * Margin content\n * This is the area to the right of the main content\n * normally covered by the in-page table of contents\n * but if `.margin` classes are present, it shows up to the right in the margin\n * and the in-page TOC is hidden.\n * On narrow screens, margin content behaves like a sidebar\n */\n\n/**\n * Variables for calculating margin widths\n */\n// We want our margin content to be 1/3 the width of the main page content\n// after factoring in a small margin\n$margin-width-relative-to-content: 36%;\n$margin-gutter: 3%;\n$margin-width-width-gutter: $margin-width-relative-to-content - $margin-gutter;\n\n// re-subtract the gutter width because we want it to leave a white-space\n// This is how much to offset the margin content.\n$margin-offset: -$margin-width-relative-to-content;\n\n/**\n * This mixin will cause something to \"pop out\" to the margin on wide screens\n */\n@mixin margin-content() {\n  // On narrow screens this is the width of margin content\n  width: 40%;\n  float: right;\n  background-color: unset;\n  font-size: 0.9em;\n  margin-left: 0.5rem;\n\n  // Wide screens and printing should force margin behavior\n  @media (min-width: $breakpoint-lg), print {\n    width: $margin-width-width-gutter;\n    margin: 0 $margin-offset 0 0;\n\n    // Prevent sequential margin content from overlapping\n    clear: right;\n\n    p.sidebar-title {\n      margin-bottom: -1rem;\n      border-bottom: none;\n      padding-left: 0px;\n\n      // Content of admonitions has fewer horizontal white space to save space\n      ~ * {\n        padding-left: 0px;\n        padding-right: 0px;\n      }\n    }\n  }\n}\n\n/**\n * Sidenotes and marginnotes\n * over-rides the \"footnote\" behavior to pop out to the margin instead.\n */\nlabel.margin-toggle {\n  margin-bottom: 0em;\n  &.marginnote-label {\n    display: none;\n  }\n  sup {\n    user-select: none;\n  }\n  @media (max-width: $breakpoint-lg) {\n    cursor: pointer;\n    color: rgb(0, 113, 188);\n    &.marginnote-label {\n      display: inline;\n      &:after {\n        content: \"\\2295\";\n      }\n    }\n  }\n}\n\ninput.margin-toggle {\n  display: none;\n  @media (max-width: $breakpoint-lg) {\n    &:checked + .sidenote,\n    &:checked + .marginnote {\n      display: block;\n      float: left;\n      left: 1rem;\n      clear: both;\n      width: 95%;\n      margin: 1rem 2.5%;\n      position: relative;\n    }\n  }\n}\n\nspan.sidenote,\nspan.marginnote {\n  z-index: $zindex-middle;\n  position: relative;\n  sup {\n    user-select: none;\n  }\n  @include margin-content();\n  border-left: none;\n  @media (max-width: $breakpoint-lg) {\n    display: none;\n  }\n}\n\naside.sidebar {\n  .note {\n    // styling for notes inside sidebars\n    margin: 1rem;\n    padding: 0rem 0rem 1rem 0rem;\n  }\n  // The titles sticking\n  .admonition-title {\n    margin: 0rem -1rem 0rem 0rem;\n  }\n}\n\n/**\n * Sidebar content in the margin.\n * This will be added by the `margin` directive or the `sidebar` directive\n */\naside.sidebar.margin {\n  // Margin content can have an empty title so this prevents an empty title block\n  // from showing up on mobile\n  .sidebar-title:empty {\n    display: none;\n  }\n\n  // On narrow screens padding is added to all sidebar content\n  // This replaces padding w/ margin for admonitions because padding is used\n  // for the title background color\n  .admonition {\n    margin: 0.5rem;\n    padding-left: 0;\n    padding-right: 0;\n\n    // Remove the title margin so it\n    .admonition-title {\n      margin-left: 0;\n      margin-right: 0;\n    }\n  }\n\n  @media (min-width: $breakpoint-lg) {\n    // No border for margin sidebars on wide screen\n    border: none;\n\n    // Admonitions don't need the extra whitespace in the margin\n    .admonition {\n      margin: 1rem 0rem;\n      padding: 0rem 0rem 1rem 0rem;\n    }\n  }\n}\n\ndiv.margin,\nfigure.margin,\naside.margin,\n.margin.docutils.container,\n.cell.tag_popout,\n.cell.tag_margin {\n  z-index: $zindex-middle;\n  position: relative;\n  @include margin-content();\n\n  // Make cell outputs take up more space if they're in the margin\n  div.cell.tag_margin .cell_output {\n    padding-left: 0;\n  }\n}\n\n// A few permutations because docutils 0.18 switched to semantic tags\ndiv.figure.margin-caption p.caption,\ndiv.figure.margin-caption figcaption,\nfigure.margin-caption figcaption {\n  z-index: $zindex-middle;\n  position: relative;\n  @include margin-content();\n}\n\n// Margin captions\n.margin-caption figcaption {\n  text-align: left;\n}\n\n/**\n * Full width content\n */\n\n// Grow 100% by the ratio of the margin to the content width.\n$content-fullwidth-width: 100% + $margin-width-relative-to-content;\n\ndiv.cell.tag_full-width,\ndiv.cell.tag_full_width,\ndiv.full_width,\ndiv.full-width {\n  z-index: $zindex-middle;\n  position: relative;\n  @media (min-width: $breakpoint-lg) {\n    max-width: $content-fullwidth-width;\n    width: $content-fullwidth-width;\n  }\n}\n","/**\n * Epigraphs and other \"special\" quote blocks\n */\n\nblockquote {\n  &.pull-quote,\n  &.epigraph,\n  &.highlights {\n    font-size: 1.25em;\n    border-left: none;\n    background-color: var(--pst-color-background);\n  }\n\n  div > p + p.attribution {\n    font-style: normal;\n    font-size: 0.9em;\n    text-align: right;\n    color: #6c757d;\n    padding-right: 2em;\n  }\n}\n","// Over-ride Sphinx default so that content fills whitespace w/ margin content\npre,\ndiv[class*=\"highlight-\"] {\n  clear: none;\n}\n","// MyST-NB and Jupyter Notebooks\ndiv.cell.tag_output_scroll div.cell_output,\ndiv.cell.tag_scroll-output div.cell_output {\n  max-height: 24em;\n  overflow-y: auto;\n}\n\ndiv.cell.tag_scroll-input div.cell_input {\n  max-height: 24em;\n  overflow-y: auto;\n}\n","/**\n * Comment JavaScript libraries\n  * Libraries supported here are:\n  * - Hypothes.is : https://web.hypothes.is/\n  * - Utteranc.es : https://utteranc.es/\n  */\n// In general we want to hide any comment libraries on print\n@media only print {\n  hypothesis-sidebar,\n  div.utterances {\n    display: none;\n  }\n}\n","/*********************************************\n* Thebe *\n* ref: https://github.com/executablebooks/thebe\n*********************************************/\n.thebelab-cell {\n  border: none !important;\n  margin-right: 0.5em !important;\n}\n\n.thebelab-cell .thebelab-input {\n  padding-left: 10px !important;\n}\n\n.cell.docutils.container {\n  padding-right: 0px !important;\n}\n\nbutton.thebe-launch-button {\n  height: 2.5em;\n  font-size: 1em;\n}\n"],"names":[],"sourceRoot":""}
\ No newline at end of file
diff --git a/docs/html/_static/styles/theme.css b/docs/html/_static/styles/theme.css
new file mode 100644
index 0000000..4519dd9
--- /dev/null
+++ b/docs/html/_static/styles/theme.css
@@ -0,0 +1,2 @@
+/* Provided by Sphinx's 'basic' theme, and included in the final set of assets */
+@import "../basic.css";
diff --git a/docs/html/_static/table1.png b/docs/html/_static/table1.png
new file mode 100644
index 0000000..c02f860
Binary files /dev/null and b/docs/html/_static/table1.png differ
diff --git a/docs/html/_static/vendor/fontawesome/6.5.2/LICENSE.txt b/docs/html/_static/vendor/fontawesome/6.5.2/LICENSE.txt
new file mode 100644
index 0000000..e69c5e3
--- /dev/null
+++ b/docs/html/_static/vendor/fontawesome/6.5.2/LICENSE.txt
@@ -0,0 +1,165 @@
+Fonticons, Inc. (https://fontawesome.com)
+
+--------------------------------------------------------------------------------
+
+Font Awesome Free License
+
+Font Awesome Free is free, open source, and GPL friendly. You can use it for
+commercial projects, open source projects, or really almost whatever you want.
+Full Font Awesome Free license: https://fontawesome.com/license/free.
+
+--------------------------------------------------------------------------------
+
+# Icons: CC BY 4.0 License (https://creativecommons.org/licenses/by/4.0/)
+
+The Font Awesome Free download is licensed under a Creative Commons
+Attribution 4.0 International License and applies to all icons packaged
+as SVG and JS file types.
+
+--------------------------------------------------------------------------------
+
+# Fonts: SIL OFL 1.1 License
+
+In the Font Awesome Free download, the SIL OFL license applies to all icons
+packaged as web and desktop font files.
+
+Copyright (c) 2024 Fonticons, Inc. (https://fontawesome.com)
+with Reserved Font Name: "Font Awesome".
+
+This Font Software is licensed under the SIL Open Font License, Version 1.1.
+This license is copied below, and is also available with a FAQ at:
+http://scripts.sil.org/OFL
+
+SIL OPEN FONT LICENSE
+Version 1.1 - 26 February 2007
+
+PREAMBLE
+The goals of the Open Font License (OFL) are to stimulate worldwide
+development of collaborative font projects, to support the font creation
+efforts of academic and linguistic communities, and to provide a free and
+open framework in which fonts may be shared and improved in partnership
+with others.
+
+The OFL allows the licensed fonts to be used, studied, modified and
+redistributed freely as long as they are not sold by themselves. The
+fonts, including any derivative works, can be bundled, embedded,
+redistributed and/or sold with any software provided that any reserved
+names are not used by derivative works. The fonts and derivatives,
+however, cannot be released under any other type of license. The
+requirement for fonts to remain under this license does not apply
+to any document created using the fonts or their derivatives.
+
+DEFINITIONS
+"Font Software" refers to the set of files released by the Copyright
+Holder(s) under this license and clearly marked as such. This may
+include source files, build scripts and documentation.
+
+"Reserved Font Name" refers to any names specified as such after the
+copyright statement(s).
+
+"Original Version" refers to the collection of Font Software components as
+distributed by the Copyright Holder(s).
+
+"Modified Version" refers to any derivative made by adding to, deleting,
+or substituting — in part or in whole — any of the components of the
+Original Version, by changing formats or by porting the Font Software to a
+new environment.
+
+"Author" refers to any designer, engineer, programmer, technical
+writer or other person who contributed to the Font Software.
+
+PERMISSION & CONDITIONS
+Permission is hereby granted, free of charge, to any person obtaining
+a copy of the Font Software, to use, study, copy, merge, embed, modify,
+redistribute, and sell modified and unmodified copies of the Font
+Software, subject to the following conditions:
+
+1) Neither the Font Software nor any of its individual components,
+in Original or Modified Versions, may be sold by itself.
+
+2) Original or Modified Versions of the Font Software may be bundled,
+redistributed and/or sold with any software, provided that each copy
+contains the above copyright notice and this license. These can be
+included either as stand-alone text files, human-readable headers or
+in the appropriate machine-readable metadata fields within text or
+binary files as long as those fields can be easily viewed by the user.
+
+3) No Modified Version of the Font Software may use the Reserved Font
+Name(s) unless explicit written permission is granted by the corresponding
+Copyright Holder. This restriction only applies to the primary font name as
+presented to the users.
+
+4) The name(s) of the Copyright Holder(s) or the Author(s) of the Font
+Software shall not be used to promote, endorse or advertise any
+Modified Version, except to acknowledge the contribution(s) of the
+Copyright Holder(s) and the Author(s) or with their explicit written
+permission.
+
+5) The Font Software, modified or unmodified, in part or in whole,
+must be distributed entirely under this license, and must not be
+distributed under any other license. The requirement for fonts to
+remain under this license does not apply to any document created
+using the Font Software.
+
+TERMINATION
+This license becomes null and void if any of the above conditions are
+not met.
+
+DISCLAIMER
+THE FONT SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTIES OF
+MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT
+OF COPYRIGHT, PATENT, TRADEMARK, OR OTHER RIGHT. IN NO EVENT SHALL THE
+COPYRIGHT HOLDER BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
+INCLUDING ANY GENERAL, SPECIAL, INDIRECT, INCIDENTAL, OR CONSEQUENTIAL
+DAMAGES, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
+FROM, OUT OF THE USE OR INABILITY TO USE THE FONT SOFTWARE OR FROM
+OTHER DEALINGS IN THE FONT SOFTWARE.
+
+--------------------------------------------------------------------------------
+
+# Code: MIT License (https://opensource.org/licenses/MIT)
+
+In the Font Awesome Free download, the MIT license applies to all non-font and
+non-icon files.
+
+Copyright 2024 Fonticons, Inc.
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of
+this software and associated documentation files (the "Software"), to deal in the
+Software without restriction, including without limitation the rights to use, copy,
+modify, merge, publish, distribute, sublicense, and/or sell copies of the Software,
+and to permit persons to whom the Software is furnished to do so, subject to the
+following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED,
+INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A
+PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
+HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
+OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
+SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+
+--------------------------------------------------------------------------------
+
+# Attribution
+
+Attribution is required by MIT, SIL OFL, and CC BY licenses. Downloaded Font
+Awesome Free files already contain embedded comments with sufficient
+attribution, so you shouldn't need to do anything additional when using these
+files normally.
+
+We've kept attribution comments terse, so we ask that you do not actively work
+to remove them from files, especially code. They're a great way for folks to
+learn about Font Awesome.
+
+--------------------------------------------------------------------------------
+
+# Brand Icons
+
+All brand icons are trademarks of their respective owners. The use of these
+trademarks does not indicate endorsement of the trademark holder by Font
+Awesome, nor vice versa. **Please do not use brand logos for any purpose except
+to represent the company, product, or service to which they refer.**
diff --git a/docs/html/_static/vendor/fontawesome/6.5.2/css/all.min.css b/docs/html/_static/vendor/fontawesome/6.5.2/css/all.min.css
new file mode 100644
index 0000000..628b2cc
--- /dev/null
+++ b/docs/html/_static/vendor/fontawesome/6.5.2/css/all.min.css
@@ -0,0 +1,5 @@
+/*!
+ * Font Awesome Free 6.5.2 by @fontawesome - https://fontawesome.com
+ * License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License)
+ * Copyright 2024 Fonticons, Inc.
+ */.fa{font-family:var(--fa-style-family,"Font Awesome 6 Free");font-weight:var(--fa-style,900)}.fa,.fa-brands,.fa-classic,.fa-regular,.fa-sharp,.fa-solid,.fab,.far,.fas{-moz-osx-font-smoothing:grayscale;-webkit-font-smoothing:antialiased;display:var(--fa-display,inline-block);font-style:normal;font-variant:normal;line-height:1;text-rendering:auto}.fa-classic,.fa-regular,.fa-solid,.far,.fas{font-family:Font Awesome\ 6 Free}.fa-brands,.fab{font-family:Font Awesome\ 6 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+/*!
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+ * Copyright 2024 Fonticons, Inc.
+ */
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new file mode 100644
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--- /dev/null
+++ b/docs/html/_static/webpack-macros.html
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+<!--
+  AUTO-GENERATED from webpack.config.js, do **NOT** edit by hand.
+  These are re-used in layout.html
+-->
+{# Load FontAwesome icons #}
+{% macro head_pre_icons() %}
+  <link href="{{ pathto('_static/vendor/fontawesome/6.5.2/css/all.min.css', 1) }}?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
+  <link rel="preload" as="font" type="font/woff2" crossorigin href="{{ pathto('_static/vendor/fontawesome/6.5.2/webfonts/fa-solid-900.woff2', 1) }}" />
+<link rel="preload" as="font" type="font/woff2" crossorigin href="{{ pathto('_static/vendor/fontawesome/6.5.2/webfonts/fa-brands-400.woff2', 1) }}" />
+<link rel="preload" as="font" type="font/woff2" crossorigin href="{{ pathto('_static/vendor/fontawesome/6.5.2/webfonts/fa-regular-400.woff2', 1) }}" />
+{% endmacro %}
+
+{% macro head_pre_assets() %}
+  <!-- Loaded before other Sphinx assets -->
+  <link href="{{ pathto('_static/styles/theme.css', 1) }}?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
+<link href="{{ pathto('_static/styles/bootstrap.css', 1) }}?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
+<link href="{{ pathto('_static/styles/pydata-sphinx-theme.css', 1) }}?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
+{% endmacro %}
+
+{% macro head_js_preload() %}
+  <!-- Pre-loaded scripts that we'll load fully later -->
+  <link rel="preload" as="script" href="{{ pathto('_static/scripts/bootstrap.js', 1) }}?digest=dfe6caa3a7d634c4db9b" />
+<link rel="preload" as="script" href="{{ pathto('_static/scripts/pydata-sphinx-theme.js', 1) }}?digest=dfe6caa3a7d634c4db9b" />
+  <script src="{{ pathto('_static/vendor/fontawesome/6.5.2/js/all.min.js', 1) }}?digest=dfe6caa3a7d634c4db9b"></script>
+{% endmacro %}
+
+{% macro body_post() %}
+  <!-- Scripts loaded after <body> so the DOM is not blocked -->
+  <script src="{{ pathto('_static/scripts/bootstrap.js', 1) }}?digest=dfe6caa3a7d634c4db9b"></script>
+<script src="{{ pathto('_static/scripts/pydata-sphinx-theme.js', 1) }}?digest=dfe6caa3a7d634c4db9b"></script>
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\ No newline at end of file
diff --git a/docs/html/acknowledgements.html b/docs/html/acknowledgements.html
new file mode 100644
index 0000000..8bb228e
--- /dev/null
+++ b/docs/html/acknowledgements.html
@@ -0,0 +1,488 @@
+
+<!DOCTYPE html>
+
+
+<html lang="en" data-content_root="./" >
+
+  <head>
+    <meta charset="utf-8" />
+    <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
+
+    <title>Acknowledgements &#8212; Addressing Uncertainty in MultiSector Dynamics Research  documentation</title>
+  
+  
+  
+  <script data-cfasync="false">
+    document.documentElement.dataset.mode = localStorage.getItem("mode") || "";
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+  </script>
+  
+  <!-- Loaded before other Sphinx assets -->
+  <link href="_static/styles/theme.css?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
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+<link href="_static/styles/pydata-sphinx-theme.css?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
+
+  
+  <link href="_static/vendor/fontawesome/6.5.2/css/all.min.css?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
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+<h1>Acknowledgements<a class="headerlink" href="#acknowledgements" title="Link to this heading">#</a></h1>
+<p>This e-book was developed by the <a class="reference external" href="https://im3.pnnl.gov">Integrated Multisector, Multiscale Modeling (IM3) project</a>, supported by the <a class="reference external" href="https://www.energy.gov/">U.S. Department of Energy</a>, <a class="reference external" href="https://www.energy.gov/science/office-science">Office of Science</a>, as part of research in the <a class="reference external" href="https://climatemodeling.science.energy.gov/program/multisector-dynamics">MultiSector Dynamics</a>, Earth and Environmental System Modeling Program.</p>
+<p>The authors would like to thank Casey Burleyson (PNNL) and Isaac Thompson (PNNL) for their insightful feedback on drafts of the text and tutorials.</p>
+<p>Open source under license <a class="reference external" href="https://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND-4.0</a>.</p>
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+<p>Our eBook is a living product that we hope continues to grow over time to stay relevant with methodological and technological advancements in the field of uncertainty quantificaiton and MultiSector Dynamics at large.  We extend an invitation to you the reader to contribute to one of our interactive Jupyter notebook tutorials.  If you feel you have a contribution that would be relevant and generalizable to the MSD community, please submit a proposal idea <a class="reference external" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/issues/new?assignees=thurber%2C+crvernon&amp;labels=triage&amp;projects=&amp;template=contribution_proposal.yml&amp;title=Contribution+Proposal">here</a>.  Your proposal will be reviewing by our team and feedback and/or a decision will be provided shortly.  Any contribution accepted will receive it’s own DOI and citation so that you may provide an independent reference to your work as you see fit.</p>
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+<h2>Using mathjax<a class="headerlink" href="#using-mathjax" title="Link to this heading">#</a></h2>
+<p>The <strong>first-order sensitivity index</strong> indicates the percent of model output variance contributed by a factor individually (i.e., the effect of varying <em>x</em><sub>i</sub>alone) and is obtained using the following (Saltelli, 2002a; Sobol, 2001):</p>
+<div class="math notranslate nohighlight">
+\[S_i^1 = \frac{V_{x_i} [E_{x\sim_i} (x_i)]}{V(y)}\]</div>
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+<li class="toctree-l1"><a class="reference internal" href="preface.html">Preface</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="acknowledgements.html">Acknowledgements</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html">2. Diagnostic Modeling Overview and Perspectives</a></li>
+<li class="toctree-l1"><a class="reference internal" href="3_sensitivity_analysis_the_basics.html">3. Sensitivity Analysis: The Basics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html">4. Sensitivity Analysis: Diagnostic &amp; Exploratory Modeling</a></li>
+<li class="toctree-l1"><a class="reference internal" href="5_conclusion.html">5. Conclusion</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
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+    <h1>Addressing Uncertainty in MultiSector Dynamics Research</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
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+                  
+  <section id="addressing-uncertainty-in-multisector-dynamics-research">
+<h1>Addressing Uncertainty in MultiSector Dynamics Research<a class="headerlink" href="#addressing-uncertainty-in-multisector-dynamics-research" title="Link to this heading">#</a></h1>
+<blockquote class="epigraph">
+<div><p>Patrick M. Reed, Antonia Hadjimichael, Keyvan Malek,
+Tina Karimi, Chris R. Vernon, Vivek Srikrishnan, Rohini S. Gupta,
+David F. Gold, Ben Lee, Klaus Keller, Travis B. Thurber, Jennie S. Rice</p>
+</div></blockquote>
+<p>A practical guide to sensitivity analysis and diagnostic model evaluation techniques for confronting the computational and conceptual challenges of multi-model, transdisciplinary workflows.</p>
+<p><strong>Topics in the Book</strong></p>
+<ul class="simple">
+<li><p><a class="reference external" href="preface.html">Preface</a></p></li>
+<li><p><a class="reference external" href="1_introduction.html">Introduction</a></p></li>
+<li><p><a class="reference external" href="2_diagnostic_modeling_overview_and_perspectives.html">An overview of diagnostic modeling and perspectives on model evaluation</a></p></li>
+<li><p><a class="reference external" href="3_sensitivity_analysis_the_basics.html">A framework for the basic methods and concepts used in sensitivity analysis</a></p></li>
+<li><p><a class="reference external" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html">Technical applications supporting diagnostic model evaluation and exploration</a></p></li>
+</ul>
+<p><strong>Interactive Tutorials</strong></p>
+<ul class="simple">
+<li><p><a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/fishery_dynamics.ipynb">Factor Discovery</a></p></li>
+<li><p><a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/hymod.ipynb">Model Calibration</a></p></li>
+<li><p><a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/sa_saltelli_sobol_ishigami.ipynb">Sobol Sensitivity Analysis</a></p></li>
+<li><p><a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/basin_users_logistic_regression.ipynb">Factor Mapping using Logistic Regression</a></p></li>
+<li><p><a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/Bedford_Greene_SD_tutorial.ipynb">Time-evolving scenario discovery for infrastructure pathways</a></p></li>
+<li><p><a class="reference external" href="https://uc-ebook.msdlive.org/user-redirect/lab/tree/notebooks/Hidden-Markov_Modeling_Approaches_to_Creating_Synthetic_Streamflow_Scenarios.ipynb">A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios</a></p></li>
+</ul>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>
+    Use the sidebar on the left to quickly navigate the eBook!
+    <br/>
+    Click or tap the <i style="padding: 0 0.25rem;" class="fas fa-bars"></i> icon to show and hide the sidebar.
+</p></div>
+<div class="admonition-info admonition">
+<p class="admonition-title">Info</p>
+<p>
+    Report a typo or just pass along something you like about the book by <a target="_blank" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/issues/new?assignees=thurber%2C+crvernon&labels=documentation%2C+triage&template=custom.md&title=Publication+Feedback">opening an issue</a> on GitHub!
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+  <section id="preface">
+<span id="id1"></span><h1>Preface<a class="headerlink" href="#preface" title="Link to this heading">#</a></h1>
+<p>This online book is meant to provide an open science “living” resource on uncertainty characterization methods for the MultiSector Dynamics (MSD) community and other technical communities confronting sustainability, climate, and energy transition challenges. The last decade has seen rapid growth in science efforts seeking to address the interconnected nature of these challenges across scales, sectors, and systems. Accompanying these advances is the growing realization that the deep integration of research from many disciplinary fields is non-trivial and raises important questions. How and why models are developed seems to have an obvious answer (“to gain understanding”). But what does it actually mean to gain understanding? What if a small change in a model or its data fundamentally changes our perceptions of what we thought we understood? What controls the outcomes of our model(s)? How do we understand the implications of model coupling, such as when one model is on the receiving end of several other models that are considered “input data”?</p>
+<p>The often quoted “All models are wrong, but some are useful.” (George Box) is a bit of a conflation trap, often used to excuse known weaknesses in complex models as just an unavoidable outcome of being a modeler. In fact, the quote actually refers to a specific class of small-scale statistical models within an application context that assures a much higher degree of understanding and data quality control than is typical for the coupled human-natural systems applications in the MSD area. Moreover, Box was actually warning readers to avoid overparameterization and emphasizing the need to better understand what underlying factors cause your model to be wrong <span id="id2">[<a class="reference internal" href="R.Bibliography.html#id204" title="G. E. P. Box. Science and Statistics. Journal of the American Statistical Association, 71(356):791–799, 1976. URL: https://www.tandfonline.com/doi/abs/10.1080/01621459.1976.10480949, doi:10.1080/01621459.1976.10480949.">1</a>]</span>.</p>
+<p>So, in short, there is a tension when attaining better performance by means of increasing the complexity of a model or model-based workflow. Box highlights that a modeler requires a clear diagnostic understanding of this performance-complexity tradeoff. If we move from small-scale models simulating readily-observed phenomena to the MSD context, things get quite a bit more complicated. How can we provide robust insights for unseen futures that emerge across a myriad of human and natural systems? Sometimes even asking, “what is a model?” or “what is data?” is complicated (e.g., data assimilated weather products, satellite-based signals translated through retrieval algorithms, demographic changes, resource demands, etc.). This MSD guidance text seeks to help readers navigate these challenges. It is meant to serve as an evolving resource that helps the MSD community learn how to better address uncertainty while working with complex chains of models bridging sectors, scales, and systems. It is not intended to be an exhaustive resource, but instead should be seen as a guided tour through state-of-the-science methods in uncertainty characterization, including global sensitivity analysis and exploratory modeling, to provide insights into complex human-natural systems interactions.</p>
+<p>To aid readers in navigating the text, the key goals for each chapter are summarized below.</p>
+<p><a class="reference internal" href="1_introduction.html#introduction"><span class="std std-numref">Chapter 1</span></a> uses the <a class="reference external" href="https://im3.pnnl.gov">Integrated Multisector Multiscale Modeling</a> project as a living lab to encapsulate the challenges that emerge in bridging disciplines to make consequential model-based insights while acknowledging the tremendous array of uncertainties that shape them.</p>
+<p><a class="reference internal" href="2_diagnostic_modeling_overview_and_perspectives.html#diagnostic-modeling"><span class="std std-numref">Chapter 2</span></a> helps the reader to better understand the importance of using diagnostic modeling to interrogate why uncertain model behaviors may emerge. The chapter also aids readers to better understand the diverse disciplinary perspectives that exist on how best to pursue consequential model-based discoveries.</p>
+<p><a class="reference internal" href="3_sensitivity_analysis_the_basics.html#sensitivity-analysis-the-basics"><span class="std std-numref">Chapter 3</span></a> is a technical tools-focused primer for readers on the key elements of uncertainty characterization that includes ensemble-based design of experiments, quantitative methods for computing global sensitivities, and a summary of existing software packages.</p>
+<p><a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#sensitivity-analysis"><span class="std std-numref">Chapter 4</span></a> narrates for readers how and why the tools from the previous chapter can be applied in a range of tasks from diagnosing model performance to formal exploratory modeling methods for making consequential model-based discoveries.</p>
+<p>The supplemental appendices provided in the text are also important resources for readers. They provide a glossary to help bridge terminology challenges, a brief summary of uncertainty quantification tools for more advanced readers, and a suite of Jupyter notebook tutorials that provide hands-on training tied to the contents of <a class="reference internal" href="3_sensitivity_analysis_the_basics.html#sensitivity-analysis-the-basics"><span class="std std-numref">Chapter 3</span></a> and <a class="reference internal" href="4_sensitivity_analysis_diagnostic_and_exploratory_modeling.html#sensitivity-analysis"><span class="std std-numref">Chapter 4</span></a>.</p>
+<p>This text was written with a number of different audiences in mind.</p>
+<p>Technical experts in uncertainty may find this to be a helpful and unique resource bridging a number of perspectives that have not been combined in prior books (e.g., formal model diagnostics, global sensitivity analysis, and exploratory modeling under deep uncertainty).</p>
+<p>Readers from different sector-specific and disciplinary-specific backgrounds can use this text to better understand potential differences and opportunities in how to make model-based insights.</p>
+<p>Academic or junior researchers can utilize this freely available text for training and teaching resources that include hands-on coding experiences.</p>
+<p>This text itself represents our strong commitment to open science and will evolve as a living resource as the communities of researchers provide feedback, innovations, and future tools.</p>
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"step-5-explore-relationship-between-uncertain-factors-and-performance"], [45, "step-5-explore-relationship-between-uncertain-factors-and-performance"]], "Step 5: Interpret our results": [[40, "step-5-interpret-our-results"], [45, "step-5-interpret-our-results"]], "Suggested Citation": [[49, "suggested-citation"]], "Summary of designs of experiments overviewed in this section. * Depends on the sample size.": [[12, "id32"], [23, "id96"]], "Synthetic Stationary Generator to Better Quantify Natural Variability": [[44, "synthetic-stationary-generator-to-better-quantify-natural-variability"], [45, "synthetic-stationary-generator-to-better-quantify-natural-variability"]], "Synthetic generation of input time series": [[11, "synthetic-generation-of-input-time-series"], [12, "synthetic-generation-of-input-time-series"], [23, "synthetic-generation-of-input-time-series"]], "Testing out a note": [[52, "testing-out-a-note"]], "The Critical Final Step: Predictive Checks": [[36, 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"Global Versus Local Sensitivity", "Why Perform Sensitivity Analysis", "One-At-a-Time (OAT)", "Full and Fractional Factorial Sampling", "Latin Hypercube Sampling (LHS)", "Low-Discrepancy Sequences", "Other types of sampling", "Synthetic generation of input time series", "Design of Experiments", "Derivative-based Methods", "Elementary Effect Methods", "Regression-based Methods", "Regional Sensitivity Analysis", "Variance-based Methods", "Analysis of Variance (ANOVA)", "Moment-Independent (Density-Based) Methods", "Sensitivity Analysis Methods", "How To Choose A Sensitivity Analysis Method: Model Traits And Dimensionality", "Software Toolkits", "<span class=\"section-number\">3. </span>Sensitivity Analysis: The Basics", "Understanding Errors: What Is Controlling Model Performance?", "Consequential Dynamics: What is Controlling Model Behaviors of Interest?", "Consequential Scenarios: What is Controlling Consequential Outcomes?", "<span class=\"section-number\">4. </span>Sensitivity Analysis: Diagnostic &amp; Exploratory Modeling", "<span class=\"section-number\">5. </span>Conclusion", "Glossary", "Introduction", "Parametric Bootstrap", "Pre-Calibration", "Markov Chain Monte Carlo", "Other Methods", "The Critical First Step: How to Choose a Prior Distribution", "The Critical Final Step: Predictive Checks", "Key Take-Home Points", "<span class=\"section-number\">1. </span>Uncertainty Quantification", "Fishery Dynamics Tutorial", "Sobol SA Tutorial", "Logistic Regression Tutorial", "HYMOD Dynamics Tutorial", "Time-evolving scenario discovery for infrastructure pathways", "A Hidden-Markov Modeling Approach to Creating Synthetic Streamflow Scenarios Tutorial", "<span class=\"section-number\">2. </span>Jupyter Notebook Tutorials", "<span class=\"section-number\">3. </span>Plotting Code Samples", "Bibliography", "Acknowledgements", "Suggested Citation", "Code of Conduct", "Contribution Guide", "Examples", "Addressing Uncertainty in MultiSector Dynamics Research", 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+	<div style="background-image: url('images/banner.png'); background-size: auto 234px; background-position: 6px 42px;" class="bg-no-repeat pl-3 lg:pl-56 py-12"><div style="text-indent:-16px;" class="text-3xl font-bold text-slate-50 text-shadow pb-1 px-3 text-center lg:text-left leading-tight backdrop-blur-sm svelte-uz1u3c">Addressing Uncertainty in<br>MultiSector Dynamics Research
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+		<div class="h-16 border-y-4 border-slate-50 border-double"></div>
+		<div class="text-md italic text-slate-50 text-shadow pt-1 px-3 text-center lg:text-left leading-tight backdrop-blur-sm svelte-uz1u3c"><div style="">Patrick M. Reed, Antonia Hadjimichael, Keyvan Malek,</div>
+			<div style="text-indent: -8px;">Tina Karimi, Chris R. Vernon, Vivek Srikrishnan, Rohini S. Gupta,
+			</div>
+			<div style="text-indent: -16px;">David F. Gold, Ben Lee, Klaus Keller, Travis B. Thurber, Jennie S. Rice</div></div></div></div>
+<div class="w-full bg-slate-50 text-xl text-slate-900 py-6 px-3 flex flex-row items-center justify-around text-center xl:rounded-md"><div class="hidden md:block font-['ornaments'] text-slate-500 text-4xl mx-6">p</div>
+	<div class="max-w-4xl">A living guide to sensitivity analysis and diagnostic model evaluation techniques for
+		confronting the computational and conceptual challenges of multi-model, transdisciplinary
+		workflows.
+	</div>
+	<div class="hidden md:block font-['ornaments'] text-slate-500 text-4xl mx-6">o</div></div>
+<div class="w-full flex flex-col items-center justify-center text-center px-3 pb-6"><div class="text-2xl font-bold text-slate-50 my-6">GET THE EBOOK</div>
+	<div class="flex flex-row flex-wrap items-center justify-around w-full border-y-4 border-double border-slate-50"><a target="_blank" href="docs/html/index.html" class="flex-none flex flex-col items-center justify-center w-64 pt-3 pb-6 group"><svg fill="none" viewBox="0 0 24 24" stroke="currentColor" aria-hidden="true" xmlns="http://www.w3.org/2000/svg" width="100%" height="100%" class="icon w-20 text-slate-50 group-hover:text-sky-500"><path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M12 6.253v13m0-13C10.832 5.477 9.246 5 7.5 5S4.168 5.477 3 6.253v13C4.168 18.477 5.754 18 7.5 18s3.332.477 4.5 1.253m0-13C13.168 5.477 14.754 5 16.5 5c1.747 0 3.332.477 4.5 1.253v13C19.832 18.477 18.247 18 16.5 18c-1.746 0-3.332.477-4.5 1.253"></path></svg>
+				<div class="text-xl font-bold text-slate-50 group-hover:text-sky-500">HTML</div>
+				<div class="text-md text-slate-50 group-hover:text-sky-500">Read in your browser!</div>
+			</a><a target="_blank" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook/releases/download/2024-07-12v99/addressinguncertaintyinmultisectordynamicsresearch.pdf" class="flex-none flex flex-col items-center justify-center w-64 pt-3 pb-6 group"><svg fill="none" viewBox="0 0 24 24" stroke="currentColor" aria-hidden="true" xmlns="http://www.w3.org/2000/svg" width="100%" height="100%" class="icon w-20 text-slate-50 group-hover:text-sky-500"><path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M12 10v6m0 0l-3-3m3 3l3-3m2 8H7a2 2 0 01-2-2V5a2 2 0 012-2h5.586a1 1 0 01.707.293l5.414 5.414a1 1 0 01.293.707V19a2 2 0 01-2 2z"></path></svg>
+				<div class="text-xl font-bold text-slate-50 group-hover:text-sky-500">PDF</div>
+				<div class="text-md text-slate-50 group-hover:text-sky-500">Download and read offline!</div>
+			</a><a target="_blank" href="https://github.com/IMMM-SFA/msd_uncertainty_ebook" class="flex-none flex flex-col items-center justify-center w-64 pt-3 pb-6 group"><svg fill="none" viewBox="0 0 24 24" stroke="currentColor" aria-hidden="true" xmlns="http://www.w3.org/2000/svg" width="100%" height="100%" class="icon w-20 text-slate-50 group-hover:text-sky-500"><path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M10 20l4-16m4 4l4 4-4 4M6 16l-4-4 4-4"></path></svg>
+				<div class="text-xl font-bold text-slate-50 group-hover:text-sky-500">Source Code</div>
+				<div class="text-md text-slate-50 group-hover:text-sky-500">Browse ebook source code!</div>
+			</a></div></div>
+<div class="w-full flex flex-col items-center justify-center text-center px-3 lg:pb-6"><div class="text-2xl font-bold text-slate-50 my-3">TOPICS IN THE BOOK</div>
+	<div class="flex flex-col lg:flex-row items-center lg:items-start justify-around mt-6 w-full"><div class="bg-slate-50 w-80 flex-none mb-12 lg:mb-0 shadow-md rounded-md"><div class="flex flex-row items-start justify-start text-left mb-3"><div class="bg-slate-900 w-20 h-20 flex-none flex items-center justify-center rounded-br-md"><div class="text-4xl font-bold text-slate-50 text-shadow svelte-uz1u3c">1
+						</div></div>
+					<div class="text-md text-slate-900 mx-3 mt-3 leading-tight">An overview of diagnostic modeling and perspectives on model evaluation.
+					</div></div>
+				<img class="w-full h-36 my-3 object-contain" src="images/idealized_uc.png" alt="">
+			</div><div class="bg-slate-50 w-80 flex-none mb-12 lg:mb-0 shadow-md rounded-md"><div class="flex flex-row items-start justify-start text-left mb-3"><div class="bg-slate-900 w-20 h-20 flex-none flex items-center justify-center rounded-br-md"><div class="text-4xl font-bold text-slate-50 text-shadow svelte-uz1u3c">2
+						</div></div>
+					<div class="text-md text-slate-900 mx-3 mt-3 leading-tight">A framework for the basic methods and concepts used in sensitivity analysis.
+					</div></div>
+				<img class="w-full h-36 my-3 object-contain" src="images/predator_prey.png" alt="">
+			</div><div class="bg-slate-50 w-80 flex-none mb-12 lg:mb-0 shadow-md rounded-md"><div class="flex flex-row items-start justify-start text-left mb-3"><div class="bg-slate-900 w-20 h-20 flex-none flex items-center justify-center rounded-br-md"><div class="text-4xl font-bold text-slate-50 text-shadow svelte-uz1u3c">3
+						</div></div>
+					<div class="text-md text-slate-900 mx-3 mt-3 leading-tight">Technical applications supporting diagnostic model evaluation and exploration.
+					</div></div>
+				<img class="w-full h-36 my-3 object-contain" src="images/fishery_output.png" alt="">
+			</div></div></div>
+<div class="w-full flex flex-col items-center justify-center text-center px-3 pb-6"><div class="text-lg text-slate-50 lg:mt-6">Share with your network!</div>
+	<div class="flex flex-row items-center justify-center mt-3 w-full"><a href="" class="rounded-lg drop-shadow p-2 mx-3 bg-sky-700 hover:bg-sky-500"><svg viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" xmlns="http://www.w3.org/2000/svg" width="100%" height="100%" class="icon w-6 text-slate-50"><path d="M23 3a10.9 10.9 0 0 1-3.14 1.53 4.48 4.48 0 0 0-7.86 3v1A10.66 10.66 0 0 1 3 4s-4 9 5 13a11.64 11.64 0 0 1-7 2c9 5 20 0 20-11.5a4.5 4.5 0 0 0-.08-.83A7.72 7.72 0 0 0 23 3z"></path></svg></a>
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+<div class="w-full flex flex-col items-center justify-center text-center px-3 pb-12"><div class="text-lg text-slate-50 lg:mt-6">Cite the eBook!</div>
+	<div class="flex flex-row items-center justify-center mt-3 w-full"><div class="flex flex-col items-center justify-center"><div class="mb-4"><select><option value="APA">APA</option><option value="BibTeX">BibTeX</option></select></div>
+    <div class="flex flex-row items-stretch justify-center"><div style="box-shadow: inset 0 2px 4px 0 rgb(0 0 0 / 0.2);" class="bg-slate-50 rounded-l-md overflow-auto"><pre class="m-4 text-sm text-slate-900 w-80 md:w-96 text-left leading-tight "></pre></div>
+        <div title="Copy to clipboard" class="rounded-r-md drop-shadow p-2 bg-sky-700 hover:bg-sky-500 hover:cursor-pointer active:bg-orange-400"><svg fill="none" viewBox="0 0 24 24" stroke="currentColor" aria-hidden="true" xmlns="http://www.w3.org/2000/svg" width="100%" height="100%" class="icon w-6 text-slate-50"><path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M8 5H6a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2v-1M8 5a2 2 0 002 2h2a2 2 0 002-2M8 5a2 2 0 012-2h2a2 2 0 012 2m0 0h2a2 2 0 012 2v3m2 4H10m0 0l3-3m-3 3l3 3"></path></svg></div></div></div></div></div>
+<div class="w-full flex flex-col items-center justify-center text-center px-3 pb-12"><div class="max-w-4xl text-sm italic text-slate-400">This e-book was developed by the <a href="https://im3.pnnl.gov" class="font-bold text-slate-500 hover:text-sky-700">Integrated MultiSector, Multiscale Modeling (IM3)</a>
+		project, supported by the
+		<a href="https://www.energy.gov/" class="font-bold text-slate-500 hover:text-sky-700">U.S. Department of Energy</a>,
+		<a href="https://www.energy.gov/science/office-science" class="font-bold text-slate-500 hover:text-sky-700">Office of Science</a>, as part of research in the
+		<a href="https://climatemodeling.science.energy.gov/program/multisector-dynamics" class="font-bold text-slate-500 hover:text-sky-700">MultiSector Dynamics</a>, Earth and Environmental System Modeling Program.
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